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Wikipedia:Branko Grünbaum#0	Branko Grünbaum (Hebrew: ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem. == Life == Grünbaum was born in Osijek, then part of the Kingdom of Yugoslavia, on 2 October 1929. His father was Jewish and his mother was Catholic, so during World War II the family survived the Holocaust by living at his Catholic grandmother's home. After the war, as a high school student, he met Zdenka Bienenstock, a Jew who had lived through the war hidden in a convent while the rest of her family were killed. Grünbaum became a student at the University of Zagreb, but grew disenchanted with the communist ideology of the Socialist Federal Republic of Yugoslavia, applied for emigration to Israel, and traveled with his family and Zdenka to Haifa in 1949. In Israel, Grünbaum found a job in Tel Aviv, but in 1950 returned to the study of mathematics, at the Hebrew University of Jerusalem. He earned a master's degree in 1954 and in the same year married Zdenka, who continued as a master's student in chemistry. He served a tour of duty as an operations researcher in the Israeli Air Force beginning in 1955, and he and Zdenka had the first of their two sons in 1956. He completed his Ph.D. in 1957; his dissertation concerned convex geometry and was supervised by Aryeh Dvoretzky. After finishing his military service in 1958, Grünbaum and his family came to the US so that Grünbaum could become a postdoctoral researcher at the Institute for Advanced Study. He then became a visiting researcher at the University of Washington in 1960. He agreed to return to Israel as a lecturer at the Hebrew University, but his plans were disrupted by the Israeli authorities determining that he was not a Jew (because his mother was not Jewish) and annulling his marriage; he and Zdenka remarried in Seattle before their return. Grünbaum remained affiliated with the Hebrew University until 1966, taking long research visits to the University of Washington and in 1965–1966 to Michigan State University. However, during the Michigan visit, learning of another case similar to their marriage annulment, he and Zdenka decided to stay in the US instead of returning to Israel, where Zdenka was still a doctoral student in chemistry. Grünbaum was given a full professorship at the University of Washington in 1966, and he remained there until retiring in 2001. == Works == Grünbaum authored over 200 papers, mostly in discrete geometry, an area in which he is known for various classification theorems. He wrote on the theory of abstract polyhedra. His paper on line arrangements may have inspired a paper by N. G. de Bruijn on quasiperiodic tilings (the most famous example of which is the Penrose tiling of the plane). This paper is also cited by the authors of a monograph on hyperplane arrangements as having inspired their research. Grünbaum also devised a multi-set generalisation of Venn diagrams. He was an editor and a frequent contributor to Geombinatorics. Grünbaum's classic monograph Convex Polytopes, first published in 1967, became the main textbook on the subject. His monograph Tilings and patterns, coauthored with G. C. Shephard, helped to rejuvenate interest in this classic field, and has proved popular with nonmathematical audiences, as well as with mathematicians. In 1976 Grünbaum won a Lester R. Ford Award for his expository article Venn diagrams and independent families of sets. In 2004, Gil Kalai and Victor Klee edited a special issue of Discrete and Computational Geometry in his honor, the "Grünbaum Festschrift". In 2005, Grünbaum was awarded the Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society. He was a Guggenheim Fellow, a Fellow of the AAAS and in 2012 he became a fellow of the American Mathematical Society. Grünbaum supervised 19 Ph.D.s and currently has at least 200 mathematical descendants. == Books == Grünbaum, Branko (2003) [1967], Kaibel, Volker; Klee, Victor; Ziegler, Günter M. (eds.), Convex Polytopes, Graduate Texts in Mathematics, vol. 221 (2nd ed.), Springer-Verlag, ISBN 0-387-00424-6. Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society, ISBN 0-8218-1659-4. Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Grünbaum, Branko (2009), Configurations of Points and Lines, Graduate Studies in Mathematics, vol. 103, American Mathematical Society, ISBN 978-0-8218-4308-6, MR 2510707. === As editor === Davis, Chandler; Grünbaum, B.; Sherk, F. A., eds. (2012) [1981], The Geometric Vein: The Coxeter Festschrift, Springer Science & Business Media, ISBN 978-1-4612-5648-9 == See also == Configuration (geometry) Convex uniform honeycomb Elongated square gyrobicupola Goldner–Harary graph Pentagram map Simplicial sphere Star coloring Star polygon Grünbaum's theorem Grünbaum–Rigby configuration == References == == Further reading == Kahle, Matthew (2019), "Branko Grünbaum in many dimensions", Geombinatorics, 28 (3): 140–146, arXiv:1901.08622, MR 3821744 Kalai, Gil; Mohar, Bojan; Novik, Isabella (June 2020), "Guest Editors' Foreword", Branko Grünbaum Memorial Issue, Discrete & Computational Geometry, 64 (2): 229–232, doi:10.1007/s00454-020-00214-y == External links == Personal web page
Wikipedia:Bra–ket notation#0	Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread. Bra–ket notation was created by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions. The name comes from the English word "bracket". == Quantum mechanics == In quantum mechanics and quantum computing, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, ⟨ {\displaystyle \langle } and ⟩ {\displaystyle \rangle } , and a vertical bar \| {\displaystyle \|} , to construct "bras" and "kets". A ket is of the form \| v ⟩ {\displaystyle \|v\rangle } . Mathematically it denotes a vector, v {\displaystyle {\boldsymbol {v}}} , in an abstract (complex) vector space V {\displaystyle V} , and physically it represents a state of some quantum system. A bra is of the form ⟨ f \| {\displaystyle \langle f\|} . Mathematically it denotes a linear form f : V → C {\displaystyle f:V\to \mathbb {C} } , i.e. a linear map that maps each vector in V {\displaystyle V} to a number in the complex plane C {\displaystyle \mathbb {C} } . Letting the linear functional ⟨ f \| {\displaystyle \langle f\|} act on a vector \| v ⟩ {\displaystyle \|v\rangle } is written as ⟨ f \| v ⟩ ∈ C {\displaystyle \langle f\|v\rangle \in \mathbb {C} } . Assume that on V {\displaystyle V} there exists an inner product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} with antilinear first argument, which makes V {\displaystyle V} an inner product space. Then with this inner product each vector ϕ ≡ \| ϕ ⟩ {\displaystyle {\boldsymbol {\phi }}\equiv \|\phi \rangle } can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: ( ϕ , ⋅ ) ≡ ⟨ ϕ \| {\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi \|} . The correspondence between these notations is then ( ϕ , ψ ) ≡ ⟨ ϕ \| ψ ⟩ {\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi \|\psi \rangle } . The linear form ⟨ ϕ \| {\displaystyle \langle \phi \|} is a covector to \| ϕ ⟩ {\displaystyle \|\phi \rangle } , and the set of all covectors forms a subspace of the dual vector space V ∨ {\displaystyle V^{\vee }} , to the initial vector space V {\displaystyle V} . The purpose of this linear form ⟨ ϕ \| {\displaystyle \langle \phi \|} can now be understood in terms of making projections onto the state ϕ , {\displaystyle {\boldsymbol {\phi }},} to find how linearly dependent two states are, etc. For the vector space C n {\displaystyle \mathbb {C} ^{n}} , kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication. If C n {\displaystyle \mathbb {C} ^{n}} has the standard Hermitian inner product ( v , w ) = v † w {\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w} , under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted † {\displaystyle \dagger } ). It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator σ ^ z {\displaystyle {\hat {\sigma }}_{z}} on a two-dimensional space Δ {\displaystyle \Delta } of spinors has eigenvalues ± 1 2 {\textstyle \pm {\frac {1}{2}}} with eigenspinors ψ + , ψ − ∈ Δ {\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta } . In bra–ket notation, this is typically denoted as ψ + = \| + ⟩ {\displaystyle {\boldsymbol {\psi }}_{+}=\|+\rangle } , and ψ − = \| − ⟩ {\displaystyle {\boldsymbol {\psi }}_{-}=\|-\rangle } . As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors. Bra–ket notation was effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of [ ϕ ∣ ψ ] {\displaystyle [\phi {\mid }\psi ]} for inner products nearly 100 years earlier. == Vector spaces == === Vectors vs kets === In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime. Such vectors are typically denoted with over arrows ( r → {\displaystyle {\vec {r}}} ), boldface ( p {\displaystyle \mathbf {p} } ) or indices ( v μ {\displaystyle v^{\mu }} ). In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element ϕ {\displaystyle \phi } of an abstract complex vector space as a ket \| ϕ ⟩ {\displaystyle \|\phi \rangle } , to refer to it as a "ket" rather than as a vector, and to pronounce it "ket- ϕ {\displaystyle \phi } " or "ket-A" for \|A⟩. Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the \| ⟩ {\displaystyle \|\ \rangle } making clear that the label indicates a vector in vector space. In other words, the symbol "\|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example, \|1⟩ + \|2⟩ is not necessarily equal to \|3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as x ^ {\displaystyle {\hat {x}}} , p ^ {\displaystyle {\hat {p}}} , L ^ z {\displaystyle {\hat {L}}_{z}} , etc. === Notation === Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example: \| A ⟩ = \| B ⟩ + \| C ⟩ \| C ⟩ = ( − 1 + 2 i ) \| D ⟩ \| D ⟩ = ∫ − ∞ ∞ e − x 2 \| x ⟩ d x . {\displaystyle {\begin{aligned}\|A\rangle &=\|B\rangle +\|C\rangle \\\|C\rangle &=(-1+2i)\|D\rangle \\\|D\rangle &=\int _{-\infty }^{\infty }e^{-x^{2}}\|x\rangle \,\mathrm {d} x\,.\end{aligned}}} Note how the last line above involves infinitely many different kets, one for each real number x. Since the ket is an element of a vector space, a bra ⟨ A \| {\displaystyle \langle A\|} is an element of its dual space, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts. A bra ⟨ ϕ \| {\displaystyle \langle \phi \|} and a ket \| ψ ⟩ {\displaystyle \|\psi \rangle } (i.e. a functional and a vector), can be combined to an operator \| ψ ⟩ ⟨ ϕ \| {\displaystyle \|\psi \rangle \langle \phi \|} of rank one with outer product \| ψ ⟩ ⟨ ϕ \| : \| ξ ⟩ ↦ \| ψ ⟩ ⟨ ϕ \| ξ ⟩ . {\displaystyle \|\psi \rangle \langle \phi \|\colon \|\xi \rangle \mapsto \|\psi \rangle \langle \phi \|\xi \rangle ~.} === Inner product and bra–ket identification on Hilbert space === The bra–ket notation is particularly useful in Hilbert spaces which have an inner product that allows Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem). The inner product on Hilbert space ( , ) {\displaystyle (\ ,\ )} (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in the bra–ket notation: for a vector ket ψ = \| ψ ⟩ {\displaystyle \psi =\|\psi \rangle } define a functional (i.e. bra) f ϕ = ⟨ ϕ \| {\displaystyle f_{\phi }=\langle \phi \|} by ( ϕ , ψ ) = ( \| ϕ ⟩ , \| ψ ⟩ ) =: f ϕ ( ψ ) = ⟨ ϕ \| ( \| ψ ⟩ ) =: ⟨ ϕ ∣ ψ ⟩ {\displaystyle (\phi ,\psi )=(\|\phi \rangle ,\|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi \|\,{\bigl (}\|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle } ==== Bras and kets as row and column vectors ==== In the simple case where we consider the vector space C n {\displaystyle \mathbb {C} ^{n}} , a ket can be identified with a column vector, and a bra as a row vector. If, moreover, we use the standard Hermitian inner product on C n {\displaystyle \mathbb {C} ^{n}} , the bra corresponding to a ket, in particular a bra ⟨m\| and a ket \|m⟩ with the same label are conjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply matrix multiplication. In particular the outer product \| ψ ⟩ ⟨ ϕ \| {\displaystyle \|\psi \rangle \langle \phi \|} of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix). For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector: ⟨ A \| B ⟩ ≐ A 1 ∗ B 1 + A 2 ∗ B 2 + ⋯ + A N ∗ B N = ( A 1 ∗ A 2 ∗ ⋯ A N ∗ ) ( B 1 B 2 ⋮ B N ) {\displaystyle \langle A\|B\rangle \doteq A_{1}^{}B_{1}+A_{2}^{}B_{2}+\cdots +A_{N}^{}B_{N}={\begin{pmatrix}A_{1}^{}&A_{2}^{}&\cdots &A_{N}^{}\end{pmatrix}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}} Based on this, the bras and kets can be defined as: ⟨ A \| ≐ ( A 1 ∗ A 2 ∗ ⋯ A N ∗ ) \| B ⟩ ≐ ( B 1 B 2 ⋮ B N ) {\displaystyle {\begin{aligned}\langle A\|&\doteq {\begin{pmatrix}A_{1}^{}&A_{2}^{}&\cdots &A_{N}^{}\end{pmatrix}}\\\|B\rangle &\doteq {\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}\end{aligned}}} and then it is understood that a bra next to a ket implies matrix multiplication. The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice versa: ⟨ A \| † = \| A ⟩ , \| A ⟩ † = ⟨ A \| {\displaystyle \langle A\|^{\dagger }=\|A\rangle ,\quad \|A\rangle ^{\dagger }=\langle A\|} because if one starts with the bra ( A 1 ∗ A 2 ∗ ⋯ A N ∗ ) , {\displaystyle {\begin{pmatrix}A_{1}^{}&A_{2}^{}&\cdots &A_{N}^{}\end{pmatrix}}\,,} then performs a complex conjugation, and then a matrix transpose, one ends up with the ket ( A 1 A 2 ⋮ A N ) {\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}} Writing elements of a finite dimensional (or mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a basis. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "\|m⟩" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "\|−⟩" and "\|+⟩". === Non-normalizable states and non-Hilbert spaces === Bra–ket notation can be used even if the vector space is not a Hilbert space. In quantum mechanics, it is common practice to write down kets which have infinite norm, i.e. non-normalizable wavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves. These do not, technically, belong to the Hilbert space itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). The bra–ket notation continues to work in an analogous way in this more general context. Banach spaces are a different generalization of Hilbert spaces. In a Banach space B, the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without a given topology, we may still notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply. == Usage in quantum mechanics == The mathematical structure of quantum mechanics is based in large part on linear algebra: Wave functions and other quantum states can be represented as vectors in a separable complex Hilbert space. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" \|ψ⟩. (Technically, the quantum states are rays of vectors in the Hilbert space, as c\|ψ⟩ corresponds to the same state for any nonzero complex number c.) Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the state ⁠1/√2⁠\|1⟩ + ⁠i/√2⁠\|2⟩ is in a quantum superposition of the states \|1⟩ and \|2⟩. Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states. Dynamics are also described by linear operators on the Hilbert space. For example, in the Schrödinger picture, there is a linear time evolution operator U with the property that if an electron is in state \|ψ⟩ right now, at a later time it will be in the state U\|ψ⟩, the same U for every possible \|ψ⟩. Wave function normalization is scaling a wave function so that its norm is 1. Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow: === Spinless position–space wave function === The Hilbert space of a spin-0 point particle can be represented in terms of a "position basis" { \|r⟩ }, where the label r extends over the set of all points in position space. These states satisfy the eigenvalue equation for the position operator: r ^ \| r ⟩ = r \| r ⟩ . {\displaystyle {\hat {\mathbf {r} }}\|\mathbf {r} \rangle =\mathbf {r} \|\mathbf {r} \rangle .} The position states are "generalized eigenvectors", not elements of the Hilbert space itself, and do not form a countable orthonormal basis. However, as the Hilbert space is separable, it does admit a countable dense subset within the domain of definition of its wavefunctions. That is, starting from any ket \|Ψ⟩ in this Hilbert space, one may define a complex scalar function of r, known as a wavefunction, Ψ ( r ) = def ⟨ r \| Ψ ⟩ . {\displaystyle \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} \|\Psi \rangle \,.} On the left-hand side, Ψ(r) is a function mapping any point in space to a complex number; on the right-hand side, \| Ψ ⟩ = ∫ d 3 r Ψ ( r ) \| r ⟩ {\displaystyle \left\|\Psi \right\rangle =\int d^{3}\mathbf {r} \,\Psi (\mathbf {r} )\left\|\mathbf {r} \right\rangle } is a ket consisting of a superposition of kets with relative coefficients specified by that function. It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by A ^ ( r ) Ψ ( r ) = def ⟨ r \| A ^ \| Ψ ⟩ . {\displaystyle {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} \|{\hat {A}}\|\Psi \rangle \,.} For instance, the momentum operator p ^ {\displaystyle {\hat {\mathbf {p} }}} has the following coordinate representation, p ^ ( r ) Ψ ( r ) = def ⟨ r \| p ^ \| Ψ ⟩ = − i ℏ ∇ Ψ ( r ) . {\displaystyle {\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} \|{\hat {\mathbf {p} }}\|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.} One occasionally even encounters an expression such as ∇ \| Ψ ⟩ {\displaystyle \nabla \|\Psi \rangle } , though this is something of an abuse of notation. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, ∇ ⟨ r \| Ψ ⟩ , {\displaystyle \nabla \langle \mathbf {r} \|\Psi \rangle \,,} even though, in the momentum basis, this operator amounts to a mere multiplication operator (by iħp). That is, to say, ⟨ r \| p ^ = − i ℏ ∇ ⟨ r \| , {\displaystyle \langle \mathbf {r} \|{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} \|~,} or p ^ = ∫ d 3 r \| r ⟩ ( − i ℏ ∇ ) ⟨ r \| . {\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~\|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} \|~.} === Overlap of states === In quantum mechanics the expression ⟨φ\|ψ⟩ is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ. Mathematically, this means the coefficient for the projection of ψ onto φ. It is also described as the projection of state ψ onto state φ. === Changing basis for a spin-1/2 particle === A stationary spin-1⁄2 particle has a two-dimensional Hilbert space. One orthonormal basis is: \| ↑ z ⟩ , \| ↓ z ⟩ {\displaystyle \|{\uparrow }_{z}\rangle \,,\;\|{\downarrow }_{z}\rangle } where \|↑z⟩ is the state with a definite value of the spin operator Sz equal to +1⁄2 and \|↓z⟩ is the state with a definite value of the spin operator Sz equal to −1⁄2. Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantum superposition) of these two states: \| ψ ⟩ = a ψ \| ↑ z ⟩ + b ψ \| ↓ z ⟩ {\displaystyle \|\psi \rangle =a_{\psi }\|{\uparrow }_{z}\rangle +b_{\psi }\|{\downarrow }_{z}\rangle } where aψ and bψ are complex numbers. A different basis for the same Hilbert space is: \| ↑ x ⟩ , \| ↓ x ⟩ {\displaystyle \|{\uparrow }_{x}\rangle \,,\;\|{\downarrow }_{x}\rangle } defined in terms of Sx rather than Sz. Again, any state of the particle can be expressed as a linear combination of these two: \| ψ ⟩ = c ψ \| ↑ x ⟩ + d ψ \| ↓ x ⟩ {\displaystyle \|\psi \rangle =c_{\psi }\|{\uparrow }_{x}\rangle +d_{\psi }\|{\downarrow }_{x}\rangle } In vector form, you might write \| ψ ⟩ ≐ ( a ψ b ψ ) or \| ψ ⟩ ≐ ( c ψ d ψ ) {\displaystyle \|\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad \|\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}} depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used. There is a mathematical relationship between a ψ {\displaystyle a_{\psi }} , b ψ {\displaystyle b_{\psi }} , c ψ {\displaystyle c_{\psi }} and d ψ {\displaystyle d_{\psi }} ; see change of basis. == Pitfalls and ambiguous uses == There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student. === Separation of inner product and vectors === A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as ψ {\displaystyle {\boldsymbol {\psi }}} , and ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} for the inner product. Consider the following dual space bra-vector in the basis { \| e n ⟩ } {\displaystyle \{\|e_{n}\rangle \}} , where { ψ n } {\displaystyle \{\psi _{n}\}} are the complex number coefficients of ⟨ ψ \| {\displaystyle \langle \psi \|} : ⟨ ψ \| = ∑ n ⟨ e n \| ψ n {\displaystyle \langle \psi \|=\sum _{n}\langle e_{n}\|\psi _{n}} It has to be determined by convention if the complex numbers { ψ n } {\displaystyle \{\psi _{n}\}} are inside or outside of the inner product, and each convention gives different results. ⟨ ψ \| ≡ ( ψ , ⋅ ) = ∑ n ( e n , ⋅ ) ψ n {\displaystyle \langle \psi \|\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}} ⟨ ψ \| ≡ ( ψ , ⋅ ) = ∑ n ( e n ψ n , ⋅ ) = ∑ n ( e n , ⋅ ) ψ n ∗ {\displaystyle \langle \psi \|\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n}\psi _{n},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}^{}} === Reuse of symbols === It is common to use the same symbol for labels and constants. For example, α ^ \| α ⟩ = α \| α ⟩ {\displaystyle {\hat {\alpha }}\|\alpha \rangle =\alpha \|\alpha \rangle } , where the symbol α {\displaystyle \alpha } is used simultaneously as the name of the operator α ^ {\displaystyle {\hat {\alpha }}} , its eigenvector \| α ⟩ {\displaystyle \|\alpha \rangle } and the associated eigenvalue α {\displaystyle \alpha } . Sometimes the hat is also dropped for operators, and one can see notation such as A \| a ⟩ = a \| a ⟩ {\displaystyle A\|a\rangle =a\|a\rangle } . === Hermitian conjugate of kets === It is common to see the usage \| ψ ⟩ † = ⟨ ψ \| {\displaystyle \|\psi \rangle ^{\dagger }=\langle \psi \|} , where the dagger ( † {\displaystyle \dagger } ) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, \| ψ ⟩ {\displaystyle \|\psi \rangle } , represents a vector in a complex Hilbert-space H {\displaystyle {\mathcal {H}}} , and the bra, ⟨ ψ \| {\displaystyle \langle \psi \|} , is a linear functional on vectors in H {\displaystyle {\mathcal {H}}} . In other words, \| ψ ⟩ {\displaystyle \|\psi \rangle } is just a vector, while ⟨ ψ \| {\displaystyle \langle \psi \|} is the combination of a vector and an inner product. === Operations inside bras and kets === This is done for a fast notation of scaling vectors. For instance, if the vector \| α ⟩ {\displaystyle \|\alpha \rangle } is scaled by 1 / 2 {\displaystyle 1/{\sqrt {2}}} , it may be denoted \| α / 2 ⟩ {\displaystyle \|\alpha /{\sqrt {2}}\rangle } . This can be ambiguous since α {\displaystyle \alpha } is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. \| α ⟩ = \| α / 2 ⟩ 1 ⊗ \| α / 2 ⟩ 2 {\displaystyle \|\alpha \rangle =\|\alpha /{\sqrt {2}}\rangle _{1}\otimes \|\alpha /{\sqrt {2}}\rangle _{2}} . == Linear operators == === Linear operators acting on kets === A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have certain properties.) In other words, if A ^ {\displaystyle {\hat {A}}} is a linear operator and \| ψ ⟩ {\displaystyle \|\psi \rangle } is a ket-vector, then A ^ \| ψ ⟩ {\displaystyle {\hat {A}}\|\psi \rangle } is another ket-vector. In an N {\displaystyle N} -dimensional Hilbert space, we can impose a basis on the space and represent \| ψ ⟩ {\displaystyle \|\psi \rangle } in terms of its coordinates as a N × 1 {\displaystyle N\times 1} column vector. Using the same basis for A ^ {\displaystyle {\hat {A}}} , it is represented by an N × N {\displaystyle N\times N} complex matrix. The ket-vector A ^ \| ψ ⟩ {\displaystyle {\hat {A}}\|\psi \rangle } can now be computed by matrix multiplication. Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time. === Linear operators acting on bras === Operators can also be viewed as acting on bras from the right hand side. Specifically, if A is a linear operator and ⟨φ\| is a bra, then ⟨φ\|A is another bra defined by the rule ( ⟨ ϕ \| A ) \| ψ ⟩ = ⟨ ϕ \| ( A \| ψ ⟩ ) , {\displaystyle {\bigl (}\langle \phi \|{\boldsymbol {A}}{\bigr )}\|\psi \rangle =\langle \phi \|{\bigl (}{\boldsymbol {A}}\|\psi \rangle {\bigr )}\,,} (in other words, a function composition). This expression is commonly written as (cf. energy inner product) ⟨ ϕ \| A \| ψ ⟩ . {\displaystyle \langle \phi \|{\boldsymbol {A}}\|\psi \rangle \,.} In an N-dimensional Hilbert space, ⟨φ\| can be written as a 1 × N row vector, and A (as in the previous section) is an N × N matrix. Then the bra ⟨φ\|A can be computed by normal matrix multiplication. If the same state vector appears on both bra and ket side, ⟨ ψ \| A \| ψ ⟩ , {\displaystyle \langle \psi \|{\boldsymbol {A}}\|\psi \rangle \,,} then this expression gives the expectation value, or mean or average value, of the observable represented by operator A for the physical system in the state \|ψ⟩. === Outer products === A convenient way to define linear operators on a Hilbert space H is given by the outer product: if ⟨ϕ\| is a bra and \|ψ⟩ is a ket, the outer product \| ϕ ⟩ ⟨ ψ \| {\displaystyle \|\phi \rangle \,\langle \psi \|} denotes the rank-one operator with the rule ( \| ϕ ⟩ ⟨ ψ \| ) ( x ) = ⟨ ψ \| x ⟩ \| ϕ ⟩ . {\displaystyle {\bigl (}\|\phi \rangle \langle \psi \|{\bigr )}(x)=\langle \psi \|x\rangle \|\phi \rangle .} For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication: \| ϕ ⟩ ⟨ ψ \| ≐ ( ϕ 1 ϕ 2 ⋮ ϕ N ) ( ψ 1 ∗ ψ 2 ∗ ⋯ ψ N ∗ ) = ( ϕ 1 ψ 1 ∗ ϕ 1 ψ 2 ∗ ⋯ ϕ 1 ψ N ∗ ϕ 2 ψ 1 ∗ ϕ 2 ψ 2 ∗ ⋯ ϕ 2 ψ N ∗ ⋮ ⋮ ⋱ ⋮ ϕ N ψ 1 ∗ ϕ N ψ 2 ∗ ⋯ ϕ N ψ N ∗ ) {\displaystyle \|\phi \rangle \,\langle \psi \|\doteq {\begin{pmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{pmatrix}}{\begin{pmatrix}\psi _{1}^{}&\psi _{2}^{}&\cdots &\psi _{N}^{}\end{pmatrix}}={\begin{pmatrix}\phi _{1}\psi _{1}^{}&\phi _{1}\psi _{2}^{}&\cdots &\phi _{1}\psi _{N}^{}\\\phi _{2}\psi _{1}^{}&\phi _{2}\psi _{2}^{}&\cdots &\phi _{2}\psi _{N}^{}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{}&\phi _{N}\psi _{2}^{}&\cdots &\phi _{N}\psi _{N}^{}\end{pmatrix}}} The outer product is an N × N matrix, as expected for a linear operator. One of the uses of the outer product is to construct projection operators. Given a ket \|ψ⟩ of norm 1, the orthogonal projection onto the subspace spanned by \|ψ⟩ is \| ψ ⟩ ⟨ ψ \| . {\displaystyle \|\psi \rangle \,\langle \psi \|\,.} This is an idempotent in the algebra of observables that acts on the Hilbert space. === Hermitian conjugate operator === Just as kets and bras can be transformed into each other (making \|ψ⟩ into ⟨ψ\|), the element from the dual space corresponding to A\|ψ⟩ is ⟨ψ\|A†, where A† denotes the Hermitian conjugate (or adjoint) of the operator A. In other words, \| ϕ ⟩ = A \| ψ ⟩ if and only if ⟨ ϕ \| = ⟨ ψ \| A † . {\displaystyle \|\phi \rangle =A\|\psi \rangle \quad {\text{if and only if}}\quad \langle \phi \|=\langle \psi \|A^{\dagger }\,.} If A is expressed as an N × N matrix, then A† is its conjugate transpose. == Properties == Bra–ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, c1 and c2 denote arbitrary complex numbers, c denotes the complex conjugate of c, A and B denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets. === Linearity === Since bras are linear functionals, ⟨ ϕ \| ( c 1 \| ψ 1 ⟩ + c 2 \| ψ 2 ⟩ ) = c 1 ⟨ ϕ \| ψ 1 ⟩ + c 2 ⟨ ϕ \| ψ 2 ⟩ . {\displaystyle \langle \phi \|{\bigl (}c_{1}\|\psi _{1}\rangle +c_{2}\|\psi _{2}\rangle {\bigr )}=c_{1}\langle \phi \|\psi _{1}\rangle +c_{2}\langle \phi \|\psi _{2}\rangle \,.} By the definition of addition and scalar multiplication of linear functionals in the dual space, ( c 1 ⟨ ϕ 1 \| + c 2 ⟨ ϕ 2 \| ) \| ψ ⟩ = c 1 ⟨ ϕ 1 \| ψ ⟩ + c 2 ⟨ ϕ 2 \| ψ ⟩ . {\displaystyle {\bigl (}c_{1}\langle \phi _{1}\|+c_{2}\langle \phi _{2}\|{\bigr )}\|\psi \rangle =c_{1}\langle \phi _{1}\|\psi \rangle +c_{2}\langle \phi _{2}\|\psi \rangle \,.} === Associativity === Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the associative property holds). For example: ⟨ ψ \| ( A \| ϕ ⟩ ) = ( ⟨ ψ \| A ) \| ϕ ⟩ = def ⟨ ψ \| A \| ϕ ⟩ ( A \| ψ ⟩ ) ⟨ ϕ \| = A ( \| ψ ⟩ ⟨ ϕ \| ) = def A \| ψ ⟩ ⟨ ϕ \| {\displaystyle {\begin{aligned}\langle \psi \|{\bigl (}A\|\phi \rangle {\bigr )}={\bigl (}\langle \psi \|A{\bigr )}\|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi \|A\|\phi \rangle \\{\bigl (}A\|\psi \rangle {\bigr )}\langle \phi \|=A{\bigl (}\|\psi \rangle \langle \phi \|{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A\|\psi \rangle \langle \phi \|\end{aligned}}} and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously because of the equalities on the left. Note that the associative property does not hold for expressions that include nonlinear operators, such as the antilinear time reversal operator in physics. === Hermitian conjugation === Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †) of expressions. The formal rules are: The Hermitian conjugate of a bra is the corresponding ket, and vice versa. The Hermitian conjugate of a complex number is its complex conjugate. The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e., ( x † ) † = x . {\displaystyle \left(x^{\dagger }\right)^{\dagger }=x\,.} Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each. These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows: Kets: ( c 1 \| ψ 1 ⟩ + c 2 \| ψ 2 ⟩ ) † = c 1 ∗ ⟨ ψ 1 \| + c 2 ∗ ⟨ ψ 2 \| . {\displaystyle {\bigl (}c_{1}\|\psi _{1}\rangle +c_{2}\|\psi _{2}\rangle {\bigr )}^{\dagger }=c_{1}^{}\langle \psi _{1}\|+c_{2}^{}\langle \psi _{2}\|\,.} Inner products: ⟨ ϕ \| ψ ⟩ ∗ = ⟨ ψ \| ϕ ⟩ . {\displaystyle \langle \phi \|\psi \rangle ^{}=\langle \psi \|\phi \rangle \,.} Note that ⟨φ\|ψ⟩ is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e., ( ⟨ ϕ \| ψ ⟩ ) † = ⟨ ϕ \| ψ ⟩ ∗ {\displaystyle {\bigl (}\langle \phi \|\psi \rangle {\bigr )}^{\dagger }=\langle \phi \|\psi \rangle ^{}} Matrix elements: ⟨ ϕ \| A \| ψ ⟩ † = ⟨ ψ \| A † \| ϕ ⟩ ⟨ ϕ \| A † B † \| ψ ⟩ † = ⟨ ψ \| B A \| ϕ ⟩ . {\displaystyle {\begin{aligned}\langle \phi \|A\|\psi \rangle ^{\dagger }&=\left\langle \psi \left\|A^{\dagger }\right\|\phi \right\rangle \\\left\langle \phi \left\|A^{\dagger }B^{\dagger }\right\|\psi \right\rangle ^{\dagger }&=\langle \psi \|BA\|\phi \rangle \,.\end{aligned}}} Outer products: ( ( c 1 \| ϕ 1 ⟩ ⟨ ψ 1 \| ) + ( c 2 \| ϕ 2 ⟩ ⟨ ψ 2 \| ) ) † = ( c 1 ∗ \| ψ 1 ⟩ ⟨ ϕ 1 \| ) + ( c 2 ∗ \| ψ 2 ⟩ ⟨ ϕ 2 \| ) . {\displaystyle {\Big (}{\bigl (}c_{1}\|\phi _{1}\rangle \langle \psi _{1}\|{\bigr )}+{\bigl (}c_{2}\|\phi _{2}\rangle \langle \psi _{2}\|{\bigr )}{\Big )}^{\dagger }={\bigl (}c_{1}^{}\|\psi _{1}\rangle \langle \phi _{1}\|{\bigr )}+{\bigl (}c_{2}^{}\|\psi _{2}\rangle \langle \phi _{2}\|{\bigr )}\,.} == Composite bras and kets == Two Hilbert spaces V and W may form a third space V ⊗ W by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.) If \|ψ⟩ is a ket in V and \|φ⟩ is a ket in W, the tensor product of the two kets is a ket in V ⊗ W. This is written in various notations: \| ψ ⟩ \| ϕ ⟩ , \| ψ ⟩ ⊗ \| ϕ ⟩ , \| ψ ϕ ⟩ , \| ψ , ϕ ⟩ . {\displaystyle \|\psi \rangle \|\phi \rangle \,,\quad \|\psi \rangle \otimes \|\phi \rangle \,,\quad \|\psi \phi \rangle \,,\quad \|\psi ,\phi \rangle \,.} See quantum entanglement and the EPR paradox for applications of this product. == The unit operator == Consider a complete orthonormal system (basis), { e i \| i ∈ N } , {\displaystyle \{e_{i}\ \|\ i\in \mathbb {N} \}\,,} for a Hilbert space H, with respect to the norm from an inner product ⟨·,·⟩. From basic functional analysis, it is known that any ket \| ψ ⟩ {\displaystyle \|\psi \rangle } can also be written as \| ψ ⟩ = ∑ i ∈ N ⟨ e i \| ψ ⟩ \| e i ⟩ , {\displaystyle \|\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}\|\psi \rangle \|e_{i}\rangle ,} with ⟨·\|·⟩ the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars, it follows that ∑ i ∈ N \| e i ⟩ ⟨ e i \| = I {\displaystyle \sum _{i\in \mathbb {N} }\|e_{i}\rangle \langle e_{i}\|=\mathbb {I} } must be the identity operator, which sends each vector to itself. This, then, can be inserted in any expression without affecting its value; for example ⟨ v \| w ⟩ = ⟨ v \| ( ∑ i ∈ N \| e i ⟩ ⟨ e i \| ) \| w ⟩ = ⟨ v \| ( ∑ i ∈ N \| e i ⟩ ⟨ e i \| ) ( ∑ j ∈ N \| e j ⟩ ⟨ e j \| ) \| w ⟩ = ⟨ v \| e i ⟩ ⟨ e i \| e j ⟩ ⟨ e j \| w ⟩ , {\displaystyle {\begin{aligned}\langle v\|w\rangle &=\langle v\|\left(\sum _{i\in \mathbb {N} }\|e_{i}\rangle \langle e_{i}\|\right)\|w\rangle \\&=\langle v\|\left(\sum _{i\in \mathbb {N} }\|e_{i}\rangle \langle e_{i}\|\right)\left(\sum _{j\in \mathbb {N} }\|e_{j}\rangle \langle e_{j}\|\right)\|w\rangle \\&=\langle v\|e_{i}\rangle \langle e_{i}\|e_{j}\rangle \langle e_{j}\|w\rangle \,,\end{aligned}}} where, in the last line, the Einstein summation convention has been used to avoid clutter. In quantum mechanics, it often occurs that little or no information about the inner product ⟨ψ\|φ⟩ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients ⟨ψ\|ei⟩ = ⟨ei\|ψ⟩* and ⟨ei\|φ⟩ of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more. For more information, see Resolution of the identity, I = ∫ d x \| x ⟩ ⟨ x \| = ∫ d p \| p ⟩ ⟨ p \| , {\displaystyle {\mathbb {I} }=\int \!dx~\|x\rangle \langle x\|=\int \!dp~\|p\rangle \langle p\|,} where \| p ⟩ = ∫ d x e i x p / ℏ \| x ⟩ 2 π ℏ . {\displaystyle \|p\rangle =\int dx{\frac {e^{ixp/\hbar }\|x\rangle }{\sqrt {2\pi \hbar }}}.} Since ⟨x′\|x⟩ = δ(x − x′), plane waves follow, ⟨ x \| p ⟩ = e i x p / ℏ 2 π ℏ . {\displaystyle \langle x\|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.} In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate \| ϖ ⟩ = lim p → 0 \| p ⟩ {\textstyle \|\varpi \rangle =\lim _{p\to 0}\|p\rangle } in the momentum representation, i.e., p ^ \| ϖ ⟩ = 0 {\displaystyle {\hat {p}}\|\varpi \rangle =0} . Consequently, the corresponding wavefunction is a constant, ⟨ x \| ϖ ⟩ 2 π ℏ = 1 {\displaystyle \langle x\|\varpi \rangle {\sqrt {2\pi \hbar }}=1} , and \| x ⟩ = δ ( x ^ − x ) \| ϖ ⟩ 2 π ℏ , {\displaystyle \|x\rangle =\delta ({\hat {x}}-x)\|\varpi \rangle {\sqrt {2\pi \hbar }},} as well as \| p ⟩ = exp ⁡ ( i p x ^ / ℏ ) \| ϖ ⟩ . {\displaystyle \|p\rangle =\exp(ip{\hat {x}}/\hbar )\|\varpi \rangle .} Typically, when all matrix elements of an operator such as ⟨ x \| A \| y ⟩ {\displaystyle \langle x\|A\|y\rangle } are available, this resolution serves to reconstitute the full operator, ∫ d x d y \| x ⟩ ⟨ x \| A \| y ⟩ ⟨ y \| = A . {\displaystyle \int dx\,dy\,\|x\rangle \langle x\|A\|y\rangle \langle y\|=A\,.} == Notation used by mathematicians == The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space). Let ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )} be a Hilbert space and h ∈ H a vector in H. What physicists would denote by \|h⟩ is the vector itself. That is, \| h ⟩ ∈ H . {\displaystyle \|h\rangle \in {\mathcal {H}}.} Let H* be the dual space of H. This is the space of linear functionals on H. The embedding Φ : H ↪ H ∗ {\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{}} is defined by Φ ( h ) = φ h {\displaystyle \Phi (h)=\varphi _{h}} , where for every h ∈ H the linear functional φ h : H → C {\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} } satisfies for every g ∈ H the functional equation φ h ( g ) = ⟨ h , g ⟩ = ⟨ h ∣ g ⟩ {\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle } . Notational confusion arises when identifying φh and g with ⟨h\| and \|g⟩ respectively. This is because of literal symbolic substitutions. Let φ h = H = ⟨ h ∣ {\displaystyle \varphi _{h}=H=\langle h\mid } and let g = G = \|g⟩. This gives φ h ( g ) = H ( g ) = H ( G ) = ⟨ h \| ( G ) = ⟨ h \| ( \| g ⟩ ) . {\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h\|(G)=\langle h\|{\bigl (}\|g\rangle {\bigr )}\,.} One ignores the parentheses and removes the double bars. Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an asterisk but an overline (which the physicists reserve for averages and the Dirac spinor adjoint) to denote complex conjugate numbers; i.e., for scalar products mathematicians usually write ⟨ ϕ , ψ ⟩ = ∫ ϕ ( x ) ψ ( x ) ¯ d x , {\displaystyle \langle \phi ,\psi \rangle =\int \phi (x){\overline {\psi (x)}}\,dx\,,} whereas physicists would write for the same quantity ⟨ ψ \| ϕ ⟩ = ∫ d x ψ ∗ ( x ) ϕ ( x ) . {\displaystyle \langle \psi \|\phi \rangle =\int dx\,\psi ^{}(x)\phi (x)~.} == See also == Angular momentum diagrams (quantum mechanics) n-slit interferometric equation Quantum state Inner product space == Notes == == References == Dirac, P. A. M. (1939). "A new notation for quantum mechanics". Mathematical Proceedings of the Cambridge Philosophical Society. 35 (3): 416–418. Bibcode:1939PCPS...35..416D. doi:10.1017/S0305004100021162. S2CID 121466183. Also see his standard text, The Principles of Quantum Mechanics, IV edition, Clarendon Press (1958), ISBN 978-0198520115 Grassmann, H. (1862). Extension Theory. History of Mathematics Sources. 2000 translation by Lloyd C. Kannenberg. American Mathematical Society, London Mathematical Society. Cajori, Florian (1929). A History Of Mathematical Notations Volume II. Open Court Publishing. p. 134. ISBN 978-0-486-67766-8. {{cite book}}: ISBN / Date incompatibility (help) Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). ISBN 0-306-44790-8. Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. Vol. III. Reading, MA: Addison-Wesley. ISBN 0-201-02118-8. Sakurai, J J; Napolitano, J (2021). Modern Quantum Mechanics (3rd ed.). Cambridge University Press. ISBN 978-1-108-42241-3. == External links == Richard Fitzpatrick, "Quantum Mechanics: A graduate level course", The University of Texas at Austin. Includes: 1. Ket space 2. Bra space 3. Operators 4. The outer product 5. Eigenvalues and eigenvectors Robert Littlejohn, Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra–ket notation. University of California, Berkeley. Gieres, F. (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". Rep. Prog. Phys. 63 (12): 1893–1931. arXiv:quant-ph/9907069. Bibcode:2000RPPh...63.1893G. doi:10.1088/0034-4885/63/12/201. S2CID 10854218.
Wikipedia:Brezis–Gallouët inequality#0	In mathematical analysis, the Brezis–Gallouët inequality, named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provides an explicit bound, which depends only logarithmically on the second derivatives. It is useful in the study of partial differential equations. Let Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} be the exterior or the interior of a bounded domain with regular boundary, or R 2 {\displaystyle \mathbb {R} ^{2}} itself. Then the Brezis–Gallouët inequality states that there exists a real C {\displaystyle C} only depending on Ω {\displaystyle \Omega } such that, for all u ∈ H 2 ( Ω ) {\displaystyle u\in H^{2}(\Omega )} which is not a.e. equal to 0, ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) ( 1 + ( log ⁡ ( 1 + ‖ u ‖ H 2 ( Ω ) ‖ u ‖ H 1 ( Ω ) ) ) 1 / 2 ) . {\displaystyle \displaystyle \\|u\\|_{L^{\infty }(\Omega )}\leq C\\|u\\|_{H^{1}(\Omega )}\left(1+{\Bigl (}\log {\bigl (}1+{\frac {\\|u\\|_{H^{2}(\Omega )}}{\\|u\\|_{H^{1}(\Omega )}}}{\bigr )}{\Bigr )}^{1/2}\right).} Noticing that, for any v ∈ H 2 ( R 2 ) {\displaystyle v\in H^{2}(\mathbb {R} ^{2})} , there holds ∫ R 2 ( ( ∂ 11 2 v ) 2 + 2 ( ∂ 12 2 v ) 2 + ( ∂ 22 2 v ) 2 ) = ∫ R 2 ( ∂ 11 2 v + ∂ 22 2 v ) 2 , {\displaystyle \int _{\mathbb {R} ^{2}}{\bigl (}(\partial _{11}^{2}v)^{2}+2(\partial _{12}^{2}v)^{2}+(\partial _{22}^{2}v)^{2}{\bigr )}=\int _{\mathbb {R} ^{2}}{\bigl (}\partial _{11}^{2}v+\partial _{22}^{2}v{\bigr )}^{2},} one deduces from the Brezis-Gallouet inequality that there exists C > 0 {\displaystyle C>0} only depending on Ω {\displaystyle \Omega } such that, for all u ∈ H 2 ( Ω ) {\displaystyle u\in H^{2}(\Omega )} which is not a.e. equal to 0, ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) ( 1 + ( log ⁡ ( 1 + ‖ Δ u ‖ L 2 ( Ω ) ‖ u ‖ H 1 ( Ω ) ) ) 1 / 2 ) . {\displaystyle \displaystyle \\|u\\|_{L^{\infty }(\Omega )}\leq C\\|u\\|_{H^{1}(\Omega )}\left(1+{\Bigl (}\log {\bigl (}1+{\frac {\\|\Delta u\\|_{L^{2}(\Omega )}}{\\|u\\|_{H^{1}(\Omega )}}}{\bigr )}{\Bigr )}^{1/2}\right).} The previous inequality is close to the way that the Brezis-Gallouet inequality is cited in. == See also == Ladyzhenskaya inequality Agmon's inequality == References ==
Wikipedia:Brian Marcus#0	Brian Marcus is an American-born mathematician who works in Canada. He is a professor in the department of mathematics at the University of British Columbia (UBC), where he is the site director of the Pacific Institute for the Mathematical Sciences (PIMS), a fellow of the AMS and the IEEE. He was the department head of mathematics at UBC from 2002 to 2007 and the deputy director of PIMS from 2016 to 2018. == Education and academic career == Marcus earned his Ph.D. in 1975 from the University of California, Berkeley (UC Berkeley); his supervisor was Rufus Bowen. He then worked as an IBM Watson Postdoctoral Fellow, an associate professor at UNC Chapel Hill and a researcher at IBM Research – Almaden. He additionally held visiting associate professor positions at UC Berkeley, University of California, Santa Cruz, and Stanford University. From 2016 to 2018, he was the deputy director of the Pacific Institute for the Mathematical Sciences, where, as of 2019, he is the UBC Site Director. He is one of the representatives of the Pacific Rim Mathematical Association. His main areas of research are ergodic theory, symbolic dynamics and information theory. He has published contributions in the theory of horocycle flows and entropy. Marcus has written over seventy research papers, some of them published in Annals of Mathematics, Inventiones Mathematicae and Journal of the AMS. His collaborators include Wolfgang Krieger, Roy Adler, Rufus Bowen, Dominique Perrin, Jack Wolf, Yuval Peres and Sheldon Newhouse. Marcus (with Doug Lind) wrote the book An Introduction to Symbolic Dynamics and Coding (currently with more than 3,000 citations on Google Scholar), and (with Susan Williams) the Scholarpedia article on symbolic dynamics. In 1993, Marcus was awarded the Leonard J. Abraham Prize Paper award of the IEEE. In 1999, he was elected as a fellow of the IEEE. He was named a fellow of the American Mathematical Society in 2018; the citation was "For contributions to dynamical systems, symbolic dynamics and applications to data storage problems, and service to the profession." == Selected publications == === Books === 1995: (with Doug Lind) An Introduction to Symbolic Dynamics and Coding, Cambridge University Press doi:10.1017/CBO9780511626302. === Research papers === Ergodic properties of horocycle flows for surfaces of negative curvature, Annals of Mathematics 105 (1977), 81-105 doi:10.2307/1971026. with Rufus Bowen: Unique ergodicity of horocycle foliations, Israel Journal of Mathematics 26 (1977) 43-67 doi:10.1007/BF03007655. The horocycle flow is mixing of all degrees, Inventiones Mathematicae 46 (1978)201-209 doi:10.1007/BF01390274 with Roy Adler: Topological entropy and equivalence of dynamical systems, Memoirs of the American Mathematical Society 219 (1979) doi:10.1090/memo/0219. Topological conjugacy of horocycle flows, American Journal of Mathematics (1983) 623-632 doi:10.2307/2374316 with Selim Tuncel: Entropy at a weight-per-symbol and an imbedding theorem for Markov chains, Inventiones Mathematicae 102 (1990), 235-266 doi:10.1007/BF01233428. with Selim Tuncel: Matrices of polynomials, positivity, and finite equivalence of Markov chains, Journal of the American Mathematical Society 6 (1993), 131- 147 doi:10.1090/S0894-0347-1993-1168959-X. == See also == Daniel Rudolph – American mathematician, contemporary of Brian Marcus == References ==
Wikipedia:Brian Swimme#0	Brian Thomas Swimme (born 1950) is a professor at the California Institute of Integral Studies, in San Francisco, where he teaches evolutionary cosmology to graduate students in the philosophy, cosmology, and consciousness program. He received his Ph.D. (1978) from the department of mathematics at the University of Oregon for work with Richard Barrar on singularity theory, with a dissertation titled Singularities in the N-Body Problem. Swimme's published work portrays the 14-billion-year trajectory of cosmogenesis "as a spellbinding drama, full of suspense, valor, tragedy, and celebration". His work includes The Universe is a Green Dragon (1984), The Universe Story, written with Thomas Berry (1992), The Hidden Heart of the Cosmos (1996), and The Journey of the Universe, written with Mary Evelyn Tucker (2011). Swimme is the producer of three DVD series: Canticle to the Cosmos (1990), The Earth's Imagination (1998), and The Powers of the Universe (2004). Swimme teamed with Mary Evelyn Tucker, David Kennard, Patsy Northcutt, and Catherine Butler to produce Journey of the Universe, a Northern California Emmy-winning film released in 2011. These works draw together scientific discoveries in astronomy, geology and biology, with humanistic insights concerning the nature of the universe. == Background == Swimme is an evolutionary cosmologist on the graduate faculty of the California Institute of Integral Studies in the philosophy, cosmology and consciousness and also ecology, spirituality, and religion programs, areas of study within the philosophy and religion program. Swimme's primary field of research is the nature of the evolutionary dynamics of the universe. He has developed an interpretation of the human as an emergent being within the universe and earth. His central concern is the role of the human within the earth community, the cultural implications of the epic of evolution, and the role of humanity in the unfolding story of earth and cosmos. Toward this goal, he founded the Center for the Story of the Universe. Swimme was featured in the television series Soul of the Universe (The BBC, 1991) and The Sacred Balance produced by David Suzuki (CBC and PBS, 2003). He is the producer of a twelve-part DVD series Canticle to the Cosmos. Other DVD programs featuring Swimme's ideas include The Earth's Imagination and The Powers of the Universe. Swimme founded the international Epic of Evolution Society in 1998. This was a result of his participation in the conference Dialogue on Science, Ethics, and Religion organized by the American Association for the Advancement of Science at the Field Museum the year before. == Philosophy == Thomas Berry introduced Swimme to the work of Pierre Teilhard de Chardin. Swimme is deeply influenced by Teilhard's ideas. Swimme described his discovery of Teilhard in his foreword to Sarah Appleton Weber's translation of Teilhard's The Human Phenomenon: He adopted Teilhard's thinking that everything in existence has a physical as well as a spiritual dimension. He believes the universe is a deep transfiguration process. Love, truth, compassion and zest—all of these qualities regarded as divine become embodied in the universe. In this way, the universe is imagined as evolving with a telos of beauty. Suzanne Taylor, founder of Mighty Companions, said Swimme is a charismatic person who seeks to place scientific technology in its context of the infancy of the Earth community as it struggles for reconnection to its sacred source. She believes that human beings are the current culmination of the still-evolving universe. Swimme tells the story of the evolution of the universe and attempts to pull people into a universe of meaning, where there is not only connectivity, but directionality as well. In Canticle to the Cosmos, Swimme says: "If you look at the disasters happening on our planet, it's because the cosmos is not understood as sacred ... a way out of our difficulty is a journey into the universe as sacred." Harvard astrophysicist Eric Chaisson wrote that Swimme, a mathematician by training, seeks a larger, warmer, more noble science story, stating that, not merely a collection of facts, science should be a student's guide to a grand world-view, including, if possible, meaning, purpose and value; he sees the cosmological perspective as one to which all modern scientists can objectively subscribe, yet the meaning and purpose of it being a subjective outgrowth of an individual's reflection upon that cosmology ... In a 2007 interview with Robert Wright, Swimme said ... if you take Buddhism and Christianity and so forth there's a kind of battle — a subtle sort of struggle taking place because they're not standing in a common ground but ... take the Earth or ecology then suddenly they can begin to explore what they have to offer. So I do think absolutely that ... there will be a flourishing of religions, not a withering away. And they will flourish to the degree that they will move into the context of planet and universe. Pacific Sun newspaper reported that Swimme was at the forefront of a new movement that integrates science and spirituality. Swimme believes there is a new story, the epic of evolution, a cosmological narrative that begins with the Big Bang, which started the whole process, and proceeds to the evolution of the universe and life on Earth. This manner of study, which engages heart and mind together, seems to teeter on the brink of religion. He believes that science, holistically, can have a great impact on people. Big History science is filled with little mysterious coincidences, upon which our entire existence rests. Swimme notes that this inspires awe and humility, and that this cosmology puts people in their proper place. He thinks that the popular view is that the Earth is like a gravel pit or a hardware store, that the Earth is just stuff to be used. He believes that consumerism has become the dominant world faith, exploiting the riches of the Earth. His fundamental aim is to present a new cosmology—one grounded in a contemporary understanding of the universe but nourished by ancient spiritual convictions that help give it meaning. In an interview in 2001, Swimme gave a basic summary of "the whole story in one line": "This is the greatest discovery of the scientific enterprise: You take hydrogen gas, and you leave it alone, and it turns into rosebushes, giraffes, and humans." Writing for the BBC in 2009, Mark Vernon said that "Swimme believes that 'the universe is attempting to be felt', which makes him a pantheist", and noted that Swimme's work "is avidly read by individuals in New Age and ecological circles". == Major publications == Manifesto for a Global Civilization (with Matthew Fox), Bear & Company, 1982, ISBN 0-939680-05-X The Universe is a Green Dragon: A Cosmic Creation Story, Bear & Company, 1984, ISBN 0-939680-14-9 The Universe Story: From the Primordial Flaring Forth to the Ecozoic Era: A Celebration of the Unfolding of the Cosmos, Harper San Francisco, 1992 (1994, ISBN 0-06-250835-0)—a culmination of a 10-year collaboration with cultural historian Thomas Berry The Hidden Heart of the Cosmos, Orbis, 1996 (revised 2019, ISBN 978-1626983434) A Walk Through Time: From Stardust to Us—The Evolution of Life on Earth (with Sidney Liebes and Elisabet Sahtouris), John Wiley & Sons, 1998, ISBN 0-471-31700-4 "Cosmological Education for Future Generations", Chapter 5 in The Thirteenth Labor: Improving Science Education, edited by Eric Chaisson and Tae-Chang Kim, CRC Press, 1999, ISBN 9057005387, doi:10.1201/9781003078357-5 Journey of the Universe (with Mary Evelyn Tucker), Yale University Press, 2011, ISBN 978-0-300-17190-7 Cosmogenesis: An Unveiling of the Expanding Universe, Counterpoint, 2022, ISBN 978-1-640-09398-0 Swimme's media work includes the video series, Canticle to the Cosmos, The Hidden Heart of the Cosmos and The Powers of the Universe. Swimme introduced Barbara Hand Clow in her books, Heart of the Christos: Starseeding from the Pleiades, Bear & Company, 1989, ISBN 0-939680-59-9, and The Pleiadian Agenda: A New Cosmology for the Age of Light, Bear & Company, 1995, ISBN 1-879181-30-4. == See also == Creation–evolution controversy Mary Evelyn Tucker § Journey of the Universe Timeline of cosmology == References == == External links == Official website Center for the Story of the Universe Journey of the Universe Brian Swimme MP3 audio from Shift in Action, sponsored by Institute of Noetic Sciences
Wikipedia:Brigitte Servatius#0	Brigitte Irma Servatius (born 1954) is a mathematician specializing in matroids and structural rigidity. She is a professor of mathematics at Worcester Polytechnic Institute, and has been the editor-in-chief of the Pi Mu Epsilon Journal since 1999. == Education and career == Servatius is originally from Graz in Austria. As a student at an all-girl gymnasium in Graz that specialized in language studies rather than mathematics, her interest in mathematics was sparked by her participation in a national mathematical olympiad, and she went on to earn master's degrees in mathematics and physics at the University of Graz. She became a high school mathematics and science teacher in Leibnitz. She moved to the US in 1981, to begin doctoral studies at Syracuse University. She completed her Ph.D. in 1987, and joined the Worcester Polytechnic Institute faculty in the same year. Her dissertation, Planar Rigidity, was supervised by Jack Graver. == Contributions == While still in Austria, Servatius began working on combinatorial group theory, and her first publication (appearing while she was a graduate student) is in that subject.[Z] She switched to the theory of structural rigidity for her doctoral research, and later became the author (with Jack Graver and Herman Servatius) of the book Combinatorial Rigidity (1993).[G] Another well-cited paper of hers in this area characterizes the planar Laman graphs, the minimally rigid graphs that can be embedded without crossings in the plane, as the graphs of pseudotriangulations, partitions of a plane region into subregions with three convex corners studied in computational geometry.[H] Servatius is also the co-editor of a book on matroid theory.[B] With Tomaž Pisanski she wrote the book Configurations from a Graphical Viewpoint (2013), on configurations of points and lines in the plane with the same number of points touching each two lines and the same number of lines touching each two points.[P] Other topics in her research include graph duality[S] and the triconnected components of infinite graphs.[D] == Selected publications == == References == == External links == Home page Brigitte Servatius publications indexed by Google Scholar
Wikipedia:Brigitte Vallée#0	Brigitte Vallée (née Salesse) (born 6 June 1950, in Courbevoie, Hauts-de-Seine, France) is a French mathematician and computer scientist. She entered the École Normale Supérieure de Jeunes Filles in 1970, and received her PhD in 1986 at the University of Caen (Lattice reduction algorithms in small dimensions). Her doctoral advisor was Jacques Stern. Vallée has been Director of Research at the French CNRS at Université de Caen, since 2001 and specialized in computational number theory and analysis of algorithms. Amongst the algorithms she studied are the celebrated LLL algorithm used for basis reductions in Euclidean lattice and the different Euclidean algorithms to determine GCD. The main tool used to achieve her results is the so-called dynamical analysis. Loosely speaking, it is a mix between analysis of algorithms and dynamical systems. Brigitte Vallée greatly contributed to the development of this method. In the early 90s, Brigitte Vallée's work on small modular squares allowed her to hold the fastest factorisation algorithm with a proved probabilistic complexity bound. Nowadays, other factorisation algorithms are faster. She was appointed a knight of the Legion of Honor by the Ministry of Higher Education and Research on 12 July 2013. == Selected publications == According to zbMath, Vallée has authored 88 publications since 1986, including 3 books. Brigitte Vallée, Generation of Elements with Small Modular Squares and Provably Fast Integer Factoring Algorithms, Mathematics of Computation, Vol. 56, No. 194 (Apr., 1991), pp. 823–849. Brigitte Vallée, Algorithmique en géométrie des nombres. Applications à la cryptographie et à la factorisation des entiers (A geometric approach to the reduction of small-scale networks), 1986 [University thesis] == References == == External links == Website List of Publications
Wikipedia:British Mathematical Olympiad#0	The British Mathematical Olympiad (BMO) forms part of the selection process for the UK International Mathematical Olympiad team and for other international maths competitions, including the European Girls' Mathematical Olympiad, the Romanian Master of Mathematics and Sciences, and the Balkan Mathematical Olympiad. It is organised by the British Mathematical Olympiad Subtrust, which is part of the United Kingdom Mathematics Trust. There are two rounds, the BMO1 and the BMO2. == BMO Round 1 == The first round of the BMO is held in November each year, and from 2006 is an open entry competition. The qualification to BMO Round 1 is through the Senior Mathematical Challenge or the Mathematical Olympiad for Girls. Students who do not make the qualification may be entered at the discretion of their school for a fee of £40. The paper lasts 3½ hours, and consists of six questions (from 2005), each worth 10 marks. The exam in the 2020-2021 cycle was adjusted to consist of two sections, first section with 4 questions each worth 5 marks (only answers required), and second section with 3 question each worth 10 marks (full solutions required). The duration of the exam had been reduced to 2½ hours, due to the difficulties of holding a 3½ hours exam under COVID-19. Candidates are required to write full proofs to the questions. An answer is marked on either a "0+" or a "10-" mark scheme, depending on whether the answer looks generally complete or not. An answer judged incomplete or unfinished is usually capped at 3 or 4, whereas for an answer judged as complete, marks may be deducted for minor errors or poor reasoning but it is likely to get a score of 7 or more. As a result, it is uncommon for an answer to score a middling mark between 4 and 6. While around 1000 gain automatic qualification to sit the BMO1 paper each year, the additional discretionary and international students means that since 2016, on average, around 1600 candidates have been entered for BMO1 each year. Although these candidates represent the very best mathematicians in their age group, the difficulty level of the BMO papers mean that many of these attain a very low score. The scores were particularly low until 2004, for example, when the median score was approximately 5-6 (out of 50). In 2005, UKMT changed the system and added an extra easier question meaning the median is now raised. In 2008, 23 students scored more than 40/60 and around 50 got over 30/60. In addition to the British students, until 2018, there was a history of about 20 students from New Zealand being invited to take part. In recent years, entries to BMO have been made from schools in Ireland, Kazakhstan, India, China, South Korea, Hong Kong, Singapore, and Thailand. BMO1 paper for the cycle 2021-22 attracted 1857 entries. Only 5 candidates scored 90% or more. A score of 21/60 was enough to earn a Distinction, awarded to top 26% of the candidates. From the results of the BMO1, around 100 top scoring students are invited to sit the BMO2. For the 2021-22 cycle, the score needed for to qualify for BMO2 was 33 for a year 13 pupil and 29 for a pupil in year 10 and below. Students who did not take the BMO1, or who did not qualify for an invitation, may be entered into the next round at the discretion of their school through payment of a fee of £50. Typically, around top 20 scoring students in BMO1 feature on what is known as the BMO leaderboard. == BMO Round 2 == BMO2 (known as the Further International Selection Test, FIST from 1972 to 1991) is normally held in late January or early February, and is significantly more difficult than BMO1. BMO2 also lasts 3½ hours, but consists of only four questions, each worth 10 marks. Like the BMO1 paper, it is not designed merely to test knowledge of advanced mathematics but rather to test the candidate's ability to apply the mathematical knowledge to solve unusual problems and is an entry point to training and selection for the international competitions. BMO2 paper for the cycle 2021-22 attracted over 200 entries. A score of 17/40 was enough to earn a Distinction, awarded to top 25% of the candidates. Only 4 candidates scored more than 30/40. Twenty-four of the top scorers from BMO2 are subsequently invited to the training camp at Trinity College, Cambridge for the first stage of the IMO UK team selection. The top 4 female scorers from BMO2 are selected to represent the UK at the European Girls' Mathematical Olympiad. == IMO Selection Papers == For more information about IMO selection in other countries, see International Mathematical Olympiad selection process Since 1985, further selection tests have been used after BMO2 to select the IMO team. (The team was selected following the single BMO paper from 1967 to 1971, then following the FIST paper for some years from 1972.) Initially these third-stage tests resulted in selection of both team and reserve; from 1993 a squad (team plus reserve) was selected following these tests with the team being separated from the reserve after further correspondence training, and after further selection tests from 2001 onwards. The third-stage tests have had names including FIST 2 (1985), Second International Selection Test (SIST), Reading Selection Test (1987), Final Selection Test (FST, 1992 to 2001) and First Selection Test (FST, from 2002); the fourth-stage tests have been Team Selection Test (TST, 2001) and Next Selection Test (NST, 2002 onwards). These tests have been held at training and selection camps in several locations, recently Trinity College, Cambridge and Oundle School. Since 2017, the tests are simply called Team Selection Tests (TSTs). Six TSTs were held in 2017 and 2018. Since 2019, the number of TSTs have been reduced to 4 or two rounds, the first round is held in late April and the second round in late May. The UK IMO squad of 10 is selected following the first round of tests with the final team of 6 to represent the UK at the International Mathematical Olympiad announced following the second round. In 2021, the first round of TSTs was held on 16–17 April and the second round on 5–6 June. The UK squad for IMO 2021 was Mohit Hulse, Isaac King, Samuel Liew, Yuka Machino, Daniel Naylor, and Jenni Voon, with Tommy Walker Mackay being the first reserve. This is only the second time over the last 54 years of the UK's participation at the International Mathematical Olympiad since 1967 that more than one female contestant has made the UK IMO team. == See also == International Mathematical Olympiad United Kingdom Mathematics Trust == References == == External links == British Mathematical Olympiad Committee site UK Mathematics Trust site for details of UK Mathematics competitions Database of people who have represented the UK at international maths competitions
Wikipedia:British Mathematical Olympiad Subtrust#0	The British Mathematical Olympiad Subtrust (BMOS) is a section of the United Kingdom Mathematics Trust which currently runs the British Mathematical Olympiad as well as the UK Mathematical Olympiad for Girls, several training camps throughout the year such as a winter camp in Hungary, an Easter camp at Trinity College, Cambridge, and other training and selection of the International Mathematical Olympiad team. Since 1999, it also organizes the UK National Mathematics Summer Schools. It was established alongside the British Mathematical Olympiad Committee (BMOC) in 1991 with the support of the Edinburgh Mathematical Society, Institute of Mathematics and its Applications, the London Mathematical Society, and the Mathematical Association, each nominated two members. The BMOS replaced some of the Mathematical Association's activities. == History == In 1996, the United Kingdom Mathematics Trust (UKMT) was set up to manage competitions of this nature, though the BMOC remained in charge of the senior olympiads. == Problems group == The 'problems group' is a subsection of the BMOS which is responsible for supplying new and interesting problems for use domestic competitions and for submission to the IMO each year. == References ==
Wikipedia:British Society for Research into Learning Mathematics#0	The British Society for Research into Learning Mathematics is a United Kingdom association for people interested in research in mathematics education. == Purpose == BSRLM organises the Special Interest Group (SIG) on mathematics education for the British Educational Research Association (BERA). It is a participating society of the Joint Mathematical Council (JMC), and has close connections with teacher associations through British Congress for Mathematics Education (BCME). === Events === BSRLM organises a day conference each academic term (three times each year) where members present reports of recently completed studies, work in progress or more speculative thinking. There are also on-going workshops run on a collaborative basis to investigate issues in different areas of interest. The summer meeting is usually preceded by a new researchers’ day and the autumn meeting includes an AGM. Many members of BSRLM are also involved in the Congress of the European Society for Research in Mathematics Education (CERME), the International Group for the Psychology of Mathematics Education (IGPME) and other international research organisations. === Publications === Proceedings of meetings of the Society are available online several weeks after each day conference. A peer reviewed journal, Research in Mathematics Education (RME) published by Taylor and Francis three times a year. Each year the Janet Duffin Award is made to the author or authors judged to making the most outstanding contribution to the journal. == Structure == The membership of BSRLM is made up mostly of researchers, students, teachers and education advisors. BSRLM is run by an elected executive committee of nine members. The current chair is Professor Jeremy Hodgen (University College London). == Origins of BSRLM == In the late 1960s and early 1970s there was a number of teacher trainers and university lecturers in the UK, principally teaching mathematics and psychology, who became involved in research in the learning and teaching of mathematics, and in the movement to establish mathematics education as an academic discipline. Two groups who were meeting during this period were the Psychology of Mathematics Education Workshop, principally in London, and the British Society for the Psychology of Learning Mathematics, in locations in the Midlands. Since many individuals attended both of these groups, they eventually came together as BSRLM in January 1985. == References == == External links == Society website
Wikipedia:Brook Taylor#0	Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician and barrister best known for several results in mathematical analysis. Taylor's most famous developments are Taylor's theorem and the Taylor series, essential in the infinitesimal approach of functions in specific points. == Life and work == Brook Taylor was born in Edmonton (former Middlesex). Taylor was the son of John Taylor, MP of Patrixbourne, Kent and Olivia Tempest, the daughter of Sir Nicholas Tempest, Baronet of Durham. He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees in LL.B. in 1709 and LL.D. in 1714. Taylor studied mathematics under John Machin and John Keill, leading to Taylor obtaining a solution to the problem of "center of oscillation". Taylor's solution remained unpublished until May 1714, when his claim to priority was disputed by Johann Bernoulli. Taylor's Methodus Incrementorum Directa et Inversa (1715) ("Direct and Indirect Methods of Incrementation") added a new branch to higher mathematics, called "calculus of finite differences". Taylor used this development to determine the form of movement in vibrating strings. Taylor also wrote the first satisfactory investigation of astronomical refraction. The same work contains the well-known Taylor's theorem, the importance of which remained unrecognized until 1772, when Joseph-Louis Lagrange realized its usefulness and termed it "the main foundation of differential calculus". In Taylor's 1715 essay Linear Perspective, Taylor set forth the principles of perspective in a more understandable form, but the work suffered from brevity and obscurity problems which plagued most of his writings, meaning the essay required further explanation in the treatises of Joshua Kirby (1754) and Daniel Fournier (1761). Taylor was elected as a fellow in the Royal Society in 1712. In the same year, Taylor sat on the committee for adjudicating the claims of Sir Isaac Newton and Gottfried Leibniz. He acted as secretary to the society from 13 January 1714 to 21 October 1718. From 1715 onward, Taylor's studies took a philosophical and religious bent. He corresponded with the Comte de Montmort on the subject of Nicolas Malebranche's tenets. Unfinished treatises written on his return from Aix-la-Chapelle in 1719, On the Jewish Sacrifices and On the Lawfulness of Eating Blood, were afterwards found among his papers. Taylor was one of few English mathematicians, along with Isaac Newton and Roger Cotes, who was capable of holding his own with the Bernoullis, but a lack of clarity affected a great part of his demonstrations and Taylor lost brevity through his failure to express his ideas fully and clearly. His health began to fail in 1717 after years of intense work. Taylor married Miss Brydges of Wallington, Surrey in 1721 without his father's approval. The marriage led to an estrangement with his father, which improved in 1723 after Taylor's wife died in childbirth while giving birth to a son. Taylor's son did not survive. He spent the next two years with his family at Bifrons, and in 1725 he married with his father's approval, Sabetta Sawbridge of Olantigh, Kent. She died in childbirth in 1730, though his only daughter, Elizabeth, survived. Taylor's father died in 1729, leaving Taylor to inherit the Bifrons estate. Taylor died at the age of 46, on 29 December 1731, at Somerset House, London. == Selected writings == Taylor's grandson, Sir William Young, printed a posthumous work entitled Contemplatio Philosophica for private circulation in 1793, (2nd Bart., 10 January 1815). The work was prefaced by a biography, and had an appendix containing letters addressed to him by Bolingbroke, Bossuet, and others. Several short papers by Taylor were published in Phil. Trans., vols. xxvii to xxxii, which including accounts of experiments in magnetism and capillary attraction. In 1719, Brook issued an improved version of his work on perspective, New Principles of Linear Perspective, which was revised by John Colson in 1749. A French translation was published in 1757. It was reprinted, with a portrait and short biography, in 1811. Taylor, Brook (1715a), Methodus Incrementorum Directa et Inversa, London: William Innys. Annotated English translation by Ian Bruce Taylor, Brook (1715b), Linear Perspective: Or, a New Method of Representing Justly All Manner of Objects as They Appear to the Eye in All Situations, London: R. Knaplock, archived from the original on 11 April 2016. == Tribute == Taylor is an impact crater located on the Moon, named in honor of Brook Taylor in 1935. == References == == Further reading == Andersen, Kirsti (1992). Brook Taylor's Work on Linear Perspective. Springer Science & Business Media. ISBN 978-1-4612-0935-5. Anderson, Marlow; Katz, Victor; Wilson, Robin (2004). Sherlock Holmes in Babylon: And Other Tales of Mathematical History. Mathematical Association of America. p. 309. ISBN 978-0-88385-546-1. Carlyle, Edward Irving (1898). "Taylor, Brook" . In Lee, Sidney (ed.). Dictionary of National Biography. Vol. 55. London: Smith, Elder & Co. Feigenbaum, Lenore (1985). "Brook Taylor and the Method of Increments". Archive for History of Exact Sciences. 34 (1–2): 1–140. doi:10.1007/BF00329903. S2CID 122105736. == External links == Media related to Brook Taylor at Wikimedia Commons O'Connor, John J.; Robertson, Edmund F., "Brook Taylor", MacTutor History of Mathematics Archive, University of St Andrews Beningbrough Hall has a painting by John Closterman of Taylor aged about 12 with his brothers and sisters. See also NPG 5320: The Children of John Taylor of Bifrons Park Archived 24 May 2008 at the Wayback Machine Brook Taylor's pedigree Taylor, a crater on the Moon named after Brook Taylor
Wikipedia:Brouwer's conjecture#0	In the mathematical field of spectral graph theory, Brouwer's conjecture is a conjecture by Andries Brouwer on upper bounds for the intermediate sums of the eigenvalues of the Laplacian of a graph in term of its number of edges. The conjecture states that if G is a simple undirected graph and L(G) its Laplacian matrix, then its eigenvalues λn(L(G)) ≤ λn−1(L(G)) ≤ ... ≤ λ1(L(G)) satisfy ∑ i = 1 t λ i ( L ( G ) ) ≤ m ( G ) + ( t + 1 2 ) , t = 1 , … , n {\displaystyle \sum _{i=1}^{t}\lambda _{i}(L(G))\leq m(G)+\left({\begin{array}{c}t+1\\2\end{array}}\right),\quad t=1,\ldots ,n} where m(G) is the number of edges of G. == State of the art == Brouwer has confirmed by computation that the conjecture is valid for all graphs with at most 10 vertices. It is also known that the conjecture is valid for any number of vertices if t = 1, 2, n − 1, and n. For certain types of graphs, Brouwer's conjecture is known to be valid for all t and for any number of vertices. In particular, it is known that is valid for trees, and for unicyclic and bicyclic graphs. It was also proved that Brouwer’s conjecture holds for two large families of graphs; the first family of graphs is obtained from a clique by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph. For certain sequences of random graphs, Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. == References ==
Wikipedia:Brownian surface#0	A Brownian surface is a fractal surface generated via a fractal elevation function. The Brownian surface is named after Brownian motion. == Example == For instance, in the three-dimensional case, where two variables X and Y are given as coordinates, the elevation function between any two points (x1, y1) and (x2, y2) can be set to have a mean or expected value that increases as the vector distance between (x1, y1) and (x2, y2). There are, however, many ways of defining the elevation function. For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces. == Generation of fractional Brownian surfaces == Efficient generation of fractional Brownian surfaces poses significant challenges. Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient method is Stein's method, which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process. The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter. The Hurst parameter is always between zero and one, with values closer to one corresponding to smoother surfaces. These surfaces were generated using a Matlab implementation of Stein's method. == See also == Wiener process Fractional Brownian motion Gaussian free field == References ==
Wikipedia:Brownian tree#0	In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a random real tree that can be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three articles published in 1991 and 1993. This tree has since then been generalized. This random tree has several equivalent definitions and constructions: using sub-trees generated by finitely many leaves, using a Brownian excursion, Poisson separating a straight line or as a limit of Galton-Watson trees. Intuitively, the Brownian tree is a binary tree whose nodes (or branching points) are dense in the tree; which is to say that for any distinct two points of the tree, there will always exist a node between them. It is a fractal object which can be approximated with computers or by physical processes with dendritic structures. == Definitions == The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles. The notions of leaf, node, branch, root are the intuitive notions on a tree (for details, see real trees). === Finite-dimensional laws === This definition gives the finite-dimensional laws of the subtrees generated by finitely many leaves. Let us consider the space of all binary trees with k {\displaystyle k} leaves numbered from 1 {\displaystyle 1} to k {\displaystyle k} . These trees have 2 k − 1 {\displaystyle 2k-1} edges with lengths ( ℓ 1 , … , ℓ 2 k − 1 ) ∈ R + 2 k − 1 {\displaystyle (\ell _{1},\dots ,\ell _{2k-1})\in \mathbb {R} _{+}^{2k-1}} . A tree is then defined by its shape τ {\displaystyle \tau } (which is to say the order of the nodes) and the edge lengths. We define a probability law P {\displaystyle \mathbb {P} } of a random variable ( T , ( L i ) 1 ≤ i ≤ 2 k − 1 ) {\displaystyle (T,(L_{i})_{1\leq i\leq 2k-1})} on this space by: P ( T = τ , L i ∈ [ ℓ i , ℓ i + d ℓ i ] , ∀ 1 ≤ i ≤ 2 k − 1 ) = s exp ⁡ ( − s 2 / 2 ) d ℓ 1 … d ℓ 2 k − 1 {\displaystyle \mathbb {P} (T=\tau \,,\,L_{i}\in [\ell _{i},\ell _{i}+d\ell _{i}],\forall 1\leq i\leq 2k-1)=s\exp(-s^{2}/2)\,d\ell _{1}\ldots d\ell _{2k-1}} where s = ∑ ℓ i {\displaystyle \textstyle s=\sum \ell _{i}} . In other words, P {\displaystyle \mathbb {P} } depends not on the shape of the tree but rather on the total sum of all the edge lengths. In other words, the Brownian tree is defined from the laws of all the finite sub-trees one can generate from it. === Continuous tree === The Brownian tree is a real tree defined from a Brownian excursion (see characterisation 4 in Real tree). Let e = ( e ( x ) , 0 ≤ x ≤ 1 ) {\displaystyle e=(e(x),0\leq x\leq 1)} be a Brownian excursion. Define a pseudometric d {\displaystyle d} on [ 0 , 1 ] {\displaystyle [0,1]} with d ( x , y ) := e ( x ) + e ( y ) − 2 min { e ( z ) ; z ∈ [ x , y ] } , {\displaystyle d(x,y):=e(x)+e(y)-2\min {\big \{}e(z)\,;z\in [x,y]{\big \}},} for any x , y ∈ [ 0 , 1 ] {\displaystyle x,y\in [0,1]} We then define an equivalence relation, noted ∼ e {\displaystyle \sim _{e}} on [ 0 , 1 ] {\displaystyle [0,1]} which relates all points x , y {\displaystyle x,y} such that d ( x , y ) = 0 {\displaystyle d(x,y)=0} . x ∼ e y ⇔ d ( x , y ) = 0. {\displaystyle x\sim _{e}y\Leftrightarrow d(x,y)=0.} d {\displaystyle d} is then a distance on the quotient space [ 0 , 1 ] / ∼ e {\displaystyle [0,1]\,/\!\sim _{e}} . It is customary to consider the excursion e / 2 {\displaystyle e/2} rather than e {\displaystyle e} . === Poisson line-breaking construction === This is also called stick-breaking construction. Consider a non-homogeneous Poisson point process N with intensity r ( t ) = t {\displaystyle r(t)=t} . In other words, for any t > 0 {\displaystyle t>0} , N t {\displaystyle N_{t}} is a Poisson variable with parameter t 2 {\displaystyle t^{2}} . Let C 1 , C 2 , … {\displaystyle C_{1},C_{2},\ldots } be the points of N {\displaystyle N} . Then the lengths of the intervals [ C i , C i + 1 ] {\displaystyle [C_{i},C_{i+1}]} are exponential variables with decreasing means. We then make the following construction: (initialisation) The first step is to pick a random point u {\displaystyle u} uniformly on the interval [ 0 , C 1 ] {\displaystyle [0,C_{1}]} . Then we glue the segment ] C 1 , C 2 ] {\displaystyle ]C_{1},C_{2}]} to u {\displaystyle u} (mathematically speaking, we define a new distance). We obtain a tree T 1 {\displaystyle T_{1}} with a root (the point 0), two leaves ( C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} ), as well as one binary branching point (the point u {\displaystyle u} ). (iteration) At step k, the segment ] C k , C k + 1 ] {\displaystyle ]C_{k},C_{k+1}]} is similarly glued to the tree T k − 1 {\displaystyle T_{k-1}} , on a uniformly random point of T k − 1 {\displaystyle T_{k-1}} . This algorithm may be used to simulate numerically Brownian trees. === Limit of Galton-Watson trees === Consider a Galton-Watson tree whose reproduction law has finite non-zero variance, conditioned to have n {\displaystyle n} nodes. Let 1 n G n {\displaystyle {\tfrac {1}{\sqrt {n}}}G_{n}} be this tree, with the edge lengths divided by n {\displaystyle {\sqrt {n}}} . In other words, each edge has length 1 n {\displaystyle {\tfrac {1}{\sqrt {n}}}} . The construction can be formalized by considering the Galton-Watson tree as a metric space or by using renormalized contour processes. Here, the limit used is the convergence in distribution of stochastic processes in the Skorokhod space (if we consider the contour processes) or the convergence in distribution defined from the Hausdorff distance (if we consider the metric spaces). == References ==
Wikipedia:Bruce M. Boghosian#0	Prof. Bruce Michael Boghosian is an American mathematician. He has been a professor of mathematics at Tufts University since 2000, and was chair of the mathematics department from 2006 to 2010. He also holds adjunct positions in the Tufts University Departments of Physics and Computer Science. == Biography == Boghosian received his bachelor's degree in physics and master's degree in nuclear engineering from the Massachusetts Institute of Technology. He earned his PhD from the Department of Applied Science at the University of California, Davis. From 1978 to 1986, he was a physicist in the Plasma Theory Group at the Lawrence Livermore National Laboratory in California. From 1986 through 1994, he was a Senior Scientist in the Mathematical Research Group at Thinking Machines Corporation in Cambridge, Massachusetts. From 1994 through 2000, prior to coming to Tufts University, Boghosian held the position of Research Associate Professor at the Center for Computational Science and Department of Physics at Boston University. From 2010 to 2014, while on leave from Tufts University, Boghosian served as the third president of the American University of Armenia in Yerevan, Armenia. Boghosian has held visiting academic positions at the Département de Mathématiques, Paris-Sud University in Orsay, France; the École normale supérieure in Paris; Peking University in Beijing; University College London; the University of California, Berkeley; the International Centre for Theoretical Physics in Trieste, Italy; the Schlumberger Cambridge Research Centre in Cambridge, United Kingdom; and the Massachusetts Institute of Technology. == Fellowships and publications == Boghosian has been a fellow of the American Physical Society since 2000, and a foreign member of the Armenian National Academy of Sciences since 2008. He is a recipient of Tufts University's Distinguished Scholar Award in 2010, and its Undergraduate Initiative in Teaching (UNITE) award in 2002. In 2014 he received the "Order of the Republic of Armenia" from the Armenian Prime Minister, and the "Gold Medal" from the Armenian Ministry of Education and Science. His 2019 article "The Inescapable Casino" was published in six languages, and included in the volume "The Best Writing on Mathematics 2020". Boghosian has over 110 publications, has given over 220 invited talks, and has one patent. He is a member of the editorial boards of the Journal of Computational Science, Physica A, and International Journal of Modern Physics C – Physics and Computers. == Research interests == Boghosian's research interests center on applied dynamical systems and applied probability theory, with an emphasis on kinetic theory, as it applies to fluids, soft condensed matter, and agent based models in the social sciences. From 2014 to the present, his work has centered on kinetic-theoretical models of wealth distribution. == References ==
Wikipedia:Bruce Morton (mathematician)#0	Bruce Morton (11 April 1926 – 15 September 2012) was an Australian/New Zealand applied mathematician. == Early life and education == Morton was born in Wellington, New Zealand and educated at Auckland Grammar School. He gained a government scholarship to attend the University of Auckland, where he completed a double degree in mathematics and physics. Whilst at the University of Auckland he was an active member of the mountaineering club and climbed with Edmund Hillary. In 1949, Morton was awarded the Rutherford fellowship to study for a BA mathematical tripos at St John's College, Cambridge. == Cambridge == In 1956 he completed his PhD in the Department of Applied Mathematics and Theoretical Physics (DAMPT) at Cambridge University under the supervision of Sir G.I. Taylor and Sir George Batchelor. His thesis work was published in paper that became a classic of the fluid dynamics literature - the much cited Morton, Taylor and Turner result. The study developed an explanation for a source of buoyancy being injected into a stratified fluid. By conserving volume, momentum and buoyancy, the study predicted the final height to which a plume of light fluid will rise in a stably stratified fluid. These predictions where then compared with laboratory experiments created using a stratified salt solution. Fellow student and co-author on the plume paper Stewart Turner recalled that Batchelor suggested that he conduct some laboratory experiments to test Morton's theory of the rise of plumes and “buoyant clouds” in stratified surroundings. The starting point was an entrainment hypothesis proposed by Taylor which assumes that the rising turbulent motion in the plume causes an inflow of environmental fluid at a rate that is proportional to the average upward velocity. When Morton and Turner wrote the manuscript documenting the agreement between theory and experiment they discovered that Taylor was also preparing a “much delayed note” on the subject. Taylor included an explanation at the end of the Morton paper explaining the circumstances. He did add some distinctive touches to the article including estimates of the height of rise of smoke from an autumn bonfire (150 ft) and a burning town (3200 m) with specified burning rates and atmospheric conditions; and as noted by Turner - the mixed units are Taylor's. == University of Manchester == After his Ph.D. Morton briefly took up an academic appointment at University College London. However, he was soon offered a position at the University of Manchester by James Lighthill where he worked until 1967. During this time he developed an interest in the propagation of bush fires. == Monash University == In 1967, Morton was appointed to a chair in applied mathematics at Monash University, Melbourne, Australia. There he established a leading research group in geophysical fluid dynamics within the department of mathematics. As well as his influential work on plumes he emphasized the importance of vorticity in the behaviour of fluids. In lectures he would often state ‘vorticity is the flow field’. He retired as chair in 1991. After his death in 2012 a special issue of the journal Australian Meteorological and Oceanographic Journal (which changed its name to Journal of Southern Hemisphere Earth Systems Science in 2016) dedicated to Morton's work and impact was published in 2014. It contained a series of invited review papers by prominent research scientists from around the world who interacted with Morton on the many topics he was involved in throughout his career. As well as plumes, he worked on tropical cyclone formation, as recognised in the special issue. == AMOS and the Morton Medal == Morton contributed to the running and organisation of the Australian Meteorological and Oceanographic Society, especially in the fostering of participation from all parts of Australia. In 2000 the Society renamed its AMOS Medal after him and commenced awarding the Morton Medal as a "biennial award recognising leadership in meteorology and/or oceanography and/or related fields, with particular emphasis on education and development of young scientists, and personal example in research". Winners of the medal include Matthias Tomczak, Gary Meyers, Andy Pitman, Ann Henderson-Sellers, David Karoly, John Church, and Matthew England. == Personal life == In 1953 Morton married Alison Gladding, who he had met in the University of Auckland mountaineering club, at the Marylebone Presbyterian Church near Marble Arch in London. They had three daughters - Clare, Janne and Anna. == Awards == 1949 - Rutherford Fellowship for study at St John's College, University of Cambridge. == References ==
Wikipedia:Bruce Reed (mathematician)#0	Bruce Alan Reed FRSC is a Canadian mathematician and computer scientist, a former Canada Research Chair in Graph Theory at McGill University. His research is primarily in graph theory. He is a distinguished research fellow of the Institute of Mathematics in the Academia Sinica, Taiwan, and an adjunct professor at the University of Victoria in Canada. == Academic career == Reed earned his Ph.D. in 1986 from McGill, under the supervision of Vašek Chvátal. Before returning to McGill as a Canada Research Chair, Reed held positions at the University of Waterloo, Carnegie Mellon University, and the French National Centre for Scientific Research. Reed was elected as a fellow of the Royal Society of Canada in 2009, and is the recipient of the 2013 CRM-Fields-PIMS Prize. In 2021 he left McGill, and subsequently became a researcher at the Academia Sinica and an adjunct professor at the University of Victoria. == Research == Reed's thesis research concerned perfect graphs. With Michael Molloy, he is the author of a book on graph coloring and the probabilistic method. Reed has also published highly cited papers on the giant component in random graphs with a given degree sequence,[MR95][MR98a] random satisfiability problems,[CR92] acyclic coloring,[AMR91] tree decomposition,[R92][R97] and constructive versions of the Lovász local lemma.[MR98b] He was an invited speaker at the International Congress of Mathematicians in 2002. His talk there concerned a proof by Reed and Benny Sudakov, using the probabilistic method, of a conjecture by Kyoji Ohba that graphs whose number of vertices and chromatic number are (asymptotically) within a factor of two of each other have equal chromatic number and list chromatic number.[RS02] == Selected publications == === Articles === === Books === == References == == External links == Home page Bruce A. Reed at DBLP Bibliography Server Bruce Reed publications indexed by Google Scholar
Wikipedia:Bruno Courcelle#0	Bruno Courcelle is a French mathematician and computer scientist, best known for Courcelle's theorem in graph theory. == Life == Courcelle earned his Ph.D. in 1976 from the French Institute for Research in Computer Science and Automation, then called IRIA, under the supervision of Maurice Nivat. He then joined the Laboratoire Bordelais de Recherche en Informatique (LaBRI) at the University of Bordeaux 1, where he remained for the rest of his career. He has been a senior member of the Institut Universitaire de France since 2007. A workshop in honor of Courcelle's retirement was held in Bordeaux in 2012. Courcelle was the first recipient of the S. Barry Cooper Prize of the Association Computability in Europe in 2020. In 2022, Courcelle was awarded the EATCS-IPEC Nerode Prize. During the COVID-19 pandemic, Courcelle protested against vaccination mandates in France. == Work == He is known for Courcelle's theorem, which combines second-order logic, the theory of formal languages, and tree decompositions of graphs to show that a wide class of algorithmic problems in graph theory have efficient solutions. Notable publications also include: Bruno Courcelle (1983). "Fundamental Properties of Infinite Trees". Theoretical Computer Science. 25: 95–169. Bruno Courcelle (1990). "Recursive Applicative Program Schemes". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 459–492. ISBN 0-444-88074-7. Bruno Courcelle (1999). "Hierarchical Graph Decompositions Defined by Grammars and Logical Formulas (invited lecture)". In Paliath Narendran; Michaël Rusinowitch (eds.). Rewriting Techniques and Applications, 10th Int. Conf., RTA-99. LNCS. Vol. 1631. Springer-Verlag. pp. 90–91. == References == == External links == Official website
Wikipedia:Bruno D'Amore#0	Bruno D’Amore (Born in Bologna, 28 September 1946) is an Italian mathematician and author. == Education == He has degrees in mathematics, pedagogy, philosophy, and a postgraduate qualification in Elementary Mathematics from a higher point of view, all obtained at the University of Bologna (Italy). D'Amore also has a Ph.D. in mathematics education from the University “Constantine the Philosopher” of Nitra in Slovakia. == Career == Formerly professor in mathematics education at the University of Bologna, D'Amore currently holds seminars and supervises doctoral theses at the Universidad Distrital “Francisco José de Caldas” of Bogotá in Colombia. He also teaches postgraduate courses at other Colombian universities. He is a member of many research groups in Italy (NRD of Bologna), Spain (GRADEM, Barcelona) and Colombia (MESCUD, Bogotá) as well as the editorial boards of numerous scientific research journals in many countries. He is also a member of the Scientific Committee of International Research Groups and International Conferences. He is the author of two collections of stories, one of which won the “Arturo Loria 2003” literary prize from the municipality of Carpi, while the other won the “Il Ceppo 2003” literary prize from the municipality of Pistoia. He has also authored numerous publications. Since 1977 he has been a member of the Association International des Critiques d’Art. He has been the secretary of a Quadriennale d’arte in the Venetian Region, the director of a private art gallery in Bologna and a consultant at private and public art galleries in Italy. In 1986 he founded the National Conference “Incontri con la Matematica” (Meetings with Mathematics) whose first edition was held in Bologna. From the month of November of the following year the conference "Meetings with math" has always done, and is held annually, during the same period, in Castel San Pietro Terme - Bologna. The event, which has collected over the years more than 20000 teachers participating, is currently headed by Professor D'Amore and by Professors Silvia Sbaragli and Martha Isabel Fandiño Pinilla (his wife). == Awards == D'Amore has been awarded various prizes for his studies and research, including “Lo Stilo d’Oro”, 2000 edition; a nomination at “Pianeta Galileo 2010”; a Ph.D. ad honorem in Social Sciences and Education from the University of Cyprus, at Nicosia, on 15 October 2013, for the international relevance of his research in mathematics education (ceremony in Cyprus, ceremony in Bogota); the “Premio a la Contribución Científica Internacional en Ciencia y Tecnología” from the University of Medellín, on 10 May 2013. He has also been awarded honorary citizenship of Castel San Pietro Terme (Bologna) on 27 September 1997, and honorary citizenship of Cerchio (L’Aquila) on 5 September 2005. == External links == Personal website Incontri con la Matematica: Official Conference Website History of the Conference History of the Congress' Proceedings Mathematics Education Research Group: NRD, Nucleo di Ricerca in Didattica della Matematica, Università di Bologna. Journal: “La matematica e la sua didattica” National and International Conferences dedicated to bruno D'Amore
Wikipedia:Bruno N Rémillard#0	Bruno N Rémillard (born July 7, 1961) is a Canadian mathematical statistician and an honorary professor at HEC Montréal. He is the 2019 Gold Medalist of the Statistical Society of Canada and was inducted as a 2019 Fellow of the Institute of Mathematical Statistics. Rémillard was President of the Statistical Society of Canada in 2022-23. == Biography == Rémillard was born in Saint-Raphaël, Quebec, Canada. He received a BSc and a MSc in mathematics from Université Laval in 1983 and 1985 and two years later a PhD in probability at Carleton University under the supervision of Donald A. Dawson. His dissertation was entitled "Large deviations and laws of the iterated logarithm for multidimensional diffusion processes with applications to diffusion processes with random coefficients." and earned him the Pierre-Robillard Award from the Statistical Society of Canada that recognizes the best PhD thesis defended in the relevant fields at a Canadian university in a given year. After working as an NSERC postdoctoral fellow at Cornell University, Bruno N Rémillard joined the Université du Québec à Trois-Rivières in 1988. He was promoted to the rank of Associate in 1992 and became full professor in 1996. He settled at HEC Montréal in 2001. == Publications == Rémillard has authored over 85 published papers, mostly in high-caliber international journals, in the fields of probability, statistics, and time series modelling. He has co-authored with Christian Genest nearly 20 papers in The Annals of Probability, The Annals of Statistics, Bernoulli, Biometrika, Journal of the American Statistical Association, Journal of Multivariate Analysis, etc. He is one of the 23 scientists to have published in the four IMS Annals. He is the author of a graduate monograph and co-authored of three other books, including two undergraduate textbooks: Corina Reischer, Raymond Leblanc, Bruno Rémillard, and Denis Larocque, Théorie des probabilités: Problèmes et solutions., Presses de l'Université du Québec, Montréal, Canada, 2002 Pierre Duchesne and Bruno Rémillard, Statistical modeling and analysis for complex data problems, GERAD 25th Anniversary Series, Springer: New York, 2005 Pierre Del Moral, Bruno Rémillard, and Sylvain Rubenthaler, Une introduction aux probabilités, Éditions Ellipses, 2006 B. Rémillard, Statistical methods for financial engineering, Chapman and Hall/CRC, 2013. == Awards and honors == Statistical Society of Canada Pierre-Robillard Award 1988 Elected Member of the International Statistical Institute 2000 Best Paper Award Canadian Journal of Statistics 2003 Pierre Laurin Award for Excellence in Research 2007 Roger Charbonneau Award 2013 Best Paper Award Econometrics Statistical Society of Canada Gold Medalist 2019 Fellow of the Institute of Mathematical Statistics 2019 == References == == External links == Personal Webpage Bruno N Rémillard publications indexed by Google Scholar
Wikipedia:Bruno Zumbo#0	Bruno D. Zumbo is a Canadian mathematical scientist trained in the tradition of research that combines mathematical analysis, statistics, and probability to develop theory and solve problems arising in measurement, testing, and surveys in the social, behavioral, and health sciences. He is currently Professor and Distinguished University Scholar, the Canada Research Chair in Psychometrics and Measurement (Tier 1), and formerly the Paragon UBC Professor of Psychometrics & Measurement at University of British Columbia. His research in the mathematical sciences reflects a wide range of topics in mathematical analysis and statistics aimed at developing and exploring the properties and applications of mathematical structures of measurement, survey design, testing, and assessment. == Education == He completed his B.Sc. at the University of Alberta (Edmonton, AB) and his MA and Ph.D. from Carleton University (Ottawa, ON). His doctoral dissertation titled "Statistical Methods to Overcome Nonindependence of Coupled Data in Significance Testing" was under the direction of Donald W. Zimmerman (Carleton University, Ottawa). == Career == Zumbo teaches in the graduate Measurement, Evaluation, & Research Methodology Program with an additional appointment in the Institute of Applied Mathematics, and earlier also in the Department of Statistics, at the University of British Columbia (UBC) in Vancouver, British Columbia, Canada. Prior to arriving at UBC in 2000, he held professorships in the Departments of Psychology and of Mathematics at the University of Northern British Columbia (1994-2000), and earlier in the Faculty of Education with an adjunct appointment in the Department of Mathematics at the University of Ottawa (1990-1994). His research interests have been focused on the mathematical sciences of measurement and scientific methodology with a blend of mathematics, social sciences like psychology, philosophy of science and measurement in science. He is known for his contributions in the fields of statistics, psychometrics, validity theory, and studies of the mathematical basis of classical test theory, item response theory, and measurement error models. His program of research is actively engaged in psychometrics for language testing, quality of life and wellbeing, and health and human development. == Awards and recognition == Distinguished University Scholar, 2017 Pioneer in the Psychometrics of Quality of Life, 2018 by the International Society for Quality of Life Studies Centenary Medal of Distinction, awarded in 2019 by the UBC School of Nursing Paragon UBC Professorship in Psychometrics and Measurement Tier 1 - Canada Research Chair in Psychometrics and Measurement, held at the University of British Columbia, awarded in 2020 == References == == External links == List of publications
Wikipedia:Bruria Kaufman#0	Bruria Kaufman (Hebrew: ברוריה קאופמן; August 21, 1918 – January 7, 2010) was an Israeli American theoretical physicist. She contributed to Albert Einstein's general theory of relativity, to statistical physics, where she used applied spinor analysis to rederive the result of Lars Onsager on the partition function of the two-dimensional Ising model, and to the study of the Mössbauer effect, on which she collaborated with John von Neumann and Harry Lipkin. == Biography == Bruria Kaufman was born in New York City to a Jewish family of Ukrainian origin. In 1926 the family immigrated to the British Mandate for Palestine, living first in Tel Aviv, and then in Jerusalem. Her main interests during her youth were music and mathematics. She studied mathematics, earning a B.Sc. from Hebrew University of Jerusalem in 1938, and a PhD from Columbia University in 1948. She married the linguist Zellig S. Harris in 1941. In 1960, she settled on Kibbutz Mishmar Ha'emek in Israel and adopted a daughter, Tami. Kaufman returned to the US in 1982. They lived in Pennsylvania, where her husband taught. He died in 1992. Kaufman moved to Arizona and married the Nobel laureate Willis Eugene Lamb in 1996, although the marriage ended in divorce. She died in January 2010 at Carmel Hospital in Haifa, following a stay at a nursing home in Kiryat Tiv'on, not far from Haifa. In keeping with her wishes, her body was cremated. == Scientific career == Kaufman was a research associate at the Institute for Advanced Study in Princeton from 1948 to 1955, where she worked with John von Neumann (1947/48) and with Albert Einstein (1950–1955). She spent the following years at the University of Pennsylvania working on a mathematical linguistics project. Kaufman returned to Israel in 1960 (with Harris) where she became professor at the Weizmann Institute of Science in Rehovot (1960–1971) and later on at the University of Haifa (1972–1988). Prof. Bruria Kaufman‘s main academic contribution was finding an elegant solution based on group theory to the two-dimensional Ising model, together with Lars Onsager in 1949. This solution was much simpler than the solution published by Onsager himself in 1944. The three dimensional Ising model was not solved until this day. In addition, Kaufman published two articles with Albert Einstein and co-authored his book on relativity. In Israel she studied the Mössbauer effect together with Professor Harri Zvi Lipkin and the two published an article together, in which Bruria combined mathematics with the physics of the effect. == Selected publications == Kaufman, Bruria (1949-10-15). "Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis". Physical Review. 76 (8). American Physical Society (APS): 1232–1243. Bibcode:1949PhRv...76.1232K. doi:10.1103/physrev.76.1232. ISSN 0031-899X. Kaufman, Bruria; Onsager, Lars (1949-10-15). "Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice". Physical Review. 76 (8). American Physical Society (APS): 1244–1252. Bibcode:1949PhRv...76.1244K. doi:10.1103/physrev.76.1244. ISSN 0031-899X. "Transition Points", Physical Society Cambridge International Conference on Low Temperatures (1946), with L. Onsager. Einstein, A.; Strauss, E. G.; Kaufman, Buria (1953). The Meaning of Relativity (4th ed.). Princeton University Press. OCLC 1724486. Kaufman's contribution is to an appendix which appeared in later editions, and was revised and published as "Algebraic Properties of the Field in the Relativistic Theory of the Asymmetric Field". Einstein, A.; Kaufman, B. (1954). "Algebraic Properties of the Field in the Relativistic Theory of the Asymmetric Field". The Annals of Mathematics. 59 (2). JSTOR: 230–244. doi:10.2307/1969690. ISSN 0003-486X. JSTOR 1969690. Einstein, A.; Kaufman, B. (1955). "A New Form of the General Relativistic Field Equations". The Annals of Mathematics. 62 (1). JSTOR: 128–138. doi:10.2307/2007103. ISSN 0003-486X. JSTOR 2007103. "Mathematical Structure of the Non-symmetric Field Theory", Proceedings of the Fiftieth Anniversary Conference on Relativity 227–238 (1955). Lifson, Shenior; Kaufman, Bruria; Lifson, Hanna (1957). "Neighbor Interactions and Symmetric Properties of Polyelectrolytes". The Journal of Chemical Physics. 27 (6). AIP Publishing: 1356–1362. Bibcode:1957JChPh..27.1356L. doi:10.1063/1.1744007. ISSN 0021-9606. Gillis, J.; Kaufman, Bruria (1962). "The Stability of a Rotating Viscous Jet". Quarterly of Applied Mathematics. 19 (4): 301–308. doi:10.1090/qam/136218. JSTOR 43634962. Kaufman, Bruria; Lipkin, Harry J (1962). "Momentum transfer to atoms bound in a crystal". Annals of Physics. 18 (2). Elsevier BV: 294–309. Bibcode:1962AnPhy..18..294K. doi:10.1016/0003-4916(62)90072-6. ISSN 0003-4916. Kaufman, Bruria; Noack, Cornelius (1965). "Unitary Symmetry of Oscillators and the Talmi Transformation". Journal of Mathematical Physics. 6 (1). AIP Publishing: 142–152. Bibcode:1965JMP.....6..142K. doi:10.1063/1.1704252. ISSN 0022-2488. Kaufman, Bruria (1966). "Special Functions of Mathematical Physics from the Viewpoint of Lie Algebra". Journal of Mathematical Physics. 7 (3). AIP Publishing: 447–457. Bibcode:1966JMP.....7..447K. doi:10.1063/1.1704953. ISSN 0022-2488. == References ==
Wikipedia:Brāhmasphuṭasiddhānta#0	The Brāhma-sphuṭa-siddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS) is a main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including the first good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta theorem. The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic equation). == Positive and negative numbers == Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers, negative numbers, and zero. For example, it notes that the sum of a positive number and a negative number is their difference or, if they are equal, zero; that subtracting a negative number is equivalent to adding a positive number; that the product of two negative numbers is positive. Some of the notions of fractions differ from the modern rational number system. For example, Brahmagupta allows division by zero resulting in a fraction with a 0 in the denominator, and defines 0/0 = 0. In modern mathematics, division by zero is undefined for any field. == Influence == Ashadhara, the son of Rihluka, wrote Graha-jnana with tables based on Brahma-sphuta-siddhanta in 1132. This work is also known by the names Graha-ganita, Brahma-tulyanayana, Bhaumadi-panchagraha-nayana, Kshanika-grahanayana, or simply Ashadhara. Harihara wrote an extended version of the Graha-jnana around 1575 CE. == References == == External links == Brahmasphutasiddhanta at GRETIL (mathematical chapters: 12, 18-20, 21.17-23) O'Connor, John J.; Robertson, Edmund F., "Brahmagupta", MacTutor History of Mathematics Archive, University of St Andrews
Wikipedia:Bubacarr Bah#0	Bubacarr Bah is a Gambian mathematician. He is (as at July 2024) Associate Professor and Head of Data Science at the MRC Unit based at Banjul, The Gambia. (Note that the unit is run by the London School of Hygiene & Tropical Medicine - but was previously managed by the Medical Research Council (United Kingdom). See COVID-19 pandemic in the Gambia). He was previously (2016 - 2022) the chair of Data Science at the African Institute for Mathematical Sciences (AIMS). He has recently been assistant professor at Stellenbosch University. == Early life and education == Bah was born in The Gambia. He studied mathematics at the University of the Gambia and graduated summa cum laude in 2004. He was awarded a Master of Science (MSc) degree from the University of Oxford, where he studied mathematical modelling as a postgraduate student of Wolfson College, Oxford. He joined the University of Edinburgh, where his PhD investigated compressed sensing and was supervised by Jared Tanner. He was a member of the Society for Industrial and Applied Mathematics (SIAM) student chapter. His work on Gaussian matrices was awarded the SIAM best student paper. == Research and career == Between 2012 and 2014 Bah was a postdoctoral researcher at the École Polytechnique Fédérale de Lausanne, where he continued his investigations into compressed sensing. He was a member of the Laboratory for Information and Inference Systems. In 2014 Bah joined the University of Texas at Austin, where he worked on signal processing, machine learning and sampling strategies in high-dimensional data. He developed a matrix for dimensionality reduction that uses bi-Lipschitz embeddings, which can exploit data redundancy. In 2016 Bah was appointed the German Chair in Mathematics at the African Institute for Mathematical Sciences (AIMS), which is supported by the Humboldt Foundation. The position was welcomed by the AIMS community, who believe Africa needs better data science infrastructure. Bah organised the software engineering for Applied Mathematical Sciences workshop, which teaches basic programming and research programming. He is responsible for connecting the AIMS in South Africa with central Africa and German universities. As of 2019, the Alexander von Humboldt Foundation supports five chairs at AIMS centres. He holds a joint position at Stellenbosch University, where he works on information theory and deep learning. In March 2019 Bah was appointed to the Google advanced technology external advisory council, a collection of experts who will consider the artificial intelligence (AI) principles of Google. Google disbanded ATEAC on April 4, 2019 following criticism by Google employees about another member, Kay Coles James. == References ==
Wikipedia:Buchberger's algorithm#0	In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same common zeros and are more convenient for extracting information on these common zeros. It was introduced by Bruno Buchberger simultaneously with the definition of Gröbner bases. The Euclidean algorithm for computing the polynomial greatest common divisor is a special case of Buchberger's algorithm restricted to polynomials of a single variable. Gaussian elimination of a system of linear equations is another special case where the degree of all polynomials equals one. For other Gröbner basis algorithms, see Gröbner basis § Algorithms and implementations. == Algorithm == A crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows: Input A set of polynomials F that generates I Output A Gröbner basis G for I G := F For every fi, fj in G, denote by gi the leading term of fi with respect to the given monomial ordering, and by aij the least common multiple of gi and gj. Choose two polynomials in G and let Sij = ⁠aij/ gi⁠ fi − ⁠aij/ gj⁠ fj (Note that the leading terms here will cancel by construction). Reduce Sij, with the multivariate division algorithm relative to the set G until the result is not further reducible. If the result is non-zero, add it to G. Repeat steps 2–4 until all possible pairs are considered, including those involving the new polynomials added in step 4. Output G The polynomial Sij is commonly referred to as the S-polynomial, where S refers to subtraction (Buchberger) or syzygy (others). The pair of polynomials with which it is associated is commonly referred to as critical pair. There are numerous ways to improve this algorithm beyond what has been stated above. For example, one could reduce all the new elements of F relative to each other before adding them. If the leading terms of fi and fj share no variables in common, then Sij will always reduce to 0 (if we use only fi and fj for reduction), so we needn't calculate it at all. The algorithm terminates because it is consistently increasing the size of the monomial ideal generated by the leading terms of our set F, and Dickson's lemma (or the Hilbert basis theorem) guarantees that any such ascending chain must eventually become constant. == Complexity == The computational complexity of Buchberger's algorithm is very difficult to estimate, because of the number of choices that may dramatically change the computation time. Nevertheless, T. W. Dubé has proved that the degrees of the elements of a reduced Gröbner basis are always bounded by 2 ( d 2 2 + d ) 2 n − 2 {\displaystyle 2\left({\frac {d^{2}}{2}}+d\right)^{2^{n-2}}} , where n is the number of variables, and d the maximal total degree of the input polynomials. This allows, in theory, to use linear algebra over the vector space of the polynomials of degree bounded by this value, for getting an algorithm of complexity d 2 n + o ( 1 ) {\displaystyle d^{2^{n+o(1)}}} . On the other hand, there are examples where the Gröbner basis contains elements of degree d 2 Ω ( n ) {\displaystyle d^{2^{\Omega (n)}}} , and the above upper bound of complexity is optimal. Nevertheless, such examples are extremely rare. Since its discovery, many variants of Buchberger's have been introduced to improve its efficiency. Faugère's F4 and F5 algorithms are presently the most efficient algorithms for computing Gröbner bases, and allow to compute routinely Gröbner bases consisting of several hundreds of polynomials, having each several hundreds of terms and coefficients of several hundreds of digits. == See also == Knuth–Bendix completion algorithm Quine–McCluskey algorithm – analogous algorithm for Boolean algebra == References == == Further reading == Buchberger, B. (August 1976). "Theoretical Basis for the Reduction of Polynomials to Canonical Forms". ACM SIGSAM Bulletin. 10 (3). ACM: 19–29. doi:10.1145/1088216.1088219. MR 0463136. S2CID 15179417. David Cox, John Little, and Donald O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer. ISBN 0-387-94680-2. Vladimir P. Gerdt, Yuri A. Blinkov (1998). Involutive Bases of Polynomial Ideals, Mathematics and Computers in Simulation, 45:519ff == External links == "Buchberger algorithm", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Buchberger's algorithm on Scholarpedia Weisstein, Eric W. "Buchberger's Algorithm". MathWorld.
Wikipedia:Buddhabrot#0	The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tika), a traditional oval crown (ushnisha), and ringlet of hair. == Discovery == The Buddhabrot rendering technique was discovered by Melinda Green, who later described it in a 1993 Usenet post to sci.fractals. Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988, Linas Vepstas relayed similar images to Cliff Pickover for inclusion in Pickover's then-forthcoming book Computers, Pattern, Chaos, and Beauty. This led directly to the discovery of Pickover stalks. Noel Griffin also implemented this idea in the 1993 "Mandelcloud" option in the Fractint renderer. However, these researchers did not filter out non-escaping trajectories required to produce the ghostly forms reminiscent of Hindu art. The inverse, "Anti-Buddhabrot" filter produces images similar to no filtering. An "Anti-Buddhabrot" is simply a Buddhabrot where only points that don't escape contribute to the image. Green first named this pattern Ganesh, since an Indian co-worker "instantly recognized it as the god 'Ganesha' which is the one with the head of an elephant." The name Buddhabrot was coined later by Lori Gardi. == Rendering method == Mathematically, the Mandelbrot set consists of the set of points c {\displaystyle c} in the complex plane for which the iteratively defined sequence z n + 1 = z n 2 + c {\displaystyle z_{n+1}=z_{n}^{2}+c} does not tend to infinity as n {\displaystyle n} goes to infinity for z 0 = 0 {\displaystyle z_{0}=0} . The Buddhabrot image can be constructed by first creating a 2-dimensional array of boxes, each corresponding to a final pixel in the image. Each box ( i , j ) {\displaystyle (i,j)} for i = 1 , … , m {\displaystyle i=1,\ldots ,m} and j = 1 , … , n {\displaystyle j=1,\ldots ,n} has size in complex coordinates of Δ x {\displaystyle \Delta x} and Δ y {\displaystyle \Delta y} , where Δ x = w / m {\displaystyle \Delta x=w/m} and Δ y = h / n {\displaystyle \Delta y=h/n} for an image of width w {\displaystyle w} and height h {\displaystyle h} . For each box, a corresponding counter is initialized to zero. Next, a random sampling of c {\displaystyle c} points are iterated through the Mandelbrot function. For points which do escape within a chosen maximum number of iterations, and therefore are not in the Mandelbrot set, the counter for each box entered during the escape to infinity is incremented by 1. In other words, for each sequence corresponding to c {\displaystyle c} that escapes, for each point z n {\displaystyle z_{n}} during the escape, the box that ( Re ( z n ) , Im ( z n ) ) {\displaystyle ({\text{Re}}(z_{n}),{\text{Im}}(z_{n}))} lies within is incremented by 1. Points which do not escape within the maximum number of iterations (and considered to be in the Mandelbrot set) are discarded. After a large number of c {\displaystyle c} values have been iterated, grayscale shades are then chosen based on the distribution of values recorded in the array. The result is a density plot highlighting regions where z n {\displaystyle z_{n}} values spend the most time on their way to infinity. == Nuances == Rendering Buddhabrot images is typically more computationally intensive than standard Mandelbrot rendering techniques. This is partly due to requiring more random points to be iterated than pixels in the image in order to build up a sharp image. Rendering highly zoomed areas requires even more computation than for standard Mandelbrot images in which a given pixel can be computed directly regardless of zoom level. Conversely, a pixel in a zoomed region of a Buddhabrot image can be affected by initial points from regions far outside the one being rendered. Without resorting to more complex probabilistic techniques, rendering zoomed portions of Buddhabrot consists of merely cropping a large full sized rendering. The maximum number of iterations chosen affects the image – higher values give sparser, more detailed appearance, as a few of the points pass through a large number of pixels before they escape, resulting in their paths being more prominent. If a lower maximum was used, these points would not escape in time and would be regarded as not escaping at all. The number of samples chosen also affects the image as not only do higher sample counts reduce the noise of the image, they can reduce the visibility of slowly moving points and small attractors, which can show up as visible streaks in a rendering of lower sample count. Some of these streaks are visible in the 1,000,000 iteration image below. Some people may even include a minimum number of iterations, where a given sample is only used if it doesn't escape before a given number of iterations, to generate more of these intricate details. Green later realized that this provided a natural way to create color Buddhabrot images by taking three such grayscale images, differing only by the maximum number of iterations used, and combining them into a single color image using the same method used by astronomers to create false color images of nebula and other celestial objects. For example, one could assign a 2,000 max iteration image to the red channel, a 200 max iteration image to the green channel, and a 20 max iteration image to the blue channel of an image in an RGB color space. Some have labelled Buddhabrot images using this technique Nebulabrots. == Relation to the logistic map == The relationship between the Mandelbrot set as defined by the iteration z 2 + c {\displaystyle z^{2}+c} , and the logistic map λ x ( 1 − x ) {\displaystyle \lambda x(1-x)} is well known. The two are related by the quadratic transformation: c r = λ ( 2 − λ ) 4 c i = 0 z r = − λ ( 2 x − 1 ) 2 z i = 0 {\displaystyle {\begin{aligned}c_{r}&={\frac {\lambda (2-\lambda )}{4}}\\c_{i}&=0\\z_{r}&=-{\frac {\lambda (2x-1)}{2}}\\z_{i}&=0\end{aligned}}} The traditional way of illustrating this relationship is aligning the logistic map and the Mandelbrot set through the relation between c r {\displaystyle c_{r}} and λ {\displaystyle \lambda } , using a common x-axis and a different y-axis, showing a one-dimensional relationship. Melinda Green discovered that the Anti-Buddhabrot paradigm fully integrates the logistic map. Both are based on tracing paths from non-escaping points, iterated from a (random) starting point, and the iteration functions are related by the transformation given above. It is then easy to see that the Anti-Buddhabrot for z 2 + c {\displaystyle z^{2}+c} , plotting paths with c = ( random , 0 ) {\displaystyle c=({\text{random}},0)} and z 0 = ( 0 , 0 ) {\displaystyle z_{0}=(0,0)} , simply generates the logistic map in the plane { c r , z r } {\displaystyle \{c_{r},z_{r}\}} , when using the given transformation. For rendering purposes we use z 0 = ( random , 0 ) {\displaystyle z_{0}=({\text{random}},0)} . In the logistic map, all z r 0 {\displaystyle z_{r0}} ultimately generate the same path. Because both the Mandelbrot set and the logistic map are an integral part of the Anti-Buddhabrot we can now show a 3D relationship between both, using the 3D axes { c r , c i , z r } {\displaystyle \{c_{r},c_{i},z_{r}\}} . The animation shows the classic Anti-Buddhabrot with c = ( random , random ) {\displaystyle c=({\text{random}},{\text{random}})} and z 0 = ( 0 , 0 ) {\displaystyle z_{0}=(0,0)} , this is the 2D Mandelbrot set in the plane { c r , c i } {\displaystyle \{c_{r},c_{i}\}} , and also the Anti-Buddhabrot with c = ( random , 0 ) {\displaystyle c=({\text{random}},0)} and z 0 = ( 0 , 0 ) {\displaystyle z_{0}=(0,0)} , this is the 2D logistic map in the plane { c r , z r } {\displaystyle \{c_{r},z_{r}\}} . We rotate the plane { c i , z r } {\displaystyle \{c_{i},z_{r}\}} around the c r {\displaystyle c_{r}} -axis, first showing { c r , c i } {\displaystyle \{c_{r},c_{i}\}} , then rotating 90° to show { c r , z r } {\displaystyle \{c_{r},z_{r}\}} , then rotating an extra 90° to show { c r , − c i } {\displaystyle \{c_{r},-c_{i}\}} . We could rotate an extra 180° but this gives the same images, mirrored around the c r {\displaystyle c_{r}} -axis. The logistic map Anti-Buddhabrot is in fact a subset of the classic Anti-Buddhabrot, situated in the plane { c r , z r } {\displaystyle \{c_{r},z_{r}\}} (or c i = 0 {\displaystyle c_{i}=0} ) of 3D { c r , c i , z r } {\displaystyle \{c_{r},c_{i},z_{r}\}} , perpendicular to the plane { c r , c i } {\displaystyle \{c_{r},c_{i}\}} . We emphasize this by showing briefly, at 90° rotation, only the projected plane c i = 0 {\displaystyle c_{i}=0} , not 'disturbed' by the projections of the planes with non-zero c i {\displaystyle c_{i}} . == References == == External links == Lobo, Albert. "Meet the Buddhabrot technique". Molecular Density. Archived from the original on 2018-09-03. Retrieved 2011-11-21. Mathologer (4 March 2016). "The dark side of the Mandelbrot set". YouTube. Archived from the original on 2021-12-22.
Wikipedia:Bunch–Nielsen–Sorensen formula#0	In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix A {\displaystyle A} and the outer product, v v T {\displaystyle vv^{T}} , of vector v {\displaystyle v} with itself. == Statement == Let λ i {\displaystyle \lambda _{i}} denote the eigenvalues of A {\displaystyle A} and λ ~ i {\displaystyle {\tilde {\lambda }}_{i}} denote the eigenvalues of the updated matrix A ~ = A + v v T {\displaystyle {\tilde {A}}=A+vv^{T}} . In the special case when A {\displaystyle A} is diagonal, the eigenvectors q ~ i {\displaystyle {\tilde {q}}_{i}} of A ~ {\displaystyle {\tilde {A}}} can be written ( q ~ i ) k = N i v k λ k − λ ~ i {\displaystyle ({\tilde {q}}_{i})_{k}={\frac {N_{i}v_{k}}{\lambda _{k}-{\tilde {\lambda }}_{i}}}} where N i {\displaystyle N_{i}} is a number that makes the vector q ~ i {\displaystyle {\tilde {q}}_{i}} normalized. == Derivation == This formula can be derived from the Sherman–Morrison formula by examining the poles of ( A − λ ~ I + v v T ) − 1 {\displaystyle (A-{\tilde {\lambda }}I+vv^{T})^{-1}} . == Remarks == The eigenvalues of A ~ {\displaystyle {\tilde {A}}} were studied by Golub. Numerical stability of the computation is studied by Gu and Eisenstat. == See also == Sherman–Morrison formula == References == == External links == Rank-One Modification of the Symmetric Eigenproblem at EUDML Some Modified Matrix Eigenvalue Problems A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem
Wikipedia:Burkard Polster#0	Burkard Polster (born 26 February 1965 in Würzburg) is a German mathematician who runs and presents the Mathologer channel on YouTube. He is a professor of mathematics at Monash University in Melbourne, Australia. == Education and career == Polster earned a doctorate from the University of Erlangen–Nuremberg in 1993 under the supervision of Karl Strambach. Other universities that Polster has been affiliated with, before joining Monash University in 2000, include the University of Würzburg, University at Albany, University of Kiel, University of California, Berkeley, University of Canterbury, and University of Adelaide. Polster's research involves topics in geometry, recreational mathematics, and the mathematics of everyday life, including how to tie shoelaces or stabilize a wobbly table. == Books == Polster is the author of multiple books including: A Geometrical Picture Book. Springer. 1998. ISBN 0-387-98437-2. Geometries on Surfaces. with Günter Steinke. Cambridge University Press. 2001. ISBN 0-521-66058-0.{{cite book}}: CS1 maint: others (link) The Mathematics of Juggling. Springer. 2003. ISBN 0-387-95513-5. Q.E.D.: Beauty in Mathematical Proof. Wooden Books. 2004. ISBN 190426350X. Included in the four-book compilation Scientia: Mathematics, Physics, Chemistry, Biology, and Astronomy for All (2011) and translated into German as Sciencia: Mathematik, Physik, Chemie, Biologie und Astronomie für alle verständlich (Librero, 2014, in German). Les Ambigrammes: l'art de symétriser les mots (in French). Editions Ecritextes. 2004. ISBN 2915633002. The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace Your Shoes. American Mathematical Society. 2006. ISBN 0821839330. Eye Twisters: Ambigrams & Other Visual Puzzles to Amaze and Entertain. Constable. 2007. ISBN 9781402757983. Math Goes to the Movies. with Marty Ross. Johns Hopkins University Press. 2012. ISBN 9781421404837.{{cite book}}: CS1 maint: others (link) A Dingo Ate My Math Book: Mathematics from Down Under. with Marty Ross. American Mathematical Society. 2017. ISBN 9781470435219.{{cite book}}: CS1 maint: others (link) Putting Two and Two Together. with Marty Ross. American Mathematical Society. 2022. ISBN 978-1-4704-6011-2.{{cite book}}: CS1 maint: others (link) == References == == External links == Mathologer, Polster's YouTube site Maths Masters, Burkard Polster and Marty Ross [1], profile at Monash University Burkard Polster publications indexed by Google Scholar
Wikipedia:Burning Ship fractal#0	The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function: z n + 1 = ( \| Re ⁡ ( z n ) \| + i \| Im ⁡ ( z n ) \| ) 2 + c , z 0 = 0 {\displaystyle z_{n+1}=(\|\operatorname {Re} \left(z_{n}\right)\|+i\|\operatorname {Im} \left(z_{n}\right)\|)^{2}+c,\quad z_{0}=0} in the complex plane C {\displaystyle \mathbb {C} } which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations. Virtually all images of the Burning Ship fractal are reflected vertically for aesthetic purposes, and some are also reflected horizontally. == Implementation == The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code. for each pixel (x, y) on the screen, do: x := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1)) y := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1)) zx := x // zx represents the real part of z zy := y // zy represents the imaginary part of z iteration := 0 max_iteration := 100 while (zxzx + zyzy < 4 and iteration < max_iteration) do xtemp := zxzx - zyzy + x zy := abs(2zxzy) + y // abs returns the absolute value zx := xtemp iteration := iteration + 1 if iteration = max_iteration then // Belongs to the set return INSIDE_COLOR return (max_iteration / iteration) × color // Assign color to pixel outside the set == Gallery == == References == == External links == About properties and symmetries of the Burning Ship fractal, featured by Theory.org Burning Ship Fractal, Description and C source code. Burning Ship with its Mset of higher powers and Julia Sets Burningship, Video, Fractal webpage includes the first representations and the original paper cited above on the Burning Ship fractal. 3D representations of the Burning Ship fractal FractalTS Mandelbrot, Burning ship and corresponding Julia set generator.
Wikipedia:Béla Szőkefalvi-Nagy#0	Béla Szőkefalvi-Nagy [beːlɒ søːkɛfɒlvi nɒɟ] (29 July 1913, Kolozsvár – 21 December 1998, Szeged) was a Hungarian mathematician. His father, Gyula Szőkefalvi-Nagy was also a famed mathematician. Szőkefalvi-Nagy collaborated with Alfréd Haar and Frigyes Riesz, founders of the Szegedian school of mathematics. He contributed to the theory of Fourier series and approximation theory. His most important achievements were made in functional analysis, especially, in the theory of Hilbert space operators. He was editor-in-chief of the Zentralblatt für Mathematik, the Acta Scientiarum Mathematicarum, and the Analysis Mathematica. He was awarded the Kossuth Prize in 1953, along with his co-author F. Riesz, for his book Leçons d'analyse fonctionnelle. He was awarded the Lomonosov Medal in 1979. The Béla Szőkefalvi-Nagy Medal honoring his memory is awarded yearly by Bolyai Institute. == His books == Béla Szőkefalvi-Nagy: Spektraldarstellung linearer Transformationen des Hilbertschen Raumes.(German) Berlin, 1942. 80 p.; 1967. 82 p. Frederic Riesz, Béla Szőkefalvi-Nagy: Leçons d'analyse fonctionnelle. (French) 2e éd. Akadémiai Kiado, Budapest, 1953, VIII+455 pp. Riesz, Frederic; Sz.-Nagy, Béla (1990) [1955]. Functional Analysis. Translated by Boron, Leo F. New York: Dover Publications. ISBN 0-486-66289-6. OCLC 21228994. Ciprian Foiaş, Béla Szőkefalvi-Nagy: Analyse harmonique des opérateurs de l'espace de Hilbert. (French) Masson et Cie, Paris; Akadémiai Kiadó, Budapest 1967 xi+373 pp. Béla Szőkefalvi-Nagy, Frederic Riesz: Funkcionálanalízis. Budapest, 1988. 534 p. (English: Functional Analysis (1990). Dover. ISBN 0-486-66289-6) == His articles == Diagonalization of matrices over H∞. Acta Scientiarum Mathematicarum. Szeged, 1976 On contractions similar to isometries and Toeplitz operators, with Ciprian Foiaş. Ann. Acad. Scient. Fennicae, 1976. The function model of a contraction and the space L’/H’, with Ciprian Foiaş. Acta Scientiarum Mathematicarum. Szeged, 1979, 1980. Toeplitz type operators and hyponormality, with Ciprian Foiaş. Operator theory. Advances and appl., 1983. Factoring compact operator-valued functions, with authors. Acta Scientiarum Mathematicarum. Szeged, 1985. Sz.-Nagy, Béla (1954), "Ein Satz über Parallelverschiebungen konvexer Körper", Acta Universitatis Szegediensis, 15: 169–177, MR 0065942, archived from the original on 2016-03-04, retrieved 2013-05-19. == Award in his honour == In 1999, Béla Szőkefalvi-Nagy's daughter Erzsébet, established the Béla Szőkefalvi-Nagy Medal to remember her father. This medal is meant to recognize distinguished mathematicians who have published significant work in Acta Scientiarum Mathematicarum. The following mathematicians have been awarded the medal: == See also == Sz.-Nagy's dilation theorem Erdős-Nagy theorem == To the memory of Béla Szőkefalvi-Nagy == Operator theory: advances and applications. Recent advances in operator theory and related topics : the Béla Szökefalvi-Nagy memorial volume : [memorial conference held August 2–6, 1999 in Szeged] / eds. László Kérchy et al. Basel; Boston; Berlin : Birkhäuser Verlag, 2001. XLIX, 670 p. (Operator theory : advances and applications; 127.) ISBN 3-7643-6607-9 == References == D.P. Zhelobenko, Bela Szokefalvi-Nagy (obituary), Russian Mathematical Surveys 54 (1999), 819-822. Szôkefalvi-Nagy Béla (in Hungarian)
Wikipedia:Bôcher Memorial Prize#0	The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five years) for a notable research work in analysis that has appeared during the past six years. The work must be published in a recognized, peer-reviewed venue. The current award is $5,000. There have been forty-one prize recipients. The first woman to win the award, Laure Saint-Raymond, did so in 2020. About eighty percent of the journal articles recognized since 2000 have been from Annals of Mathematics, the Journal of the American Mathematical Society, Inventiones Mathematicae, and Acta Mathematica. == Past winners == Source: 1923 George David Birkhoff for Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18 (1917), 119-300. 1924 Eric Temple Bell for Arithmetical paraphrases. I, II. Trans. Amer. Math. Soc. 22 (1921), 1-30, 198-219. 1924 Solomon Lefschetz for On certain numerical invariants with applications to Abelian varieties. Trans. Amer. Math. Soc. 22 (1921), 407-482. 1928 James W. Alexander II for Combinatorial analysis situs. Trans. Amer. Math. Soc. 28 (1926), 301-329. 1933 Marston Morse for The foundations of a theory of the calculus of variations in the large in m-space. Trans. Amer. Math. Soc. 31 (1929), 379-404. 1933 Norbert Wiener for Tauberian theorems. Ann. Math. 33 (1932), 1-100. 1938 John von Neumann for Almost periodic functions. I. Trans. Amer. Math. Soc. 36 (1934), 445-294 Almost periodic functions. II. Trans. Amer. Math. Soc. 37 (1935), 21-50 1943 Jesse Douglas for Green's function and the problem of Plateau. Amer. J. Math. 61 (1939), 545-589 The most general form of the problem of Plateau. Amer. J. Math. 61 (1939), 590-608 Solution of the inverse problem of the calculus of variations. Proc. Natl. Acad. Sci. U.S.A. 25 (1939), 631-637. 1948 Albert Schaeffer and Donald Spencer for Coefficients of schlicht functions. I. Duke Math. J. 10 (1943), 611-635 Coefficients of schlicht functions. II. Duke Math. J. 12 (1945), 107-125 Coefficients of schlicht functions. III. Proc. Natl. Acad. Sci. U.S.A. 32 (!946), 111-116 Coefficients of schlicht functions. IV. Proc. Natl. Acad. Sci. U.S.A. 35 (1949), 143-150. 1953 Norman Levinson for "his contributions to the theory of linear, nonlinear, ordinary, and partial differential equations contained in his papers of recent years" 1959 Louis Nirenberg for "his work in partial differential equations" 1964 Paul Cohen for On a conjecture of Littlewood and idempotent measures. Amer. J. Math. 82 (1960), 191-212. 1969 Isadore Singer for "his work on the index problem" and especially The index of elliptic operators. I. Ann. of Math. (2) 87 (1968), 484-530 The index of elliptic operators. III. Ann. of Math. (2) 87 (1968), 546-604 both written with Michael Atiyah. 1974 Donald Samuel Ornstein for Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337-352. 1979 Alberto Calderón for "his fundamental work on the theory of singular integrals and partial differential equations" and in particular Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. U.S.A. 74 (1977), 1324-1327. 1984 Luis Caffarelli for "his deep and fundamental work in nonlinear partial differential equations, in particular his work on free boundary problems, vortex theory and regularity theory" 1984 Richard Melrose for "his solution of several outstanding problems in diffraction theory and scattering theory and for developing the analytical tools needed for their resolution" 1989 Richard Schoen for "his work on the application of partial differential equations to differential geometry," in particular Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. 20 (1984), 479-495. 1994 Leon Simon for: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. (2) 118 (1983), no. 3, 525–571. Cylindrical tangent cones and the singular set of minimal submanifolds. J. Diff. Geom. 38 (1993), no. 3, 585–652. Rectifiability of the singular set of energy minimizing maps. Calc. Var. Partial Differential Equations 3 (1995), no. 1, 1–65. 1999 Demetrios Christodoulou for: The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. [written with Sergiu Klainerman] Examples of naked singularity formation in the gravitational collapse of a scalar field. Ann. of Math. (2) 140 (1994), no. 3, 607–653. The instability of naked singularities in the gravitational collapse of a scalar field. Ann. of Math. (2) 149 (1999), no. 1, 183–217 1999 Sergiu Klainerman for: The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. [written with Demetrios Christodoulou] Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268 [with Matei Machedon] Smoothing estimates for null forms and applications. Duke Math. J. 81 (1995), no. 1, 99–133 [with Matei Machedon] 1999 Thomas Wolff for "his work in harmonic analysis," "harmonic measure, and unique continuation," including Counterexamples with harmonic gradients in ℝ3. Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 321–384, Princeton Math. Ser., 42, Princeton Univ. Press, Princeton, NJ, 1995 An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. A Kakeya-type problem for circles. Amer. J. Math. 119 (1997), no. 5, 985–1026 2002 Daniel Tătaru for On global existence and scattering for the wave maps equations. Amer. J. Math. 123 (2001) no. 1, 37–77 in addition to his "important work on Strichartz estimates for wave equations with rough coefficients and applications to quasilinear wave equations, as well as his many deep contributions to unique continuation problems" 2002 Terence Tao for Global regularity of wave maps I. Small critical Sobolev norm in high dimensions. Internat. Math. Res. Notices (2001), no. 6, 299–328 Global regularity of wave maps II. Small energy in two dimensions. Comm. Math. Phys. 2244 (2001), no. 2, 443–544. in addition to "his remarkable series of papers, written in collaboration with J. Colliander, M. Keel, G. Staffilani, and H. Takaoka, on global regularity in optimal Sobolev spaces for KdV and other equations, as well as his many deep contributions to Strichartz and bilinear estimates." 2002 Lin Fanghua for Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49 (1996), no. 4, 323–359. Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. of Math. (2) 149 (1999), no. 3, 785–829. in addition to other "fundamental contributions to our understanding of the Ginzburg-Landau (GL) equations with a small parameter" and "many deep contributions to harmonic maps and liquid crystals." 2005 Frank Merle for "his fundamental work in the analysis of nonlinear dispersive equations" including: Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. of Math. (2) 155 (2002), no. 1, 235–280 [written with Yvan Martel] Blow up in finite time and dynamics of blow up solutions for the L2-critical generalized KdV equation. J. Amer. Math. Soc. 15 (2002), no. 3, 617–664 [written with Yvan Martel] On universality of blow-up profile for L2 critical nonlinear Schrödinger equation. Invent. Math. 156 (2004), no. 3, 565–672 [with Pierre Raphael] 2008 Alberto Bressan for: Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. xii+250 pp. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. of Math. (2) 161 (2005), no. 1, 223–342 [written with Stefano Bianchini] 2008 Charles Fefferman for "his many fundamental contributions to different areas of analysis" including A sharp form of Whitney's extension theorem. Ann. of Math. (2) 161 (2005), no. 1, 509–577 Whitney's extension problem for Cm. Ann. of Math. (2) 164 (2006), no. 1, 313–359. 2008 Carlos Kenig for "his important contributions to harmonic analysis, partial differential equations, and nonlinear dispersive PDE" including: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620 [written with Gustavo Ponce and Luis Vega] Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Amer. Math. Soc. 20 (2007), no. 3, 753–798 [written with Alexandru Ionescu] Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201 (2008), no. 2, 147–212 [written with Frank Merle] 2011 Assaf Naor for "introducing new invariants of metric spaces and for applying his new understanding of the distortion between various metric structures to theoretical computer science" and his "remarkable work [...] on a lower bound in the sparsest cut problem" including On metric Ramsey-type phenomena. Ann. of Math. (2) 162 (2005), no. 2, 643–709 [written with Yair Bartal, Nathan Linial, and Manor Mendel] Metric cotype. Ann. of Math. (2) 168 (2008), no. 1, 247–298 [written with Manor Mendel] Euclidean distortion and the sparsest cut. J. Amer. Math. Soc. 21 (2008), no. 1, 1–21 [written with Sanjeev Arora and James R. Lee] Compression bounds for Lipschitz maps from the Heisenberg group to L1. Acta Math. 207 (2011), no. 2, 291–373 [written with Jeff Cheeger and Bruce Kleiner] 2011 Gunther Uhlmann for "his fundamental work on inverse problems" including The Calderón problem with partial data. Ann. of Math. (2) 165 (2007), no. 2, 567–591 [written with Carlos Kenig and Johannes Sjöstrand] The Calderón problem with partial data in two dimensions. J. Amer. Math. Soc. 23 (2010), no. 3, 655–691 [written with Oleg Imanuvilov and Masahiro Yamamoto] as well as "incisive work on boundary rigidity with L. Pestov and with P. Stepanov and on nonuniqueness (also known as cloaking) with A. Greenleaf, Y. Kurylev, and M. Lassas." 2014 Simon Brendle for "his outstanding solutions of long standing problems in geometric analysis", including Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22 (2009), no. 1, 287–307. [written with Richard Schoen] Embedded minimal tori in S3 and the Lawson conjecture. Acta Math. 211 (2013), no. 2, 177–190. in addition to "his deep contributions to the study of the Yamabe equation." 2017 András Vasy for Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces. Invent. Math. 194 (2013), 381-513. 2020 Camillo De Lellis for "his innovative point of view on the construction of continuous dissipative solutions of the Euler equations, which ultimately led to Isett's full solution of the Onsager conjecture, and his spectacular work in the regularity theory of minimal surfaces, where he completed and improved Almgren's program" including: Dissipative continuous Euler flows. Invent. Math. 193 (2013), no. 2, 377–407 [written with László Székelyhidi] Regularity of area minimizing currents III: blow-up. Ann. of Math. (2) 183 (2016), no. 2, 577–617 [written with Emanuele Spadaro] 2020 Lawrence Guth for "his deep and influential development of algebraic and topological methods for partitioning the Euclidean space and multi-scale organization of data, and his powerful applications of these tools in harmonic analysis, incidence geometry, analytic number theory, and partial differential equations" including: A restriction estimate using polynomial partitioning. J. Amer. Math. Soc. 29 (2016), no. 2, 371–413 A sharp Schrödinger maximal estimate in ℝ2. Ann. of Math. (2) 186 (2017), no. 2, 607–640 [written with Xiumin Du and Xiaochun Li] 2020 Laure Saint-Raymond for "her transformative contributions to kinetic theory, fluid dynamics, and Hilbert's sixth problem on 'developing mathematically the limiting processes...which lead from the atomistic view to the laws of motion of continua,'" including: The Brownian motion as the limit of a deterministic system of hard-spheres. Invent. Math. 203 (2016), no. 2, 493–553 [written with Thierry Bodineau and Isabelle Gallagher] Mathematical study of degenerate boundary layers: a large scale ocean circulation problem. Mem. Amer. Math. Soc. 253 (2018), no. 1206, vi+105 pp. [written with Anne-Laure Dalibard] 2023 Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jérémie Szeftel for "their groundbreaking work establishing the existence of blow-up solutions to the defocusing NLS equation in some supercritical regimes and to the compressible Euler and Navier-Stokes equations" including On the implosion of a compressible fluid I: smooth self-similar inviscid profiles. Annals of Mathematics 196 (2022); On the implosion of a compressible fluid II: singularity formation. Annals of Mathematics 196 (2022); and On blow up for the energy super critical defocusing nonlinear Schrödinger equations. Inventiones Mathematicae 227 (2022). == See also == List of mathematics awards == References == == External links == AMS Prize - Bôcher Memorial Prize MacTutor History of Mathematics - Winners of the Bôcher Prize
Wikipedia:Böttcher's equation#0	Böttcher's equation, named after Lucjan Böttcher, is the functional equation F ( h ( z ) ) = ( F ( z ) ) n {\displaystyle F(h(z))=(F(z))^{n}} where h is a given analytic function with a superattracting fixed point of order n at a, (that is, h ( z ) = a + c ( z − a ) n + O ( ( z − a ) n + 1 ) , {\displaystyle h(z)=a+c(z-a)^{n}+O((z-a)^{n+1})~,} in a neighbourhood of a), with n ≥ 2 F is a sought function. The logarithm of this functional equation amounts to Schröder's equation. == Solution == Solution of functional equation is a function in implicit form. Lucian Emil Böttcher sketched a proof in 1904 on the existence of solution: an analytic function F in a neighborhood of the fixed point a, such that: F ( a ) = 0 {\displaystyle F(a)=0} This solution is sometimes called: the Böttcher coordinate the Böttcher function the Boettcher map. The complete proof was published by Joseph Ritt in 1920, who was unaware of the original formulation. Böttcher's coordinate (the logarithm of the Schröder function) conjugates h(z) in a neighbourhood of the fixed point to the function zn. An especially important case is when h(z) is a polynomial of degree n, and a = ∞ . === Explicit === One can explicitly compute Böttcher coordinates for: power maps z → z d {\displaystyle z\to z^{d}} Chebyshev polynomials ==== Examples ==== For the function h and n=2 h ( x ) = x 2 1 − 2 x 2 {\displaystyle h(x)={\frac {x^{2}}{1-2x^{2}}}} the Böttcher function F is: F ( x ) = x 1 + x 2 {\displaystyle F(x)={\frac {x}{1+x^{2}}}} == Applications == Böttcher's equation plays a fundamental role in the part of holomorphic dynamics which studies iteration of polynomials of one complex variable. Global properties of the Böttcher coordinate were studied by Fatou and Douady and Hubbard. == See also == Schröder's equation External ray == References ==
Wikipedia:Børge Jessen#0	Børge Christian Jessen (19 June 1907 – 20 March 1993) was a Danish mathematician best known for his work in analysis, specifically on the Riemann zeta function, and in geometry, specifically on Hilbert's third problem. == Early years == Jessen was born on 19 June 1907 in Copenhagen to Hans Jessen and Christine Jessen (née Larsen). He attended Skt. Jørgens Gymnasium, where he was taught by the Hungarian mathematician Julius Pal during his first year. In 1925, Jessen graduated from the gymnasium and enrolled at the University of Copenhagen. During his time at the university he got to know Harald Bohr, then a leading figure in Danish mathematics. In 1928, Bohr established a collaboration with Jessen, which would last until Bohr's death in 1951. After receiving his master's degree in the spring of 1929, Jessen embarked on a stay abroad. Supported by the Carlsberg Foundation, he spent the fall of 1929 at the University of Szeged, where he met Frigyes Riesz, Alfréd Haar, and Lipót Fejér. He then spent the winter semester of 1929–30 at the University of Göttingen, where he attended lectures by David Hilbert and Edmund Landau while working on his PhD thesis. On 1 May 1930 Jessen defended his thesis in Copenhagen. He later elaborated the thesis into an article that was published in Acta Mathematica in 1934. The same year, he was appointed as a docent at The Royal Veterinary and Agricultural University in Denmark. In 1931, Jessen married Ellen Pedersen (1903–1979), cand. mag. in mathematics and the daughter of Peder Oluf Pedersen. Jessen continued to travel frequently in the early 1930s, visiting Paris, Cambridge, England, the Institute for Advanced Study, Yale and Harvard University in America. == Career == Jessen was a professor of descriptive geometry at the Technical University of Denmark from 1935 till 1942, when he moved back to the University of Copenhagen where he was professor from 1942 to 1977 when he retired. He was the president of the Carlsberg Foundation in 1955-1963 and one of the founders of the Hans Christian Ørsted Institute. He was the Secretary of the Interim Executive Committee of the International Mathematical Union (1950–1952), and in September 1951 he officially declared the founding of the Union, with its first domicile in Copenhagen. He was also active in the Danish Mathematical Society. After his death, the society named an award in his honor (Børge Jessen Diploma Award). == See also == Jessen's icosahedron Jessen–Wintner theorem == References == == External links == Bernard Bru and Salah Eid "Jessen’s theorem and Lévy’s lemma" in JEHPS June 2009 A short biography Børge Jessen at the Mathematics Genealogy Project
Wikipedia:Bījapallava#0	Bījapallava (or Bījapallavaṃ) is a commentary in Sanskrit of Bhaskara II's Bījagaṇita composed by the 16th-17th century astrologer-mathematician Kṛṣṇa Daivajña. This work is also known by several other names: Kalpālatāvatāra, Bījānkura and Nāvāakura. A manuscript of the work, copied in 1601, has survived to the present day indicating that the work must have been composed earlier than 1601. The Bījapallava commentary is written in prose. Commentaries composed in prose, since they are not constrained by considerations of conforming to a particular meter, generally contain more information, more detailed explanations and often original material not found in the work on which the commentary is written. Bījapallava also follows this general pattern. T. Hayashi, a Japanese historian of Indian mathematics, in his forward to the critical edition of Bījapallava, writes: ". . . he [Kṛṣṇa Daivajña] goes on to discuss the mathematical contents in great detail, giving proofs (upapattis) for the rules and step-by-step solutions for the examples; but when the solution is easy, he merely refers to Bhaskara's auto-commentary. His discussions, often in the form of disputations between an imaginary opponent and himself, go deep into the nature of important mathematical concepts such as negative quantity, zero and unknown quantity, into the raison d'être of particular steps of the algorithms, and into various conditions for solubility of the mathematical problems treated in the Bijaganita." The general style of the commentary can be summarized thus. For each stanza of the original text, the commentator gives explanations of the words used in the stanza, then the derivations of the word, synonyms and syntactic combinations of the word are given. He also gives alternate readings of the text and points out which one of them is preferable. What is of greatest interest to historians of mathematics is that he also gives detailed proofs of the rules enunciated in the original text and the detailed step-by-step solutions of the illustrative examples. This has helped translators of Bījagaṇita to understand the real import of the various rules stated therein. For example, H. T. Colebrooke while translating Bījagaṇita has extensively referred to Bījapallava seeking additional clarifications. Though there are large number of commentaries on Bhāskara II's Līlāvatī, there are not many commentaries on his Bījagaṇita. In fact, chronologically, Bījapallava is the second known commentary on Bījagaṇita the first one being a commentary called Sūryaprakāśa composed by Sūryadāsa in 1538, a native of Parthapura. Even though Sūryaprakāśa contains explanations of almost every verse in Bījagaṇita, the explanations in Bījapallava are more informative and more elaborate with additional original ideas and examples. == Salient features == Here are some of the salient features of Bījapallava: The concept of "number line" and its application to explain addition and subtraction of positive and negative numbers. Detailed proof of the Kuṭṭaka method for solving linear Diophantine equations. Proof of Bhāskara II's rule for solving quadratic equations. Proofs of the rules for solving linear equations in several unknowns, equations with higher powers of unknowns and equations with products of unknowns. == Full text of the work == Full text of Bījapallavaṃ, Kṛṣṇa Daivajña's commentary on the Bījagaṇita of Bhāskara II: Kṛṣṇa Daivajña (1958). Bijapallavam edited with Introduction by T. V. Radhakrishna Sastri. T. M. S. S. M. Library, Tanjore: S. Gopalan. Retrieved 22 June 2024. Full text of a critical study on Bījapallavaṃ: Sita Sundar Ram (2012). Bijapallava of Kṛṣṇa Daivajña: Algebra in Sixteenth Century India, a Critical Study. Chennai: The Kuppuswami Sastri Research Institute. Retrieved 22 June 2024. == References ==
Wikipedia:C. J. Eliezer#0	Christie Jayaratnam Eliezer (Tamil: கிரிஸ்டி ஜெயரத்தினம் எலியேசர், romanized: Kirisṭi Jeyarattiṉam Eliyēcar; 12 June 1918 – 10 March 2001) was a Ceylon Tamil mathematician, physicist and academic. == Early life and family == Eliezer was born on 12 June 1918 in Navatkuli in northern Ceylon. He was the son of Jacob Richard Eliezer and Elizabeth Ponnammah Vairakiam. Both of his parents died when he was young. Eliezer was educated at the Wesleyan Mission School, Puloly and Hartley College, Point Pedro (1926–33) where he passed the Cambridge Local Examinations with honours and distinction. He then spent a year studying at St. Joseph's College, Colombo before joining Ceylon University College in 1935, graduating with a first class honours B.Sc. special degree in mathematics. Eliezer married Ranee, daughter of Rev. John Handy. They had five children (Dhamayanthi, Ratna, Anandhi, Renuka and Tamara). == Career == Eliezer worked at Ceylon University College as a visiting lecturer in 1938 before proceeding to Christ's College, Cambridge (1939–43) on a scholarship to study mathematics and theoretical physics. He received a first class mathematics tripos from Christ's College in 1941. He received a Ph.D. degree from Cambridge in 1946 after producing a thesis, supervised by Paul Dirac, on spinning electron and electromagnetic field. Returning to Ceylon Eliezer lectured at the University of Ceylon for a brief period before rejoining Christ's College as a fellow (1946–49). He received a D.Sc. degree in 1949. He was called to the bar at the Middle Temple in 1949. Eliezer was appointed professor of mathematics at University of Ceylon in 1949. During his ten years at the university he was dean of the Faculty of Science from 1954 to 1957 and deputy pro-vice chancellor in 1955. Eliezer was a scholar at the Institute for Advanced Study from 1955 to 1956, working with J. Robert Oppenheimer. He spent some time at the University of Chicago. Following the passing of the Sinhala Only Act in 1956, Sinhalese nationalists at the University of Ceylon, led by vice-chancellor Nicholas Attygalle and chancellor Dudley Senanayake, attempted to remove Tamil as a medium of instruction at the university but this was thwarted by Eliezer and A. M. A. Azeez, a member of the university's council. Eliezer was appointed foundation professor of mathematics at the University of Malaya in 1959. The appointment was only meant be for two years but the deteriorating situation in Ceylon meant that Eliezer decided to stay in Malaya. During his nine years at the university he was dean of the Faculty of Science from 1959 to 1963 and deputy principal and vice-chancellor for another three years. Eliezer became the first professor of applied mathematics at La Trobe University in 1968. During his 15 years at the university he was dean of the School of Physical Sciences (1969–71 and 1982–83) and deputy vice-chancellor for a period. After retiring in 1983 he was appointed emeritus professor at La Trobe. Eliezer had been president of the Ceylon Association for the Advancement of Science. He was a Fellow of the Institute of Mathematics and its Applications. He received an honorary D.Sc.Inf. degree from the University of Jaffna in 1981. He was made a Member of the Order of Australia in 1996. He was awarded the Maamanithar (Great Man) honour by the rebel Liberation Tigers of Tamil Eelam on 19 October 1997. Eliezer was vice-president of the Colombo branch of the Young Men’s Christian Association and a member of the Jaffna College board. He was president of the Ceylon Tamil Association of Victoria and chairman of the Australian Federation of Tamil Associations (1984-2001). Eliezer helped Tamil refugees fleeing to Australia following the 1983 anti-Tamil Black July riots. He co-hosted a Tamil language programme on the Special Broadcasting Service. Eliezer died on 10 March 2001 in Melbourne. == Works == Eliezer wrote several books and articles including: Concise Vector Analysis (1963, Pergamon Press) A Modern Text-book on Statics: For Students of Applied Mathematics, Physics and Engineering (1964, Pergamon Press) Mathematics : Queen of the Arts, Handmaiden of the Sciences (1969, La Trobe University) The First Integrals of Some Differential Equations of Dynamics (1978, La Trobe University) Mechanics for Year Eleven (1988, co-author J. G. Barton) == References == == External links == One Hundred Tamils of the 20th Century, Tamil Nation
Wikipedia:C. S. Yogananda#0	C. S. Yogananda is a mathematician and Professor of Mathematics at the J.S.S Science and Technology University in Mysore, India. He is the founder of Sriranga Digital Software Technologies and author who has published several writings on mathematics. == Education == Yogananda received his Ph.D. from the Institute of Mathematical Sciences, Chennai under advisor Ramachandran Balasubramanian. == Mathematical Olympiads == Yogananda has been involved with the Mathematical Olympiad movement in India since 1989. He has participated as a resource person and evaluated answer books for various regional (RMO) and national (INMO) Olympiads. He was also an academic coordinator for the Problem Coordinators workshops organized by the National Board for Higher Mathematics. He was a member of the Core Faculty at the International Mathematical Olympiad (IMO)Training Camps and was also involved in the selection of the Indian Team since 1989. Yogananda was a member of the Organizing Committee (Computer Committee/Publications Committee/Problem Selection Committee) when India hosted the IMO in July 1996 in Mumbai. Yogananda was an Observer and Deputy Leader for the Indian team participating in the IMO in the years 1993, 1995, and 1998. During this time, India won 3 Gold, 10 Silver and 4 Bronze medals in total. == Sriranga Digital Software Technologies == Yogananda established Sriranga Digital Software Technologies in 2003, with the primary intention of bringing digital technologies to the service of Indian languages. == Advaita Sharada == Yogananda also created a digital archive of Sri Shankaracharya's works. The Advaita Sharada initiative has been recognized by the Jagadgurus of Sri Sringeri Sharada Peetham. The archive is a collection of Prasthanatraya Bhashya of Sri Shankara, his prakarana granthas and also the commentaries by later commentators of Sri Shankara. The archive can help in understanding the works of Sri Shankara in more depth since it allows a complete search of the texts. == Honorary positions == Honorary Secretary, Leelavati Trust (Regd.), Bangalore. Honorary Joint Director, Jawaharlal Nehru Planetarium, Bangalore from November 1, 2000, to October 31, 2003. Member Secretary, Steering Committee for Informatics Olympiad. Chairman, TeX Users Group India (TUGIndia). == References == == External links == DVK Murthy Prakashana Number Theory IMO 1998 list
Wikipedia:CLRg property#0	In mathematics, the notion of “common limit in the range” property denoted by CLRg property is a theorem that unifies, generalizes, and extends the contractive mappings in fuzzy metric spaces, where the range of the mappings does not necessarily need to be a closed subspace of a non-empty set X {\displaystyle X} . Suppose X {\displaystyle X} is a non-empty set, and d {\displaystyle d} is a distance metric; thus, ( X , d ) {\displaystyle (X,d)} is a metric space. Now suppose we have self mappings f , g : X → X . {\displaystyle f,g:X\to X.} These mappings are said to fulfil CLRg property if lim k → ∞ f x k = lim k → ∞ g x k = g x , {\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=gx,} for some x ∈ X . {\displaystyle x\in X.} Next, we give some examples that satisfy the CLRg property. == Examples == Source: === Example 1 === Suppose ( X , d ) {\displaystyle (X,d)} is a usual metric space, with X = [ 0 , ∞ ) . {\displaystyle X=[0,\infty ).} Now, if the mappings f , g : X → X {\displaystyle f,g:X\to X} are defined respectively as follows: f x = x 4 {\displaystyle fx={\frac {x}{4}}} g x = 3 x 4 {\displaystyle gx={\frac {3x}{4}}} for all x ∈ X . {\displaystyle x\in X.} Now, if the following sequence { x k } = { 1 / k } {\displaystyle \{x_{k}\}=\{1/k\}} is considered. We can see that lim k → ∞ f x k = lim k → ∞ g x k = g 0 = 0 , {\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=g0=0,} thus, the mappings f {\displaystyle f} and g {\displaystyle g} fulfilled the CLRg property. Another example that shades more light to this CLRg property is given below === Example 2 === Let ( X , d ) {\displaystyle (X,d)} is a usual metric space, with X = [ 0 , ∞ ) . {\displaystyle X=[0,\infty ).} Now, if the mappings f , g : X → X {\displaystyle f,g:X\to X} are defined respectively as follows: f x = x + 1 {\displaystyle fx=x+1} g x = 2 x {\displaystyle gx=2x} for all x ∈ X . {\displaystyle x\in X.} Now, if the following sequence { x k } = { 1 + 1 / k } {\displaystyle \{x_{k}\}=\{1+1/k\}} is considered. We can easily see that lim k → ∞ f x k = lim k → ∞ g x k = g 1 = 2 , {\displaystyle \lim _{k\to \infty }fx_{k}=\lim _{k\to \infty }gx_{k}=g1=2,} hence, the mappings f {\displaystyle f} and g {\displaystyle g} fulfilled the CLRg property. == References ==
Wikipedia:CSS code#0	In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of stabilizer code constructed from classical codes with some special properties. An example of a CSS code is the Steane code. == Construction == Let C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} be two (classical) [ n , k 1 ] {\displaystyle [n,k_{1}]} , [ n , k 2 ] {\displaystyle [n,k_{2}]} codes such, that C 2 ⊂ C 1 {\displaystyle C_{2}\subset C_{1}} and C 1 , C 2 ⊥ {\displaystyle C_{1},C_{2}^{\perp }} both have minimal distance ≥ 2 t + 1 {\displaystyle \geq 2t+1} , where C 2 ⊥ {\displaystyle C_{2}^{\perp }} is the code dual to C 2 {\displaystyle C_{2}} . Then define CSS ( C 1 , C 2 ) {\displaystyle {\text{CSS}}(C_{1},C_{2})} , the CSS code of C 1 {\displaystyle C_{1}} over C 2 {\displaystyle C_{2}} as an [ n , k 1 − k 2 , d ] {\displaystyle [n,k_{1}-k_{2},d]} code, with d ≥ 2 t + 1 {\displaystyle d\geq 2t+1} as follows: Define for x ∈ C 1 : \| x + C 2 ⟩ := {\displaystyle x\in C_{1}:{\|}x+C_{2}\rangle :=} 1 / \| C 2 \| {\displaystyle 1/{\sqrt {{\|}C_{2}{\|}}}} ∑ y ∈ C 2 \| x + y ⟩ {\displaystyle \sum _{y\in C_{2}}{\|}x+y\rangle } , where + {\displaystyle +} is bitwise addition modulo 2. Then CSS ( C 1 , C 2 ) {\displaystyle {\text{CSS}}(C_{1},C_{2})} is defined as { \| x + C 2 ⟩ ∣ x ∈ C 1 } {\displaystyle \{{\|}x+C_{2}\rangle \mid x\in C_{1}\}} . == References == Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 844974180. == External links ==
Wikipedia:Cabiria Andreian Cazacu#0	Cabiria Andreian Cazacu (February 19, 1928 – May 22, 2018) was a Romanian mathematician known for her work in complex analysis. She held the chair in mathematical analysis at the University of Bucharest from 1973 to 1975, and was dean of the faculty of mathematics at the University of Bucharest from 1976 to 1984. == Life == Andreian Cazacu was born on February 19, 1928, in Iași, the daughter of mathematics teacher Ioan T. Ardeleanu. Towards the end of World War II, her family became refugees in Bucharest, where she completed her high school studies in 1945. She then enrolled in the Faculty of Mathematics at the University of Bucharest, graduating with a B.S. in 1949; her undergraduate thesis, on Generalized nilpotent groups, was written under the guidance of Dan Barbilian. She then continued at the university, first as a teaching assistant and then as a lecturer starting in 1950. She became a student of Simion Stoilow, completing a doctorate in 1955 under his supervision, with the dissertation Normally exhaustible Riemann surfaces. After being named associate professor in 1955, she completed a habilitation in 1967, with the habilitation thesis Classes of Riemann coverings, and was promoted to full professor in 1968. From 1951 to 1969 Andreian Cazacu held a research position at the Institute of Mathematics of the Romanian Academy, where she was a leading participant in Stoilow's seminar on complex analysis. She held visiting positions at the Free University of Berlin, Université libre de Bruxelles, the University of Helsinki, the University of Łódź, and Université de Moncton. Between 1976 and 2010 she supervised the Ph.D. theses of 15 students. Andreian Cazacu was one of the main organizers of eleven editions of the Romanian-Finnish Seminar on complex analysis and potential theory, founded by Rolf Nevanlinna and Stoilow; the proceedings of four of these seminars, for which she was an editor, appeared in the Springer Lecture Notes in Mathematics series as four separate volumes. She died on May 22, 2018, in Bucharest and was buried at Ghencea Cemetery. == Publications == Andreian Cazacu wrote "approximately 100 scientific papers and six books". The books include: Andreian Cazacu, Cabiria; Constantinescu, Corneliu; Jurchescu, Martin (1965). Probleme moderne de teoria funcțiilor [Modern problems of the theory of functions] (in Romanian). Bucharest: Editura Academiei Republicii Populare Române. MR 0200105. OCLC 6014733. Andreian Cazacu, Cabiria; Deleanu, Aristide; Jurchescu, Martin (1966). Topologie, categorii, suprafețe riemanniene [Topology, categories, Riemann surfaces] (in Romanian). Bucharest: Editura Academiei Republicii Populare Române. MR 0215663. OCLC 4666024. Andreian Cazacu, Cabiria (1975). Theorie der Funktionen mehrerer komplexer Veränderlicher [Theory of functions of several complex variables]. Lehrbücher und Monographien aus dem Gebiet der Exakten Wissenschaften, Mathematische Reihe (in German). Vol. 51. Basel-Stuttgart: Birkhäuser. ISBN 3-7643-0770-6. MR 0473209. OCLC 2090281. == Recognition == Andreian Cazacu won the Simion Stoilow Prize of the Romanian Academy in 1966. In 1998 the University of Craiova gave her an honorary doctorate. She was awarded in 2000 the National Order of Faithful Service, Officer rank by President Ion Iliescu, while in 2011 she was awarded the National Order of Faithful Service, Commander rank by President Traian Băsescu. She became an honorary member of the Romanian Academy in 2006. In 2010, the journal Complex Variables and Elliptic Equations published a special issue in honor of her 80th birthday. == References ==
Wikipedia:Caccioppoli set#0	In mathematics, a Caccioppoli set is a subset of R n {\displaystyle \mathbb {R} ^{n}} whose boundary is (in a suitable sense) measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation, and its perimeter is the total variation of the characteristic function. == History == The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper (Caccioppoli 1927): considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, provided this quantity was bounded. The measure of the boundary of a set was defined as a functional, precisely a set function, for the first time: also, being defined on open sets, it can be defined on all Borel sets and its value can be approximated by the values it takes on an increasing net of subsets. Another clearly stated (and demonstrated) property of this functional was its lower semi-continuity. In the paper (Caccioppoli 1928), he precised by using a triangular mesh as an increasing net approximating the open domain, defining positive and negative variations whose sum is the total variation, i.e. the area functional. His inspiring point of view, as he explicitly admitted, was those of Giuseppe Peano, as expressed by the Peano-Jordan Measure: to associate to every portion of a surface an oriented plane area in a similar way as an approximating chord is associated to a curve. Also, another theme found in this theory was the extension of a functional from a subspace to the whole ambient space: the use of theorems generalizing the Hahn–Banach theorem is frequently encountered in Caccioppoli research. However, the restricted meaning of total variation in the sense of Tonelli added much complication to the formal development of the theory, and the use of a parametric description of the sets restricted its scope. Lamberto Cesari introduced the "right" generalization of functions of bounded variation to the case of several variables only in 1936: perhaps, this was one of the reasons that induced Caccioppoli to present an improved version of his theory only nearly 24 years later, in the talk (Caccioppoli 1953) at the IV UMI Congress in October 1951, followed by five notes published in the Rendiconti of the Accademia Nazionale dei Lincei. These notes were sharply criticized by Laurence Chisholm Young in the Mathematical Reviews. In 1952 Ennio De Giorgi presented his first results, developing the ideas of Caccioppoli, on the definition of the measure of boundaries of sets at the Salzburg Congress of the Austrian Mathematical Society: he obtained this results by using a smoothing operator, analogous to a mollifier, constructed from the Gaussian function, independently proving some results of Caccioppoli. Probably he was led to study this theory by his teacher and friend Mauro Picone, who had also been the teacher of Caccioppoli and was likewise his friend. De Giorgi met Caccioppoli in 1953 for the first time: during their meeting, Caccioppoli expressed a profound appreciation of his work, starting their lifelong friendship. The same year he published his first paper on the topic i.e. (De Giorgi 1953): however, this paper and the closely following one did not attracted much interest from the mathematical community. It was only with the paper (De Giorgi 1954), reviewed again by Laurence Chisholm Young in the Mathematical Reviews, that his approach to sets of finite perimeter became widely known and appreciated: also, in the review, Young revised his previous criticism on the work of Caccioppoli. The last paper of De Giorgi on the theory of perimeters was published in 1958: in 1959, after the death of Caccioppoli, he started to call sets of finite perimeter "Caccioppoli sets". Two years later Herbert Federer and Wendell Fleming published their paper (Federer & Fleming 1960), changing the approach to the theory. Basically they introduced two new kind of currents, respectively normal currents and integral currents: in a subsequent series of papers and in his famous treatise, Federer showed that Caccioppoli sets are normal currents of dimension n {\displaystyle n} in n {\displaystyle n} -dimensional euclidean spaces. However, even if the theory of Caccioppoli sets can be studied within the framework of theory of currents, it is customary to study it through the "traditional" approach using functions of bounded variation, as the various sections found in a lot of important monographs in mathematics and mathematical physics testify. == Formal definition == In what follows, the definition and properties of functions of bounded variation in the n {\displaystyle n} -dimensional setting will be used. === Caccioppoli definition === Definition 1. Let Ω {\displaystyle \Omega } be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and let E {\displaystyle E} be a Borel set. The perimeter of E {\displaystyle E} in Ω {\displaystyle \Omega } is defined as follows P ( E , Ω ) = V ( χ E , Ω ) := sup { ∫ Ω χ E ( x ) d i v ϕ ( x ) d x : ϕ ∈ C c 1 ( Ω , R n ) , ‖ ϕ ‖ L ∞ ( Ω ) ≤ 1 } {\displaystyle P(E,\Omega )=V\left(\chi _{E},\Omega \right):=\sup \left\{\int _{\Omega }\chi _{E}(x)\mathrm {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x:{\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \\|{\boldsymbol {\phi }}\\|_{L^{\infty }(\Omega )}\leq 1\right\}} where χ E {\displaystyle \chi _{E}} is the characteristic function of E {\displaystyle E} . That is, the perimeter of E {\displaystyle E} in an open set Ω {\displaystyle \Omega } is defined to be the total variation of its characteristic function on that open set. If Ω = R n {\displaystyle \Omega =\mathbb {R} ^{n}} , then we write P ( E ) = P ( E , R n ) {\displaystyle P(E)=P(E,\mathbb {R} ^{n})} for the (global) perimeter. Definition 2. The Borel set E {\displaystyle E} is a Caccioppoli set if and only if it has finite perimeter in every bounded open subset Ω {\displaystyle \Omega } of R n {\displaystyle \mathbb {R} ^{n}} , i.e. P ( E , Ω ) < + ∞ {\displaystyle P(E,\Omega )<+\infty } whenever Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} is open and bounded. Therefore, a Caccioppoli set has a characteristic function whose total variation is locally bounded. From the theory of functions of bounded variation it is known that this implies the existence of a vector-valued Radon measure D χ E {\displaystyle D\chi _{E}} such that ∫ Ω χ E ( x ) d i v ϕ ( x ) d x = ∫ E d i v ϕ ( x ) d x = − ∫ Ω ⟨ ϕ , D χ E ( x ) ⟩ ∀ ϕ ∈ C c 1 ( Ω , R n ) {\displaystyle \int _{\Omega }\chi _{E}(x)\mathrm {div} {\boldsymbol {\phi }}(x)\mathrm {d} x=\int _{E}\mathrm {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-\int _{\Omega }\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle \qquad \forall {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})} As noted for the case of general functions of bounded variation, this vector measure D χ E {\displaystyle D\chi _{E}} is the distributional or weak gradient of χ E {\displaystyle \chi _{E}} . The total variation measure associated with D χ E {\displaystyle D\chi _{E}} is denoted by \| D χ E \| {\displaystyle \|D\chi _{E}\|} , i.e. for every open set Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} we write \| D χ E \| ( Ω ) {\displaystyle \|D\chi _{E}\|(\Omega )} for P ( E , Ω ) = V ( χ E , Ω ) {\displaystyle P(E,\Omega )=V(\chi _{E},\Omega )} . === De Giorgi definition === In his papers (De Giorgi 1953) and (De Giorgi 1954), Ennio De Giorgi introduces the following smoothing operator, analogous to the Weierstrass transform in the one-dimensional case W λ χ E ( x ) = ∫ R n g λ ( x − y ) χ E ( y ) d y = ( π λ ) − n 2 ∫ E e − ( x − y ) 2 λ d y {\displaystyle W_{\lambda }\chi _{E}(x)=\int _{\mathbb {R} ^{n}}g_{\lambda }(x-y)\chi _{E}(y)\mathrm {d} y=(\pi \lambda )^{-{\frac {n}{2}}}\int _{E}e^{-{\frac {(x-y)^{2}}{\lambda }}}\mathrm {d} y} As one can easily prove, W λ χ ( x ) {\displaystyle W_{\lambda }\chi (x)} is a smooth function for all x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , such that lim λ → 0 W λ χ E ( x ) = χ E ( x ) {\displaystyle \lim _{\lambda \to 0}W_{\lambda }\chi _{E}(x)=\chi _{E}(x)} also, its gradient is everywhere well defined, and so is its absolute value ∇ W λ χ E ( x ) = g r a d W λ χ E ( x ) = D W λ χ E ( x ) = ( ∂ W λ χ E ( x ) ∂ x 1 ⋮ ∂ W λ χ E ( x ) ∂ x n ) ⟺ \| D W λ χ E ( x ) \| = ∑ k = 1 n \| ∂ W λ χ E ( x ) ∂ x k \| 2 {\displaystyle \nabla W_{\lambda }\chi _{E}(x)=\mathrm {grad} W_{\lambda }\chi _{E}(x)=DW_{\lambda }\chi _{E}(x)={\begin{pmatrix}{\frac {\partial W_{\lambda }\chi _{E}(x)}{\partial x_{1}}}\\\vdots \\{\frac {\partial W_{\lambda }\chi _{E}(x)}{\partial x_{n}}}\\\end{pmatrix}}\Longleftrightarrow \left\|DW_{\lambda }\chi _{E}(x)\right\|={\sqrt {\sum _{k=1}^{n}\left\|{\frac {\partial W_{\lambda }\chi _{E}(x)}{\partial x_{k}}}\right\|^{2}}}} Having defined this function, De Giorgi gives the following definition of perimeter: Definition 3. Let Ω {\displaystyle \Omega } be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and let E {\displaystyle E} be a Borel set. The perimeter of E {\displaystyle E} in Ω {\displaystyle \Omega } is the value P ( E , Ω ) = lim λ → 0 ∫ Ω \| D W λ χ E ( x ) \| d x {\displaystyle P(E,\Omega )=\lim _{\lambda \to 0}\int _{\Omega }\|DW_{\lambda }\chi _{E}(x)\|\mathrm {d} x} Actually De Giorgi considered the case Ω = R n {\displaystyle \Omega =\mathbb {R} ^{n}} : however, the extension to the general case is not difficult. It can be proved that the two definitions are exactly equivalent: for a proof see the already cited De Giorgi's papers or the book (Giusti 1984). Now having defined what a perimeter is, De Giorgi gives the same definition 2 of what a set of (locally) finite perimeter is. == Basic properties == The following properties are the ordinary properties which the general notion of a perimeter is supposed to have: If Ω ⊆ Ω 1 {\displaystyle \Omega \subseteq \Omega _{1}} then P ( E , Ω ) ≤ P ( E , Ω 1 ) {\displaystyle P(E,\Omega )\leq P(E,\Omega _{1})} , with equality holding if and only if the closure of E {\displaystyle E} is a compact subset of Ω {\displaystyle \Omega } . For any two Cacciopoli sets E 1 {\displaystyle E_{1}} and E 2 {\displaystyle E_{2}} , the relation P ( E 1 ∪ E 2 , Ω ) ≤ P ( E 1 , Ω ) + P ( E 2 , Ω 1 ) {\displaystyle P(E_{1}\cup E_{2},\Omega )\leq P(E_{1},\Omega )+P(E_{2},\Omega _{1})} holds, with equality holding if and only if d ( E 1 , E 2 ) > 0 {\displaystyle d(E_{1},E_{2})>0} , where d {\displaystyle d} is the distance between sets in euclidean space. If the Lebesgue measure of E {\displaystyle E} is 0 {\displaystyle 0} , then P ( E ) = 0 {\displaystyle P(E)=0} : this implies that if the symmetric difference E 1 △ E 2 {\displaystyle E_{1}\triangle E_{2}} of two sets has zero Lebesgue measure, the two sets have the same perimeter i.e. P ( E 1 ) = P ( E 2 ) {\displaystyle P(E_{1})=P(E_{2})} . == Notions of boundary == For any given Caccioppoli set E ⊂ R n {\displaystyle E\subset \mathbb {R} ^{n}} there exist two naturally associated analytic quantities: the vector-valued Radon measure D χ E {\displaystyle D\chi _{E}} and its total variation measure \| D χ E \| {\displaystyle \|D\chi _{E}\|} . Given that P ( E , Ω ) = ∫ Ω \| D χ E \| {\displaystyle P(E,\Omega )=\int _{\Omega }\|D\chi _{E}\|} is the perimeter within any open set Ω {\displaystyle \Omega } , one should expect that D χ E {\displaystyle D\chi _{E}} alone should somehow account for the perimeter of E {\displaystyle E} . === The topological boundary === It is natural to try to understand the relationship between the objects D χ E {\displaystyle D\chi _{E}} , \| D χ E \| {\displaystyle \|D\chi _{E}\|} , and the topological boundary ∂ E {\displaystyle \partial E} . There is an elementary lemma that guarantees that the support (in the sense of distributions) of D χ E {\displaystyle D\chi _{E}} , and therefore also \| D χ E \| {\displaystyle \|D\chi _{E}\|} , is always contained in ∂ E {\displaystyle \partial E} : Lemma. The support of the vector-valued Radon measure D χ E {\displaystyle D\chi _{E}} is a subset of the topological boundary ∂ E {\displaystyle \partial E} of E {\displaystyle E} . Proof. To see this choose x 0 ∉ ∂ E {\displaystyle x_{0}\notin \partial E} : then x 0 {\displaystyle x_{0}} belongs to the open set R n ∖ ∂ E {\displaystyle \mathbb {R} ^{n}\setminus \partial E} and this implies that it belongs to an open neighborhood A {\displaystyle A} contained in the interior of E {\displaystyle E} or in the interior of R n ∖ E {\displaystyle \mathbb {R} ^{n}\setminus E} . Let ϕ ∈ C c 1 ( A ; R n ) {\displaystyle \phi \in C_{c}^{1}(A;\mathbb {R} ^{n})} . If A ⊆ ( R n ∖ E ) ∘ = R n ∖ E − {\displaystyle A\subseteq (\mathbb {R} ^{n}\setminus E)^{\circ }=\mathbb {R} ^{n}\setminus E^{-}} where E − {\displaystyle E^{-}} is the closure of E {\displaystyle E} , then χ E ( x ) = 0 {\displaystyle \chi _{E}(x)=0} for x ∈ A {\displaystyle x\in A} and ∫ Ω ⟨ ϕ , D χ E ( x ) ⟩ = − ∫ A χ E ( x ) div ⁡ ϕ ( x ) d x = 0 {\displaystyle \int _{\Omega }\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle =-\int _{A}\chi _{E}(x)\,\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=0} Likewise, if A ⊆ E ∘ {\displaystyle A\subseteq E^{\circ }} then χ E ( x ) = 1 {\displaystyle \chi _{E}(x)=1} for x ∈ A {\displaystyle x\in A} so ∫ Ω ⟨ ϕ , D χ E ( x ) ⟩ = − ∫ A div ⁡ ϕ ( x ) d x = 0 {\displaystyle \int _{\Omega }\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle =-\int _{A}\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=0} With ϕ ∈ C c 1 ( A , R n ) {\displaystyle \phi \in C_{c}^{1}(A,\mathbb {R} ^{n})} arbitrary it follows that x 0 {\displaystyle x_{0}} is outside the support of D χ E {\displaystyle D\chi _{E}} . === The reduced boundary === The topological boundary ∂ E {\displaystyle \partial E} turns out to be too crude for Caccioppoli sets because its Hausdorff measure overcompensates for the perimeter P ( E ) {\displaystyle P(E)} defined above. Indeed, the Caccioppoli set E = { ( x , y ) : 0 ≤ x , y ≤ 1 } ∪ { ( x , 0 ) : − 1 ≤ x ≤ 1 } ⊂ R 2 {\displaystyle E=\{(x,y):0\leq x,y\leq 1\}\cup \{(x,0):-1\leq x\leq 1\}\subset \mathbb {R} ^{2}} representing a square together with a line segment sticking out on the left has perimeter P ( E ) = 4 {\displaystyle P(E)=4} , i.e. the extraneous line segment is ignored, while its topological boundary ∂ E = { ( x , 0 ) : − 1 ≤ x ≤ 1 } ∪ { ( x , 1 ) : 0 ≤ x ≤ 1 } ∪ { ( x , y ) : x ∈ { 0 , 1 } , 0 ≤ y ≤ 1 } {\displaystyle \partial E=\{(x,0):-1\leq x\leq 1\}\cup \{(x,1):0\leq x\leq 1\}\cup \{(x,y):x\in \{0,1\},0\leq y\leq 1\}} has one-dimensional Hausdorff measure H 1 ( ∂ E ) = 5 {\displaystyle {\mathcal {H}}^{1}(\partial E)=5} . The "correct" boundary should therefore be a subset of ∂ E {\displaystyle \partial E} . We define: Definition 4. The reduced boundary of a Caccioppoli set E ⊂ R n {\displaystyle E\subset \mathbb {R} ^{n}} is denoted by ∂ ∗ E {\displaystyle \partial ^{}E} and is defined to be equal to be the collection of points x {\displaystyle x} at which the limit: ν E ( x ) := lim ρ ↓ 0 D χ E ( B ρ ( x ) ) \| D χ E \| ( B ρ ( x ) ) ∈ R n {\displaystyle \nu _{E}(x):=\lim _{\rho \downarrow 0}{\frac {D\chi _{E}(B_{\rho }(x))}{\|D\chi _{E}\|(B_{\rho }(x))}}\in \mathbb {R} ^{n}} exists and has length equal to one, i.e. \| ν E ( x ) \| = 1 {\displaystyle \|\nu _{E}(x)\|=1} . One can remark that by the Radon-Nikodym Theorem the reduced boundary ∂ ∗ E {\displaystyle \partial ^{}E} is necessarily contained in the support of D χ E {\displaystyle D\chi _{E}} , which in turn is contained in the topological boundary ∂ E {\displaystyle \partial E} as explained in the section above. That is: ∂ ∗ E ⊆ support ⁡ D χ E ⊆ ∂ E {\displaystyle \partial ^{}E\subseteq \operatorname {support} D\chi _{E}\subseteq \partial E} The inclusions above are not necessarily equalities as the previous example shows. In that example, ∂ E {\displaystyle \partial E} is the square with the segment sticking out, support ⁡ D χ E {\displaystyle \operatorname {support} D\chi _{E}} is the square, and ∂ ∗ E {\displaystyle \partial ^{}E} is the square without its four corners. === De Giorgi's theorem === For convenience, in this section we treat only the case where Ω = R n {\displaystyle \Omega =\mathbb {R} ^{n}} , i.e. the set E {\displaystyle E} has (globally) finite perimeter. De Giorgi's theorem provides geometric intuition for the notion of reduced boundaries and confirms that it is the more natural definition for Caccioppoli sets by showing P ( E ) ( = ∫ \| D χ E \| ) = H n − 1 ( ∂ ∗ E ) {\displaystyle P(E)\left(=\int \|D\chi _{E}\|\right)={\mathcal {H}}^{n-1}(\partial ^{}E)} i.e. that its Hausdorff measure equals the perimeter of the set. The statement of the theorem is quite long because it interrelates various geometric notions in one fell swoop. Theorem. Suppose E ⊂ R n {\displaystyle E\subset \mathbb {R} ^{n}} is a Caccioppoli set. Then at each point x {\displaystyle x} of the reduced boundary ∂ ∗ E {\displaystyle \partial ^{}E} there exists a multiplicity one approximate tangent space T x {\displaystyle T_{x}} of \| D χ E \| {\displaystyle \|D\chi _{E}\|} , i.e. a codimension-1 subspace T x {\displaystyle T_{x}} of R n {\displaystyle \mathbb {R} ^{n}} such that lim λ ↓ 0 ∫ R n f ( λ − 1 ( z − x ) ) \| D χ E \| ( z ) = ∫ T x f ( y ) d H n − 1 ( y ) {\displaystyle \lim _{\lambda \downarrow 0}\int _{\mathbb {R} ^{n}}f(\lambda ^{-1}(z-x))\|D\chi _{E}\|(z)=\int _{T_{x}}f(y)\,d{\mathcal {H}}^{n-1}(y)} for every continuous, compactly supported f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } . In fact the subspace T x {\displaystyle T_{x}} is the orthogonal complement of the unit vector ν E ( x ) = lim ρ ↓ 0 D χ E ( B ρ ( x ) ) \| D χ E \| ( B ρ ( x ) ) ∈ R n {\displaystyle \nu _{E}(x)=\lim _{\rho \downarrow 0}{\frac {D\chi _{E}(B_{\rho }(x))}{\|D\chi _{E}\|(B_{\rho }(x))}}\in \mathbb {R} ^{n}} defined previously. This unit vector also satisfies lim λ ↓ 0 { λ − 1 ( z − x ) : z ∈ E } → { y ∈ R n : y ⋅ ν E ( x ) > 0 } {\displaystyle \lim _{\lambda \downarrow 0}\left\{\lambda ^{-1}(z-x):z\in E\right\}\to \left\{y\in \mathbb {R} ^{n}:y\cdot \nu _{E}(x)>0\right\}} locally in L 1 {\displaystyle L^{1}} , so it is interpreted as an approximate inward pointing unit normal vector to the reduced boundary ∂ ∗ E {\displaystyle \partial ^{}E} . Finally, ∂ ∗ E {\displaystyle \partial ^{}E} is (n-1)-rectifiable and the restriction of (n-1)-dimensional Hausdorff measure H n − 1 {\displaystyle {\mathcal {H}}^{n-1}} to ∂ ∗ E {\displaystyle \partial ^{}E} is \| D χ E \| {\displaystyle \|D\chi _{E}\|} , i.e. \| D χ E \| ( A ) = H n − 1 ( A ∩ ∂ ∗ E ) {\displaystyle \|D\chi _{E}\|(A)={\mathcal {H}}^{n-1}(A\cap \partial ^{}E)} for all Borel sets A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} . In other words, up to H n − 1 {\displaystyle {\mathcal {H}}^{n-1}} -measure zero the reduced boundary ∂ ∗ E {\displaystyle \partial ^{}E} is the smallest set on which D χ E {\displaystyle D\chi _{E}} is supported. == Applications == === A Gauss–Green formula === From the definition of the vector Radon measure D χ E {\displaystyle D\chi _{E}} and from the properties of the perimeter, the following formula holds true: ∫ E div ⁡ ϕ ( x ) d x = − ∫ ∂ E ⟨ ϕ , D χ E ( x ) ⟩ ϕ ∈ C c 1 ( Ω , R n ) {\displaystyle \int _{E}\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-\int _{\partial E}\langle {\boldsymbol {\phi }},D\chi _{E}(x)\rangle \qquad {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})} This is one version of the divergence theorem for domains with non smooth boundary. De Giorgi's theorem can be used to formulate the same identity in terms of the reduced boundary ∂ ∗ E {\displaystyle \partial ^{}E} and the approximate inward pointing unit normal vector ν E {\displaystyle \nu _{E}} . Precisely, the following equality holds ∫ E div ⁡ ϕ ( x ) d x = − ∫ ∂ ∗ E ϕ ( x ) ⋅ ν E ( x ) d H n − 1 ( x ) ϕ ∈ C c 1 ( Ω , R n ) {\displaystyle \int _{E}\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-\int _{\partial ^{*}E}{\boldsymbol {\phi }}(x)\cdot \nu _{E}(x)\,\mathrm {d} {\mathcal {H}}^{n-1}(x)\qquad {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})} == See also == == Notes == == References == == External links == O'Neil, Toby Christopher (2001) [1994], "Geometric measure theory", Encyclopedia of Mathematics, EMS Press Zagaller, Victor Abramovich (2001) [1994], "Perimeter", Encyclopedia of Mathematics, EMS Press Function of bounded variation at Encyclopedia of Mathematics
Wikipedia:Cahiers de Topologie et Géométrie Différentielle Catégoriques#0	The Cahiers de Topologie et Géométrie Différentielle Catégoriques (French: Notebooks of categorical topology and categorical differential geometry) is a French mathematical scientific journal established by Charles Ehresmann in 1957. It concentrates on category theory "and its applications, [e]specially in topology and differential geometry". Its older papers (two years or more after publication) are freely available on the internet through the French NUMDAM service. It was originally published by the Institut Henri Poincaré under the name Cahiers de Topologie; after the first volume, Ehresmann changed the publisher to the Institut Henri Poincaré and later Dunod/Bordas. In the eighth volume he changed the name to Cahiers de Topologie et Géométrie Différentielle. After Ehresmann's death in 1979 the editorship passed to his wife Andrée Ehresmann; in 1984, at the suggestion of René Guitart, the name was changed again, to add "Catégoriques". == References == == External links == Official website as of January 2018; previous official website Archive at Numdam: Volumes 1 (1957) - 7 (1965) : Séminaire Ehresmann. Topologie et géométrie différentielle; Volumes 8 (1966) - 52 (2011) : Cahiers de Topologie et Géométrie Différentielle Catégoriques Table of Contents for Volumes 38 (1997) through 57 (2016) maintained at the electronic journal Theory and Applications of Categories
Wikipedia:Caius Iacob#0	Caius Iacob (March 29, 1912 – February 6, 1992) was a Romanian mathematician, professor at the University of Bucharest, and titular member of the Romanian Academy. After the fall of communism in 1989, he was elected to the Senate of Romania. == Biography == He was born in Arad, the son of Lazăr Iacob and Camelia, née Moldovan. His father was professor of Canon Law and served as delegate for Arad at the Great National Assembly of Alba Iulia of 1 December 1918. Caius Iacob attended the Moise Nicoară High School in his native city, and then completed his secondary education at the Emanuil Gojdu High School in Oradea. After passing his baccalaureate examination with the highest mark in the nation, he was admitted in 1928 at the Faculty of Sciences of the University of Bucharest, from where he graduated in 1931, aged nineteen. Iacob continued his studies at the Faculty of Sciences of the University of Paris, with thesis advisor Henri Villat. He defended his thesis, Sur la détermination des fonctions harmoniques par certaines conditions aux limites: applications à l'hydrodynamique, on 24 June 1935. His most important work was in the studies of classical hydrodynamics, fluid mechanics, mathematical analysis, and compressible-flow theory. Iacob started his academic career in 1935 at Politehnica University of Timișoara, after which he became a professor at the University of Bucharest and at Babeș-Bolyai University in Cluj. In 1955, he was elected a corresponding member of the Romanian Academy, becoming a titular member in 1963. From 1980 to the end of his life he served as President of the Mathematics section of the Romanian Academy. He was awarded several prizes for his work: the Henri de Parville Prize by the French Academy of Sciences (1940), the State Prize of the Romanian People's Republic (1952), and the Order of the Star of the Romanian People's Republic, 3rd class (1964). In May 1990, he was elected senator for the Christian Democratic National Peasants' Party — the only member of the party to be elected to the upper chamber of the Parliament of Romania that year. He died in Bucharest in February 1992. == Legacy == Iacob was one of the founders of the Institute of Applied Mathematics of the Romanian Academy in 1991. Ten years later, the institute merged with the Center for Mathematical Statistics of the Academy (that had been founded by Gheorghe Mihoc in 1964), becoming the current Gheorghe Mihoc–Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy. A high school and a middle school, as well as a street and a plaza in Arad also bear his name. == References ==
Wikipedia:Calculus on Manifolds (book)#0	Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates. == Description == Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy–Green, Ostrogradsky–Gauss (divergence theorem), and Kelvin–Stokes, in the language of differential forms on differentiable manifolds embedded in Euclidean space, and as corollaries of the generalized Stokes theorem on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results: The cover of Calculus on Manifolds features snippets of a July 2, 1850 letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the Kelvin–Stokes theorem). == Reception == Calculus on Manifolds aims to present the topics of multivariable and vector calculus in the manner in which they are seen by a modern working mathematician, yet simply and selectively enough to be understood by undergraduate students whose previous coursework in mathematics comprises only one-variable calculus and introductory linear algebra. While Spivak's elementary treatment of modern mathematical tools is broadly successful—and this approach has made Calculus on Manifolds a standard introduction to the rigorous theory of multivariable calculus—the text is also well known for its laconic style, lack of motivating examples, and frequent omission of non-obvious steps and arguments. For example, in order to state and prove the generalized Stokes' theorem on chains, a profusion of unfamiliar concepts and constructions (e.g., tensor products, differential forms, tangent spaces, pullbacks, exterior derivatives, cube and chains) are introduced in quick succession within the span of 25 pages. Moreover, careful readers have noted a number of nontrivial oversights throughout the text, including missing hypotheses in theorems, inaccurately stated theorems, and proofs that fail to handle all cases. == Other textbooks == A more recent textbook which also covers these topics at an undergraduate level is the text Analysis on Manifolds by James Munkres (366 pp.). At more than twice the length of Calculus on Manifolds, Munkres's work presents a more careful and detailed treatment of the subject matter at a leisurely pace. Nevertheless, Munkres acknowledges the influence of Spivak's earlier text in the preface of Analysis on Manifolds. Spivak's five-volume textbook A Comprehensive Introduction to Differential Geometry states in its preface that Calculus on Manifolds serves as a prerequisite for a course based on this text. In fact, several of the concepts introduced in Calculus on Manifolds reappear in the first volume of this classic work in more sophisticated settings. == See also == == Footnotes == === Notes === === Citations === == References == Auslander, Louis (1967), "Review of Calculus on manifolds—a modern approach to classical theorems of advanced calculus", Quarterly of Applied Mathematics, 24 (4): 388–389 Botts, Truman (1966), "Reviewed Work: Calculus on Manifolds by Michael Spivak", Science, 153 (3732): 164–165, doi:10.1126/science.153.3732.164-a Hubbard, John H.; Hubbard, Barbara Burke (2009) [1998], Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (4th ed.), Upper Saddle River, N.J.: Prentice Hall (4th edition by Matrix Editions (Ithaca, N.Y.)), ISBN 978-0-9715766-5-0 [An elementary approach to differential forms with an emphasis on concrete examples and computations] Katz, Victor J. (1979), "The History of Stokes' Theorem", Mathematics Magazine, 52 (3), Mathematical Association of America: 146–156, doi:10.2307/2690275 Loomis, Lynn Harold; Sternberg, Shlomo (2014) [1968], Advanced Calculus (Revised ed.), Reading, Mass.: Addison-Wesley (revised edition by Jones and Bartlett (Boston); reprinted by World Scientific (Hackensack, N.J.)), pp. 305–567, ISBN 978-981-4583-93-0 [A general treatment of differential forms, differentiable manifolds, and selected applications to mathematical physics for advanced undergraduates] Munkres, James (1968), "Review of Calculus on Manifolds", The American Mathematical Monthly, 75 (5): 567–568, doi:10.2307/2314769, JSTOR 2314769 Munkres, James (1991), Analysis on Manifolds, Redwood City, Calif.: Addison-Wesley (reprinted by Westview Press (Boulder, Colo.)), ISBN 978-0-201-31596-7 [An undergraduate treatment of multivariable and vector calculus with coverage similar to Calculus on Manifolds, with mathematical ideas and proofs presented in greater detail] Nickerson, Helen K.; Spencer, Donald C.; Steenrod, Norman E. (1959), Advanced Calculus, Princeton, N.J.: Van Nostrand, ISBN 978-0-486-48090-9 {{citation}}: ISBN / Date incompatibility (help) [A unified treatment of linear and multilinear algebra, multivariable calculus, differential forms, and introductory algebraic topology for advanced undergraduates] Rudin, Walter (1976) [1953], Principles of Mathematical Analysis (3rd ed.), New York: McGraw Hill, pp. 204–299, ISBN 978-0-07-054235-8 [An unorthodox though rigorous approach to differential forms that avoids many of the usual algebraic constructions] Spivak, Michael (2018) [1965], Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (Mathematics Monograph Series), New York: W. A. Benjamin, Inc. (reprinted by Addison-Wesley (Reading, Mass.) and Westview Press (Boulder, Colo.)), ISBN 978-0-8053-9021-6 [A brief, rigorous, and modern treatment of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates] Spivak, Michael (1999) [1970], A Comprehensive Introduction to Differential Geometry, Vol. 1 (3rd ed.), Houston, Tex.: Publish or Perish, Inc., ISBN 978-0-9140-9870-6 [A thorough account of differentiable manifolds at the graduate level; contains a more sophisticated reframing and extensions of Chapters 4 and 5 of Calculus on Manifolds] Tu, Loring W. (2011) [2008], An Introduction to Manifolds (2nd ed.), New York: Springer, ISBN 978-1-4419-7399-3 [A standard treatment of the theory of smooth manifolds at the 1st year graduate level]
Wikipedia:Calcutta Mathematical Society#0	The Calcutta Mathematical Society (CalMathSoc) is an association of professional mathematicians dedicated to the interests of mathematical research and education in India. The Society has its head office located at Kolkata, India. == History == Calcutta Mathematical Society was established on 6 September 1908 under the stewardship of Sir Asutosh Mookerjee, the then Vice-Chancellor of Calcutta University. He was the founder president of the Society, and was assisted by Sir Gurudas Banerjee, Prof. C.E. Cullis and Prof. Gauri Sankar Dey as Vice Presidents and Prof. Phanindra Lal Ganguly as the Founder Secretary of the organization. It is said that the founders were inspired by the structure and operations of the London Mathematical Society while forming this organization. Over more than the last 100 years, the Society has fostered teaching and research of theoretical and applied mathematical sciences through several pedagogic and technical activities. It is honored to be associated with legends like Albert Einstein, Anil Kumar Gain, S. Chandrasekhar, Abdus Salam and many more eminent scientists and researchers across the globe. == Activities == The main academic activities of the Society can broadly be classified under the following three heads: Memorial Lectures, Special Lectures and Regular Seminars and Symposiums. The Memorial Lectures are organized by the Society every year in honor of great academics who were once associates and patrons of the organization. The Special Lectures are given on request by eminent researchers and scientists who visit Kolkata from time to time. The Seminars and Symposiums are generally held on an annual basis, focusing on the Pedagogic and Technical topics as well as topics of popular interest. 'International Symposium on Mathematical Physics in memory of S. Chandrasekhar with a special session on Abdus Salam' is notable one. National seminars on 'Theory & Methodology of Mathematics Teaching', 'Contribution of René Descartes','Contribution of Gottfried Leibniz', National Seminar on 'Power generation, Environment Pollution and related Mathematical Equations' were the most notable programme. Director of all those programs was Professor N. C. Ghosh. Professor C. G. Chakraborty was Director of National seminar on 'Satyndranath Bose & his contribution', Professor B. N. Mondal was Director of the 'Workshop on Mathematics Teaching Research & Training' organised at Calcutta Mathematical Society. == Publications == In terms of publishing substantial academic work, Calcutta Mathematical Society is the 1st Mathematical Society in India and Asia, and is the 13th in the whole world. The main publication of the Society is the Bulletin of Calcutta Mathematical Society, which commenced its journey back in 1909 and has been of great repute in the global scenario of mathematics for more than 100 years. The major publications of the Society are its four journals and bulletins as follows. Bulletin of Calcutta Mathematical Society Journal of Calcutta Mathematical Society Review of Calcutta Mathematical Society News Bulletin of Calcutta Mathematical Society == Structure == The main governing body of the Society is its Council, which is composed of the President, Vice-Presidents, Secretary, Treasurer, Editorial Secretary and Assistant Secretary (if required). These posts are nominated and elected by the members at the Annual General Meeting. == See also == Indian Mathematical Society Kerala Mathematical Association London Mathematical Society American Mathematical Society Canadian Mathematical Society Australian Mathematical Society European Mathematical Society List of Mathematical Societies == References == == External links == Official website
Wikipedia:Cami Sawyer#0	Cameron Cunningham (Cami) Sawyer is an American mathematician who has worked in New Zealand at Massey University and the Ministry of Education. Trained in algebraic topology, her work in New Zealand has focused on mathematics education, educational technology, distance learning, and the needs of Māori students in mathematics. == Education and career == Sawyer has a postgraduate diploma in teaching from Texas State University, and completed a PhD in 1999 at the University of North Texas. Her dissertation, On the Cohomology of the Complement of a Toral Arrangement, was supervised by J. Matthew Douglass. She had already visited New Zealand under a Rotary Scholarship in the 1980s, and in the early 2000s emigrated there with her husband and children. She became a secondary school teacher before joining the Massey University staff as a senior tutor in mathematics in the Institute of Fundamental Sciences. Since 2015, she has also been associated with the Pūhoro STEM Academy, a program hosted by Massey for encouraging secondary-school Māori to continue their science and technology education. In 2021 she moved from Massey to the Ministry of Education, as Learning Area Lead of Mathematics and Statistics. == Recognition and service == Sawyer is a Fellow of the New Zealand Mathematical Society (NZMS), and has chaired the NZMS Education Group. In 2019 she won a Sustained Excellence in Tertiary Teaching Award in the Te Whatu Kairangi Awards of the Ako Aotearoa National Centre for Tertiary Teaching Excellence, a government-funded organisation for the support and promotion of tertiary-level education. == References == == External links == Staff profile at Massey
Wikipedia:Cancelling out#0	Cancelling out is a mathematical process used for removing subexpressions from a mathematical expression, when this removal does not change the meaning or the value of the expression because the subexpressions have equal and opposing effects. For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. As another example, if a×b=a×c, then the multiplicative term a can be canceled out if a≠0, resulting in the equivalent expression b=c; this is equivalent to dividing through by a. == Cancelling == If the subexpressions are not identical, then it may still be possible to cancel them out partly. For example, in the simple equation 3 + 2y = 8y, both sides actually contain 2y (because 8y is the same as 2y + 6y). Therefore, the 2y on both sides can be cancelled out, leaving 3 = 6y, or y = 0.5. This is equivalent to subtracting 2y from both sides. At times, cancelling out can introduce limited changes or extra solutions to an equation. For example, given the inequality ab ≥ 3b, it looks like the b on both sides can be cancelled out to give a ≥ 3 as the solution. But cancelling 'naively' like this, will mean we don't get all the solutions (sets of (a, b) satisfying the inequality). This is because if b were a negative number then dividing by a negative would change the ≥ relationship into a ≤ relationship. For example, although 2 is more than 1, –2 is less than –1. Also if b were zero then zero times anything is zero and cancelling out would mean dividing by zero in that case which cannot be done. So in fact, while cancelling works, cancelling out correctly will lead us to three sets of solutions, not just one we thought we had. It will also tell us that our 'naive' solution is only a solution in some cases, not all cases: If b > 0: we can cancel out to get a ≥ 3. If b < 0: then cancelling out gives a ≤ 3 instead, because we would have to reverse the relationship in this case. If b is exactly zero: then the equation is true for any value of a, because both sides would be zero, and 0 ≥ 0. So some care may be needed to ensure that cancelling out is done correctly and no solutions are overlooked or incorrect. Our simple inequality has three sets of solutions, which are: b > 0 and a ≥ 3. (For example b = 5 and a = 6 is a solution because 6 x 5 is 30 and 3 x 5 is 15, and 30 ≥ 15)or b < 0 and a ≤ 3 (For example b = –5 and a = 2 is a solution because 2 x (–5) is –10 and 3 x (–5) is –15, and –10 ≥ –15)or b = 0 (and a can be any number) (because anything x zero ≥ 3 x zero) Our 'naïve' solution (that a ≥ 3) would also be wrong sometimes. For example, if b = –5 then a = 4 is not a solution even though 4 ≥ 3, because 4 × (–5) is –20, and 3 x (–5) is –15, and –20 is not ≥ –15. == In advanced and abstract algebra, and infinite series == In more advanced mathematics, cancelling out can be used in the context of infinite series, whose terms can be cancelled out to get a finite sum or a convergent series. In this case, the term telescoping is often used. Considerable care and prevention of errors is often necessary to ensure the amended equation will be valid, or to establish the bounds within which it will be valid, because of the nature of such series. == Related concepts and use in other fields == In computational science, cancelling out is often used for improving the accuracy and the execution time of numerical algorithms. == See also == Al-Jabr Elementary algebra Equation == References ==
Wikipedia:Canonical basis#0	In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. In a polynomial ring, it refers to its standard basis given by the monomials, ( X i ) i {\displaystyle (X^{i})_{i}} . For finite extension fields, it means the polynomial basis. In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix A {\displaystyle A} , if the set is composed entirely of Jordan chains. In representation theory, it refers to the basis of the quantum groups introduced by Lusztig. == Representation theory == The canonical basis for the irreducible representations of a quantized enveloping algebra of type A D E {\displaystyle ADE} and also for the plus part of that algebra was introduced by Lusztig by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology). Specializing the parameter q {\displaystyle q} to q = 1 {\displaystyle q=1} yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter q {\displaystyle q} to q = 0 {\displaystyle q=0} yields something like a shadow of a basis. This shadow (but not the basis itself) for the case of irreducible representations was considered independently by Kashiwara; it is sometimes called the crystal basis. The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara (by an algebraic method) and by Lusztig (by a topological method). There is a general concept underlying these bases: Consider the ring of integral Laurent polynomials Z := Z [ v , v − 1 ] {\displaystyle {\mathcal {Z}}:=\mathbb {Z} \left[v,v^{-1}\right]} with its two subrings Z ± := Z [ v ± 1 ] {\displaystyle {\mathcal {Z}}^{\pm }:=\mathbb {Z} \left[v^{\pm 1}\right]} and the automorphism ⋅ ¯ {\displaystyle {\overline {\cdot }}} defined by v ¯ := v − 1 {\displaystyle {\overline {v}}:=v^{-1}} . A precanonical structure on a free Z {\displaystyle {\mathcal {Z}}} -module F {\displaystyle F} consists of A standard basis ( t i ) i ∈ I {\displaystyle (t_{i})_{i\in I}} of F {\displaystyle F} , An interval finite partial order on I {\displaystyle I} , that is, ( − ∞ , i ] := { j ∈ I ∣ j ≤ i } {\displaystyle (-\infty ,i]:=\{j\in I\mid j\leq i\}} is finite for all i ∈ I {\displaystyle i\in I} , A dualization operation, that is, a bijection F → F {\displaystyle F\to F} of order two that is ⋅ ¯ {\displaystyle {\overline {\cdot }}} -semilinear and will be denoted by ⋅ ¯ {\displaystyle {\overline {\cdot }}} as well. If a precanonical structure is given, then one can define the Z ± {\displaystyle {\mathcal {Z}}^{\pm }} submodule F ± := ∑ Z ± t j {\textstyle F^{\pm }:=\sum {\mathcal {Z}}^{\pm }t_{j}} of F {\displaystyle F} . A canonical basis of the precanonical structure is then a Z {\displaystyle {\mathcal {Z}}} -basis ( c i ) i ∈ I {\displaystyle (c_{i})_{i\in I}} of F {\displaystyle F} that satisfies: c i ¯ = c i {\displaystyle {\overline {c_{i}}}=c_{i}} and c i ∈ ∑ j ≤ i Z + t j and c i ≡ t i mod v F + {\displaystyle c_{i}\in \sum _{j\leq i}{\mathcal {Z}}^{+}t_{j}{\text{ and }}c_{i}\equiv t_{i}\mod vF^{+}} for all i ∈ I {\displaystyle i\in I} . One can show that there exists at most one canonical basis for each precanonical structure. A sufficient condition for existence is that the polynomials r i j ∈ Z {\displaystyle r_{ij}\in {\mathcal {Z}}} defined by t j ¯ = ∑ i r i j t i {\textstyle {\overline {t_{j}}}=\sum _{i}r_{ij}t_{i}} satisfy r i i = 1 {\displaystyle r_{ii}=1} and r i j ≠ 0 ⟹ i ≤ j {\displaystyle r_{ij}\neq 0\implies i\leq j} . A canonical basis induces an isomorphism from F + ∩ F + ¯ = ∑ i Z c i {\displaystyle \textstyle F^{+}\cap {\overline {F^{+}}}=\sum _{i}\mathbb {Z} c_{i}} to F + / v F + {\displaystyle F^{+}/vF^{+}} . === Hecke algebras === Let ( W , S ) {\displaystyle (W,S)} be a Coxeter group. The corresponding Iwahori-Hecke algebra H {\displaystyle H} has the standard basis ( T w ) w ∈ W {\displaystyle (T_{w})_{w\in W}} , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by T w ¯ := T w − 1 − 1 {\displaystyle {\overline {T_{w}}}:=T_{w^{-1}}^{-1}} . This is a precanonical structure on H {\displaystyle H} that satisfies the sufficient condition above and the corresponding canonical basis of H {\displaystyle H} is the Kazhdan–Lusztig basis C w ′ = ∑ y ≤ w P y , w ( v 2 ) T w {\displaystyle C_{w}'=\sum _{y\leq w}P_{y,w}(v^{2})T_{w}} with P y , w {\displaystyle P_{y,w}} being the Kazhdan–Lusztig polynomials. == Linear algebra == If we are given an n × n matrix A {\displaystyle A} and wish to find a matrix J {\displaystyle J} in Jordan normal form, similar to A {\displaystyle A} , we are interested only in sets of linearly independent generalized eigenvectors. A matrix in Jordan normal form is an "almost diagonal matrix," that is, as close to diagonal as possible. A diagonal matrix D {\displaystyle D} is a special case of a matrix in Jordan normal form. An ordinary eigenvector is a special case of a generalized eigenvector. Every n × n matrix A {\displaystyle A} possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If λ {\displaystyle \lambda } is an eigenvalue of A {\displaystyle A} of algebraic multiplicity μ {\displaystyle \mu } , then A {\displaystyle A} will have μ {\displaystyle \mu } linearly independent generalized eigenvectors corresponding to λ {\displaystyle \lambda } . For any given n × n matrix A {\displaystyle A} , there are infinitely many ways to pick the n linearly independent generalized eigenvectors. If they are chosen in a particularly judicious manner, we can use these vectors to show that A {\displaystyle A} is similar to a matrix in Jordan normal form. In particular, Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains. Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors x m − 1 , x m − 2 , … , x 1 {\displaystyle \mathbf {x} _{m-1},\mathbf {x} _{m-2},\ldots ,\mathbf {x} _{1}} that are in the Jordan chain generated by x m {\displaystyle \mathbf {x} _{m}} are also in the canonical basis. === Computation === Let λ i {\displaystyle \lambda _{i}} be an eigenvalue of A {\displaystyle A} of algebraic multiplicity μ i {\displaystyle \mu _{i}} . First, find the ranks (matrix ranks) of the matrices ( A − λ i I ) , ( A − λ i I ) 2 , … , ( A − λ i I ) m i {\displaystyle (A-\lambda _{i}I),(A-\lambda _{i}I)^{2},\ldots ,(A-\lambda _{i}I)^{m_{i}}} . The integer m i {\displaystyle m_{i}} is determined to be the first integer for which ( A − λ i I ) m i {\displaystyle (A-\lambda _{i}I)^{m_{i}}} has rank n − μ i {\displaystyle n-\mu _{i}} (n being the number of rows or columns of A {\displaystyle A} , that is, A {\displaystyle A} is n × n). Now define ρ k = rank ⁡ ( A − λ i I ) k − 1 − rank ⁡ ( A − λ i I ) k ( k = 1 , 2 , … , m i ) . {\displaystyle \rho _{k}=\operatorname {rank} (A-\lambda _{i}I)^{k-1}-\operatorname {rank} (A-\lambda _{i}I)^{k}\qquad (k=1,2,\ldots ,m_{i}).} The variable ρ k {\displaystyle \rho _{k}} designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue λ i {\displaystyle \lambda _{i}} that will appear in a canonical basis for A {\displaystyle A} . Note that rank ⁡ ( A − λ i I ) 0 = rank ⁡ ( I ) = n . {\displaystyle \operatorname {rank} (A-\lambda _{i}I)^{0}=\operatorname {rank} (I)=n.} Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see generalized eigenvector). === Example === This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order. The matrix A = ( 4 1 1 0 0 − 1 0 4 2 0 0 1 0 0 4 1 0 0 0 0 0 5 1 0 0 0 0 0 5 2 0 0 0 0 0 4 ) {\displaystyle A={\begin{pmatrix}4&1&1&0&0&-1\\0&4&2&0&0&1\\0&0&4&1&0&0\\0&0&0&5&1&0\\0&0&0&0&5&2\\0&0&0&0&0&4\end{pmatrix}}} has eigenvalues λ 1 = 4 {\displaystyle \lambda _{1}=4} and λ 2 = 5 {\displaystyle \lambda _{2}=5} with algebraic multiplicities μ 1 = 4 {\displaystyle \mu _{1}=4} and μ 2 = 2 {\displaystyle \mu _{2}=2} , but geometric multiplicities γ 1 = 1 {\displaystyle \gamma _{1}=1} and γ 2 = 1 {\displaystyle \gamma _{2}=1} . For λ 1 = 4 , {\displaystyle \lambda _{1}=4,} we have n − μ 1 = 6 − 4 = 2 , {\displaystyle n-\mu _{1}=6-4=2,} ( A − 4 I ) {\displaystyle (A-4I)} has rank 5, ( A − 4 I ) 2 {\displaystyle (A-4I)^{2}} has rank 4, ( A − 4 I ) 3 {\displaystyle (A-4I)^{3}} has rank 3, ( A − 4 I ) 4 {\displaystyle (A-4I)^{4}} has rank 2. Therefore m 1 = 4. {\displaystyle m_{1}=4.} ρ 4 = rank ⁡ ( A − 4 I ) 3 − rank ⁡ ( A − 4 I ) 4 = 3 − 2 = 1 , {\displaystyle \rho _{4}=\operatorname {rank} (A-4I)^{3}-\operatorname {rank} (A-4I)^{4}=3-2=1,} ρ 3 = rank ⁡ ( A − 4 I ) 2 − rank ⁡ ( A − 4 I ) 3 = 4 − 3 = 1 , {\displaystyle \rho _{3}=\operatorname {rank} (A-4I)^{2}-\operatorname {rank} (A-4I)^{3}=4-3=1,} ρ 2 = rank ⁡ ( A − 4 I ) 1 − rank ⁡ ( A − 4 I ) 2 = 5 − 4 = 1 , {\displaystyle \rho _{2}=\operatorname {rank} (A-4I)^{1}-\operatorname {rank} (A-4I)^{2}=5-4=1,} ρ 1 = rank ⁡ ( A − 4 I ) 0 − rank ⁡ ( A − 4 I ) 1 = 6 − 5 = 1. {\displaystyle \rho _{1}=\operatorname {rank} (A-4I)^{0}-\operatorname {rank} (A-4I)^{1}=6-5=1.} Thus, a canonical basis for A {\displaystyle A} will have, corresponding to λ 1 = 4 , {\displaystyle \lambda _{1}=4,} one generalized eigenvector each of ranks 4, 3, 2 and 1. For λ 2 = 5 , {\displaystyle \lambda _{2}=5,} we have n − μ 2 = 6 − 2 = 4 , {\displaystyle n-\mu _{2}=6-2=4,} ( A − 5 I ) {\displaystyle (A-5I)} has rank 5, ( A − 5 I ) 2 {\displaystyle (A-5I)^{2}} has rank 4. Therefore m 2 = 2. {\displaystyle m_{2}=2.} ρ 2 = rank ⁡ ( A − 5 I ) 1 − rank ⁡ ( A − 5 I ) 2 = 5 − 4 = 1 , {\displaystyle \rho _{2}=\operatorname {rank} (A-5I)^{1}-\operatorname {rank} (A-5I)^{2}=5-4=1,} ρ 1 = rank ⁡ ( A − 5 I ) 0 − rank ⁡ ( A − 5 I ) 1 = 6 − 5 = 1. {\displaystyle \rho _{1}=\operatorname {rank} (A-5I)^{0}-\operatorname {rank} (A-5I)^{1}=6-5=1.} Thus, a canonical basis for A {\displaystyle A} will have, corresponding to λ 2 = 5 , {\displaystyle \lambda _{2}=5,} one generalized eigenvector each of ranks 2 and 1. A canonical basis for A {\displaystyle A} is { x 1 , x 2 , x 3 , x 4 , y 1 , y 2 } = { ( − 4 0 0 0 0 0 ) , ( − 27 − 4 0 0 0 0 ) , ( 25 − 25 − 2 0 0 0 ) , ( 0 36 − 12 − 2 2 − 1 ) , ( 3 2 1 1 0 0 ) , ( − 8 − 4 − 1 0 1 0 ) } . {\displaystyle \left\{\mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3},\mathbf {x} _{4},\mathbf {y} _{1},\mathbf {y} _{2}\right\}=\left\{{\begin{pmatrix}-4\\0\\0\\0\\0\\0\end{pmatrix}},{\begin{pmatrix}-27\\-4\\0\\0\\0\\0\end{pmatrix}},{\begin{pmatrix}25\\-25\\-2\\0\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\36\\-12\\-2\\2\\-1\end{pmatrix}},{\begin{pmatrix}3\\2\\1\\1\\0\\0\end{pmatrix}},{\begin{pmatrix}-8\\-4\\-1\\0\\1\\0\end{pmatrix}}\right\}.} x 1 {\displaystyle \mathbf {x} _{1}} is the ordinary eigenvector associated with λ 1 {\displaystyle \lambda _{1}} . x 2 , x 3 {\displaystyle \mathbf {x} _{2},\mathbf {x} _{3}} and x 4 {\displaystyle \mathbf {x} _{4}} are generalized eigenvectors associated with λ 1 {\displaystyle \lambda _{1}} . y 1 {\displaystyle \mathbf {y} _{1}} is the ordinary eigenvector associated with λ 2 {\displaystyle \lambda _{2}} . y 2 {\displaystyle \mathbf {y} _{2}} is a generalized eigenvector associated with λ 2 {\displaystyle \lambda _{2}} . A matrix J {\displaystyle J} in Jordan normal form, similar to A {\displaystyle A} is obtained as follows: M = ( x 1 x 2 x 3 x 4 y 1 y 2 ) = ( − 4 − 27 25 0 3 − 8 0 − 4 − 25 36 2 − 4 0 0 − 2 − 12 1 − 1 0 0 0 − 2 1 0 0 0 0 2 0 1 0 0 0 − 1 0 0 ) , {\displaystyle M={\begin{pmatrix}\mathbf {x} _{1}&\mathbf {x} _{2}&\mathbf {x} _{3}&\mathbf {x} _{4}&\mathbf {y} _{1}&\mathbf {y} _{2}\end{pmatrix}}={\begin{pmatrix}-4&-27&25&0&3&-8\\0&-4&-25&36&2&-4\\0&0&-2&-12&1&-1\\0&0&0&-2&1&0\\0&0&0&2&0&1\\0&0&0&-1&0&0\end{pmatrix}},} J = ( 4 1 0 0 0 0 0 4 1 0 0 0 0 0 4 1 0 0 0 0 0 4 0 0 0 0 0 0 5 1 0 0 0 0 0 5 ) , {\displaystyle J={\begin{pmatrix}4&1&0&0&0&0\\0&4&1&0&0&0\\0&0&4&1&0&0\\0&0&0&4&0&0\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix}},} where the matrix M {\displaystyle M} is a generalized modal matrix for A {\displaystyle A} and A M = M J {\displaystyle AM=MJ} . == See also == Canonical form Change of basis Normal basis Normal form (disambiguation) Polynomial basis == Notes == == References == Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 Deng, Bangming; Ju, Jie; Parshall, Brian; Wang, Jianpan (2008), Finite Dimensional Algebras and Quantum Groups, Mathematical surveys and monographs, vol. 150, Providence, R.I.: American Mathematical Society, ISBN 9780821875315 Kashiwara, Masaki (1990), "Crystalizing the q-analogue of universal enveloping algebras", Communications in Mathematical Physics, 133 (2): 249–260, Bibcode:1990CMaPh.133..249K, doi:10.1007/bf02097367, ISSN 0010-3616, MR 1090425, S2CID 121695684 Kashiwara, Masaki (1991), "On crystal bases of the q-analogue of universal enveloping algebras", Duke Mathematical Journal, 63 (2): 465–516, doi:10.1215/S0012-7094-91-06321-0, ISSN 0012-7094, MR 1115118 Lusztig, George (1990), "Canonical bases arising from quantized enveloping algebras", Journal of the American Mathematical Society, 3 (2): 447–498, doi:10.2307/1990961, ISSN 0894-0347, JSTOR 1990961, MR 1035415 Lusztig, George (1991), "Quivers, perverse sheaves and quantized enveloping algebras", Journal of the American Mathematical Society, 4 (2): 365–421, doi:10.2307/2939279, ISSN 0894-0347, JSTOR 2939279, MR 1088333 Lusztig, George (1993), Introduction to quantum groups, Boston, MA: Birkhauser Boston, ISBN 0-8176-3712-5, MR 1227098 Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646
Wikipedia:Cantor function#0	In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905). == Definition == To define the Cantor function c : [ 0 , 1 ] → [ 0 , 1 ] {\displaystyle c:[0,1]\to [0,1]} , let x {\displaystyle x} be any number in [ 0 , 1 ] {\displaystyle [0,1]} and obtain c ( x ) {\displaystyle c(x)} by the following steps: Express x {\displaystyle x} in base 3, using digits 0, 1, 2. If the base-3 representation of x {\displaystyle x} contains a 1, replace every digit strictly after the first 1 with 0. Replace any remaining 2s with 1s. Interpret the result as a binary number. The result is c ( x ) {\displaystyle c(x)} . For example: 1 4 {\displaystyle {\tfrac {1}{4}}} has the ternary representation 0.02020202... There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... This is the binary representation of 1 3 {\displaystyle {\tfrac {1}{3}}} , so c ( 1 4 ) = 1 3 {\displaystyle c({\tfrac {1}{4}})={\tfrac {1}{3}}} . 1 5 {\displaystyle {\tfrac {1}{5}}} has the ternary representation 0.01210121... The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since it has no 2s. This is the binary representation of 1 4 {\displaystyle {\tfrac {1}{4}}} , so c ( 1 5 ) = 1 4 {\displaystyle c({\tfrac {1}{5}})={\tfrac {1}{4}}} . 200 243 {\displaystyle {\tfrac {200}{243}}} has the ternary representation 0.21102 (or 0.211012222...). The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. This is the binary representation of 3 4 {\displaystyle {\tfrac {3}{4}}} , so c ( 200 243 ) = 3 4 {\displaystyle c({\tfrac {200}{243}})={\tfrac {3}{4}}} . Equivalently, if C {\displaystyle {\mathcal {C}}} is the Cantor set on [0,1], then the Cantor function c : [ 0 , 1 ] → [ 0 , 1 ] {\displaystyle c:[0,1]\to [0,1]} can be defined as c ( x ) = { ∑ n = 1 ∞ a n 2 n , if x = ∑ n = 1 ∞ 2 a n 3 n ∈ C for a n ∈ { 0 , 1 } ; sup y ≤ x , y ∈ C c ( y ) , if x ∈ [ 0 , 1 ] ∖ C . {\displaystyle c(x)={\begin{cases}\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{2^{n}}},&\displaystyle {\text{if }}x=\sum _{n=1}^{\infty }{\frac {2a_{n}}{3^{n}}}\in {\mathcal {C}}\ {\text{for}}\ a_{n}\in \{0,1\};\\\displaystyle \sup _{y\leq x,\,y\in {\mathcal {C}}}c(y),&\displaystyle {\text{if }}x\in [0,1]\setminus {\mathcal {C}}.\end{cases}}} This formula is well-defined, since every member of the Cantor set has a unique base 3 representation that only contains the digits 0 or 2. (For some members of C {\displaystyle {\mathcal {C}}} , the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example, 1 3 {\displaystyle {\tfrac {1}{3}}} = 0.13 = 0.02222...3 is a member of the Cantor set). Since c ( 0 ) = 0 {\displaystyle c(0)=0} and c ( 1 ) = 1 {\displaystyle c(1)=1} , and c {\displaystyle c} is monotonic on C {\displaystyle {\mathcal {C}}} , it is clear that 0 ≤ c ( x ) ≤ 1 {\displaystyle 0\leq c(x)\leq 1} also holds for all x ∈ [ 0 , 1 ] ∖ C {\displaystyle x\in [0,1]\smallsetminus {\mathcal {C}}} . == Properties == The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 3 ⁡ ( 2 ) {\displaystyle \alpha =\log _{3}(2)} ) but not absolutely continuous. It is constant on intervals of the form (0.x1x2x3...xn022222..., 0.x1x2x3...xn200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no derivative at any point in an uncountable subset of the Cantor set containing the interval endpoints described above. The Cantor function can also be seen as the cumulative probability distribution function of the 1/2-1/2 Bernoulli measure μ supported on the Cantor set: c ( x ) = μ ( [ 0 , x ] ) {\textstyle c(x)=\mu ([0,x])} . This probability distribution, called the Cantor distribution, has no discrete part. That is, the corresponding measure is atomless. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure. However, no non-constant part of the Cantor function can be represented as an integral of a probability density function; integrating any putative probability density function that is not almost everywhere zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as Vitali (1905) pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere. The Cantor function is the standard example of a singular function. The Cantor function is also a standard example of a function with bounded variation but, as mentioned above, is not absolutely continuous. However, every absolutely continuous function is continuous with bounded variation. The Cantor function is non-decreasing, and so in particular its graph defines a rectifiable curve. Scheeffer (1884) showed that the arc length of its graph is 2. Note that the graph of any nondecreasing function such that f ( 0 ) = 0 {\displaystyle f(0)=0} and f ( 1 ) = 1 {\displaystyle f(1)=1} has length not greater than 2. In this sense, the Cantor function is extremal. === Lack of absolute continuity === The Lebesgue measure of the Cantor set is 0. Therefore, for any positive ε < 1 and any δ > 0, there exists a finite sequence of pairwise disjoint sub-intervals with total length < δ over which the Cantor function cumulatively rises more than ε. In fact, for every δ > 0 there are finitely many pairwise disjoint intervals (xk,yk) (1 ≤ k ≤ M) with ∑ k = 1 M ( y k − x k ) < δ {\displaystyle \sum \limits _{k=1}^{M}(y_{k}-x_{k})<\delta } and ∑ k = 1 M ( c ( y k ) − c ( x k ) ) = 1 {\displaystyle \sum \limits _{k=1}^{M}(c(y_{k})-c(x_{k}))=1} . == Alternative definitions == === Iterative construction === Below we define a sequence ( f n ) n {\displaystyle (f_{n})_{n}} of functions on the unit interval that converges to the Cantor function. Let f 0 ( x ) = x {\displaystyle f_{0}(x)=x} . Then, for every integer n ≥ 0 {\displaystyle n\geq 0} , the next function f n + 1 ( x ) {\displaystyle f_{n+1}(x)} will be defined in terms of f n ( x ) {\displaystyle f_{n}(x)} as follows: f n + 1 ( x ) = { 1 2 f n ( 3 x ) if 0 ≤ x ≤ 1 3 1 2 if 1 3 ≤ x ≤ 2 3 1 2 + 1 2 f n ( 3 x − 2 ) if 2 3 ≤ x ≤ 1 {\displaystyle f_{n+1}(x)={\begin{cases}\displaystyle {\frac {1}{2}}f_{n}(3x)&{\text{if }}0\leq x\leq {\frac {1}{3}}\\\displaystyle {\frac {1}{2}}&{\text{if }}{\frac {1}{3}}\leq x\leq {\frac {2}{3}}\\\displaystyle {\frac {1}{2}}+{\frac {1}{2}}f_{n}(3x-2)&{\text{if }}{\frac {2}{3}}\leq x\leq 1\end{cases}}} The three definitions are compatible at the end-points 1 3 {\displaystyle {\tfrac {1}{3}}} and 2 3 {\displaystyle {\tfrac {2}{3}}} , because f n ( 0 ) = 0 {\displaystyle f_{n}(0)=0} and f n ( 1 ) = 1 {\displaystyle f_{n}(1)=1} for every n {\displaystyle n} , by induction. One may check that ( f n ) n {\displaystyle (f_{n})_{n}} converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of f n + 1 {\displaystyle f_{n+1}} , one sees that max x ∈ [ 0 , 1 ] \| f n + 1 ( x ) − f n ( x ) \| ≤ 1 2 max x ∈ [ 0 , 1 ] \| f n ( x ) − f n − 1 ( x ) \| , n ≥ 1. {\displaystyle \max _{x\in [0,1]}\|f_{n+1}(x)-f_{n}(x)\|\leq {\frac {1}{2}}\,\max _{x\in [0,1]}\|f_{n}(x)-f_{n-1}(x)\|,\quad n\geq 1.} If f {\displaystyle f} denotes the limit function, it follows that, for every n ≥ 0 {\displaystyle n\geq 0} , max x ∈ [ 0 , 1 ] \| f ( x ) − f n ( x ) \| ≤ 2 − n + 1 max x ∈ [ 0 , 1 ] \| f 1 ( x ) − f 0 ( x ) \| . {\displaystyle \max _{x\in [0,1]}\|f(x)-f_{n}(x)\|\leq 2^{-n+1}\,\max _{x\in [0,1]}\|f_{1}(x)-f_{0}(x)\|.} === Fractal volume === The Cantor function is closely related to the Cantor set. The Cantor set C can be defined as the set of those numbers in the interval [0, 1] that do not contain the digit 1 in their base-3 (triadic) expansion, except if the 1 is followed by zeros only (in which case the tail 1000 … {\displaystyle \ldots } can be replaced by 0222 … {\displaystyle \ldots } to get rid of any 1). It turns out that the Cantor set is a fractal with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the D-dimensional volume H D {\displaystyle H_{D}} (in the sense of a Hausdorff-measure) takes a finite value, where D = log 3 ⁡ ( 2 ) {\displaystyle D=\log _{3}(2)} is the fractal dimension of C. We may define the Cantor function alternatively as the D-dimensional volume of sections of the Cantor set f ( x ) = H D ( C ∩ ( 0 , x ) ) . {\displaystyle f(x)=H_{D}(C\cap (0,x)).} == Self-similarity == The Cantor function possesses several symmetries. For 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} , there is a reflection symmetry c ( x ) = 1 − c ( 1 − x ) {\displaystyle c(x)=1-c(1-x)} and a pair of magnifications, one on the left and one on the right: c ( x 3 ) = c ( x ) 2 {\displaystyle c\left({\frac {x}{3}}\right)={\frac {c(x)}{2}}} and c ( x + 2 3 ) = 1 + c ( x ) 2 {\displaystyle c\left({\frac {x+2}{3}}\right)={\frac {1+c(x)}{2}}} The magnifications can be cascaded; they generate the dyadic monoid. This is exhibited by defining several helper functions. Define the reflection as r ( x ) = 1 − x {\displaystyle r(x)=1-x} The first self-symmetry can be expressed as r ∘ c = c ∘ r {\displaystyle r\circ c=c\circ r} where the symbol ∘ {\displaystyle \circ } denotes function composition. That is, ( r ∘ c ) ( x ) = r ( c ( x ) ) = 1 − c ( x ) {\displaystyle (r\circ c)(x)=r(c(x))=1-c(x)} and likewise for the other cases. For the left and right magnifications, write the left-mappings L D ( x ) = x 2 {\displaystyle L_{D}(x)={\frac {x}{2}}} and L C ( x ) = x 3 {\displaystyle L_{C}(x)={\frac {x}{3}}} Then the Cantor function obeys L D ∘ c = c ∘ L C {\displaystyle L_{D}\circ c=c\circ L_{C}} Similarly, define the right mappings as R D ( x ) = 1 + x 2 {\displaystyle R_{D}(x)={\frac {1+x}{2}}} and R C ( x ) = 2 + x 3 {\displaystyle R_{C}(x)={\frac {2+x}{3}}} Then, likewise, R D ∘ c = c ∘ R C {\displaystyle R_{D}\circ c=c\circ R_{C}} The two sides can be mirrored one onto the other, in that L D ∘ r = r ∘ R D {\displaystyle L_{D}\circ r=r\circ R_{D}} and likewise, L C ∘ r = r ∘ R C {\displaystyle L_{C}\circ r=r\circ R_{C}} These operations can be stacked arbitrarily. Consider, for example, the sequence of left-right moves L R L L R . {\displaystyle LRLLR.} Adding the subscripts C and D, and, for clarity, dropping the composition operator ∘ {\displaystyle \circ } in all but a few places, one has: L D R D L D L D R D ∘ c = c ∘ L C R C L C L C R C {\displaystyle L_{D}R_{D}L_{D}L_{D}R_{D}\circ c=c\circ L_{C}R_{C}L_{C}L_{C}R_{C}} Arbitrary finite-length strings in the letters L and R correspond to the dyadic rationals, in that every dyadic rational can be written as both y = n / 2 m {\displaystyle y=n/2^{m}} for integer n and m and as finite length of bits y = 0. b 1 b 2 b 3 ⋯ b m {\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}} with b k ∈ { 0 , 1 } . {\displaystyle b_{k}\in \{0,1\}.} Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function. Some notational rearrangements can make the above slightly easier to express. Let g 0 {\displaystyle g_{0}} and g 1 {\displaystyle g_{1}} stand for L and R. Function composition extends this to a monoid, in that one can write g 010 = g 0 g 1 g 0 {\displaystyle g_{010}=g_{0}g_{1}g_{0}} and generally, g A g B = g A B {\displaystyle g_{A}g_{B}=g_{AB}} for some binary strings of digits A, B, where AB is just the ordinary concatenation of such strings. The dyadic monoid M is then the monoid of all such finite-length left-right moves. Writing γ ∈ M {\displaystyle \gamma \in M} as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function: γ D ∘ c = c ∘ γ C {\displaystyle \gamma _{D}\circ c=c\circ \gamma _{C}} The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite binary tree; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on de Rham curves. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the modular group S L ( 2 , Z ) . {\displaystyle SL(2,\mathbb {Z} ).} Note that the Cantor function bears more than a passing resemblance to Minkowski's question-mark function. In particular, it obeys the exact same symmetry relations, although in an altered form. == Generalizations == Let y = ∑ k = 1 ∞ b k 2 − k {\displaystyle y=\sum _{k=1}^{\infty }b_{k}2^{-k}} be the dyadic (binary) expansion of the real number 0 ≤ y ≤ 1 in terms of binary digits bk ∈ {0,1}. This expansion is discussed in greater detail in the article on the dyadic transformation. Then consider the function C z ( y ) = ∑ k = 1 ∞ b k z k . {\displaystyle C_{z}(y)=\sum _{k=1}^{\infty }b_{k}z^{k}.} For z = 1/3, the inverse of the function x = 2 C1/3(y) is the Cantor function. That is, y = y(x) is the Cantor function. In general, for any z < 1/2, Cz(y) looks like the Cantor function turned on its side, with the width of the steps getting wider as z approaches zero. As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focuses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of fractal dimension, with the Hausdorff dimension the most popular choice. This line of research was started in the 1990s by Darst, who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, ( log 3 ⁡ ( 2 ) ) 2 {\displaystyle (\log _{3}(2))^{2}} . Subsequently Falconer showed that this squaring relationship holds for all Ahlfors's regular, singular measures, i.e. dim H ⁡ { x : f ′ ( x ) = lim h → 0 + μ ( [ x , x + h ] ) h does not exist } = ( dim H ⁡ supp ⁡ ( μ ) ) 2 {\displaystyle \dim _{H}\left\{x:f'(x)=\lim _{h\to 0^{+}}{\frac {\mu ([x,x+h])}{h}}{\text{ does not exist}}\right\}=\left(\dim _{H}\operatorname {supp} (\mu )\right)^{2}} Later, Troscheit obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and self-similar sets. Hermann Minkowski's question mark function loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The question mark function has the interesting property of having vanishing derivatives at all rational numbers. == See also == Dyadic transformation Weierstrass function, a function that is continuous everywhere but differentiable nowhere. == Notes == == References == Bass, Richard Franklin (2013) [2011]. Real analysis for graduate students (Second ed.). Createspace Independent Publishing. ISBN 978-1-4818-6914-0. Cantor, G. (1884). "De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur" [The power of perfect sets of points: Extract from a letter addressed to the editor]. Acta Mathematica. 4. International Press of Boston: 381–392. doi:10.1007/bf02418423. ISSN 0001-5962. Reprinted in: E. Zermelo (Ed.), Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, New York, 1980. Darst, Richard B.; Palagallo, Judith A.; Price, Thomas E. (2010), Curious curves, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., ISBN 978-981-4291-28-6, MR 2681574 Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. (2006). "The Cantor function". Expositiones Mathematicae. 24 (1). Elsevier BV: 1–37. doi:10.1016/j.exmath.2005.05.002. ISSN 0723-0869. MR 2195181. Fleron, Julian F. (1994-04-01). "A Note on the History of the Cantor Set and Cantor Function". Mathematics Magazine. 67 (2). Informa UK Limited: 136–140. doi:10.2307/2690689. ISSN 0025-570X. JSTOR 2690689. Lebesgue, H. (1904), Leçons sur l'intégration et la recherche des fonctions primitives [Lessons on integration and search for primitive functions], Paris: Gauthier-Villars Leoni, Giovanni (2017). A first course in Sobolev spaces. Vol. 181 (2nd ed.). Providence, Rhode Island: American Mathematical Society. p. 734. ISBN 978-1-4704-2921-8. OCLC 976406106. Scheeffer, Ludwig (1884). "Allgemeine Untersuchungen über Rectification der Curven" [General investigations on rectification of the curves]. Acta Mathematica. 5. International Press of Boston: 49–82. doi:10.1007/bf02421552. ISSN 0001-5962. Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008) [2001]. Elementary real analysis (Second ed.). ClassicalRealAnalysis.com. ISBN 978-1-4348-4367-8. Vestrup, E.M. (2003). The theory of measures and integration. Wiley series in probability and statistics. John Wiley & sons. ISBN 978-0471249771. Vitali, A. (1905), "Sulle funzioni integrali" [On the integral functions], Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 40: 1021–1034 == External links == Cantor ternary function at Encyclopaedia of Mathematics Cantor Function by Douglas Rivers, the Wolfram Demonstrations Project. Weisstein, Eric W. "Cantor Function". MathWorld.
Wikipedia:Cantor tree surface#0	In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of handles. == See also == Jacob's ladder surface Loch Ness monster surface == References ==
Wikipedia:Cantor–Zassenhaus algorithm#0	In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by David G. Cantor and Hans Zassenhaus in 1981. It is arguably the dominant algorithm for solving the problem, having replaced the earlier Berlekamp's algorithm of 1967. It is currently implemented in many computer algebra systems like PARI/GP. == Overview == === Background === The Cantor–Zassenhaus algorithm takes as input a square-free polynomial f ( x ) {\displaystyle f(x)} (i.e. one with no repeated factors) of degree n with coefficients in a finite field F q {\displaystyle \mathbb {F} _{q}} whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, f ( x ) / gcd ( f ( x ) , f ′ ( x ) ) {\displaystyle f(x)/\gcd(f(x),f'(x))} is a squarefree polynomial with the same factors as f ( x ) {\displaystyle f(x)} , so that the Cantor–Zassenhaus algorithm can be used to factor arbitrary polynomials). It gives as output a polynomial g ( x ) {\displaystyle g(x)} with coefficients in the same field such that g ( x ) {\displaystyle g(x)} divides f ( x ) {\displaystyle f(x)} . The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of f ( x ) {\displaystyle f(x)} into powers of irreducible polynomials (recalling that the ring of polynomials over any field is a unique factorisation domain). All possible factors of f ( x ) {\displaystyle f(x)} are contained within the factor ring R = F q [ x ] ⟨ f ( x ) ⟩ {\displaystyle R={\frac {\mathbb {F} _{q}[x]}{\langle f(x)\rangle }}} . If we suppose that f ( x ) {\displaystyle f(x)} has irreducible factors p 1 ( x ) , p 2 ( x ) , … , p s ( x ) {\displaystyle p_{1}(x),p_{2}(x),\ldots ,p_{s}(x)} , all of degree d, then this factor ring is isomorphic to the direct product of factor rings S = ∏ i = 1 s F q [ x ] ⟨ p i ( x ) ⟩ {\displaystyle S=\prod _{i=1}^{s}{\frac {\mathbb {F} _{q}[x]}{\langle p_{i}(x)\rangle }}} . The isomorphism from R to S, say ϕ {\displaystyle \phi } , maps a polynomial g ( x ) ∈ R {\displaystyle g(x)\in R} to the s-tuple of its reductions modulo each of the p i ( x ) {\displaystyle p_{i}(x)} , i.e. if: g ( x ) ≡ g 1 ( x ) ( mod p 1 ( x ) ) , g ( x ) ≡ g 2 ( x ) ( mod p 2 ( x ) ) , ⋮ g ( x ) ≡ g s ( x ) ( mod p s ( x ) ) , {\displaystyle {\begin{aligned}g(x)&{}\equiv g_{1}(x){\pmod {p_{1}(x)}},\\g(x)&{}\equiv g_{2}(x){\pmod {p_{2}(x)}},\\&{}\ \ \vdots \\g(x)&{}\equiv g_{s}(x){\pmod {p_{s}(x)}},\end{aligned}}} then ϕ ( g ( x ) + ⟨ f ( x ) ⟩ ) = ( g 1 ( x ) + ⟨ p 1 ( x ) ⟩ , … , g s ( x ) + ⟨ p s ( x ) ⟩ ) {\displaystyle \phi (g(x)+\langle f(x)\rangle )=(g_{1}(x)+\langle p_{1}(x)\rangle ,\ldots ,g_{s}(x)+\langle p_{s}(x)\rangle )} . It is important to note the following at this point, as it shall be of critical importance later in the algorithm: Since the p i ( x ) {\displaystyle p_{i}(x)} are each irreducible, each of the factor rings in this direct sum is in fact a field. These fields each have degree q d {\displaystyle q^{d}} . === Core result === The core result underlying the Cantor–Zassenhaus algorithm is the following: If a ( x ) ∈ R {\displaystyle a(x)\in R} is a polynomial satisfying: a ( x ) ≠ 0 , ± 1 {\displaystyle a(x)\neq 0,\pm 1} a i ( x ) ∈ { 0 , − 1 , 1 } for i = 1 , 2 , … , s , {\displaystyle a_{i}(x)\in \{0,-1,1\}{\text{ for }}i=1,2,\ldots ,s,} where a i ( x ) {\displaystyle a_{i}(x)} is the reduction of a ( x ) {\displaystyle a(x)} modulo p i ( x ) {\displaystyle p_{i}(x)} as before, and if any two of the following three sets is non-empty: A = { i ∣ a i ( x ) = 0 } , {\displaystyle A=\{i\mid a_{i}(x)=0\},} B = { i ∣ a i ( x ) = − 1 } , {\displaystyle B=\{i\mid a_{i}(x)=-1\},} C = { i ∣ a i ( x ) = 1 } , {\displaystyle C=\{i\mid a_{i}(x)=1\},} then there exist the following non-trivial factors of f ( x ) {\displaystyle f(x)} : gcd ( f ( x ) , a ( x ) ) = ∏ i ∈ A p i ( x ) , {\displaystyle \gcd(f(x),a(x))=\prod _{i\in A}p_{i}(x),} gcd ( f ( x ) , a ( x ) + 1 ) = ∏ i ∈ B p i ( x ) , {\displaystyle \gcd(f(x),a(x)+1)=\prod _{i\in B}p_{i}(x),} gcd ( f ( x ) , a ( x ) − 1 ) = ∏ i ∈ C p i ( x ) . {\displaystyle \gcd(f(x),a(x)-1)=\prod _{i\in C}p_{i}(x).} === Algorithm === The Cantor–Zassenhaus algorithm computes polynomials of the same type as a ( x ) {\displaystyle a(x)} above using the isomorphism discussed in the Background section. It proceeds as follows, in the case where the field F q {\displaystyle \mathbb {F} _{q}} is of odd-characteristic (the process can be generalised to characteristic 2 fields in a fairly straightforward way. Select a random polynomial b ( x ) ∈ R {\displaystyle b(x)\in R} such that b ( x ) ≠ 0 , ± 1 {\displaystyle b(x)\neq 0,\pm 1} . Set m = ( q d − 1 ) / 2 {\displaystyle m=(q^{d}-1)/2} and compute b ( x ) m {\displaystyle b(x)^{m}} . Since ϕ {\displaystyle \phi } is an isomorphism, we have (using our now-established notation): ϕ ( b ( x ) m ) = ( b 1 m ( x ) + ⟨ p 1 ( x ) ⟩ , … , b s m ( x ) + ⟨ p s ( x ) ⟩ ) . {\displaystyle \phi (b(x)^{m})=(b_{1}^{m}(x)+\langle p_{1}(x)\rangle ,\ldots ,b_{s}^{m}(x)+\langle p_{s}(x)\rangle ).} Now, each b i ( x ) + ⟨ p i ( x ) ⟩ {\displaystyle b_{i}(x)+\langle p_{i}(x)\rangle } is an element of a field of order q d {\displaystyle q^{d}} , as noted earlier. The multiplicative subgroup of this field has order q d − 1 {\displaystyle q^{d}-1} and so, unless b i ( x ) = 0 {\displaystyle b_{i}(x)=0} , we have b i ( x ) q d − 1 = 1 {\displaystyle b_{i}(x)^{q^{d}-1}=1} for each i and hence b i ( x ) m = ± 1 {\displaystyle b_{i}(x)^{m}=\pm 1} for each i. If b i ( x ) = 0 {\displaystyle b_{i}(x)=0} , then of course b i ( x ) m = 0 {\displaystyle b_{i}(x)^{m}=0} . Hence b ( x ) m {\displaystyle b(x)^{m}} is a polynomial of the same type as a ( x ) {\displaystyle a(x)} above. Further, since b ( x ) ≠ 0 , ± 1 {\displaystyle b(x)\neq 0,\pm 1} , at least two of the sets A , B {\displaystyle A,B} and C are non-empty and by computing the above GCDs we may obtain non-trivial factors. Since the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm. == Applications == One important application of the Cantor–Zassenhaus algorithm is in computing discrete logarithms over finite fields of prime-power order. Computing discrete logarithms is an important problem in public key cryptography. For a field of prime-power order, the fastest known method is the index calculus method, which involves the factorisation of field elements. If we represent the prime-power order field in the usual way – that is, as polynomials over the prime order base field, reduced modulo an irreducible polynomial of appropriate degree – then this is simply polynomial factorisation, as provided by the Cantor–Zassenhaus algorithm. == Implementation in computer algebra systems == The Cantor–Zassenhaus algorithm is implemented in the PARI/GP computer algebra system as the factormod() function (formerly factorcantor()). == See also == Polynomial factorization Factorization of polynomials over finite fields == References == == External links == https://web.archive.org/web/20200301213349/http://blog.fkraiem.org/2013/12/01/polynomial-factorisation-over-finite-fields-part-3-final-splitting-cantor-zassenhaus-in-odd-characteristic/
Wikipedia:Capelli's identity#0	In mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process. == Statement == Suppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization operator E i j = ∑ a = 1 n x i a ∂ ∂ x j a . {\displaystyle E_{ij}=\sum _{a=1}^{n}x_{ia}{\frac {\partial }{\partial x_{ja}}}.} The Capelli identity states that the following differential operators, expressed as determinants, are equal: \| E 11 + n − 1 ⋯ E 1 , n − 1 E 1 n ⋮ ⋱ ⋮ ⋮ E n − 1 , 1 ⋯ E n − 1 , n − 1 + 1 E n − 1 , n E n 1 ⋯ E n , n − 1 E n n + 0 \| = \| x 11 ⋯ x 1 n ⋮ ⋱ ⋮ x n 1 ⋯ x n n \| \| ∂ ∂ x 11 ⋯ ∂ ∂ x 1 n ⋮ ⋱ ⋮ ∂ ∂ x n 1 ⋯ ∂ ∂ x n n \| . {\displaystyle {\begin{vmatrix}E_{11}+n-1&\cdots &E_{1,n-1}&E_{1n}\\\vdots &\ddots &\vdots &\vdots \\E_{n-1,1}&\cdots &E_{n-1,n-1}+1&E_{n-1,n}\\E_{n1}&\cdots &E_{n,n-1}&E_{nn}+0\end{vmatrix}}={\begin{vmatrix}x_{11}&\cdots &x_{1n}\\\vdots &\ddots &\vdots \\x_{n1}&\cdots &x_{nn}\end{vmatrix}}{\begin{vmatrix}{\frac {\partial }{\partial x_{11}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{n1}}}&\cdots &{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}.} Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as det ( A ) = ∑ σ ∈ S n sgn ⁡ ( σ ) A σ ( 1 ) , 1 A σ ( 2 ) , 2 ⋯ A σ ( n ) , n , {\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )A_{\sigma (1),1}A_{\sigma (2),2}\cdots A_{\sigma (n),n},} where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant. The operators Eij can be written in a matrix form: E = X D t , {\displaystyle E=XD^{t},} where E , X , D {\displaystyle E,X,D} are matrices with elements Eij, xij, ∂ ∂ x i j {\displaystyle {\frac {\partial }{\partial x_{ij}}}} respectively. If all elements in these matrices would be commutative then clearly det ( E ) = det ( X ) det ( D t ) {\displaystyle \det(E)=\det(X)\det(D^{t})} . The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutativity is a small correction: ( n − i ) δ i j {\displaystyle (n-i)\delta _{ij}} on the left hand side. For generic noncommutative matrices formulas like det ( A B ) = det ( A ) det ( B ) {\displaystyle \det(AB)=\det(A)\det(B)} do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for n = 2, but is already long for n = 3. == Relations with representation theory == Consider the following slightly more general context. Suppose that n {\displaystyle n} and m {\displaystyle m} are two integers and x i j {\displaystyle x_{ij}} for i = 1 , … , n , j = 1 , … , m {\displaystyle i=1,\dots ,n,\ j=1,\dots ,m} , be commuting variables. Redefine E i j {\displaystyle E_{ij}} by almost the same formula: E i j = ∑ a = 1 m x i a ∂ ∂ x j a . {\displaystyle E_{ij}=\sum _{a=1}^{m}x_{ia}{\frac {\partial }{\partial x_{ja}}}.} with the only difference that summation index a {\displaystyle a} ranges from 1 {\displaystyle 1} to m {\displaystyle m} . One can easily see that such operators satisfy the commutation relations: [ E i j , E k l ] = δ j k E i l − δ i l E k j . {\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}.~~~~~~~~~} Here [ a , b ] {\displaystyle [a,b]} denotes the commutator a b − b a {\displaystyle ab-ba} . These are the same commutation relations which are satisfied by the matrices e i j {\displaystyle e_{ij}} which have zeros everywhere except the position ( i , j ) {\displaystyle (i,j)} , where 1 stands. ( e i j {\displaystyle e_{ij}} are sometimes called matrix units). Hence we conclude that the correspondence π : e i j ↦ E i j {\displaystyle \pi :e_{ij}\mapsto E_{ij}} defines a representation of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} in the vector space of polynomials of x i j {\displaystyle x_{ij}} . === Case m = 1 and representation Sk Cn === It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi: E i j = x i ∂ ∂ x j . {\displaystyle E_{ij}=x_{i}{\frac {\partial }{\partial x_{j}}}.} In particular, for the polynomials of the first degree it is seen that: E i j x k = δ j k x i . {\displaystyle E_{ij}x_{k}=\delta _{jk}x_{i}.~~~~~~~~~~~~~~} Hence the action of E i j {\displaystyle E_{ij}} restricted to the space of first-order polynomials is exactly the same as the action of matrix units e i j {\displaystyle e_{ij}} on vectors in C n {\displaystyle \mathbb {C} ^{n}} . So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentation of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} , which we identified with the standard representation in C n {\displaystyle \mathbb {C} ^{n}} . Going further, it is seen that the differential operators E i j {\displaystyle E_{ij}} preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentation of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} . One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power S k C n {\displaystyle S^{k}\mathbb {C} ^{n}} of the standard representation C n {\displaystyle \mathbb {C} ^{n}} . One can also easily identify the highest weight structure of these representations. The monomial x 1 k {\displaystyle x_{1}^{k}} is a highest weight vector, indeed: E i j x 1 k = 0 {\displaystyle E_{ij}x_{1}^{k}=0} for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: E i i x 1 k = k δ i 1 x 1 k {\displaystyle E_{ii}x_{1}^{k}=k\delta _{i1}x_{1}^{k}} . Such representation is sometimes called bosonic representation of g l n {\displaystyle {\mathfrak {gl}}_{n}} . Similar formulas E i j = ψ i ∂ ∂ ψ j {\displaystyle E_{ij}=\psi _{i}{\frac {\partial }{\partial \psi _{j}}}} define the so-called fermionic representation, here ψ i {\displaystyle \psi _{i}} are anti-commuting variables. Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to Λ k C n {\displaystyle \Lambda ^{k}\mathbb {C} ^{n}} i.e. anti-symmetric tensor power of C n {\displaystyle \mathbb {C} ^{n}} . Highest weight of such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representations of g l n {\displaystyle {\mathfrak {gl}}_{n}} . ==== Capelli identity for m = 1 ==== Let us return to the Capelli identity. One can prove the following: det ( E + ( n − i ) δ i j ) = 0 , n > 1 {\displaystyle \det(E+(n-i)\delta _{ij})=0,\qquad n>1} the motivation for this equality is the following: consider E i j c = x i p j {\displaystyle E_{ij}^{c}=x_{i}p_{j}} for some commuting variables x i , p j {\displaystyle x_{i},p_{j}} . The matrix E c {\displaystyle E^{c}} is of rank one and hence its determinant is equal to zero. Elements of matrix E {\displaystyle E} are defined by the similar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity: det ( E c ) = 0 {\displaystyle \det(E^{c})=0} can be preserved for the small price of correcting matrix E {\displaystyle E} by ( n − i ) δ i j {\displaystyle (n-i)\delta _{ij}} . Let us also mention that similar identity can be given for the characteristic polynomial: det ( t + E + ( n − i ) δ i j ) = t [ n ] + T r ( E ) t [ n − 1 ] , {\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\mathrm {Tr} (E)t^{[n-1]},~~~~} where t [ k ] = t ( t + 1 ) ⋯ ( t + k − 1 ) {\displaystyle t^{[k]}=t(t+1)\cdots (t+k-1)} . The commutative counterpart of this is a simple fact that for rank = 1 matrices the characteristic polynomial contains only the first and the second coefficients. Consider an example for n = 2. \| t + E 11 + 1 E 12 E 21 t + E 22 \| = \| t + x 1 ∂ 1 + 1 x 1 ∂ 2 x 2 ∂ 1 t + x 2 ∂ 2 \| = ( t + x 1 ∂ 1 + 1 ) ( t + x 2 ∂ 2 ) − x 2 ∂ 1 x 1 ∂ 2 = t ( t + 1 ) + t ( x 1 ∂ 1 + x 2 ∂ 2 ) + x 1 ∂ 1 x 2 ∂ 2 + x 2 ∂ 2 − x 2 ∂ 1 x 1 ∂ 2 {\displaystyle {\begin{aligned}&{\begin{vmatrix}t+E_{11}+1&E_{12}\\E_{21}&t+E_{22}\end{vmatrix}}={\begin{vmatrix}t+x_{1}\partial _{1}+1&x_{1}\partial _{2}\\x_{2}\partial _{1}&t+x_{2}\partial _{2}\end{vmatrix}}\\[8pt]&=(t+x_{1}\partial _{1}+1)(t+x_{2}\partial _{2})-x_{2}\partial _{1}x_{1}\partial _{2}\\[6pt]&=t(t+1)+t(x_{1}\partial _{1}+x_{2}\partial _{2})+x_{1}\partial _{1}x_{2}\partial _{2}+x_{2}\partial _{2}-x_{2}\partial _{1}x_{1}\partial _{2}\end{aligned}}} Using ∂ 1 x 1 = x 1 ∂ 1 + 1 , ∂ 1 x 2 = x 2 ∂ 1 , x 1 x 2 = x 2 x 1 {\displaystyle \partial _{1}x_{1}=x_{1}\partial _{1}+1,\partial _{1}x_{2}=x_{2}\partial _{1},x_{1}x_{2}=x_{2}x_{1}} we see that this is equal to: t ( t + 1 ) + t ( x 1 ∂ 1 + x 2 ∂ 2 ) + x 2 x 1 ∂ 1 ∂ 2 + x 2 ∂ 2 − x 2 x 1 ∂ 1 ∂ 2 − x 2 ∂ 2 = t ( t + 1 ) + t ( x 1 ∂ 1 + x 2 ∂ 2 ) = t [ 2 ] + t T r ( E ) . {\displaystyle {\begin{aligned}&{}\quad t(t+1)+t(x_{1}\partial _{1}+x_{2}\partial _{2})+x_{2}x_{1}\partial _{1}\partial _{2}+x_{2}\partial _{2}-x_{2}x_{1}\partial _{1}\partial _{2}-x_{2}\partial _{2}\\[8pt]&=t(t+1)+t(x_{1}\partial _{1}+x_{2}\partial _{2})=t^{[2]}+t\,\mathrm {Tr} (E).\end{aligned}}} === The universal enveloping algebra === U ( g l n ) {\displaystyle U({\mathfrak {gl}}_{n})} and its center An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is, the commutator [ E i j , det ( E + ( n − i ) δ i j ) ] = 0 {\displaystyle [E_{ij},\det(E+(n-i)\delta _{ij})]=0} is equal to zero. It can be generalized: Consider any elements Eij in any ring, such that they satisfy the commutation relation [ E i j , E k l ] = δ j k E i l − δ i l E k j {\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}} , (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows: det ( t + E + ( n − i ) δ i j ) = t [ n ] + ∑ k = n − 1 , … , 0 t [ k ] C k , {\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\sum _{k=n-1,\dots ,0}t^{[k]}C_{k},~~~~~} where t [ k ] = t ( t + 1 ) ⋯ ( t + k − 1 ) , {\displaystyle t^{[k]}=t(t+1)\cdots (t+k-1),} then: elements Ck commute with all elements Eij elements Ck can be given by the formulas similar to the commutative case: C k = ∑ I = ( i 1 < i 2 < ⋯ < i k ) det ( E + ( k − i ) δ i j ) I I , {\displaystyle C_{k}=\sum _{I=(i_{1}<i_{2}<\cdots <i_{k})}\det(E+(k-i)\delta _{ij})_{II},} i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction + ( k − i ) δ i j {\displaystyle +(k-i)\delta _{ij}} . In particular element C0 is the Capelli determinant considered above. These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the direct few lines short proof does not seem to exist, despite the simplicity of the formulation. The universal enveloping algebra U ( g l n ) {\displaystyle U({\mathfrak {gl}}_{n})} can be defined as an algebra generated by Eij subject to the relations [ E i j , E k l ] = δ j k E i l − δ i l E k j {\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}} alone. The proposition above shows that elements Ck belong to the center of U ( g l n ) {\displaystyle U({\mathfrak {gl}}_{n})} . It can be shown that they actually are free generators of the center of U ( g l n ) {\displaystyle U({\mathfrak {gl}}_{n})} . They are sometimes called Capelli generators. The Capelli identities for them will be discussed below. Consider an example for n = 2. \| t + E 11 + 1 E 12 E 21 t + E 22 \| = ( t + E 11 + 1 ) ( t + E 22 ) − E 21 E 12 = t ( t + 1 ) + t ( E 11 + E 22 ) + E 11 E 22 − E 21 E 12 + E 22 . {\displaystyle {\begin{aligned}{}\quad {\begin{vmatrix}t+E_{11}+1&E_{12}\\E_{21}&t+E_{22}\end{vmatrix}}&=(t+E_{11}+1)(t+E_{22})-E_{21}E_{12}\\&=t(t+1)+t(E_{11}+E_{22})+E_{11}E_{22}-E_{21}E_{12}+E_{22}.\end{aligned}}} It is immediate to check that element ( E 11 + E 22 ) {\displaystyle (E_{11}+E_{22})} commute with E i j {\displaystyle E_{ij}} . (It corresponds to an obvious fact that the identity matrix commute with all other matrices). More instructive is to check commutativity of the second element with E i j {\displaystyle E_{ij}} . Let us do it for E 12 {\displaystyle E_{12}} : [ E 12 , E 11 E 22 − E 21 E 12 + E 22 ] {\displaystyle [E_{12},E_{11}E_{22}-E_{21}E_{12}+E_{22}]} = [ E 12 , E 11 ] E 22 + E 11 [ E 12 , E 22 ] − [ E 12 , E 21 ] E 12 − E 21 [ E 12 , E 12 ] + [ E 12 , E 22 ] {\displaystyle =[E_{12},E_{11}]E_{22}+E_{11}[E_{12},E_{22}]-[E_{12},E_{21}]E_{12}-E_{21}[E_{12},E_{12}]+[E_{12},E_{22}]} = − E 12 E 22 + E 11 E 12 − ( E 11 − E 22 ) E 12 − 0 + E 12 {\displaystyle =-E_{12}E_{22}+E_{11}E_{12}-(E_{11}-E_{22})E_{12}-0+E_{12}} = − E 12 E 22 + E 22 E 12 + E 12 = − E 12 + E 12 = 0. {\displaystyle =-E_{12}E_{22}+E_{22}E_{12}+E_{12}=-E_{12}+E_{12}=0.} We see that the naive determinant E 11 E 22 − E 21 E 12 {\displaystyle E_{11}E_{22}-E_{21}E_{12}} will not commute with E 12 {\displaystyle E_{12}} and the Capelli's correction + E 22 {\displaystyle +E_{22}} is essential to ensure the centrality. === General m and dual pairs === Let us return to the general case: E i j = ∑ a = 1 m x i a ∂ ∂ x j a , {\displaystyle E_{ij}=\sum _{a=1}^{m}x_{ia}{\frac {\partial }{\partial x_{ja}}},} for arbitrary n and m. Definition of operators Eij can be written in a matrix form: E = X D t {\displaystyle E=XD^{t}} , where E {\displaystyle E} is n × n {\displaystyle n\times n} matrix with elements E i j {\displaystyle E_{ij}} ; X {\displaystyle X} is n × m {\displaystyle n\times m} matrix with elements x i j {\displaystyle x_{ij}} ; D {\displaystyle D} is n × m {\displaystyle n\times m} matrix with elements ∂ ∂ x i j {\displaystyle {\frac {\partial }{\partial x_{ij}}}} . Capelli–Cauchy–Binet identities For general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements of these matrices would commute then one knows that the determinant of E can be expressed by the so-called Cauchy–Binet formula via minors of X and D. An analogue of this formula also exists for matrix E again for the same mild price of the correction E → ( E + ( n − i ) δ i j ) {\displaystyle E\rightarrow (E+(n-i)\delta _{ij})} : det ( E + ( n − i ) δ i j ) = ∑ I = ( 1 ≤ i 1 < i 2 < ⋯ < i n ≤ m ) det ( X I ) det ( D I t ) {\displaystyle \det(E+(n-i)\delta _{ij})=\sum _{I=(1\leq i_{1}<i_{2}<\cdots <i_{n}\leq m)}\det(X_{I})\det(D_{I}^{t})} , In particular (similar to the commutative case): if m < n, then det ( E + ( n − i ) δ i j ) = 0 {\displaystyle \det(E+(n-i)\delta _{ij})=0} ; if m = n we return to the identity above. Let us also mention that similar to the commutative case (see Cauchy–Binet for minors), one can express not only the determinant of E, but also its minors via minors of X and D: det ( E + ( s − i ) δ i j ) K L = ∑ I = ( 1 ≤ i 1 < i 2 < ⋯ < i s ≤ m ) det ( X K I ) det ( D I L t ) {\displaystyle \det(E+(s-i)\delta _{ij})_{KL}=\sum _{I=(1\leq i_{1}<i_{2}<\cdots <i_{s}\leq m)}\det(X_{KI})\det(D_{IL}^{t})} , Here K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usually M K L {\displaystyle M_{KL}} denotes a submatrix of M formed by the elements M kalb. Pay attention that the Capelli correction now contains s, not n as in previous formula. Note that for s=1, the correction (s − i) disappears and we get just the definition of E as a product of X and transpose to D. Let us also mention that for generic K,L corresponding minors do not commute with all elements Eij, so the Capelli identity exists not only for central elements. As a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention the following: det ( t + E + ( n − i ) δ i j ) = t [ n ] + ∑ k = n − 1 , … , 0 t [ k ] ∑ I , J det ( X I J ) det ( D J I t ) , {\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\sum _{k=n-1,\dots ,0}t^{[k]}\sum _{I,J}\det(X_{IJ})\det(D_{JI}^{t}),} where I = ( 1 ≤ i 1 < ⋯ < i k ≤ n ) , {\displaystyle I=(1\leq i_{1}<\cdots <i_{k}\leq n),} J = ( 1 ≤ j 1 < ⋯ < j k ≤ n ) {\displaystyle J=(1\leq j_{1}<\cdots <j_{k}\leq n)} . This formula is similar to the commutative case, modula + ( n − i ) δ i j {\displaystyle +(n-i)\delta _{ij}} at the left hand side and t[n] instead of tn at the right hand side. Relation to dual pairs Modern interest in these identities has been much stimulated by Roger Howe who considered them in his theory of reductive dual pairs (also known as Howe duality). To make the first contact with these ideas, let us look more precisely on operators E i j {\displaystyle E_{ij}} . Such operators preserve the degree of polynomials. Let us look at the polynomials of degree 1: E i j x k l = x i l δ j k {\displaystyle E_{ij}x_{kl}=x_{il}\delta _{jk}} , we see that index l is preserved. One can see that from the representation theory point of view polynomials of the first degree can be identified with direct sum of the representations C n ⊕ ⋯ ⊕ C n {\displaystyle \mathbb {C} ^{n}\oplus \cdots \oplus \mathbb {C} ^{n}} , here l-th subspace (l=1...m) is spanned by x i l {\displaystyle x_{il}} , i = 1, ..., n. Let us give another look on this vector space: C n ⊕ ⋯ ⊕ C n = C n ⊗ C m . {\displaystyle \mathbb {C} ^{n}\oplus \cdots \oplus \mathbb {C} ^{n}=\mathbb {C} ^{n}\otimes \mathbb {C} ^{m}.} Such point of view gives the first hint of symmetry between m and n. To deepen this idea consider: E i j dual = ∑ a = 1 n x a i ∂ ∂ x a j . {\displaystyle E_{ij}^{\text{dual}}=\sum _{a=1}^{n}x_{ai}{\frac {\partial }{\partial x_{aj}}}.} These operators are given by the same formulas as E i j {\displaystyle E_{ij}} modula renumeration i ↔ j {\displaystyle i\leftrightarrow j} , hence by the same arguments we can deduce that E i j dual {\displaystyle E_{ij}^{\text{dual}}} form a representation of the Lie algebra g l m {\displaystyle {\mathfrak {gl}}_{m}} in the vector space of polynomials of xij. Before going further we can mention the following property: differential operators E i j dual {\displaystyle E_{ij}^{\text{dual}}} commute with differential operators E k l {\displaystyle E_{kl}} . The Lie group G L n × G L m {\displaystyle GL_{n}\times GL_{m}} acts on the vector space C n ⊗ C m {\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{m}} in a natural way. One can show that the corresponding action of Lie algebra g l n × g l m {\displaystyle {\mathfrak {gl}}_{n}\times {\mathfrak {gl}}_{m}} is given by the differential operators E i j {\displaystyle E_{ij}~~~~} and E i j dual {\displaystyle E_{ij}^{\text{dual}}} respectively. This explains the commutativity of these operators. The following deeper properties actually hold true: The only differential operators which commute with E i j {\displaystyle E_{ij}~~~~} are polynomials in E i j dual {\displaystyle E_{ij}^{\text{dual}}} , and vice versa. Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible representations of G L n {\displaystyle GL_{n}} and G L m {\displaystyle GL_{m}} can be given as follows: C [ x i j ] = S ( C n ⊗ C m ) = ∑ D ρ n D ⊗ ρ m D ′ . {\displaystyle \mathbb {C} [x_{ij}]=S(\mathbb {C} ^{n}\otimes \mathbb {C} ^{m})=\sum _{D}\rho _{n}^{D}\otimes \rho _{m}^{D'}.} The summands are indexed by the Young diagrams D, and representations ρ D {\displaystyle \rho ^{D}} are mutually non-isomorphic. And diagram D {\displaystyle {D}} determine D ′ {\displaystyle {D'}} and vice versa. In particular the representation of the big group G L n × G L m {\displaystyle GL_{n}\times GL_{m}} is multiplicity free, that is each irreducible representation occurs only one time. One easily observe the strong similarity to Schur–Weyl duality. == Generalizations == Much work have been done on the identity and its generalizations. Approximately two dozens of mathematicians and physicists contributed to the subject, to name a few: R. Howe, B. Kostant Fields medalist A. Okounkov A. Sokal, D. Zeilberger. It seems historically the first generalizations were obtained by Herbert Westren Turnbull in 1948, who found the generalization for the case of symmetric matrices (see for modern treatments). The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view. Such generalizations consist of changing Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} to simple Lie algebras and their super (q), and current versions. As well as identity can be generalized for different reductive dual pairs. And finally one can consider not only the determinant of the matrix E, but its permanent, trace of its powers and immanants. Let us mention few more papers; still the list of references is incomplete. It has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purely algebraic generalization of the identity have been found in 2008 by S. Caracciolo, A. Sportiello, A. D. Sokal which has nothing to do with any Lie algebras. === Turnbull's identity for symmetric matrices === Consider symmetric matrices X = \| x 11 x 12 x 13 ⋯ x 1 n x 12 x 22 x 23 ⋯ x 2 n x 13 x 23 x 33 ⋯ x 3 n ⋮ ⋮ ⋮ ⋱ ⋮ x 1 n x 2 n x 3 n ⋯ x n n \| , D = \| 2 ∂ ∂ x 11 ∂ ∂ x 12 ∂ ∂ x 13 ⋯ ∂ ∂ x 1 n ∂ ∂ x 12 2 ∂ ∂ x 22 ∂ ∂ x 23 ⋯ ∂ ∂ x 2 n ∂ ∂ x 13 ∂ ∂ x 23 2 ∂ ∂ x 33 ⋯ ∂ ∂ x 3 n ⋮ ⋮ ⋮ ⋱ ⋮ ∂ ∂ x 1 n ∂ ∂ x 2 n ∂ ∂ x 3 n ⋯ 2 ∂ ∂ x n n \| {\displaystyle X={\begin{vmatrix}x_{11}&x_{12}&x_{13}&\cdots &x_{1n}\\x_{12}&x_{22}&x_{23}&\cdots &x_{2n}\\x_{13}&x_{23}&x_{33}&\cdots &x_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\x_{1n}&x_{2n}&x_{3n}&\cdots &x_{nn}\end{vmatrix}},D={\begin{vmatrix}2{\frac {\partial }{\partial x_{11}}}&{\frac {\partial }{\partial x_{12}}}&{\frac {\partial }{\partial x_{13}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\[6pt]{\frac {\partial }{\partial x_{12}}}&2{\frac {\partial }{\partial x_{22}}}&{\frac {\partial }{\partial x_{23}}}&\cdots &{\frac {\partial }{\partial x_{2n}}}\\[6pt]{\frac {\partial }{\partial x_{13}}}&{\frac {\partial }{\partial x_{23}}}&2{\frac {\partial }{\partial x_{33}}}&\cdots &{\frac {\partial }{\partial x_{3n}}}\\[6pt]\vdots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{1n}}}&{\frac {\partial }{\partial x_{2n}}}&{\frac {\partial }{\partial x_{3n}}}&\cdots &2{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}} Herbert Westren Turnbull in 1948 discovered the following identity: det ( X D + ( n − i ) δ i j ) = det ( X ) det ( D ) {\displaystyle \det(XD+(n-i)\delta _{ij})=\det(X)\det(D)} Combinatorial proof can be found in the paper, another proof and amusing generalizations in the paper, see also discussion below. === The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices === Consider antisymmetric matrices X = \| 0 x 12 x 13 ⋯ x 1 n − x 12 0 x 23 ⋯ x 2 n − x 13 − x 23 0 ⋯ x 3 n ⋮ ⋮ ⋮ ⋱ ⋮ − x 1 n − x 2 n − x 3 n ⋯ 0 \| , D = \| 0 ∂ ∂ x 12 ∂ ∂ x 13 ⋯ ∂ ∂ x 1 n − ∂ ∂ x 12 0 ∂ ∂ x 23 ⋯ ∂ ∂ x 2 n − ∂ ∂ x 13 − ∂ ∂ x 23 0 ⋯ ∂ ∂ x 3 n ⋮ ⋮ ⋮ ⋱ ⋮ − ∂ ∂ x 1 n − ∂ ∂ x 2 n − ∂ ∂ x 3 n ⋯ 0 \| . {\displaystyle X={\begin{vmatrix}0&x_{12}&x_{13}&\cdots &x_{1n}\\-x_{12}&0&x_{23}&\cdots &x_{2n}\\-x_{13}&-x_{23}&0&\cdots &x_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\-x_{1n}&-x_{2n}&-x_{3n}&\cdots &0\end{vmatrix}},D={\begin{vmatrix}0&{\frac {\partial }{\partial x_{12}}}&{\frac {\partial }{\partial x_{13}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\[6pt]-{\frac {\partial }{\partial x_{12}}}&0&{\frac {\partial }{\partial x_{23}}}&\cdots &{\frac {\partial }{\partial x_{2n}}}\\[6pt]-{\frac {\partial }{\partial x_{13}}}&-{\frac {\partial }{\partial x_{23}}}&0&\cdots &{\frac {\partial }{\partial x_{3n}}}\\[6pt]\vdots &\vdots &\vdots &\ddots &\vdots \\[6pt]-{\frac {\partial }{\partial x_{1n}}}&-{\frac {\partial }{\partial x_{2n}}}&-{\frac {\partial }{\partial x_{3n}}}&\cdots &0\end{vmatrix}}.} Then det ( X D + ( n − i ) δ i j ) = det ( X ) det ( D ) . {\displaystyle \det(XD+(n-i)\delta _{ij})=\det(X)\det(D).} === The Caracciolo–Sportiello–Sokal identity for Manin matrices === Consider two matrices M and Y over some associative ring which satisfy the following condition [ M i j , Y k l ] = − δ j k Q i l {\displaystyle [M_{ij},Y_{kl}]=-\delta _{jk}Q_{il}~~~~~} for some elements Qil. Or ”in words”: elements in j-th column of M commute with elements in k-th row of Y unless j = k, and in this case commutator of the elements Mik and Ykl depends only on i, l, but does not depend on k. Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements). Then for the square matrix case det ( M Y + Q d i a g ( n − 1 , n − 2 , … , 1 , 0 ) ) = det ( M ) det ( Y ) . {\displaystyle \det(MY+Q\,\mathrm {diag} (n-1,n-2,\dots ,1,0))=\det(M)\det(Y).~~~~~~~} Here Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n − 1, n − 2, ..., 1, 0 on the diagonal. See proposition 1.2' formula (1.15) page 4, our Y is transpose to their B. Obviously the original Cappeli's identity the particular case of this identity. Moreover from this identity one can see that in the original Capelli's identity one can consider elements ∂ ∂ x i j + f i j ( x 11 , … , x k l , … ) {\displaystyle {\frac {\partial }{\partial x_{ij}}}+f_{ij}(x_{11},\dots ,x_{kl},\dots )} for arbitrary functions fij and the identity still will be true. === The Mukhin–Tarasov–Varchenko identity and the Gaudin model === ==== Statement ==== Consider matrices X and D as in Capelli's identity, i.e. with elements x i j {\displaystyle x_{ij}} and ∂ i j {\displaystyle \partial _{ij}} at position (ij). Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complex numbers. det ( ∂ ∂ z − A − X 1 z − B D t ) {\displaystyle \det \left({\frac {\partial }{\partial _{z}}}-A-X{\frac {1}{z-B}}D^{t}\right)} = det Put all x and z on the left, while all derivations on the right calculate as if all commute {\displaystyle ={\det }_{{\text{Put all }}x{\text{ and }}z{\text{ on the left, while all derivations on the right}}}^{\text{calculate as if all commute}}} ( ∂ ∂ z − A − X 1 z − B D t ) {\displaystyle \left({\frac {\partial }{\partial _{z}}}-A-X{\frac {1}{z-B}}D^{t}\right)} Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries. The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, while derivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics). ==== The Gaudin quantum integrable system and Talalaev's theorem ==== The matrix L ( z ) = A + X 1 z − B D t {\displaystyle L(z)=A+X{\frac {1}{z-B}}D^{t}} is a Lax matrix for the Gaudin quantum integrable spin chain system. D. Talalaev solved the long-standing problem of the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model, discovering the following theorem. Consider det ( ∂ ∂ z − L ( z ) ) = ∑ i = 0 n H i ( z ) ( ∂ ∂ z ) i . {\displaystyle \det \left({\frac {\partial }{\partial _{z}}}-L(z)\right)=\sum _{i=0}^{n}H_{i}(z)\left({\frac {\partial }{\partial _{z}}}\right)^{i}.} Then for all i,j,z,w [ H i ( z ) , H j ( w ) ] = 0 , {\displaystyle [H_{i}(z),H_{j}(w)]=0,~~~~~~~~} i.e. Hi(z) are generating functions in z for the differential operators in x which all commute. So they provide quantum commuting conservation laws for the Gaudin model. === Permanents, immanants, traces – "higher Capelli identities" === The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanents, immanants and traces. Based on the combinatorial approach paper by S.G. Williamson was one of the first results in this direction. ==== Turnbull's identity for permanents of antisymmetric matrices ==== Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of the HUKS identity above. Then p e r m ( X t D − ( n − i ) δ i j ) = p e r m Put all x on the left, with all derivations on the right Calculate as if all commute ( X t D ) . {\displaystyle \mathrm {perm} (X^{t}D-(n-i)\delta _{ij})=\mathrm {perm} _{{\text{Put all }}x{\text{ on the left, with all derivations on the right}}}^{\text{Calculate as if all commute}}(X^{t}D).} Let us cite: "...is stated without proof at the end of Turnbull’s paper". The authors themselves follow Turnbull – at the very end of their paper they write: "Since the proof of this last identity is very similar to the proof of Turnbull’s symmetric analog (with a slight twist), we leave it as an instructive and pleasant exercise for the reader.". The identity is deeply analyzed in paper . == References == == Further reading ==
Wikipedia:Carathéodory's existence theorem#0	In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory. == Introduction == Consider the differential equation y ′ ( t ) = f ( t , y ( t ) ) {\displaystyle y'(t)=f(t,y(t))} with initial condition y ( t 0 ) = y 0 , {\displaystyle y(t_{0})=y_{0},} where the function ƒ is defined on a rectangular domain of the form R = { ( t , y ) ∈ R × R n : \| t − t 0 \| ≤ a , \| y − y 0 \| ≤ b } . {\displaystyle R=\{(t,y)\in \mathbf {R} \times \mathbf {R} ^{n}\,:\,\|t-t_{0}\|\leq a,\|y-y_{0}\|\leq b\}.} Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition. However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation y ′ ( t ) = H ( t ) , y ( 0 ) = 0 , {\displaystyle y'(t)=H(t),\quad y(0)=0,} where H denotes the Heaviside function defined by H ( t ) = { 0 , if t ≤ 0 ; 1 , if t > 0. {\displaystyle H(t)={\begin{cases}0,&{\text{if }}t\leq 0;\\1,&{\text{if }}t>0.\end{cases}}} It makes sense to consider the ramp function y ( t ) = ∫ 0 t H ( s ) d s = { 0 , if t ≤ 0 ; t , if t > 0 {\displaystyle y(t)=\int _{0}^{t}H(s)\,\mathrm {d} s={\begin{cases}0,&{\text{if }}t\leq 0;\\t,&{\text{if }}t>0\end{cases}}} as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at t = 0 {\displaystyle t=0} , because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition. A function y is called a solution in the extended sense of the differential equation y ′ = f ( t , y ) {\displaystyle y'=f(t,y)} with initial condition y ( t 0 ) = y 0 {\displaystyle y(t_{0})=y_{0}} if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition. The absolute continuity of y implies that its derivative exists almost everywhere. == Statement of the theorem == Consider the differential equation y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},} with f {\displaystyle f} defined on the rectangular domain R = { ( t , y ) \| \| t − t 0 \| ≤ a , \| y − y 0 \| ≤ b } {\displaystyle R=\{(t,y)\,\|\,\|t-t_{0}\|\leq a,\|y-y_{0}\|\leq b\}} . If the function f {\displaystyle f} satisfies the following three conditions: f ( t , y ) {\displaystyle f(t,y)} is continuous in y {\displaystyle y} for each fixed t {\displaystyle t} , f ( t , y ) {\displaystyle f(t,y)} is measurable in t {\displaystyle t} for each fixed y {\displaystyle y} , there is a Lebesgue-integrable function m : [ t 0 − a , t 0 + a ] → [ 0 , ∞ ) {\displaystyle m:[t_{0}-a,t_{0}+a]\to [0,\infty )} such that \| f ( t , y ) \| ≤ m ( t ) {\displaystyle \|f(t,y)\|\leq m(t)} for all ( t , y ) ∈ R {\displaystyle (t,y)\in R} , then the differential equation has a solution in the extended sense in a neighborhood of the initial condition. A mapping f : R → R n {\displaystyle f\colon R\to \mathbf {R} ^{n}} is said to satisfy the Carathéodory conditions on R {\displaystyle R} if it fulfills the condition of the theorem. == Uniqueness of a solution == Assume that the mapping f {\displaystyle f} satisfies the Carathéodory conditions on R {\displaystyle R} and there is a Lebesgue-integrable function k : [ t 0 − a , t 0 + a ] → [ 0 , ∞ ) {\displaystyle k:[t_{0}-a,t_{0}+a]\to [0,\infty )} , such that \| f ( t , y 1 ) − f ( t , y 2 ) \| ≤ k ( t ) \| y 1 − y 2 \| , {\displaystyle \|f(t,y_{1})-f(t,y_{2})\|\leq k(t)\|y_{1}-y_{2}\|,} for all ( t , y 1 ) ∈ R , ( t , y 2 ) ∈ R . {\displaystyle (t,y_{1})\in R,(t,y_{2})\in R.} Then, there exists a unique solution y ( t ) = y ( t , t 0 , y 0 ) {\displaystyle y(t)=y(t,t_{0},y_{0})} to the initial value problem y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}.} Moreover, if the mapping f {\displaystyle f} is defined on the whole space R × R n {\displaystyle \mathbf {R} \times \mathbf {R} ^{n}} and if for any initial condition ( t 0 , y 0 ) ∈ R × R n {\displaystyle (t_{0},y_{0})\in \mathbf {R} \times \mathbf {R} ^{n}} , there exists a compact rectangular domain R ( t 0 , y 0 ) ⊂ R × R n {\displaystyle R_{(t_{0},y_{0})}\subset \mathbf {R} \times \mathbf {R} ^{n}} such that the mapping f {\displaystyle f} satisfies all conditions from above on R ( t 0 , y 0 ) {\displaystyle R_{(t_{0},y_{0})}} . Then, the domain E ⊂ R 2 + n {\displaystyle E\subset \mathbf {R} ^{2+n}} of definition of the function y ( t , t 0 , y 0 ) {\displaystyle y(t,t_{0},y_{0})} is open and y ( t , t 0 , y 0 ) {\displaystyle y(t,t_{0},y_{0})} is continuous on E {\displaystyle E} . == Example == Consider a linear initial value problem of the form y ′ ( t ) = A ( t ) y ( t ) + b ( t ) , y ( t 0 ) = y 0 . {\displaystyle y'(t)=A(t)y(t)+b(t),\quad y(t_{0})=y_{0}.} Here, the components of the matrix-valued mapping A : R → R n × n {\displaystyle A\colon \mathbf {R} \to \mathbf {R} ^{n\times n}} and of the inhomogeneity b : R → R n {\displaystyle b\colon \mathbf {R} \to \mathbf {R} ^{n}} are assumed to be integrable on every finite interval. Then, the right hand side of the differential equation satisfies the Carathéodory conditions and there exists a unique solution to the initial value problem. == See also == Picard–Lindelöf theorem Cauchy–Kowalevski theorem == Notes == == References == Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill. Hale, Jack K. (1980), Ordinary Differential Equations (2nd ed.), Malabar: Robert E. Krieger Publishing Company, ISBN 0-89874-011-8. Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157.
Wikipedia:Carl Anton Bjerknes#0	Carl Anton Bjerknes ( BYURK-niss, Norwegian: [ˈbjæ̂rkneːs]; 24 October 1825 – 20 March 1903) was a Norwegian mathematician and physicist. Bjerknes' earlier work was in pure mathematics, but he is principally known for his studies in hydrodynamics. == Biography == Carl Anton Bjerknes was born in Oslo, Norway. His father was Abraham Isaksen Bjerknes and his mother Elen Birgitte Holmen. Bjerknes studied mining at the University of Oslo, and after that mathematics at the University of Göttingen and the University of Paris. In 1866 he held a chair for applied mathematics and in 1869 for mathematics. Over a fifty-year time period, Bjerknes taught mathematics at the University of Oslo and at the military college. A pupil of Peter Gustav Lejeune Dirichlet, Gabriel Lamé and Augustin-Louis Cauchy Bjerknes worked for the rest of his life in the field of hydrodynamics. He tried to explain the electrodynamics of James Clerk Maxwell by hydrodynamical analogies and similarly he proposed a mechanical explanation of gravitation. Although he did not succeed in his attempts to explain all those things, his findings in the field of hydrodynamics were important. His experiments were shown at the first International Exposition of Electricity in Paris that ran from August 15, 1881 through to November 15, 1881 at the Palais de l'Industrie on the Champs-Élysées and at the Scandinavian naturalist meeting in Stockholm. John Charles Fields the founder of the Fields Medal for outstanding achievement in mathematics had this to say about the great minds that Norway had produced since it gained independence: ...for the number of great men which Norway has produced within the comparatively short period of its national existence is quite remarkable. Niels Henrik Abel was the first of a succession of eminent mathematicians, and it is not alone in mathematics that Norwegians have distinguished themselves ... [Among those] are to be found such men as Bjerknes, Peter Ludwig Mejdell Sylow and Sophus Lie in mathematics, Bjørnstjerne Bjørnson and Henrik Ibsen in literature, Edvard Grieg and Christian Sinding in music. == International Exposition of Electricity == When at the 1881 Paris International Electric Exhibition, he (Carl Anton) and his son (Vilhelm Bjerknes), demonstrated instruments that reproduced hydrodynamic analogies, few observers could ignore these baffling phenomena. Such celebrities as Hermann von Helmholtz, Gustav Kirchhoff, William Thomson (Lord Kelvin), the Siemens brothers, and the Marquis of Salisbury visited the small Norwegian exhibit booth and watched with amazement as a system of pulsating spheres and similar devices appeared to reproduce well-known electric and magnetic phenomena. For many observers the Bjerknes apparatus seemed to illustrate that the mysterious nature of electricity could perhaps be revealed. British observers allegedly exclaimed, "Maxwell should have seen this!" Of the eleven diplômes d'honneur, seven went to non-French exhibitors, including Werner Siemens, Thomas Edison, Alexander Graham Bell and William Thomson. Professor Carl Anton Bjerknes, representing Norway, joined their ranks. == Family == On June 30, 1859, after returning from his foreign travels, Bjerknes married Wilhelmine Dorothea Koren (10.11.1837–21.10.1923) whose father was a minister in the Church in West Norway. His son Norwegian physicist and meteorologist, Vilhelm Bjerknes continued the work of his father. == Death == Bjerknes died suddenly of a stroke on 20 March 1903 at the age of 77. == Selected works == Niels Henrik Abel. En skildring af hans liv og videnskabelige virksomhed (A description of his life and scientific activity) (Stockholm. 1880) == References == == Other sources == O'Connor, John J.; Robertson, Edmund F., "Carl Anton Bjerknes", MacTutor History of Mathematics Archive, University of St Andrews Bjerknes, V. (1904), Carl Anton Bjerknes: Gedächtnisrede, Leipzig: J. A. Barth, p. 31 Wilson, E.B. (1904), "Review: Carl Anton Bjerknes: Gedächtnisrede", Bull. Amer. Math. Soc., 10 (10): 516, doi:10.1090/S0002-9904-1904-01166-0 Author profile in the database zbMATH
Wikipedia:Carl Christoffer Georg Andræ#0	Carl Christopher Georg Andræ (14 October 1812 – 2 February 1893) was a Danish politician and mathematician. From 1842 until 1854, he was professor of mathematics and mechanics at the national military college. He was elected to the Royal Danish Academy of Sciences and Letters in 1853. Andræ was by royal appointment a member of the 1848 Danish Constituent Assembly. In 1854, he became Finance Minister in the Cabinet of Bang before also becoming Council President of Denmark 1856-1857 as leader of the Cabinet of Andræ. After being replaced as Council President by Carl Christian Hall in 1857 Andræ continued as Finance Minister in the Cabinet of Hall I until 1858. Being an individualist he, after the defeat of the National Liberals, never formally joined any political group but remained for the rest of his life a sceptical de facto conservative spectator of the 'Constitutional Struggle'. == Early life and education == Andræ was born in Hjertebjerg Rectory on the island of Møn. His parents were captain at the Third Jutland Infantry Regiment Johann Georg Andræ (1775–1814) Nicoline Christine Holm (1789–1862). He enrolled at Landkadetakademiet in 1825. In 1829, he was appointed to Second Lieutenant in the Road Corps. He followed a course in mathematics under Hans Christian Ørsted at the College of Applied Sciences before enrolling at the new Militære Højskole in 1830. He graduated with honours in December 1834 and was then made a First Lieutenant in the Engineering Corps. He completed two study trips to Paris in 1835–38, and he made significant contributions to the field of geodesy. == Single transferable vote == Andræ developed a multi-winner ranked voting system, a version of what is now called the single transferable vote (STV), which was used in Danish elections from 1855. This was two years before Thomas Hare published his first description of an STV system, without reference to Andræ. Thoroughly convinced of the soundness of his method of electing representatives and ready to defend it in the cabinet or the parliament, he saw it used successfully in Danish elections. But he made no effort to bring it to the attention of scientific men and statesmen in other countries, much less to defend his claim as an inventor. == Personal life == In 1842, Andræ married Hansine Pouline Schack, an early feminist, who commented on his political views in her diaries, published from 1914 to 1920 as Geheimeraadinde Andræs politiske Dagbøger. He died on 2 February 1893. He is buried in Assistens Cemetery in Copenhagen. == Notes == == External links == Author profile in the database zbMATH == References == Zachariæ, G. (1887–1905). "Andræ, Carl Christopher Georg". In Carl Frederik Bricka (ed.). Dansk biografisk Lexikon (in Danish). Vol. I (Aaberg–Beaumelle). Runeberg. pp. 258–264. Retrieved 9 September 2013.
Wikipedia:Carl Fabian Björling#0	Carl Fabian Emanuel Björling (30 November 1839 – 6 May 1910) was a Swedish mathematician and meteorologist. == Life == He was born on 30 November 1839 in Västerås, Sweden, and died on 6 May 1910. He was the son of mathematician Emanuel Björling and father of lawyer Carl Georg Björling. == Career == He attained his Ph.D. from Uppsala University in 1863. In 1863, he became an associate professor of mathematics at Uppsala University. In 1867 he was appointed a lecturer of mathematics and physics at the Halmstad grammar school. From 1873 to 1904 he was the professor of mathematics at Lund University. In 1886, he became a member of the Royal Swedish Academy of Sciences. == References == "571–572 (Nordisk familjebok / Uggleupplagan. 3. Bergsvalan – Branstad)". runeberg.org. Retrieved 2014-01-31. == External links == "The Mathematics Genealogy Project – Carl Björling". genealogy.math.ndsu.nodak.edu. Retrieved 2014-01-31. Author profile in the database zbMATH
Wikipedia:Carl Jensen Burrau#0	Carl Jensen Burrau (29 July 1867 – 8 October 1944) was a Danish mathematician who worked on problems relating to physics and astronomy while also working as an actuary. Burrau was born in Helsingör (Elsinore), Denmark and was educated at Copenhagen University. He worked as an astronomy assistant at the university observatory from 1893 to 1898. He is known for his work on a three-body problem, examining the orbits of two equal masses revolving about each other. His collaborations with Törvald Thiele led to the so-called Thiele–Burrau method. His dissertation of 1895 examines methods of identifying constants from photographs of star positions using Bessel's classic method. He then worked on actuarial mathematics, writing a book on the subject, Forsikringsstatistikens Grundlag (1925), and taught applied mathematics at Copenhagen university. == References ==
Wikipedia:Carl S. Herz#0	Carl Samuel Herz (10 April 1930 – 1 May 1995) was an American-Canadian mathematician, specializing in harmonic analysis. His name is attached to the Herz–Schur multiplier. He held professorships at Cornell University and McGill University, where he was Peter Redpath Professor of Mathematics at the time of his death. == Education and career == Herz received his bachelor's degree from Cornell University in 1950 and continued on as a mathematics graduate student at Princeton University. There he received a Ph.D. under the supervision of Salomon Bochner in 1953 with the dissertation "Bessel Functions of Matrix Argument". According to Tom H. Koornwinder, Herz's dissertation (published in the Annals of Mathematics in May 1955) "was a pioneering paper in the field of special functions in several variables associated with Lie groups and with root systems." Herz returned to Cornell as an instructor, rising in rank to assistant professor in 1955, associate professor in 1958, and full professor in 1963. He remained at Cornell until 1969. During the academic year 1969–1970 he worked at Brandeis University and then in 1970 joined the faculty of McGill University as full professor, where he remained until his death in 1995. During the academic year 1962–1963 Herz was a Sloan Fellow at Université de Paris-Sud at Orsay, where he established close ties with mathematicians there that led to frequent academic visits at Orsay of a month or two each year. In the academic years 1957–1958 and 1976–1977 he was a visiting scholar at the Institute for Advanced Study. Herz did mathematical research on spectral synthesis, positive-definite functions, Fourier transforms on convex sets, potential theory, Hp, and BMO. According to Nicholas Varopoulos, Herz made contributions "to the theory of symmetric spaces, Lie groups and the heat kernel on these; among other things he succeeded in classifying all faithful representations of Lie groups by contact transformations of a compact manifold." In 1978 he was elected of a Fellow of the Royal Society of Canada. In 1986 he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. He was the president of the Canadian Mathematical Society in 1987–1989. The Institut des sciences mathématiques, a consortium of eight Quebec universities of which Herz was Director at the time of his death, established the Carl Herz Prize in his honor. == Personal life == Herz met Judith Scherer, a Professor of English, while teaching at Cornell. They were married in 1960. They had two children, Rachel and Nathaniel. Rachel is a psychologist and cognitive neuroscientist, specializing in the psychology of smell. Nathaniel is a trained lawyer working as web developer. == Selected publications == Herz, C. S. (1954). "On the mean inversion of Fourier and Hankel transforms". Proc Natl Acad Sci U S A. 40 (10): 996–999. Bibcode:1954PNAS...40..996H. doi:10.1073/pnas.40.10.996. PMC 534209. PMID 16589594. Herz, Carl S. (May 1955). "Bessel functions of matrix argument". Annals of Mathematics. 61 (3): 474–523. doi:10.2307/1969810. JSTOR 1969810. Herz, C. S. (1956). "Spectral synthesis for the Cantor set". Proc Natl Acad Sci U S A. 42 (1): 42–43. Bibcode:1956PNAS...42...42H. doi:10.1073/pnas.42.1.42. PMC 534229. PMID 16589812. "A note on summability methods and spectral theory". Trans. Amer. Math. Soc. 86: 506–510. 1957. doi:10.1090/s0002-9947-1957-0093685-2. MR 0093685. "A note on the span of translations in Lp". Proc. Amer. Math. Soc. 8: 724–727. 1957. doi:10.1090/s0002-9939-1957-0088604-4. MR 0088604. Herz, Carl S. (1960). "The spectral theory of bounded functions". Trans. Amer. Math. Soc. 94 (2): 181–232. doi:10.1090/s0002-9947-1960-0131779-3. MR 0131779. Herz, C. S. (1961). "A maximal theorem". Proc. Amer. Math. Soc. 12 (2): 229–233. doi:10.1090/s0002-9939-1961-0151861-0. MR 0151861. Herz, C. S. (1963). "A class of negative-definite functions". Proc. Amer. Math. Soc. 14 (4): 670–676. doi:10.1090/s0002-9939-1963-0158251-7. MR 0158251. Herz, Carl (1971). "The theory of p-spaces with an application to convolution operators". Trans. Amer. Math. Soc. 154: 69–82. doi:10.1090/s0002-9947-1971-0272952-0. MR 0272952. Herz, Carl (1974). "Bounded mean oscillation and regulated martingales". Trans. Amer. Math. Soc. 193: 199–215. doi:10.1090/s0002-9947-1974-0353447-5. MR 0353447. Herz, Carl (1991). "The derivative of the exponential map". Proc. Amer. Math. Soc. 112 (3): 909–911. doi:10.1090/s0002-9939-1991-1086328-8. MR 1086328. == References ==
Wikipedia:Carl Størmer#0	Fredrik Carl Mülertz Størmer (Norwegian pronunciation: [fʁɛdʁɪk kaːl ˈmʏlɐːt͡s ̍ˈʃtøːmɐː]) (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of π and Størmer's theorem on consecutive smooth numbers. In physics, he is known for studying the movement of charged particles in the magnetosphere and the formation of aurorae, and for his book on these subjects, From the Depths of Space to the Heart of the Atom. He worked for many years as a professor of mathematics at the University of Oslo in Norway. A crater on the far side of the Moon is named after him. == Personal life and career == Størmer was born on 3 September 1874 in Skien, the only child of a pharmacist Georg Ludvig Størmer (1842–1930) and Elisabeth Amalie Johanne Henriette Mülertz (1844–1916). His uncle was the entrepreneur and inventor Henrik Christian Fredrik Størmer. Størmer studied mathematics at the Royal Frederick University in Kristiania, Norway (now the University of Oslo, in Oslo) from 1892 to 1897, earning the rank of candidatus realium in 1898. He then studied with Picard, Poincaré, Painlevé, Jordan, Darboux, and Goursat at the Sorbonne in Paris from 1898 to 1900. He returned to Kristiania in 1900 as a research fellow in mathematics, visited the University of Göttingen in 1902, and returned to Kristiania in 1903, where he was appointed as a professor of mathematics, a position he held for 43 years. After he received a permanent position in Kristiania, Størmer published his subsequent writings under a shortened version of his name, Carl Størmer. In 1918, he was elected as the first president of the newly formed Norwegian Mathematical Society. He participated regularly in Scandinavian mathematical congresses, and was president of the 1936 International Congress of Mathematicians in Oslo (from 1924 the new name of Kristiania). Størmer was also affiliated with the Institute of Theoretical Astrophysics at the University of Oslo, which was founded in 1934. He died on 13 August 1957, at Blindern. He was also an amateur street photographer, beginning in his student days. In the years 1893-1897, he documented daily life on the streets of Oslo using a miniature CP Stirn spy camera. Between 1942 and 1943, he shared a small portion of his works with the public. However, it was not until he was 70 years old that he organized an exhibition showcasing all his historical photographs of celebrities that he had taken over the years. For instance it included one of Henrik Ibsen strolling down Karl Johans gate, the main road in Oslo. Most of these can now be viewed in Norway's Digitalt Museum. He was also a supervisory council member of the insurance company Forsikringsselskapet Norden. In February 1900 he married consul's daughter Ada Clauson (1877–1973), with whom he eventually had five children. Their son Leif Størmer became a professor of historical geology at the University of Oslo. His daughter Henny married landowner Carl Otto Løvenskiold. Carl Størmer is also the grandfather of the mathematician Erling Størmer. == Mathematical research == Størmer's first mathematical publication, published when he was a beginning student at the age of 18, concerned trigonometric series generalizing the Taylor expansion of the arcsine function. He revisited this problem a few years later. Next, he systematically investigated Machin-like formula by which the number π may be represented as a rational combination of the so-called "Gregory numbers" of the form arctan 1/n. Machin's original formula, π = 16 arctan ⁡ 1 5 − 4 arctan ⁡ 1 239 , {\displaystyle \textstyle \pi =16\arctan {\frac {1}{5}}-4\arctan {\frac {1}{239}},} is of this type, and Størmer showed that there were three other ways of representing π as a rational combination of two Gregory numbers. He then investigated combinations of three Gregory numbers, and found 102 representations of π of this form, but was unable to determine whether there might be additional solutions of this type. These representations led to fast algorithms for computing numerical approximations of π. In particular, a four-term representation found by Størmer, π = 176 arctan ⁡ 1 57 + 28 arctan ⁡ 1 239 − 48 arctan ⁡ 1 682 + 96 arctan ⁡ 1 12943 {\displaystyle \textstyle \pi =176\arctan {\frac {1}{57}}+28\arctan {\frac {1}{239}}-48\arctan {\frac {1}{682}}+96\arctan {\frac {1}{12943}}} was used in a record-setting calculation of π to 1,241,100,000,000 decimal digits in 2002 by Yasumasa Kanada. Størmer is also noted for the Størmer numbers, which arose from the decomposition of Gregory numbers in Størmer's work. Størmer's theorem, which he proved in 1897, shows that, for any finite set P of prime numbers, there are only finitely many pairs of consecutive integers having only the numbers from P as their prime factors. In addition, Størmer describes an algorithm for finding all such pairs. The superparticular ratios generated by these consecutive pairs are of particular importance in music theory. Størmer proves this theorem by reducing the problem to a finite set of Pell equations, and the theorem itself can also be interpreted as describing the possible factorizations of solutions to Pell's equation. Chapman quotes Louis Mordell as saying "His result is very pretty, and there are many applications of it." Additional subjects of Størmer's mathematical research included Lie groups, the gamma function, and Diophantine approximation of algebraic numbers and of the transcendental numbers arising from elliptic functions. From 1905 Størmer was an editor of the journal Acta Mathematica, and he was also an editor of the posthumously-published mathematical works of Niels Henrik Abel and Sophus Lie. == Astrophysical research == From 1903, when Størmer first observed Kristian Birkeland's experimental attempts to explain the aurora borealis, he was fascinated by aurorae and related phenomena. His first work on the subject attempted to model mathematically the paths taken by charged particles perturbed by the influence of a magnetized sphere, and Størmer eventually published over 48 papers on the motion of charged particles. By modeling the problem using differential equations and polar coordinates, Størmer was able to show that the radius of curvature of any particle's path is proportional to the square of its distance from the sphere's center. To solve the resulting differential equations numerically, he used Verlet integration, which is therefore also known as Störmer's method. Ernst Brüche and Willard Harrison Bennett verified experimentally Størmer's predicted particle motions; Bennett called his experimental apparatus "Störmertron" in honor of Størmer. Størmer's calculations showed that small variations in the trajectories of particles approaching the earth would be magnified by the effects of the Earth's magnetic field, explaining the convoluted shapes of aurorae. Størmer also considered the possibility that particles might be trapped within the geomagnetic field, and worked out the orbits of these trapped particles. Størmer's work on this subject applies to what are today called the magnetospheric ring current and Van Allen radiation belts. As well as modeling these phenomena mathematically, Størmer took many photographs of aurorae, from 20 different observatories across Norway. He measured their heights and latitudes by triangulation from multiple observatories, and showed that the aurora are typically as high as 100 kilometers above ground. He classified them by their shapes, and discovered in 1926 the "solar-illuminated aurora", a phenomenon that can occur at twilight when the upper parts of an aurora are lit by the sun; these aurorae can be as high as 1000 km above ground. Størmer's book, From the Depths of Space to the Heart of the Atom, describing his work in this area, was translated into five different languages from the original Norwegian. A second book, The Polar Aurora (Oxford Press, 1955), contains both his experimental work on aurorae and his mathematical attempts to model them. In his review of this book, Canadian astronomer John F. Heard calls Størmer "the acknowledged authority" on aurorae. Heard writes, "The Polar Aurora will undoubtedly remain for many years a standard reference book; it belongs on the desk of anyone whose work or interest is involved with aurorae." Other astrophysical phenomena investigated by Størmer include pulsations of the earth's magnetic field, echoing in radio transmissions, nacreous clouds and noctilucent clouds, zodiacal light, meteor trails, the solar corona and solar vortices, and cosmic rays. == Awards and honors == Størmer was a Foreign Member of the Royal Society (ForMemRS) and a Corresponding Member of the French Academy of Sciences. He was also a member of the Norwegian Academy of Science and Letters from 1900. He was given honorary degrees by Oxford University (in 1947), the University of Copenhagen (1951), and the Sorbonne (1953), and in 1922 the French Academy awarded him their Janssen Medal. Three times Størmer was a plenary speaker in the International Congress of Mathematicians (1908 in Rome, 1924 in Toronto, and 1936 in Oslo); he was an invited speaker of the ICM in 1920 in Strasbourg and in 1932 in Zurich. In 1971, the crater Störmer on the far side of the Moon was named after him. The Lie-Størmer Center at UiT - The Arctic University of Norway is named after him In 1902, Størmer was decorated with King Oscar II's Medal of Merit in gold. He was also decorated as a Knight, First Order of the Order of St. Olav in 1939. He was upgraded to Grand Cross of the Order of St. Olav in 1954. == References ==
Wikipedia:Carl Wilhelm Borchardt#0	Carl Wilhelm Borchardt (22 February 1817 – 27 June 1880) was a German mathematician. Borchardt was born to a Jewish family in Berlin. His father, Moritz, was a respected merchant, and his mother was Emma Heilborn. Borchardt studied under a number of tutors, including Julius Plücker and Jakob Steiner. He studied at the University of Berlin under Lejeune Dirichlet in 1836 and at the University of Königsberg in 1839. In 1848 he began teaching at the University of Berlin. He did research in the area of the arithmetic-geometric mean, continuing work by Gauss and Lagrange. He generalised the results of Kummer on diagonalising symmetric matrices, using determinants and Sturm functions. He was also an editor of Crelle's Journal from 1856 to 1880, during which time it was known as Borchardt's Journal. He died in Rüdersdorf, Germany. His grave is preserved in the Protestant Friedhof III der Jerusalems- und Neuen Kirchengemeinde (Cemetery No. III of the congregations of Jerusalem's Church and New Church) in Berlin-Kreuzberg, south of Hallesches Tor. == See also == Cayley's formula == References ==
Wikipedia:Carleman linearization#0	In mathematics, Carleman linearization (or Carleman embedding) is a technique to transform a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. It was introduced by the Swedish mathematician Torsten Carleman in 1932. Carleman linearization is related to composition operator and has been widely used in the study of dynamical systems. It also been used in many applied fields, such as in control theory and in quantum computing. == Procedure == Consider the following autonomous nonlinear system: x ˙ = f ( x ) + ∑ j = 1 m g j ( x ) d j ( t ) {\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)d_{j}(t)} where x ∈ R n {\displaystyle x\in R^{n}} denotes the system state vector. Also, f {\displaystyle f} and g i {\displaystyle g_{i}} 's are known analytic vector functions, and d j {\displaystyle d_{j}} is the j t h {\displaystyle j^{th}} element of an unknown disturbance to the system. At the desired nominal point, the nonlinear functions in the above system can be approximated by Taylor expansion f ( x ) ≃ f ( x 0 ) + ∑ k = 1 η 1 k ! ∂ f [ k ] ∣ x = x 0 ( x − x 0 ) [ k ] {\displaystyle f(x)\simeq f(x_{0})+\sum _{k=1}^{\eta }{\frac {1}{k!}}\partial f_{[k]}\mid _{x=x_{0}}(x-x_{0})^{[k]}} where ∂ f [ k ] ∣ x = x 0 {\displaystyle \partial f_{[k]}\mid _{x=x_{0}}} is the k t h {\displaystyle k^{th}} partial derivative of f ( x ) {\displaystyle f(x)} with respect to x {\displaystyle x} at x = x 0 {\displaystyle x=x_{0}} and x [ k ] {\displaystyle x^{[k]}} denotes the k t h {\displaystyle k^{th}} Kronecker product. Without loss of generality, we assume that x 0 {\displaystyle x_{0}} is at the origin. Applying Taylor approximation to the system, we obtain x ˙ ≃ ∑ k = 0 η A k x [ k ] + ∑ j = 1 m ∑ k = 0 η B j k x [ k ] d j {\displaystyle {\dot {x}}\simeq \sum _{k=0}^{\eta }A_{k}x^{[k]}+\sum _{j=1}^{m}\sum _{k=0}^{\eta }B_{jk}x^{[k]}d_{j}} where A k = 1 k ! ∂ f [ k ] ∣ x = 0 {\displaystyle A_{k}={\frac {1}{k!}}\partial f_{[k]}\mid _{x=0}} and B j k = 1 k ! ∂ g j [ k ] ∣ x = 0 {\displaystyle B_{jk}={\frac {1}{k!}}\partial g_{j[k]}\mid _{x=0}} . Consequently, the following linear system for higher orders of the original states are obtained: d ( x [ i ] ) d t ≃ ∑ k = 0 η − i + 1 A i , k x [ k + i − 1 ] + ∑ j = 1 m ∑ k = 0 η − i + 1 B j , i , k x [ k + i − 1 ] d j {\displaystyle {\frac {d(x^{[i]})}{dt}}\simeq \sum _{k=0}^{\eta -i+1}A_{i,k}x^{[k+i-1]}+\sum _{j=1}^{m}\sum _{k=0}^{\eta -i+1}B_{j,i,k}x^{[k+i-1]}d_{j}} where A i , k = ∑ l = 0 i − 1 I n [ l ] ⊗ A k ⊗ I n [ i − 1 − l ] {\displaystyle A_{i,k}=\sum _{l=0}^{i-1}I_{n}^{[l]}\otimes A_{k}\otimes I_{n}^{[i-1-l]}} , and similarly B j , i , κ = ∑ l = 0 i − 1 I n [ l ] ⊗ B j , κ ⊗ I n [ i − 1 − l ] {\displaystyle B_{j,i,\kappa }=\sum _{l=0}^{i-1}I_{n}^{[l]}\otimes B_{j,\kappa }\otimes I_{n}^{[i-1-l]}} . Employing Kronecker product operator, the approximated system is presented in the following form x ˙ ⊗ ≃ A x ⊗ + ∑ j = 1 m [ B j x ⊗ d j + B j 0 d j ] + A r {\displaystyle {\dot {x}}_{\otimes }\simeq Ax_{\otimes }+\sum _{j=1}^{m}[B_{j}x_{\otimes }d_{j}+B_{j0}d_{j}]+A_{r}} where x ⊗ = [ x T x [ 2 ] T . . . x [ η ] T ] T {\displaystyle x_{\otimes }={\begin{bmatrix}x^{T}&x^{{[2]}^{T}}&...&x^{{[\eta ]}^{T}}\end{bmatrix}}^{T}} , and A , B j , A r {\displaystyle A,B_{j},A_{r}} and B j , 0 {\displaystyle B_{j,0}} matrices are defined in (Hashemian and Armaou 2015). == See also == Carleman matrix Composition operator == References == == External links == A lecture about Carleman linearization by Igor Mezić
Wikipedia:Carleman matrix#0	In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains. == Definition == The Carleman matrix of an infinitely differentiable function f ( x ) {\displaystyle f(x)} is defined as: M [ f ] j k = 1 k ! [ d k d x k ( f ( x ) ) j ] x = 0 , {\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}~,} so as to satisfy the (Taylor series) equation: ( f ( x ) ) j = ∑ k = 0 ∞ M [ f ] j k x k . {\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.} For instance, the computation of f ( x ) {\displaystyle f(x)} by f ( x ) = ∑ k = 0 ∞ M [ f ] 1 , k x k . {\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.~} simply amounts to the dot-product of row 1 of M [ f ] {\displaystyle M[f]} with a column vector [ 1 , x , x 2 , x 3 , . . . ] τ {\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }} . The entries of M [ f ] {\displaystyle M[f]} in the next row give the 2nd power of f ( x ) {\displaystyle f(x)} : f ( x ) 2 = ∑ k = 0 ∞ M [ f ] 2 , k x k , {\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}~,} and also, in order to have the zeroth power of f ( x ) {\displaystyle f(x)} in M [ f ] {\displaystyle M[f]} , we adopt the row 0 containing zeros everywhere except the first position, such that f ( x ) 0 = 1 = ∑ k = 0 ∞ M [ f ] 0 , k x k = 1 + ∑ k = 1 ∞ 0 ⋅ x k . {\displaystyle f(x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0\cdot x^{k}~.} Thus, the dot product of M [ f ] {\displaystyle M[f]} with the column vector [ 1 , x , x 2 , . . . ] T {\displaystyle {\begin{bmatrix}1,x,x^{2},...\end{bmatrix}}^{T}} yields the column vector [ 1 , f ( x ) , f ( x ) 2 , . . . ] T {\displaystyle \left[1,f(x),f(x)^{2},...\right]^{T}} , i.e., M [ f ] [ 1 x x 2 x 3 ⋮ ] = [ 1 f ( x ) ( f ( x ) ) 2 ( f ( x ) ) 3 ⋮ ] . {\displaystyle M[f]{\begin{bmatrix}1\\x\\x^{2}\\x^{3}\\\vdots \end{bmatrix}}={\begin{bmatrix}1\\f(x)\\(f(x))^{2}\\(f(x))^{3}\\\vdots \end{bmatrix}}.} == Generalization == A generalization of the Carleman matrix of a function can be defined around any point, such as: M [ f ] x 0 = M x [ x − x 0 ] M [ f ] M x [ x + x 0 ] {\displaystyle M[f]_{x_{0}}=M_{x}[x-x_{0}]M[f]M_{x}[x+x_{0}]} or M [ f ] x 0 = M [ g ] {\displaystyle M[f]_{x_{0}}=M[g]} where g ( x ) = f ( x + x 0 ) − x 0 {\displaystyle g(x)=f(x+x_{0})-x_{0}} . This allows the matrix power to be related as: ( M [ f ] x 0 ) n = M x [ x − x 0 ] M [ f ] n M x [ x + x 0 ] {\displaystyle (M[f]_{x_{0}})^{n}=M_{x}[x-x_{0}]M[f]^{n}M_{x}[x+x_{0}]} === General Series === Another way to generalize it even further is think about a general series in the following way: Let h ( x ) = ∑ n c n ( h ) ⋅ ψ n ( x ) {\displaystyle h(x)=\sum _{n}c_{n}(h)\cdot \psi _{n}(x)} be a series approximation of h ( x ) {\displaystyle h(x)} , where { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} is a basis of the space containing h ( x ) {\displaystyle h(x)} Assuming that { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} is also a basis for f ( x ) {\displaystyle f(x)} , We can define G [ f ] m n = c n ( ψ m ∘ f ) {\displaystyle G[f]_{mn}=c_{n}(\psi _{m}\circ f)} , therefore we have ψ m ∘ f = ∑ n c n ( ψ m ∘ f ) ⋅ ψ n = ∑ n G [ f ] m n ⋅ ψ n {\displaystyle \psi _{m}\circ f=\sum _{n}c_{n}(\psi _{m}\circ f)\cdot \psi _{n}=\sum _{n}G[f]_{mn}\cdot \psi _{n}} , now we can prove that G [ g ∘ f ] = G [ g ] ⋅ G [ f ] {\displaystyle G[g\circ f]=G[g]\cdot G[f]} , if we assume that { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} is also a basis for g ( x ) {\displaystyle g(x)} and g ( f ( x ) ) {\displaystyle g(f(x))} . Let g ( x ) {\displaystyle g(x)} be such that ψ l ∘ g = ∑ m G [ g ] l m ⋅ ψ m {\displaystyle \psi _{l}\circ g=\sum _{m}G[g]_{lm}\cdot \psi _{m}} where G [ g ] l m = c m ( ψ l ∘ g ) {\displaystyle G[g]_{lm}=c_{m}(\psi _{l}\circ g)} . Now ∑ n G [ g ∘ f ] l n ψ n = ψ l ∘ ( g ∘ f ) = ( ψ l ∘ g ) ∘ f = ∑ m G [ g ] l m ( ψ m ∘ f ) = ∑ m G [ g ] l m ∑ n G [ f ] m n ψ n = ∑ n , m G [ g ] l m G [ f ] m n ψ n = ∑ n ( ∑ m G [ g ] l m G [ f ] m n ) ψ n {\displaystyle {\begin{aligned}\sum _{n}G[g\circ f]_{ln}\psi _{n}=\psi _{l}\circ (g\circ f)&=(\psi _{l}\circ g)\circ f\\&=\sum _{m}G[g]_{lm}(\psi _{m}\circ f)\\&=\sum _{m}G[g]_{lm}\sum _{n}G[f]_{mn}\psi _{n}\\&=\sum _{n,m}G[g]_{lm}G[f]_{mn}\psi _{n}\\&=\sum _{n}(\sum _{m}G[g]_{lm}G[f]_{mn})\psi _{n}\end{aligned}}} Comparing the first and the last term, and from { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} being a base for f ( x ) {\displaystyle f(x)} , g ( x ) {\displaystyle g(x)} and g ( f ( x ) ) {\displaystyle g(f(x))} it follows that G [ g ∘ f ] = ∑ m G [ g ] l m G [ f ] m n = G [ g ] ⋅ G [ f ] {\displaystyle G[g\circ f]=\sum _{m}G[g]_{lm}G[f]_{mn}=G[g]\cdot G[f]} ==== Examples ==== ===== Rederive (Taylor) Carleman Matrix ===== If we set ψ n ( x ) = x n {\displaystyle \psi _{n}(x)=x^{n}} we have the Carleman matrix. Because h ( x ) = ∑ n c n ( h ) ⋅ ψ n ( x ) = ∑ n c n ( h ) ⋅ x n {\displaystyle h(x)=\sum _{n}c_{n}(h)\cdot \psi _{n}(x)=\sum _{n}c_{n}(h)\cdot x^{n}} then we know that the n-th coefficient c n ( h ) {\displaystyle c_{n}(h)} must be the nth-coefficient of the taylor series of h {\displaystyle h} . Therefore c n ( h ) = 1 n ! h ( n ) ( 0 ) {\displaystyle c_{n}(h)={\frac {1}{n!}}h^{(n)}(0)} Therefore G [ f ] m n = c n ( ψ m ∘ f ) = c n ( f ( x ) m ) = 1 n ! [ d n d x n ( f ( x ) ) m ] x = 0 {\displaystyle G[f]_{mn}=c_{n}(\psi _{m}\circ f)=c_{n}(f(x)^{m})={\frac {1}{n!}}\left[{\frac {d^{n}}{dx^{n}}}(f(x))^{m}\right]_{x=0}} Which is the Carleman matrix given above. (It's important to note that this is not an orthornormal basis) ===== Carleman Matrix For Orthonormal Basis ===== If { e n ( x ) } n {\displaystyle \{e_{n}(x)\}_{n}} is an orthonormal basis for a Hilbert Space with a defined inner product ⟨ f , g ⟩ {\displaystyle \langle f,g\rangle } , we can set ψ n = e n {\displaystyle \psi _{n}=e_{n}} and c n ( h ) {\displaystyle c_{n}(h)} will be ⟨ h , e n ⟩ {\displaystyle {\displaystyle \langle h,e_{n}\rangle }} . Then G [ f ] m n = c n ( e m ∘ f ) = ⟨ e m ∘ f , e n ⟩ {\displaystyle G[f]_{mn}=c_{n}(e_{m}\circ f)=\langle e_{m}\circ f,e_{n}\rangle } . ===== Carleman Matrix for Fourier Series ===== If e n ( x ) = e i n x {\displaystyle e_{n}(x)=e^{inx}} we have the analogous for Fourier Series. Let c ^ n {\displaystyle {\hat {c}}_{n}} and G ^ {\displaystyle {\hat {G}}} represent the carleman coefficient and matrix in the fourier basis. Because the basis is orthogonal, we have. c ^ n ( h ) = ⟨ h , e n ⟩ = 1 2 π ∫ − π π h ( x ) ⋅ e − i n x d x {\displaystyle {\hat {c}}_{n}(h)=\langle h,e_{n}\rangle ={\cfrac {1}{2\pi }}\int _{-\pi }^{\pi }\displaystyle h(x)\cdot e^{-inx}dx} . Then, therefore, G ^ [ f ] m n = c n ^ ( e m ∘ f ) = ⟨ e m ∘ f , e n ⟩ {\displaystyle {\hat {G}}[f]_{mn}={\hat {c_{n}}}(e_{m}\circ f)=\langle e_{m}\circ f,e_{n}\rangle } which is G ^ [ f ] m n = 1 2 π ∫ − π π e i m f ( x ) ⋅ e − i n x d x {\displaystyle {\hat {G}}[f]_{mn}={\cfrac {1}{2\pi }}\int _{-\pi }^{\pi }\displaystyle e^{imf(x)}\cdot e^{-inx}dx} == Properties == Carleman matrices satisfy the fundamental relationship M [ f ∘ g ] = M [ f ] M [ g ] , {\displaystyle M[f\circ g]=M[f]M[g]~,} which makes the Carleman matrix M a (direct) representation of f ( x ) {\displaystyle f(x)} . Here the term f ∘ g {\displaystyle f\circ g} denotes the composition of functions f ( g ( x ) ) {\displaystyle f(g(x))} . Other properties include: M [ f n ] = M [ f ] n {\displaystyle \,M[f^{n}]=M[f]^{n}} , where f n {\displaystyle \,f^{n}} is an iterated function and M [ f − 1 ] = M [ f ] − 1 {\displaystyle \,M[f^{-1}]=M[f]^{-1}} , where f − 1 {\displaystyle \,f^{-1}} is the inverse function (if the Carleman matrix is invertible). == Examples == The Carleman matrix of a constant is: M [ a ] = ( 1 0 0 ⋯ a 0 0 ⋯ a 2 0 0 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[a]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&0&0&\cdots \\a^{2}&0&0&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} The Carleman matrix of the identity function is: M x [ x ] = ( 1 0 0 ⋯ 0 1 0 ⋯ 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[x]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&1&0&\cdots \\0&0&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} The Carleman matrix of a constant addition is: M x [ a + x ] = ( 1 0 0 ⋯ a 1 0 ⋯ a 2 2 a 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[a+x]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&1&0&\cdots \\a^{2}&2a&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} The Carleman matrix of the successor function is equivalent to the Binomial coefficient: M x [ 1 + x ] = ( 1 0 0 0 ⋯ 1 1 0 0 ⋯ 1 2 1 0 ⋯ 1 3 3 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[1+x]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&1&0&0&\cdots \\1&2&1&0&\cdots \\1&3&3&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ 1 + x ] j k = ( j k ) {\displaystyle M_{x}[1+x]_{jk}={\binom {j}{k}}} The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials: M x [ log ⁡ ( 1 + x ) ] = ( 1 0 0 0 0 ⋯ 0 1 − 1 2 1 3 − 1 4 ⋯ 0 0 1 − 1 11 12 ⋯ 0 0 0 1 − 3 2 ⋯ 0 0 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[\log(1+x)]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&-{\frac {1}{2}}&{\frac {1}{3}}&-{\frac {1}{4}}&\cdots \\0&0&1&-1&{\frac {11}{12}}&\cdots \\0&0&0&1&-{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ log ⁡ ( 1 + x ) ] j k = s ( k , j ) j ! k ! {\displaystyle M_{x}[\log(1+x)]_{jk}=s(k,j){\frac {j!}{k!}}} The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials: M x [ − log ⁡ ( 1 − x ) ] = ( 1 0 0 0 0 ⋯ 0 1 1 2 1 3 1 4 ⋯ 0 0 1 1 11 12 ⋯ 0 0 0 1 3 2 ⋯ 0 0 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[-\log(1-x)]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&\cdots \\0&0&1&1&{\frac {11}{12}}&\cdots \\0&0&0&1&{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ − log ⁡ ( 1 − x ) ] j k = \| s ( k , j ) \| j ! k ! {\displaystyle M_{x}[-\log(1-x)]_{jk}=\|s(k,j)\|{\frac {j!}{k!}}} The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials: M x [ exp ⁡ ( x ) − 1 ] = ( 1 0 0 0 0 ⋯ 0 1 1 2 1 6 1 24 ⋯ 0 0 1 1 7 12 ⋯ 0 0 0 1 3 2 ⋯ 0 0 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[\exp(x)-1]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&{\frac {1}{2}}&{\frac {1}{6}}&{\frac {1}{24}}&\cdots \\0&0&1&1&{\frac {7}{12}}&\cdots \\0&0&0&1&{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ exp ⁡ ( x ) − 1 ] j k = S ( k , j ) j ! k ! {\displaystyle M_{x}[\exp(x)-1]_{jk}=S(k,j){\frac {j!}{k!}}} The Carleman matrix of exponential functions is: M x [ exp ⁡ ( a x ) ] = ( 1 0 0 0 ⋯ 1 a a 2 2 a 3 6 ⋯ 1 2 a 2 a 2 4 a 3 3 ⋯ 1 3 a 9 a 2 2 9 a 3 2 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[\exp(ax)]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&a&{\frac {a^{2}}{2}}&{\frac {a^{3}}{6}}&\cdots \\1&2a&2a^{2}&{\frac {4a^{3}}{3}}&\cdots \\1&3a&{\frac {9a^{2}}{2}}&{\frac {9a^{3}}{2}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ exp ⁡ ( a x ) ] j k = ( j a ) k k ! {\displaystyle M_{x}[\exp(ax)]_{jk}={\frac {(ja)^{k}}{k!}}} The Carleman matrix of a constant multiple is: M x [ c x ] = ( 1 0 0 ⋯ 0 c 0 ⋯ 0 0 c 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&c&0&\cdots \\0&0&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} The Carleman matrix of a linear function is: M x [ a + c x ] = ( 1 0 0 ⋯ a c 0 ⋯ a 2 2 a c c 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[a+cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&c&0&\cdots \\a^{2}&2ac&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} The Carleman matrix of a function f ( x ) = ∑ k = 1 ∞ f k x k {\displaystyle f(x)=\sum _{k=1}^{\infty }f_{k}x^{k}} is: M [ f ] = ( 1 0 0 ⋯ 0 f 1 f 2 ⋯ 0 0 f 1 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&f_{1}&f_{2}&\cdots \\0&0&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} The Carleman matrix of a function f ( x ) = ∑ k = 0 ∞ f k x k {\displaystyle f(x)=\sum _{k=0}^{\infty }f_{k}x^{k}} is: M [ f ] = ( 1 0 0 ⋯ f 0 f 1 f 2 ⋯ f 0 2 2 f 0 f 1 f 1 2 + 2 f 0 f 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\f_{0}&f_{1}&f_{2}&\cdots \\f_{0}^{2}&2f_{0}f_{1}&f_{1}^{2}+2f_{0}f_{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} == Related matrices == The Bell matrix or the Jabotinsky matrix of a function f ( x ) {\displaystyle f(x)} is defined as B [ f ] j k = 1 j ! [ d j d x j ( f ( x ) ) k ] x = 0 , {\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}~,} so as to satisfy the equation ( f ( x ) ) k = ∑ j = 0 ∞ B [ f ] j k x j , {\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}~,} These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials. It is the transpose of the Carleman matrix and satisfy B [ f ∘ g ] = B [ g ] B [ f ] , {\displaystyle B[f\circ g]=B[g]B[f]~,} which makes the Bell matrix B an anti-representation of f ( x ) {\displaystyle f(x)} . == See also == Carleman linearization Composition operator Function composition Schröder's equation Bell polynomials == Notes == == References == R Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices, World Scientific, 2001. (preview) R. Aldrovandi, L. P. Freitas, Continuous Iteration of Dynamical Maps, online preprint, 1997. P. Gralewicz, K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, online preprint, 2000. K Kowalski and W-H Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientific, 1991. (preview)
Wikipedia:Carleman's condition#0	In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure μ {\displaystyle \mu } satisfies Carleman's condition, there is no other measure ν {\displaystyle \nu } having the same moments as μ . {\displaystyle \mu .} The condition was discovered by Torsten Carleman in 1922. == Hamburger moment problem == For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following: Let μ {\displaystyle \mu } be a measure on R {\displaystyle \mathbb {R} } such that all the moments m n = ∫ − ∞ + ∞ x n d μ ( x ) , n = 0 , 1 , 2 , ⋯ {\displaystyle m_{n}=\int _{-\infty }^{+\infty }x^{n}\,d\mu (x)~,\quad n=0,1,2,\cdots } are finite. If ∑ n = 1 ∞ m 2 n − 1 2 n = + ∞ , {\displaystyle \sum _{n=1}^{\infty }m_{2n}^{-{\frac {1}{2n}}}=+\infty ,} then the moment problem for ( m n ) {\displaystyle (m_{n})} is determinate; that is, μ {\displaystyle \mu } is the only measure on R {\displaystyle \mathbb {R} } with ( m n ) {\displaystyle (m_{n})} as its sequence of moments. == Stieltjes moment problem == For the Stieltjes moment problem, the sufficient condition for determinacy is ∑ n = 1 ∞ m n − 1 2 n = + ∞ . {\displaystyle \sum _{n=1}^{\infty }m_{n}^{-{\frac {1}{2n}}}=+\infty .} == Generalized Carleman's condition == In, Nasiraee et al. showed that, despite previous assumptions, when the integrand is an arbitrary function, Carleman's condition is not sufficient, as demonstrated by a counter-example. In fact, the example violates the bijection, i.e. determinacy, property in the probability sum theorem. When the integrand is an arbitrary function, they further establish a sufficient condition for the determinacy of the moment problem, referred to as the generalized Carleman's condition. == Notes == == References == Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd. Chapter 3.3, Durrett, Richard. Probability: Theory and Examples. 5th ed. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge ; New York, NY: Cambridge University Press, 2019.
Wikipedia:Carlo Ignazio Giulio#0	Carlo Ignazio Giulio (11 August 1803 – 29 June 1859) was an Italian mathematician, mechanical engineer and politician. == Bibliography == Giulio, Carlo Ignazio (1846). Quattro lezioni sul sistema metrico decimale dette da C.I. Giulio nella scuola di meccanica applicata alle arti le sere delli 20, 25, 27 e 30 giugno 1846 (in Italian). Presso G. Pomba E C. Editori. Sunti delle lezioni di meccanica applicata alle arti (in Italian). Torino: Pomba. 1846. Giulio, Carlo Ignazio (1854). Elementi di cinematica applicata alle arti esposti da C.I. Giulio: ad uso delle scuole universitarie, speciali e tecniche, e degli ingegneri, capi di officine e macchinisti (in Italian). Stamperia reale. == References ==
Wikipedia:Carlo Rosati#0	Carlo Rosati (Livorno, 24 April 1876 – Pisa, 19 August 1929) was an Italian mathematician working on algebraic geometry who introduced the Rosati involution. == Notes == == References == Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290 Rosati, Carlo (1918), "Sulle corrispondenze algebriche fra i punti di due curve algebriche", Annali di Matematica Pura ed Applicata (in Italian), 3 (28): 35–60, doi:10.1007/BF02419717, S2CID 121620469 == External links == Carlo Rosati in Mathematica Italiana Carlo Rosati Archived 2011-09-29 at the Wayback Machine
Wikipedia:Carlos Benjamin de Lyra#0	Carlos Benjamin de Lyra (Pernambuco, 23 November 1927 – São Paulo, 21 July 1974) was a prominent Brazilian mathematician, a pioneer in algebraic topology in Brazil and professor at the University of São Paulo. Born in Recife, Pernambuco, he came from a family of sugarcane plantation owners and his dad was the owner of the Diário de Pernambuco, a newspaper that was known nationwide. Lyra was an important mathematician in his area, his course Introdução à Topologia Algébrica was taught in the first Colóquio Brasileiro de Matemática and would become the first text in this field written in Brazilian Portuguese. After the death of his father, his mother married a Wall Street stockbroker and, together, the couple moved to New York City with Lyra and his younger brother. When he was 15, in the suburbs of the city where he lived, he met Richard Courant. The founder of the presently named Courant Institute of Mathematical Sciences was responsible for inspiring de Lyra to study mathematics. Lyra made a substantial career for himself throughout his life. Beginning as associate professor at the University of São Paulo alongside Elza Gomide, he helped to organize and administrate a course in the 1° Colóquio Brasileiro de Matemática, he became a doctor in Mathematics with his thesis Sobre os espaços de mesmo tipo de homotopia que o dos poliedros, he was one of the founders of the Sociedade Brasileira de Matemática, he was involved in the creation of the Instituto de Matemática e Estatística at the University of São Paulo (IME-USP), taught as a professor in a variety of courses, and participated in the restructuring of the undergraduate and postgraduate courses in Mathematics at the University of São Paulo. On the 21st of July 1974, Carlos Benjamin de Lyra died due to a brain tumour. His thesis H-equivalencia de grupos topológicos, was revised and published by his friend Peter Hilton. In his honor, the library at the IME-USP bears his name, along with a road in the Chácara São João neighbourhood, in the capital of São Paulo. == Early life == === Early years and education === Carlos Benjamin de Lyra was born in the city of Recife in Pernambuco on the 23rd of November 1927. His parents were Carlos de Lyra Filho, a sugarcane plantation owner and owner of the Pernambuco Daily newspaper which received a lot of national attention, and Elizabeth Lau de Lyra, a German woman that came to Brazil with her family. The marriage between them was arranged due to how close the two families were in business and faith. Carlos de Lyra Filho came from an earlier marriage, where he had five sons, where after the death of his first wife he remarried with Elizabeth, where they had two children: Carlos and George. Carlos Benjamin de Lyra's father died when he was 9 years old. Widowed, his mother married Paul Nortz, a stockbroker at Wall Street. Together, the family, including the children, moved to New York City. Lyra did not go to school during his early years, he was instead home schooled by a tutor who taught German in this time. In the United States, he studied in Catholic schools, and finished his primary education and middle school at Iona High School, New Rochelle. Around the age of 15, Lyra met the researcher and founder of the Institute of New York University, Richard Courant (1888–1972), during one of his daily train journeys. It was Courant that provided Lyra his interest in Mathematics, whereas his earlier stated interest was in astronomy. === Return to Brazil === By the end of middle school, Lyra had the opportunity to obtain US citizenship and follow the wishes of his stepfather who wanted to see him work in Wall Street and study in a university which would accept him, including Yale. But he refused this, as he had an attachment to Brazil and was critical of US society and culture. In 1945, returning to Brazil after the end of the Second World War, Lyra enlisted himself in the Brazilian Army to continue his path to citizenship and spend some time living with his brothers from his father's earlier marriage as guardians until he turned 18. The following year, he would move to São Paulo, living in the home of Manuel Tavares, a lawyer and friend of the family, and soon begin his study of Mathematics. === Family and early career === In São Paulo, 1947, Lyra began studying Mathematics under the former Faculdade de Filosofia, Ciências e Letras of the University of São Paulo (FFCL). During this time, he would meet French mathematicians André Weil (1906–1998) and Jean Dieudonné (1906–1992), and have lectures with Professor Cândido Lima da Silva Dias (1913–1998) on the topic of algebraic topology, the area of study he would pursue during the rest of his career. After graduating in 1950, Lyra developed an interest in algebraic topology when he attended a course on the Theory of Simplicial Homology, which was administrated by Professor Cândido Lima, on the year of his graduation. With this, he travelled to France in 1952 to take part in a postgraduate program with funding from the Conselho Nacional de Pesquisas (CNPq). He met and studied with Henri Cartan and participated in seminars and watched Hurewicz's lectures on homotopy in the Collège de France in Paris. During his time in France, Lyra met Leda Lacerda, a woman from Rio de Janeiro and graduated in physics from the Faculdade Nacional de Filosofia. Lacerda was in Europe at that time to study in the European Institute of Physics. After meeting Lyra, they began their relationship and they moved to the Hotel des Grands Hommes, in Paris, where they would remain until the end of their stay in France. A short time later. Leda became pregnant with their first son and returned to Brazil, with Lyra following soon after completing his postgraduation. Together in Brazil, they were married in the home of Lasar Segall, father of the artist Maurício Segall, a friend of the couple. In 1954, Jorge Lacerda de Lyra was born. In all, Carlos and Leda had 3 children: Jorge, Sylvia (1956) and Eduardo (1958). In that same year, Carlos was hired by the FFCL, together with Elza Gomide, and became associate professor in the Cátedra de Análise Matemática which was headed by Professor Omar Catunda. This began his academic career in the University of São Paulo. == Academic career == === University of São Paulo === Lyra continued as associate professor in the Cátedra de Análise Matemática until 1958. Influenced by a visiting professor at the USP, Alexander Grothendieck, Carlos learnt about cohomology, furthering his interest of topology. In 1956, Carlos organized a weekly seminar on algebraic topology, where they would discuss important topics from this field. He defended his doctorate thesis in 1958, titled Sobre os espaços de mesmo tipo de homotopia que o dos poliedros, with Cândido Lima da Silva Dias advising him. He received funding from the Rockefeller Foundation in 1961 and moved his family to New Jersey, where he visited the Institute for Advanced Study at Princeton. This advanced his understanding of cohomological operations and, through the advising of John Milnor (1931 -), he developed his study in differential topology. Concerned that he would not be allowed to return to Brazil due to his socialist affiliation, Lyra returned after one year in New Jersey out of the two years the original funding provided for. After receiving his familial inheritance, he bought a house in Vila Mariana. In 1963, he became the coordinator of the night course of Mathematical analysis in the Departamento de Matemática of the USP, until 1970. Carlos was a fundamental figure in the creation of the Instituto de Matemática e Estatística of USP (IME-USP) in January 1970. In 1972, he was hired as a Senior Professor. He was recognised as a professor and researcher who was dedicated and inspired students and colleagues. In 1968 he defended his postgraduate thesis, titled H-Equivalência de grupos topológicos, leading to him obtaining the title of Professor in Complements in Geometry and Superior Geometry at the FFCL. === Colóquios de Matemática === Lyra as a professor frequently participated in events such as the Colóquios de Matemática. It was in the first Colóquio Brasileiro de Matemática in which he led a course based on his doctorate (Introdução à Topologia Algébrica), and on the fifth and seventh he presented once again. He also participated on the Comissão Organizadora of the reunion in his first year, coordinated a commission in the Colóquio, and participated once again in the sixth. In 1967, during the sixth Colóquio Brasileiro de Matemática, Lyra integrated a commission whose aim was to define the scientific character and the organization of the Escola Latino-Americana de Matemática. This commission was composed of the professors: Emilio Lluis (Mexico), Orlando Villamayor (Argentina) and Carlos Benjamin de Lyra (Brazil). Working together, they produced a document which was then approved by the plenary. === Mathematical Societies === Between 1966 and 1967, Lyra was the elected president of the Sociedade de Matemática de São Paulo, however it did not last for very long and, before its end, he had one published work in the Diário Oficial do Estado de São Paulo. Years later, in 1969, he was one of the founders of the Sociedade Brasileira de Matemática, and was elected as one of four advisors. His mandate lasted until 1972, and later, he became the Secretary General. == Death == In his final years, Lyra continued to be active in his work. After the creation of the IME-USP in 1970, he was chosen by the CNPq to be a conference professor and, two years later, nominated, by the president of the CNPq, to be the Brazilian representative in the Permanent Commission of the Escola Latino-Americana de Matemática (ELAM). However, in the period, Lyra lost one of his sons. Eduardo Lacerda de Lyra died in 1971, 12 years-old, in an accident at the beach. In his final year, Lyra passed the exam for becoming a professor, wrote and turned in his Memorial to become an adjunct professor of the IME. Curiously, Carlos had the habit of telling his friends that he would die early and, according to Jorge and Sylvia, the men in their family lived relatively short lives, this be confirmed with his own early death. On the 21st of July 1974, 46 years-old, Carlos Benjamin de Lyra died. After feeling ill, he was hospitalized for four days and diagnosed with having a brain tumour. According to Leda, Carlos returned home on the condition that he would return to the hospital in three days. This period at home gave him sufficient time to finish an article he was in the process of writing and nothing more. His thesis for becoming a professor was revised and submitted postmortem for publishing by his friend Peter Hilton who also brought with him the article that Lyra left in his office to England to be published under the title "SHM-maps of CW-groups". == Political views == Lyra was described as a socialist intellectual and, between the years of 1950 and 1955, was a member of the Partido Socialista Brasileiro, having such colleagues as Paul Singer, Febus Gikovate, among others. One of the reasons for not pursuing US citizenship was because of his critical views of the country. He was also an acting member of the Commissão de Elaboração do Livro Branco, on the outcomes of the event that occurred in the Maria Antônia street of São Paulo, acting as a rapporteur. The events of that day, caused by conflicts between students of the University of São Paulo and students of the Instituto Presbiteriyear Mackenzie during Brazil's military dictatorship, came to be known as the Battle of Maria Antônia. == Major works == == Scientific societies == Academia Brasileira de Ciências American Mathematical Society Sociedade Brasileira de Matemática Sociedade Brasileira para o Progresso da Ciência Sociedade de Matemática de São Paulo == See also == Institute of Mathematics and Statistics, University of São Paulo Partido Socialista Brasileiro Anais da Academia Brasileira de Ciências == References ==
Wikipedia:Carlos Castillo-Chavez#0	Carlos Castillo-Chavez (born March 29, 1952) is a Mexican-American mathematician. He held positions as a Regents Professor and the Joaquín Bustoz Jr. Professor of Mathematical Biology at Arizona State University. Castillo-Chavez founded the Mathematical and Theoretical Biology Institute (MTBI) at Cornell University in 1996. His research and publications focus on mathematics, social structures, and epidemiology. == Biography == Carlos Castillo-Chavez was born on March 29, 1952, in Mexico. He immigrated to the United States, from Mexico, in 1974, at age 22. He worked in a cheese factory in Wisconsin before continuing his studies. He attended the University of Wisconsin–Stevens Point, graduating in 1976 with degrees in mathematics and Spanish literature. He earned a MS in Mathematics from the University of Wisconsin–Milwaukee. He earned a Ph.D. in mathematics from the University of Wisconsin–Madison in 1984. Before joining Arizona State University in 2004, he was a professor at Cornell University for 18 years. He has published scientific articles and books, and served on committees for organizations such as the National Science Foundation, the Alfred P. Sloan Foundation, the National Institutes of Health, the Society for Industrial and Applied Mathematics, and the American Mathematical Society. From 2016 to 2018, he served as rector of Yachay Tech University in Ecuador. As a mathematical epidemiologist researcher, his interests include the mechanisms underlying the spread of diseases and their containment, prevention, and elimination. In 2006, Arizona State University described him as an expert in epidemiological modeling, and a contributor to the literature on the progression of diseases. According to a September 2020 update, his 52 PhD students included 21 women, 29 from U.S. underrepresented groups, and 7 from Latin America. He also mentored over 500 undergraduates, primarily through the Mathematical and Theoretical Biology Institute. He has been recognized for work aimed at enhancing academic success, and for providing research opportunities for underrepresented groups in mathematics and biology. According to the Mathematics Genealogy Project, Castillo-Chavez is listed as one of the top doctoral advisors in mathematics, and is noted as the only Latino mathematician in their top 250 list. In 2020, he retired from Arizona State University, after resigning from his posts the previous year. An ASU investigation substantiated a graduate student report that he created a hostile environment and engaged in harassment, but the matter was closed without further action upon his retirement. According to reporting on the investigation, his "tough love" approach was cited by both supporters and detractors as a source of support and conflict, respectively. Castillo-Chavez founded the Applied Mathematics in the Life and Social Sciences BS and PhD programs (2008) at the Simon A. Levin Mathematical, Computational, and Modeling Sciences Center at Arizona State University. Castillo-Chavez established the Mathematical and Theoretical Biology Institute (MTBI) at Cornell University in 1996. It moved to Arizona State University in the spring of 2004. From 1996 to 2004, MTBI received funding from Cornell University and Los Alamos National Laboratory (T-Division). The National Science Foundation, the National Security Agency, and Arizona State University have also provided partial funding. As of 2021-2022, it was renamed the Quantitative Research in the Life and Social Sciences Program (QRLSSP). Castillo-Chavez was also the director of the Institute for Strengthening and the Joaquin Bustoz Math-Science Honors Program (JBMSHP), a summer residential mathematics program for students interested in academic careers requiring mathematics, science, or engineering-based coursework, particularly those from underrepresented groups. == Research == Castillo-Chavez has co-authored over 560 publications and a dozen books, including textbooks, research monographs, and edited volumes. His research explores the intersection of the mathematical, natural, and social sciences, focusing on how dynamic social landscapes affect disease dispersal, evolution, and control, as well as the impact of environmental risk, social structures, and human behavior on disease dynamics, including addiction. He and his collaborators have introduced mathematical models for the spread of scientific concepts, ideas, or media-driven information, such as the social contagion effect in recurrent mass killings and school shootings. They have also studied the role of behavior and mobility in the dynamics of emergent and re-emergent diseases, including Ebola, influenza, tuberculosis, and Zika virus. His publications have also included models and frameworks for collaborative learning based on the activities of the ASU Mathematical and Theoretical Biology Institute. == Awards and recognition == His awards and recognition include: Three White House Awards (1992, 1997, and 2011). His MTBI program received the Presidential Awards for Excellence in Science, Mathematics and Engineering Mentoring (PAESMEM). The 12th American Mathematical Society Distinguished Public Service Award in 2010. The 2007 Mentor Award from the American Association for the Advancement of Science (AAAS). The 17th recipient of the SIAM Prize for Distinguished Service to the Profession. Member of the Board of Higher Education at the National Academy of Sciences (2009-2015) and served on President Barack Obama's Committee on the National Medal of Science (2010-2015). Fellow of the American Association for the Advancement of Science; Society for Industrial and Applied Mathematics; Founding Fellow of the American Mathematical Society; and American College of Epidemiology. Held honorary Professorships at Xi’an Jiatong University in China, the Universidad de Belgrano in Argentina, and East Tennessee State University. Past appointments include a Stanislaw M. Ulam Distinguished Scholar at Los Alamos National Laboratory, a Cátedra Patrimonial at UNAM in México, and a Martin Luther King Jr. Professorship at MIT. On February 24, 2016, the University Francisco Gavidia inaugurated the Centro de Modelaje Matemático Carlos Castillo-Chavez in the City of San Salvador, El Salvador. Served on NSF’s Advisory Committee for Education and Human Resources (2016-2019) and on NSF’s Cyber Infrastructure Advisory Boards (2016-2019). The inaugural recipient of the William Yslas Velez Outstanding STEM Award, co-sponsored by the Victoria Foundation and the Pasqua Yaqui Tribe of Arizona (2015). Elected as a Member-at-Large of the Section on Mathematics of the AAAS (2016–2020). In April 2017, Brown University invited Castillo-Chavez to present a lecture in the Series "Thinking Out Loud," titled "The Role of Contagion in the Building and Sustainability of Communities." Pete C. Garcia, Victoria Foundation - Higher Education Award. Outstanding Latina/o Faculty: Research in Higher Education Award. September 4, 2019. == Appointments == Primary Affiliations have included: School of Human Evolution and Social Change, Arizona State University Global Institute of Sustainability, Distinguished Sustainability Scientist, Arizona State University Founding Director Simon A. Levin Mathematical, Computational & Modeling Sciences Center, Arizona State University ASU-SFI Center for Biosocial Complex Systems Center for Gender Equity in Science and Technology, Arizona State University External Santa Fe Institute, External Faculty Member Biological Statistics and Computational Biology, Cornell University - Adjunct Faculty == Selected publications == Books (selected) Carlos Castillo-Chavez, Fred Brauer, Zhilan Feng (2019). Mathematical Models in Epidemiology. New York: Springer. ISBN 9781493998265 Carlos Castillo-Chavez, Fred Brauer (2013). Mathematical Models for Communicable Diseases. SIAM. ISBN 9781611972412 Clemence, Dominic; Gumel, Abba; Castillo-Chávez, Carlos; Mickens, Ronald E. (2006). Mathematical studies on human disease dynamics: emerging paradigms and challenges: AMS-IMS-SIAM Joint Summer Research Conference, competitive mathematical models of disease dynamics: emerging paradigms and challenges, July 17–21, 2005, Snowbird, Utah. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-3775-3. Castillo-Chávez, Carlos (2003). Bioterrorism: mathematical modeling applications in homeland security. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-549-0. Blower, Sally; Castillo-Chávez, Carlos (Ed) (2002). Mathematical approaches for emerging and reemerging infectious diseases: an introduction. Berlin: Springer. ISBN 0-387-95354-X. Castillo-Chávez, Carlos; Brauer, Fred (2001). Mathematical models in population biology and epidemiology. Berlin: Springer. ISBN 0-387-98902-1. Carlos Castillo-Chavez (editor) (1989). Mathematical and Statistical Approaches to AIDS Epidemiology. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-52174-7 Scientific articles (selected/most cited out of more than 500 publications) Castillo-Chavez Carlos, Derdei Bichara, and Benjamin R Morin. Perspectives on the role of mobility, behavior, and time scales in the spread of diseases. Proceedings of the National Academy of Sciences, 113(51):14582–14588, 2016. Chowell, D., C. Castillo-Chavez, S Krishna, X Qiu, Modelling the effect of early detection of Ebola- The Lancet Infectious Diseases, 15(2): 148--149, 2015 Carlos Castillo-Chavez, Roy Curtiss, Peter Daszak, Simon A. Levin, Oscar Patterson-Lomba, Charles Perrings, George Poste, and Sherry Towers. Beyond Ebola: lessons to mitigate future pandemics. The Lancet Global Health 3 (7), e354-e355. 2015 Eli P. Fenichel, Carlos Castillo-Chavez, M. G. Ceddia, Gerardo Chowell, Paula A. Gonzalez Parra, Graham J. Hickling, Garth Holloway, Richard Horan, Benjamin Morin, Charles Perrings, Michael Springborn, Leticia Velazquez, and Cristina Villalobos, "Adaptive human behavior in epidemiological models", Proceedings of the National Academy of Sciences PNAS, USA 2011; 108:6306-11 Castillo-Chavez, C. and B. Song: "Dynamical Models of Tuberculosis and applications", Journal of Mathematical Biosciences and Engineering, 1(2): 361-404, 2004. Castillo-Chavez C., Z. Feng and W. Huang. "On the computation Ro and its role on global stability", Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125, 229-250, Springer-Verlag, Berlin-Heidelberg-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002. Chowell, G., Hengartner, N.W., Castillo-Chavez, C., Fenimore, P.W., Hyman, J.M. "The Basic Reproductive Number of Ebola and the Effects of Public Health Measures: The Cases of Congo and Uganda". Journal of Theoretical Biology, 229(1): 119-126 (July 2004) == References == == External links == Mathematical and Theoretical Biology Institute and Institute for Strengthening the Understanding of Mathematics and Science Archived 2019-10-22 at the Wayback Machine Carlos Castillo-Chavez at the SACNAS Biography Project Interview with Carlos Castillo-Chavez at AAAS Simon A. Levin Mathematical, Computational and Modeling Sciences Center
Wikipedia:Carlos Lousto#0	Carlos O. Lousto is a Distinguished Professor in the School of Mathematical Sciences in Rochester Institute of Technology, known for his work on black hole collisions. == Professional career == Lousto is a Distinguished Professor in the RIT's School of Mathematical Sciences and co-director of the Center for Computational Relativity and Gravitation. He holds two PhDs, one in Astronomy (studying accretion disks around black holes and the structure of neutron stars) from the National University of La Plata, and one in Physics from the University of Buenos Aires (on Quantum Field Theory in curved spacetimes), received in 1987 and 1992. Carlos Lousto has an extensive research experience which ranges from observational astronomy to black hole perturbation theory and numerical relativity to string theory and quantum gravity. He has authored and co-authored over 250 papers , including several reviews and book chapters. His research is funded by NSF and NASA grants and supercomputing allocations in national labs. Lousto is a key author of the breakthrough on binary black hole simulations and his research discovered that supermassive black holes can be ejected from most galaxies at speeds of up to 5000 km/s. He recently performed challenging simulations of small mass ratio black hole binaries up to 100:1 and at separations up to 100M and for flip-flopping black holes. Lousto has designed the Funes (UTB), NewHorizon, BlueSky, and GreenPrairies (RIT) supercomputer clusters to perform binary black hole simulations and used them to support the first detection of gravitational waves from the merger of two black holes. === Distinctions === In 1991, Carlos Lousto was honored with an Alexander von Humboldt Foundation fellowship. In 2006 and in 2016 his research was acknowledged in the US congressional records. In 2012, Carlos Lousto was elected a Fellow of the American Physical Society "For his important contributions at the interface between perturbation theory and numerical relativity and in understanding how to simulate binary black holes". 2016 Special Breakthrough Prize in Fundamental Physics "For the observation of gravitational waves, opening new horizons in astronomy and physics". 2019 Edward A. Bouchet Award Recipient "For contributions to both numerical relativity, conducive to the solution of the binary black hole problem, and the understanding of the first detection of gravitational waves and service to the Hispanic scientific community, including the establishment of the Center for Gravitational Wave Astronomy, the University of Texas at Brownsville in 2003". == Selected bibliography == Highlights per year: Lousto, Carlos O.; Mazzitelli, Francisco D. (1997-09-15). "Exact self-consistent gravitational shock wave in semiclassical gravity". Physical Review D. 56 (6): 3471–3477. arXiv:gr-qc/9611009. Bibcode:1997PhRvD..56.3471L. doi:10.1103/physrevd.56.3471. ISSN 0556-2821. S2CID 5075915. Lousto, Carlos O. (2000-06-05). "Pragmatic Approach to Gravitational Radiation Reaction in Binary Black Holes". Physical Review Letters. 84 (23): 5251–5254. arXiv:gr-qc/9912017. Bibcode:2000PhRvL..84.5251L. doi:10.1103/physrevlett.84.5251. ISSN 0031-9007. PMID 10990916. S2CID 1788277. Baker, J.; Brügmann, B.; Campanelli, M.; Lousto, C. O.; Takahashi, R. (2001-08-31). "Plunge Waveforms from Inspiralling Binary Black Holes". Physical Review Letters. 87 (12): 121103. arXiv:gr-qc/0102037. Bibcode:2001PhRvL..87l1103B. doi:10.1103/physrevlett.87.121103. ISSN 0031-9007. PMID 11580497. S2CID 39434471. Campanelli, M.; Lousto, C. O.; Marronetti, P.; Zlochower, Y. (2006-03-22). "Accurate Evolutions of Orbiting Black-Hole Binaries without Excision". Physical Review Letters. 96 (11): 11101. arXiv:gr-qc/0511048. Bibcode:2006PhRvL..96k1101C. doi:10.1103/physrevlett.96.111101. ISSN 0031-9007. PMID 16605808. S2CID 5954627. Campanelli, M.; Lousto, C. O.; Zlochower, Y. (2006-08-16). "Spinning-black-hole binaries: The orbital hang-up". Physical Review D. 74 (4): 041501. arXiv:gr-qc/0604012. Bibcode:2006PhRvD..74d1501C. doi:10.1103/physrevd.74.041501. ISSN 1550-7998. S2CID 119376905. Campanelli, Manuela; Lousto, Carlos; Zlochower, Yosef; Merritt, David (2007-03-05). "Large Merger Recoils and Spin Flips from Generic Black Hole Binaries". The Astrophysical Journal. 659 (1). American Astronomical Society: L5 – L8. arXiv:gr-qc/0701164. Bibcode:2007ApJ...659L...5C. doi:10.1086/516712. ISSN 0004-637X. S2CID 9761881. Campanelli, Manuela; Lousto, Carlos O.; Zlochower, Yosef; Merritt, David (2007-06-07). "Maximum Gravitational Recoil". Physical Review Letters. 98 (23): 231102. arXiv:gr-qc/0702133. Bibcode:2007PhRvL..98w1102C. doi:10.1103/physrevlett.98.231102. ISSN 0031-9007. PMID 17677894. S2CID 29246347. Lousto, Carlos O.; Zlochower, Yosef (2011-01-24). "Orbital Evolution of Extreme-Mass-Ratio Black-Hole Binaries with Numerical Relativity". Physical Review Letters. 106 (4): 041101. arXiv:1009.0292. Bibcode:2011PhRvL.106d1101L. doi:10.1103/physrevlett.106.041101. ISSN 0031-9007. PMID 21405317. S2CID 46427657. Lousto, Carlos O.; Zlochower, Yosef (2011-12-02). "Hangup Kicks: Still Larger Recoils by Partial Spin-Orbit Alignment of Black-Hole Binaries". Physical Review Letters. 107 (23): 231102. arXiv:1108.2009. Bibcode:2011PhRvL.107w1102L. doi:10.1103/physrevlett.107.231102. ISSN 0031-9007. PMID 22182078. S2CID 15546595. Lousto, Carlos O.; Healy, James (2015-04-06). "Flip-Flopping Binary Black Holes". Physical Review Letters. 114 (14): 141101. arXiv:1410.3830. Bibcode:2015PhRvL.114n1101L. doi:10.1103/physrevlett.114.141101. ISSN 0031-9007. PMID 25910104. S2CID 36664234. Abbott, B. P.; Abbott, R.; Abbott, T. D.; Abernathy, M. R.; Acernese, F.; et al. (LIGO collaboration and Virgo collaboration) (2016-02-11). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6): 061102. arXiv:1602.03837. Bibcode:2016PhRvL.116f1102A. doi:10.1103/physrevlett.116.061102. ISSN 0031-9007. PMID 26918975. == References == == External links == (in Spanish) Radio Interview (in Spanish) TV interview, P1, TV interview, P2
Wikipedia:Carlos Matheus#0	Carlos Matheus Silva Santos (born May 1, 1984 in Aracaju) is a Brazilian mathematician working in dynamical systems, analysis and geometry. He is research director at the CNRS, in Paris. He earned his Ph.D. from the Instituto de Matemática Pura e Aplicada (IMPA) in 2004 under the supervision of Marcelo Viana, at the age of 19. == Selected publications == with G. Forni, and A. Zorich: "Square-tiled cyclic covers", Journal of Modern Dynamics, vol. 5, no. 2, pp. 285–318 (2011). with A. Avila, and J.-C. Yoccoz: "SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are regular", Geometric and Functional Analysis, vol. 23, no. 6, pp. 1705–1729 (2013). with M. Möller, and J.-C. Yoccoz: "A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces", Inventiones mathematicae (2014). == Further reading == Época – Os segredos das ilhas de excelência (by Carlos Rydlewski, in Portuguese) == References == == External links == Matheus' home-page at the IMPA Matheus' personal blog "Disquisitiones Mathematicae"
Wikipedia:Carlotta Longo#0	Carlotta Longo (27 June 1895 – after 1959) born Carlotta Bresolin, was an Italian mathematical physicist who wrote a doctoral dissertation in 1918 related to general relativity, and then became a high school teacher in Rome. Longo's thesis, advised by Tullio Levi-Civita, presented what Ludwik Silberstein called a "geometrically elegant investigation" of electrostatics in general relativity. Her second marriage was to the Afro-Italian actor Lodovico Longo, who played a minor character in the film Harlem. == References ==
Wikipedia:Carlson's theorem#0	In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem. Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions. == Statement == Assume that f satisfies the following three conditions. The first two conditions bound the growth of f at infinity, whereas the third one states that f vanishes on the non-negative integers. f(z) is an entire function of exponential type, meaning that \| f ( z ) \| ≤ C e τ \| z \| , z ∈ C {\displaystyle \|f(z)\|\leq Ce^{\tau \|z\|},\quad z\in \mathbb {C} } for some real values C, τ. There exists c < π such that \| f ( i y ) \| ≤ C e c \| y \| , y ∈ R {\displaystyle \|f(iy)\|\leq Ce^{c\|y\|},\quad y\in \mathbb {R} } f(n) = 0 for every non-negative integer n. Then f is identically zero. == Sharpness == === First condition === The first condition may be relaxed: it is enough to assume that f is analytic in Re z > 0, continuous in Re z ≥ 0, and satisfies \| f ( z ) \| ≤ C e τ \| z \| , Re ⁡ z > 0 {\displaystyle \|f(z)\|\leq Ce^{\tau \|z\|},\quad \operatorname {Re} z>0} for some real values C, τ. === Second condition === To see that the second condition is sharp, consider the function f(z) = sin(πz). It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of c = π, and indeed it is not identically zero. === Third condition === A result, due to Rubel (1956), relaxes the condition that f vanish on the integers. Namely, Rubel showed that the conclusion of the theorem remains valid if f vanishes on a subset A ⊂ {0, 1, 2, ...} of upper density 1, meaning that lim sup n → ∞ \| A ∩ { 0 , 1 , … , n − 1 } \| n = 1. {\displaystyle \limsup _{n\to \infty }{\frac {\left\|A\cap \{0,1,\ldots ,n-1\}\right\|}{n}}=1.} This condition is sharp, meaning that the theorem fails for sets A of upper density smaller than 1. == Applications == Suppose f(z) is a function that possesses all finite forward differences Δ n f ( 0 ) {\displaystyle \Delta ^{n}f(0)} . Consider then the Newton series g ( z ) = ∑ n = 0 ∞ ( z n ) Δ n f ( 0 ) {\displaystyle g(z)=\sum _{n=0}^{\infty }{z \choose n}\,\Delta ^{n}f(0)} with ( z n ) {\textstyle {z \choose n}} is the binomial coefficient and Δ n f ( 0 ) {\displaystyle \Delta ^{n}f(0)} is the n-th forward difference. By construction, one then has that f(k) = g(k) for all non-negative integers k, so that the difference h(k) = f(k) − g(k) = 0. This is one of the conditions of Carlson's theorem; if h obeys the others, then h is identically zero, and the finite differences for f uniquely determine its Newton series. That is, if a Newton series for f exists, and the difference satisfies the Carlson conditions, then f is unique. == See also == Newton series Mahler's theorem Table of Newtonian series == References == F. Carlson, Sur une classe de séries de Taylor, (1914) Dissertation, Uppsala, Sweden, 1914. Riesz, M. (1920). "Sur le principe de Phragmén–Lindelöf". Proceedings of the Cambridge Philosophical Society. 20: 205–107., cor 21(1921) p. 6. Hardy, G.H. (1920). "On two theorems of F. Carlson and S. Wigert". Acta Mathematica. 42: 327–339. doi:10.1007/bf02404414. E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press (See section 5.81) R. P. Boas, Jr., Entire functions, (1954) Academic Press, New York. DeMar, R. (1962). "Existence of interpolating functions of exponential type". Trans. Amer. Math. Soc. 105 (3): 359–371. doi:10.1090/s0002-9947-1962-0141920-6. DeMar, R. (1963). "Vanishing Central Differences". Proc. Amer. Math. Soc. 14: 64–67. doi:10.1090/s0002-9939-1963-0143907-2. Rubel, L. A. (1956), "Necessary and sufficient conditions for Carlson's theorem on entire functions", Trans. Amer. Math. Soc., 83 (2): 417–429, doi:10.1090/s0002-9947-1956-0081944-8, JSTOR 1992882, MR 0081944, PMC 528143, PMID 16578453
Wikipedia:Carlyle circle#0	In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. == Definition == Given the quadratic equation x2 − sx + p = 0 the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation. == Defining property == The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is x(x − s) + (y − 1)(y − p) = 0. The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle) x2 − sx + p = 0. == Construction of regular polygons == === Regular pentagon === The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation z5 − 1 = 0. One root of this equation is z0 = 1 which corresponds to the point P0(1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation z4 + z3 + z2 + z + 1 = 0. These roots can be represented in the form ω, ω2, ω3, ω4 where ω = exp (2iπ/5). Let these correspond to the points P1, P2, P3, P4. Letting p1 = ω + ω4, p2 = ω2 + ω3 we have p1 + p2 = −1, p1p2 = −1. (These can be quickly shown to be true by direct substitution into the quartic above and noting that ω6 = ω, and ω7 = ω2.) So p1 and p2 are the roots of the quadratic equation x2 + x − 1 = 0. The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (−1, −1) and center at (−1/2, 0). Carlyle circles are used to construct p1 and p2. From the definitions of p1 and p2 it also follows that p1 = 2 cos(2π/5), p2 = 2 cos(4π/5). These are then used to construct the points P1, P2, P3, P4. This detailed procedure involving Carlyle circles for the construction of regular pentagons is given below. Draw a circle in which to inscribe the pentagon and mark the center point O. Draw a horizontal line through the center of the circle. Mark one intersection with the circle as point B. Construct a vertical line through the center. Mark one intersection with the circle as point A. Construct the point M as the midpoint of O and B. Draw a circle centered at M through the point A. This is the Carlyle circle for x2 + x − 1 = 0. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V. These are the points p1 and p2 mentioned above. Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon. The fifth vertex is the intersection of the horizontal axis with the original circle. === Regular heptadecagon === There is a similar method involving Carlyle circles to construct regular heptadecagons. The figure to the right illustrates the procedure. === Regular 257-gon === To construct a regular 257-gon using Carlyle circles, as many as 24 Carlyle circles are to be constructed. One of these is the circle to solve the quadratic equation x2 + x − 64 = 0. === Regular 65537-gon === There is a procedure involving Carlyle circles for the construction of a regular 65537-gon. However there are practical problems for the implementation of the procedure; for example, it requires the construction of the Carlyle circle for the solution of the quadratic equation x2 + x − 214 = 0. == History == According to Howard Eves (1911–2004), the mathematician John Leslie (1766–1832) described the geometric construction of roots of a quadratic equation with a circle in his book Elements of Geometry and noted that this idea was provided by his former student Thomas Carlyle (1795–1881). However while the description in Leslie's book contains an analogous circle construction, it was presented solely in elementary geometric terms without the notion of a Cartesian coordinate system or a quadratic function and its roots: To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle. In 1867 the Austrian engineer Eduard Lill published a graphical method to determine the roots of a polynomial (Lill's method). If it is applied on a quadratic function, then it yields the trapezoid figure from Carlyle's solution to Leslie's problem (see graphic) with one of its sides being the diameter of the Carlyle circle. In an article from 1925 G. A. Miller pointed out that a slight modification of Lill's method applied to a normed quadratic function yields a circle that allows the geometric construction of the roots of that function and gave the explicit modern definition of what was later to be called Carlyle circle. Eves used the circle in the modern sense in one of the exercises of his book Introduction to the History of Mathematics (1953) and pointed out the connection to Leslie and Carlyle. Later publications started to adopt the names Carlyle circle, Carlyle method or Carlyle algorithm, though in German speaking countries the term Lill circle (Lill-Kreis) is used as well. DeTemple used in 1989 and 1991 Carlyle circles to devise compass-and-straightedge constructions for regular polygons, in particular the pentagon, the heptadecagon, the 257-gon and the 65537-gon. Ladislav Beran described in 1999 how the Carlyle circle can be used to construct the complex roots of a normed quadratic function. == References ==
Wikipedia:Carmichael function#0	In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} holds for every integer a coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n. As this is a finite abelian group, there must exist an element whose order equals the exponent, λ(n). Such an element is called a primitive λ-root modulo n. The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo n is φ(n), where φ is Euler's totient function. Since the order of an element of a finite group divides the order of the group, λ(n) divides φ(n). The following table compares the first 36 values of λ(n) (sequence A002322 in the OEIS) and φ(n) (in bold if they are different; the ns such that they are different are listed in OEIS: A033949). == Numerical examples == n = 5. The set of numbers less than and coprime to 5 is {1,2,3,4}. Hence Euler's totient function has value φ(5) = 4 and the value of Carmichael's function, λ(5), must be a divisor of 4. The divisor 1 does not satisfy the definition of Carmichael's function since a 1 ≢ 1 ( mod 5 ) {\displaystyle a^{1}\not \equiv 1{\pmod {5}}} except for a ≡ 1 ( mod 5 ) {\displaystyle a\equiv 1{\pmod {5}}} . Neither does 2 since 2 2 ≡ 3 2 ≡ 4 ≢ 1 ( mod 5 ) {\displaystyle 2^{2}\equiv 3^{2}\equiv 4\not \equiv 1{\pmod {5}}} . Hence λ(5) = 4. Indeed, 1 4 ≡ 2 4 ≡ 3 4 ≡ 4 4 ≡ 1 ( mod 5 ) {\displaystyle 1^{4}\equiv 2^{4}\equiv 3^{4}\equiv 4^{4}\equiv 1{\pmod {5}}} . Both 2 and 3 are primitive λ-roots modulo 5 and also primitive roots modulo 5. n = 8. The set of numbers less than and coprime to 8 is {1,3,5,7} . Hence φ(8) = 4 and λ(8) must be a divisor of 4. In fact λ(8) = 2 since 1 2 ≡ 3 2 ≡ 5 2 ≡ 7 2 ≡ 1 ( mod 8 ) {\displaystyle 1^{2}\equiv 3^{2}\equiv 5^{2}\equiv 7^{2}\equiv 1{\pmod {8}}} . The primitive λ-roots modulo 8 are 3, 5, and 7. There are no primitive roots modulo 8. == Recurrence for λ(n) == The Carmichael lambda function of a prime power can be expressed in terms of the Euler totient. Any number that is not 1 or a prime power can be written uniquely as the product of distinct prime powers, in which case λ of the product is the least common multiple of the λ of the prime power factors. Specifically, λ(n) is given by the recurrence λ ( n ) = { φ ( n ) if n is 1, 2, 4, or an odd prime power, 1 2 φ ( n ) if n = 2 r , r ≥ 3 , lcm ⁡ ( λ ( n 1 ) , λ ( n 2 ) , … , λ ( n k ) ) if n = n 1 n 2 … n k where n 1 , n 2 , … , n k are powers of distinct primes. {\displaystyle \lambda (n)={\begin{cases}\varphi (n)&{\text{if }}n{\text{ is 1, 2, 4, or an odd prime power,}}\\{\tfrac {1}{2}}\varphi (n)&{\text{if }}n=2^{r},\ r\geq 3,\\\operatorname {lcm} {\Bigl (}\lambda (n_{1}),\lambda (n_{2}),\ldots ,\lambda (n_{k}){\Bigr )}&{\text{if }}n=n_{1}n_{2}\ldots n_{k}{\text{ where }}n_{1},n_{2},\ldots ,n_{k}{\text{ are powers of distinct primes.}}\end{cases}}} Euler's totient for a prime power, that is, a number pr with p prime and r ≥ 1, is given by φ ( p r ) = p r − 1 ( p − 1 ) . {\displaystyle \varphi (p^{r}){=}p^{r-1}(p-1).} == Carmichael's theorems == Carmichael proved two theorems that, together, establish that if λ(n) is considered as defined by the recurrence of the previous section, then it satisfies the property stated in the introduction, namely that it is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} for all a relatively prime to n. This implies that the order of every element of the multiplicative group of integers modulo n divides λ(n). Carmichael calls an element a for which a λ ( n ) {\displaystyle a^{\lambda (n)}} is the least power of a congruent to 1 (mod n) a primitive λ-root modulo n. (This is not to be confused with a primitive root modulo n, which Carmichael sometimes refers to as a primitive φ {\displaystyle \varphi } -root modulo n.) If g is one of the primitive λ-roots guaranteed by the theorem, then g m ≡ 1 ( mod n ) {\displaystyle g^{m}\equiv 1{\pmod {n}}} has no positive integer solutions m less than λ(n), showing that there is no positive m < λ(n) such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} for all a relatively prime to n. The second statement of Theorem 2 does not imply that all primitive λ-roots modulo n are congruent to powers of a single root g. For example, if n = 15, then λ(n) = 4 while φ ( n ) = 8 {\displaystyle \varphi (n)=8} and φ ( λ ( n ) ) = 2 {\displaystyle \varphi (\lambda (n))=2} . There are four primitive λ-roots modulo 15, namely 2, 7, 8, and 13 as 1 ≡ 2 4 ≡ 8 4 ≡ 7 4 ≡ 13 4 {\displaystyle 1\equiv 2^{4}\equiv 8^{4}\equiv 7^{4}\equiv 13^{4}} . The roots 2 and 8 are congruent to powers of each other and the roots 7 and 13 are congruent to powers of each other, but neither 7 nor 13 is congruent to a power of 2 or 8 and vice versa. The other four elements of the multiplicative group modulo 15, namely 1, 4 (which satisfies 4 ≡ 2 2 ≡ 8 2 ≡ 7 2 ≡ 13 2 {\displaystyle 4\equiv 2^{2}\equiv 8^{2}\equiv 7^{2}\equiv 13^{2}} ), 11, and 14, are not primitive λ-roots modulo 15. For a contrasting example, if n = 9, then λ ( n ) = φ ( n ) = 6 {\displaystyle \lambda (n)=\varphi (n)=6} and φ ( λ ( n ) ) = 2 {\displaystyle \varphi (\lambda (n))=2} . There are two primitive λ-roots modulo 9, namely 2 and 5, each of which is congruent to the fifth power of the other. They are also both primitive φ {\displaystyle \varphi } -roots modulo 9. == Properties of the Carmichael function == In this section, an integer n {\displaystyle n} is divisible by a nonzero integer m {\displaystyle m} if there exists an integer k {\displaystyle k} such that n = k m {\displaystyle n=km} . This is written as m ∣ n . {\displaystyle m\mid n.} === A consequence of minimality of λ(n) === Suppose am ≡ 1 (mod n) for all numbers a coprime with n. Then λ(n) \| m. Proof: If m = kλ(n) + r with 0 ≤ r < λ(n), then a r = 1 k ⋅ a r ≡ ( a λ ( n ) ) k ⋅ a r = a k λ ( n ) + r = a m ≡ 1 ( mod n ) {\displaystyle a^{r}=1^{k}\cdot a^{r}\equiv \left(a^{\lambda (n)}\right)^{k}\cdot a^{r}=a^{k\lambda (n)+r}=a^{m}\equiv 1{\pmod {n}}} for all numbers a coprime with n. It follows that r = 0 since r < λ(n) and λ(n) is the minimal positive exponent for which the congruence holds for all a coprime with n. === λ(n) divides φ(n) === This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. λ(n) is the exponent of the multiplicative group of integers modulo n while φ(n) is the order of that group. In particular, the two must be equal in the cases where the multiplicative group is cyclic due to the existence of a primitive root, which is the case for odd prime powers. We can thus view Carmichael's theorem as a sharpening of Euler's theorem. === Divisibility === a \| b ⇒ λ ( a ) \| λ ( b ) {\displaystyle a\,\|\,b\Rightarrow \lambda (a)\,\|\,\lambda (b)} Proof. By definition, for any integer k {\displaystyle k} with gcd ( k , b ) = 1 {\displaystyle \gcd(k,b)=1} (and thus also gcd ( k , a ) = 1 {\displaystyle \gcd(k,a)=1} ), we have that b \| ( k λ ( b ) − 1 ) {\displaystyle b\,\|\,(k^{\lambda (b)}-1)} , and therefore a \| ( k λ ( b ) − 1 ) {\displaystyle a\,\|\,(k^{\lambda (b)}-1)} . This establishes that k λ ( b ) ≡ 1 ( mod a ) {\displaystyle k^{\lambda (b)}\equiv 1{\pmod {a}}} for all k relatively prime to a. By the consequence of minimality proved above, we have λ ( a ) \| λ ( b ) {\displaystyle \lambda (a)\,\|\,\lambda (b)} . === Composition === For all positive integers a and b it holds that λ ( l c m ( a , b ) ) = l c m ( λ ( a ) , λ ( b ) ) {\displaystyle \lambda (\mathrm {lcm} (a,b))=\mathrm {lcm} (\lambda (a),\lambda (b))} . This is an immediate consequence of the recurrence for the Carmichael function. === Exponential cycle length === If r m a x = max i { r i } {\displaystyle r_{\mathrm {max} }=\max _{i}\{r_{i}\}} is the biggest exponent in the prime factorization n = p 1 r 1 p 2 r 2 ⋯ p k r k {\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}} of n, then for all a (including those not coprime to n) and all r ≥ rmax, a r ≡ a λ ( n ) + r ( mod n ) . {\displaystyle a^{r}\equiv a^{\lambda (n)+r}{\pmod {n}}.} In particular, for square-free n ( rmax = 1), for all a we have a ≡ a λ ( n ) + 1 ( mod n ) . {\displaystyle a\equiv a^{\lambda (n)+1}{\pmod {n}}.} === Average value === For any n ≥ 16: 1 n ∑ i ≤ n λ ( i ) = n ln ⁡ n e B ( 1 + o ( 1 ) ) ln ⁡ ln ⁡ n / ( ln ⁡ ln ⁡ ln ⁡ n ) {\displaystyle {\frac {1}{n}}\sum _{i\leq n}\lambda (i)={\frac {n}{\ln n}}e^{B(1+o(1))\ln \ln n/(\ln \ln \ln n)}} (called Erdős approximation in the following) with the constant B := e − γ ∏ p ∈ P ( 1 − 1 ( p − 1 ) 2 ( p + 1 ) ) ≈ 0.34537 {\displaystyle B:=e^{-\gamma }\prod _{p\in \mathbb {P} }\left({1-{\frac {1}{(p-1)^{2}(p+1)}}}\right)\approx 0.34537} and γ ≈ 0.57721, the Euler–Mascheroni constant. The following table gives some overview over the first 226 – 1 = 67108863 values of the λ function, for both, the exact average and its Erdős-approximation. Additionally given is some overview over the more easily accessible “logarithm over logarithm” values LoL(n) := ⁠ln λ(n)/ln n⁠ with LoL(n) > ⁠4/5⁠ ⇔ λ(n) > n⁠4/5⁠. There, the table entry in row number 26 at column % LoL > ⁠4/5⁠ → 60.49 indicates that 60.49% (≈ 40000000) of the integers 1 ≤ n ≤ 67108863 have λ(n) > n⁠4/5⁠ meaning that the majority of the λ values is exponential in the length l := log2(n) of the input n, namely ( 2 4 5 ) l = 2 4 l 5 = ( 2 l ) 4 5 = n 4 5 . {\displaystyle \left(2^{\frac {4}{5}}\right)^{l}=2^{\frac {4l}{5}}=\left(2^{l}\right)^{\frac {4}{5}}=n^{\frac {4}{5}}.} === Prevailing interval === For all numbers N and all but o(N) positive integers n ≤ N (a "prevailing" majority): λ ( n ) = n ( ln ⁡ n ) ln ⁡ ln ⁡ ln ⁡ n + A + o ( 1 ) {\displaystyle \lambda (n)={\frac {n}{(\ln n)^{\ln \ln \ln n+A+o(1)}}}} with the constant A := − 1 + ∑ p ∈ P ln ⁡ p ( p − 1 ) 2 ≈ 0.2269688 {\displaystyle A:=-1+\sum _{p\in \mathbb {P} }{\frac {\ln p}{(p-1)^{2}}}\approx 0.2269688} === Lower bounds === For any sufficiently large number N and for any Δ ≥ (ln ln N)3, there are at most N exp ⁡ ( − 0.69 ( Δ ln ⁡ Δ ) 1 3 ) {\displaystyle N\exp \left(-0.69(\Delta \ln \Delta )^{\frac {1}{3}}\right)} positive integers n ≤ N such that λ(n) ≤ ne−Δ. === Minimal order === For any sequence n1 < n2 < n3 < ⋯ of positive integers, any constant 0 < c < ⁠1/ln 2⁠, and any sufficiently large i: λ ( n i ) > ( ln ⁡ n i ) c ln ⁡ ln ⁡ ln ⁡ n i . {\displaystyle \lambda (n_{i})>\left(\ln n_{i}\right)^{c\ln \ln \ln n_{i}}.} === Small values === For a constant c and any sufficiently large positive A, there exists an integer n > A such that λ ( n ) < ( ln ⁡ A ) c ln ⁡ ln ⁡ ln ⁡ A . {\displaystyle \lambda (n)<\left(\ln A\right)^{c\ln \ln \ln A}.} Moreover, n is of the form n = ∏ q ∈ P ( q − 1 ) \| m ⁡ q {\displaystyle n=\mathop {\prod _{q\in \mathbb {P} }} _{(q-1)\|m}q} for some square-free integer m < (ln A)c ln ln ln A. === Image of the function === The set of values of the Carmichael function has counting function x ( ln ⁡ x ) η + o ( 1 ) , {\displaystyle {\frac {x}{(\ln x)^{\eta +o(1)}}},} where η = 1 − 1 + ln ⁡ ln ⁡ 2 ln ⁡ 2 ≈ 0.08607 {\displaystyle \eta =1-{\frac {1+\ln \ln 2}{\ln 2}}\approx 0.08607} == Use in cryptography == The Carmichael function is important in cryptography due to its use in the RSA encryption algorithm. == Proof of Theorem 1 == For n = p, a prime, Theorem 1 is equivalent to Fermat's little theorem: a p − 1 ≡ 1 ( mod p ) for all a coprime to p . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}\qquad {\text{for all }}a{\text{ coprime to }}p.} For prime powers pr, r > 1, if a p r − 1 ( p − 1 ) = 1 + h p r {\displaystyle a^{p^{r-1}(p-1)}=1+hp^{r}} holds for some integer h, then raising both sides to the power p gives a p r ( p − 1 ) = 1 + h ′ p r + 1 {\displaystyle a^{p^{r}(p-1)}=1+h'p^{r+1}} for some other integer h ′ {\displaystyle h'} . By induction it follows that a φ ( p r ) ≡ 1 ( mod p r ) {\displaystyle a^{\varphi (p^{r})}\equiv 1{\pmod {p^{r}}}} for all a relatively prime to p and hence to pr. This establishes the theorem for n = 4 or any odd prime power. === Sharpening the result for higher powers of two === For a coprime to (powers of) 2 we have a = 1 + 2h2 for some integer h2. Then, a 2 = 1 + 4 h 2 ( h 2 + 1 ) = 1 + 8 ( h 2 + 1 2 ) =: 1 + 8 h 3 {\displaystyle a^{2}=1+4h_{2}(h_{2}+1)=1+8{\binom {h_{2}+1}{2}}=:1+8h_{3}} , where h 3 {\displaystyle h_{3}} is an integer. With r = 3, this is written a 2 r − 2 = 1 + 2 r h r . {\displaystyle a^{2^{r-2}}=1+2^{r}h_{r}.} Squaring both sides gives a 2 r − 1 = ( 1 + 2 r h r ) 2 = 1 + 2 r + 1 ( h r + 2 r − 1 h r 2 ) =: 1 + 2 r + 1 h r + 1 , {\displaystyle a^{2^{r-1}}=\left(1+2^{r}h_{r}\right)^{2}=1+2^{r+1}\left(h_{r}+2^{r-1}h_{r}^{2}\right)=:1+2^{r+1}h_{r+1},} where h r + 1 {\displaystyle h_{r+1}} is an integer. It follows by induction that a 2 r − 2 = a 1 2 φ ( 2 r ) ≡ 1 ( mod 2 r ) {\displaystyle a^{2^{r-2}}=a^{{\frac {1}{2}}\varphi (2^{r})}\equiv 1{\pmod {2^{r}}}} for all r ≥ 3 {\displaystyle r\geq 3} and all a coprime to 2 r {\displaystyle 2^{r}} . === Integers with multiple prime factors === By the unique factorization theorem, any n > 1 can be written in a unique way as n = p 1 r 1 p 2 r 2 ⋯ p k r k {\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}} where p1 < p2 < ... < pk are primes and r1, r2, ..., rk are positive integers. The results for prime powers establish that, for 1 ≤ j ≤ k {\displaystyle 1\leq j\leq k} , a λ ( p j r j ) ≡ 1 ( mod p j r j ) for all a coprime to n and hence to p i r i . {\displaystyle a^{\lambda \left(p_{j}^{r_{j}}\right)}\equiv 1{\pmod {p_{j}^{r_{j}}}}\qquad {\text{for all }}a{\text{ coprime to }}n{\text{ and hence to }}p_{i}^{r_{i}}.} From this it follows that a λ ( n ) ≡ 1 ( mod p j r j ) for all a coprime to n , {\displaystyle a^{\lambda (n)}\equiv 1{\pmod {p_{j}^{r_{j}}}}\qquad {\text{for all }}a{\text{ coprime to }}n,} where, as given by the recurrence, λ ( n ) = lcm ⁡ ( λ ( p 1 r 1 ) , λ ( p 2 r 2 ) , … , λ ( p k r k ) ) . {\displaystyle \lambda (n)=\operatorname {lcm} {\Bigl (}\lambda \left(p_{1}^{r_{1}}\right),\lambda \left(p_{2}^{r_{2}}\right),\ldots ,\lambda \left(p_{k}^{r_{k}}\right){\Bigr )}.} From the Chinese remainder theorem one concludes that a λ ( n ) ≡ 1 ( mod n ) for all a coprime to n . {\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}\qquad {\text{for all }}a{\text{ coprime to }}n.} == See also == Carmichael number == Notes == == References == Erdős, Paul; Pomerance, Carl; Schmutz, Eric (1991). "Carmichael's lambda function". Acta Arithmetica. 58 (4): 363–385. doi:10.4064/aa-58-4-363-385. ISSN 0065-1036. MR 1121092. Zbl 0734.11047. Friedlander, John B.; Pomerance, Carl; Shparlinski, Igor E. (2001). "Period of the power generator and small values of the Carmichael function". Mathematics of Computation. 70 (236): 1591–1605, 1803–1806. doi:10.1090/s0025-5718-00-01282-5. ISSN 0025-5718. MR 1836921. Zbl 1029.11043. Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36, 193–195. ISBN 978-1-4020-2546-4. Zbl 1079.11001. Carmichael, Robert D. [1914]. The Theory of Numbers at Project Gutenberg
Wikipedia:Carol Walker#0	Carol Lee Walker (born 1935) is a retired American mathematician and mathematics textbook author. Walker's early mathematical research, in the 1960s and 1970s, concerned the theory of abelian groups. In the 1990s, her interests shifted to fuzzy logic and fuzzy control systems. == Education and career == Walker was born in Martinez, California on August 19, 1935, and went to high school in Montrose, Colorado. She studied music education at the University of Colorado Boulder, with a year off to work as a primary-school music teacher in Colorado, and graduated in 1957. Next, she went to the University of Denver for graduate study in mathematics, but after one year transferred to New Mexico State University, where she earned a master's degree in 1961 and completed her PhD in 1963. Her dissertation, On p α {\displaystyle p^{\alpha }} -pure sequences of abelian groups, was supervised by David Kent Harrison. After postdoctoral research at the Institute for Advanced Study, she returned to Mexico State University as an assistant professor in 1964, and quickly earned tenure as an associate professor in 1966. She was promoted to full professor in 1972. She chaired the Department of Mathematical Sciences from 1979 to 1993, and served as associate dean of arts and sciences from 1993 until her retirement in 1996. == Books == Walker is the coauthor of books including: Mathematics for the Liberal Arts Student (with Fred Richman and Robert J. Wisner, Brooks-Cole, 1967; 2nd ed., 1973; 3rd ed., with James Brewer, Prentice-Hall, 2000; 4th ed., 2003) Doing Mathematics with Scientific WorkPlace (with Darel Hardy, Brooks-Cole, 1995; multiple editions) A First Course in Fuzzy and Neural Control (with Hung T. Nguyen, Radipuram Prasad, and Elbert Walker, CRC Press, 2003) Applied Algebra: Codes, Ciphers, and Discrete Algorithms (with Darel Hardy, Prentice-Hall, 2003) Calculus: Understanding Its Concepts and Methods (with Darel Hardy, Fred Richman, and Robert J. Wisner, MacKichan Software, 2006) == Recognition == The New Mexico State University alumni gave Walker their Distinguished Alumni Award in 2001. == Personal life == Walker was married to Elbert Walker (1930–2018), another mathematician who joined the New Mexico State University faculty in 1957. == References ==
Wikipedia:Carola-Bibiane Schönlieb#0	Carola-Bibiane Schönlieb (born 1979) is an Austrian mathematician who works in image processing and partial differential equations. She is a Fellow of Jesus College, Cambridge and Professor of Applied Mathematics in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. She is the author of the book Partial Differential Equation Methods for Image Inpainting (Cambridge University Press, 2015), on methods for using the solutions to partial differential equations to fill in gaps in digital images. Schönlieb earned a master's degree in mathematics from the University of Salzburg in 2004. She completed her Ph.D. at Cambridge in 2009. Her dissertation, Modern PDE Techniques for Image Inpainting, was supervised by Peter Markowich. After postdoctoral study at the University of Göttingen she returned to Cambridge as a lecturer in 2010. In 2016 Schönlieb won the Whitehead Prize of the London Mathematical Society "for her spectacular contributions to the mathematics of image analysis". She won a Philip Leverhulme Prize in 2017, and is the 2018 Mary Cartwright Lecturer of the London Mathematical Society. Since 2016 she has also been a Fellow of the Alan Turing Institute. Carola is Director of the Cantab Capital Institute for the Mathematics of Information and she is Co-Director of the Centre for Mathematical Imaging in Healthcare—both are based at the University of Cambridge. In 2021 she was elected Council Members-at-Large for SIAM for a term running January 1, 2022 - December 31, 2024. In 2022, she received a honorary doctorate from the University of Klagenfurt. She is a SIAM Fellow, in the 2024 class of fellows. == References == == External links == Home page Carola-Bibiane Schönlieb publications indexed by Google Scholar
Wikipedia:Carolina Araujo (mathematician)#0	Carolina Bhering de Araujo (born in 1976) is a Brazilian mathematician specializing in algebraic geometry, including birational geometry, Fano varieties, and foliations. Other than her research in mathematics, she is also known for her efforts for improving the conditions for women mathematicians. == Education and career == Araujo was born and raised in Rio de Janeiro, Brazil. She did her undergraduate studies in Brazil, completing a degree in mathematics in 1998 from the Pontifical Catholic University of Rio de Janeiro. She earned her PhD in 2004 at Princeton University, where her dissertation, supervised by János Kollár, was titled The Variety of Tangents to Rational Curves. She is currently a researcher at the Instituto Nacional de Matemática Pura e Aplicada in Brazil (IMPA), and the only woman (as of 2018) on the permanent research staff at IMPA. She is also a Simons Associate at the Abdus Salam International Centre for Theoretical Physics (ICTP). She is the vice-president of the Committee for Women in Mathematics at the International Mathematical Union. During and after her PhD, Araujo developed techniques related to Japanese mathematician Shigefumi Mori's proposed theory of rational curves of minimal degree, which she published in 2008.[A] == Recognition == Araujo won the L'Oreal Award for Women in Science in Brazil in 2008. Araujo was both an organizer and an invited speaker at the 2018 International Congress of Mathematicians. She led the inaugural World Meeting for Women in Mathematics (WM)2 in August 2018. She was also one of the female mathematicians profiled in the short documentary called Journeys of Women in Mathematics, funded by the Simons Foundation. Araujo was awarded the 2020 Ramanujan Prize from the International Centre for Theoretical Physics. She is included in a deck of playing cards featuring notable women mathematicians published by the Association of Women in Mathematics. == Selected bibliography == == References == == External links == Faculty profile page for Carolina Araujo at IMPA
Wikipedia:Carolyn A. Maher#0	Carolyn A. Maher is the Distinguished Professor of Mathematics Education and Director of the Robert B. Davis Institute for Learning. She received the 2022 National Council of Teachers of Mathematics (NCTM) Lifetime Achievement Award. == Early life and education == Maher received an Ed.D. in Mathematics Education (1972), M.Ed. in Education (1965) and B.A. (1962) from Rutgers University with a major in Mathematics Education and a minor in Statistics. == Career and research == Maher worked as an elementary school teacher from 1962-1967 in the Matawan Regional School District, Augusta, and Scotch Plains. In 1992 she became a professor of mathematics education at Rutgers University and became the Distinguished Professor of Mathematics Education in 2007. Her work focuses on different studies of student’s mathematical reasoning and argumentation. Her work was inspired by her prior experience as a teacher and she wanted to understand how students learn to develop their intellectual curiosity and their need to learn. She has given several invited talks throughout the world, including in South Africa, Brazil and Mozambique. Maher's work addresses how to provide equal access to STEM while improving both learning and teaching. She helps teachers with teaching methods that consider student differences and strength and their diversity of backgrounds, language, culture, and ethnicity. Her work has been funded by the National Science Foundation for over three decades. Maher has served in various professional organizations. From 1998 she has been the editor-in-chief of The Journal of Mathematical Behavior and has been a part of the editorial boards of The British Journal of Educational Studies and the Journal for Research in Mathematics Education. She has been the President for North American Group of the Psychology of Mathematics Education; Chair for the American Education Research Association; and an elected member to the Holmdel Public Schools Board of Education. == References ==
Wikipedia:Carolyn Kieran#0	Carolyn Kieran is a Canadian mathematics educator known for her studies of how students learn algebra. She is a professor emerita of mathematics at the Université du Québec à Montréal. == Education and career == Kieran has bachelor's degrees from Marianopolis College and the Université de Montréal, a master's degree from Concordia University, and a doctorate from McGill University. She joined the mathematics department at the Université du Québec à Montréal in 1983 and became a full professor there in 1991. She retired in 2008 and was named a professor emerita in 2010. == Books == Kieran is a co-author, with J. Pang, D. Schifter, and S. F. Ng, of Early Algebra: Research into its Nature, its Learning, its Teaching (Springer Open, 2016). She is a co-editor of volumes including: Research Issues in the Learning and Teaching of Algebra, Vol. 4 (1989) Selected Lectures from the Seventh International Congress on Mathematical Education (1994) Approaches to Algebra: Perspectives for Research and Teaching Computer Algebra Systems in Secondary School Mathematics Education (2003). == References == == External links == Carolyn Kieran publications indexed by Google Scholar }
Wikipedia:Carolyn Yackel#0	Carolyn Yackel is an American mathematician who has been Professor of Mathematics at Mercer University in Macon, Georgia since 2001. From 1998 to 2001 she was Max Zorn Visiting Assistant Professor of Mathematics at Indiana University. Yackel's mother, Erna Beth Yackel, was a mathematics educator on the faculty at Purdue University Northwest. Originally trained as a commutative algebraist, her current interests center on mathematics education and mathematics in art, particularly as applied to fiber art. She specializes in the realization of geometric and topological structures through quilting, cross-stitching, crocheting, knitting, and embroidery. She is on the Board of the Gathering 4 Gardner and also has a long association with The Bridges Conference. == Early life and career == Yackel was born in West Lafayette, Indiana. She received her S.B. in mathematics from the University of Chicago (1992) and her M.S. in mathematics from the University of Michigan (1994). She completed her PhD with the dissertation “Asymptotic Behavior of Annihilator Lengths in Certain Quotient Rings” under Melvin Hochster at the University of Michigan (1998). Combining her interests in mathematics, quilting and knitting she is one of 24 mathematicians and artists who make up the Mathemalchemy Team. == Books == Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture, edited by Carolyn Yackel, Robert Bosch, Eve Torrence, and Kristof Fenyvesi, Tessellations Publishing, Phoenix, AZ 2020. Figuring Fibers, edited by belcastro, s-m and Yackel, C. A.. Providence, RI: American Mathematics Society, 2018. Crafting by Concepts: fiber arts and mathematics, edited by belcastro, s-m and Yackel, C. A., Natick, MA: AK Peters, 2011. Making Mathematics with Needlework: Ten Papers and Ten Projects, edited by belcastro, s. m. and Yackel, C. A.. Wellesley, MA: AK Peters, 2007. == Selected papers == Taalman, L. and Yackel, C. A. “Wallpaper Patterns for Lattice Designs” In Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture, 223–230, Tessellations Publishing, Phoenix, AZ 2020. Yackel, C. A. “Rhombic Triacontahedron” In Illustrating Mathematics, 26–27. American Mathematics Society, 2020. Yackel, C. A. “Treating Templeton Squares Like Truchet Tiles” In Figuring Fibers, Providence, RI: AMS, 2018. Yackel, C. “Report: The 2015 Joint Mathematics Meetings exhibition of mathematical art” Journal of Mathematics and the Arts. 10(1–4) (2016) 9–13. Yackel, C. “Teaching Temari: Geometrically Embroidered Spheres in the Classroom” In Proceedings of the 2012 Bridges Towson Conference, 563–566. Tessellations Publishing, Phoenix, AZ, USA. 2012. Yackel, C. A. “In Pursuit of Dancing Squares” Math Horizons September 2011, 19. Yackel, C. A. with belcastro, s-m. “Spherical Symmetries of Temari” In Crafting by Concepts, 151–185. AK Peters, 2011. Shepherd, M. with belcastro, s-m and Yackel, C. A. “Group Actions in Cross-stitch” In Crafting by Concepts, 151–185. AK Peters, 2011. == References == == External links == Carolyn Yackel's Home page Carolyn Yackel at the Mathematics Genealogy Project
Wikipedia:Carré du champ operator#0	The carré du champ operator (French for square of a field operator) is a bilinear, symmetric operator from analysis and probability theory. The carré du champ operator measures how far an infinitesimal generator is from being a derivation. The operator was introduced in 1969 by Hiroshi Kunita and independently discovered in 1976 by Jean-Pierre Roth in his doctoral thesis. The name "carré du champ" comes from electrostatics. == Carré du champ operator for a Markov semigroup == Let ( X , E , μ ) {\displaystyle (X,{\mathcal {E}},\mu )} be a σ-finite measure space, { P t } t ≥ 0 {\displaystyle \{P_{t}\}_{t\geq 0}} a Markov semigroup of non-negative operators on L 2 ( X , μ ) {\displaystyle L^{2}(X,\mu )} , A {\displaystyle A} the infinitesimal generator of { P t } t ≥ 0 {\displaystyle \{P_{t}\}_{t\geq 0}} and A {\displaystyle {\mathcal {A}}} the algebra of functions in D ( A ) {\displaystyle {\mathcal {D}}(A)} , i.e. a vector space such that for all f , g ∈ A {\displaystyle f,g\in {\mathcal {A}}} also f g ∈ A {\displaystyle fg\in {\mathcal {A}}} . === Carré du champ operator === The carré du champ operator of a Markovian semigroup { P t } t ≥ 0 {\displaystyle \{P_{t}\}_{t\geq 0}} is the operator Γ : A × A → R {\displaystyle \Gamma :{\mathcal {A}}\times {\mathcal {A}}\to \mathbb {R} } defined (following P. A. Meyer) as Γ ( f , g ) = 1 2 ( A ( f g ) − f A ( g ) − g A ( f ) ) {\displaystyle \Gamma (f,g)={\frac {1}{2}}\left(A(fg)-fA(g)-gA(f)\right)} for all f , g ∈ A {\displaystyle f,g\in {\mathcal {A}}} . === Properties === From the definition, it follows that Γ ( f , g ) = lim t → 0 1 2 t ( P t ( f g ) − P t f P t g ) . {\displaystyle \Gamma (f,g)=\lim \limits _{t\to 0}{\frac {1}{2t}}\left(P_{t}(fg)-P_{t}fP_{t}g\right).} For f ∈ A {\displaystyle f\in {\mathcal {A}}} we have P t ( f 2 ) ≥ ( P t f ) 2 {\displaystyle P_{t}(f^{2})\geq (P_{t}f)^{2}} and thus A ( f 2 ) ≥ 2 f A f {\displaystyle A(f^{2})\geq 2fAf} and Γ ( f ) := Γ ( f , f ) ≥ 0 , ∀ f ∈ A {\displaystyle \Gamma (f):=\Gamma (f,f)\geq 0,\quad \forall f\in {\mathcal {A}}} therefore the carré du champ operator is positive. The domain is D ( A ) := { f ∈ L 2 ( X , μ ) ; lim t ↓ 0 P t f − f t exists and is in L 2 ( X , μ ) } . {\displaystyle {\mathcal {D}}(A):=\left\{f\in L^{2}(X,\mu );\;\lim \limits _{t\downarrow 0}{\frac {P_{t}f-f}{t}}{\text{ exists and is in }}L^{2}(X,\mu )\right\}.} === Remarks === The definition in Roth's thesis is slightly different. == Bibliography == Ledoux, Michel (2000). "The geometry of Markov diffusion generators". Annales de la Faculté des sciences de Toulouse: Mathématiques. Série 6. 9 (2): 305–366. doi:10.5802/afst.962. hdl:20.500.11850/146400. Meyer, Paul-André (1976). "L'Operateur carré du champ". Séminaire de Probabilités X Université de Strasbourg. Lecture Notes in Mathematics (in French). Vol. 511. Berlin, Heidelberg: Springer. pp. 142–161. doi:10.1007/BFb0101102. ISBN 978-3-540-07681-0. == References ==
Wikipedia:Cartan formula#0	In mathematics, the Cartan formula can mean: one in differential geometry: L X = d ι X + ι X d {\displaystyle {\mathcal {L}}_{X}=\mathrm {d} \,\iota _{X}+\iota _{X}\mathrm {d} } , where L X , d {\displaystyle {\mathcal {L}}_{X},\mathrm {d} } , and ι X {\displaystyle \iota _{X}} are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms. See interior product for the detail. It is also called the Cartan homotopy formula or Cartan magic formula. This formula is named after Élie Cartan. one in algebraic topology, which is one of the five axioms of Steenrod algebra. It reads: S q n ( x ⌣ y ) = ∑ i + j = n ( S q i x ) ⌣ ( S q j y ) or P n ( x ⌣ y ) = ∑ i + j = n ( P i x ) ⌣ ( P j y ) {\displaystyle {\begin{aligned}Sq^{n}(x\smile y)&=\sum _{i+j=n}(Sq^{i}x)\smile (Sq^{j}y)\quad {\text{or}}\\P^{n}(x\smile y)&=\sum _{i+j=n}(P^{i}x)\smile (P^{j}y)\end{aligned}}} . See Steenrod algebra for the detail. The name derives from Henri Cartan, son of Élie. == Footnotes == == See also == List of things named after Élie Cartan
Wikipedia:Cartan–Kuranishi prolongation theorem#0	Given an exterior differential system defined on a manifold M, the Cartan–Kuranishi prolongation theorem says that after a finite number of prolongations the system is either in involution (admits at least one 'large' integral manifold), or is impossible. == History == The theorem is named after Élie Cartan and Masatake Kuranishi. Cartan made several attempts in 1946 to prove the result, but it was in 1957 that Kuranishi provided a proof of Cartan's conjecture. == Applications == This theorem is used in infinite-dimensional Lie theory. == See also == Cartan-Kähler theorem == References == M. Kuranishi, On É. Cartan's prolongation theorem of exterior differential systems, Amer. J. Math., vol. 79, 1957, p. 1–47 "Partial differential equations on a manifold", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Wikipedia:Cartan–Kähler theorem#0	In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I {\displaystyle I} . It is named for Élie Cartan and Erich Kähler. == Meaning == It is not true that merely having d I {\displaystyle dI} contained in I {\displaystyle I} is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution. == Statement == Let ( M , I ) {\displaystyle (M,I)} be a real analytic EDS. Assume that P ⊆ M {\displaystyle P\subseteq M} is a connected, k {\displaystyle k} -dimensional, real analytic, regular integral manifold of I {\displaystyle I} with r ( P ) ≥ 0 {\displaystyle r(P)\geq 0} (i.e., the tangent spaces T p P {\displaystyle T_{p}P} are "extendable" to higher dimensional integral elements). Moreover, assume there is a real analytic submanifold R ⊆ M {\displaystyle R\subseteq M} of codimension r ( P ) {\displaystyle r(P)} containing P {\displaystyle P} and such that T p R ∩ H ( T p P ) {\displaystyle T_{p}R\cap H(T_{p}P)} has dimension k + 1 {\displaystyle k+1} for all p ∈ P {\displaystyle p\in P} . Then there exists a (locally) unique connected, ( k + 1 ) {\displaystyle (k+1)} -dimensional, real analytic integral manifold X ⊆ M {\displaystyle X\subseteq M} of I {\displaystyle I} that satisfies P ⊆ X ⊆ R {\displaystyle P\subseteq X\subseteq R} . == Proof and assumptions == The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary. == References == Jean Dieudonné, Eléments d'analyse, vol. 4, (1977) Chapt. XVIII.13 R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Springer Verlag, New York, 1991. == External links == Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", Encyclopedia of Mathematics, EMS Press R. Bryant, "Nine Lectures on Exterior Differential Systems", 1999 E. Cartan, "On the integration of systems of total differential equations," transl. by D. H. Delphenich E. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich
Wikipedia:Cartesian tensor#0	In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product. Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system. == Cartesian basis and related terminology == === Vectors in three dimensions === In 3D Euclidean space, R 3 {\displaystyle \mathbb {R} ^{3}} , the standard basis is ex, ey, ez. Each basis vector points along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so the basis is orthonormal. Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice, see orientation (vector space) for details. For Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as a linear combination of the basis vectors ex, ey, ez: a = a x e x + a y e y + a z e z {\displaystyle \mathbf {a} =a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}} where the coordinates of the vector with respect to the Cartesian basis are denoted ax, ay, az. It is common and helpful to display the basis vectors as column vectors e x = ( 1 0 0 ) , e y = ( 0 1 0 ) , e z = ( 0 0 1 ) {\displaystyle \mathbf {e} _{\text{x}}={\begin{pmatrix}1\\0\\0\end{pmatrix}}\,,\quad \mathbf {e} _{\text{y}}={\begin{pmatrix}0\\1\\0\end{pmatrix}}\,,\quad \mathbf {e} _{\text{z}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}} when we have a coordinate vector in a column vector representation: a = ( a x a y a z ) {\displaystyle \mathbf {a} ={\begin{pmatrix}a_{\text{x}}\\a_{\text{y}}\\a_{\text{z}}\end{pmatrix}}} A row vector representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see Einstein notation and covariance and contravariance of vectors for why. The term "component" of a vector is ambiguous: it could refer to: a specific coordinate of the vector such as az (a scalar), and similarly for x and y, or the coordinate scalar-multiplying the corresponding basis vector, in which case the "y-component" of a is ayey (a vector), and similarly for x and z. A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors ex ↦ e1, ey ↦ e2, ez ↦ e3 and coordinates ax ↦ a1, ay ↦ a2, az ↦ a3. In general, the notation e1, e2, e3 refers to any basis, and a1, a2, a3 refers to the corresponding coordinate system; although here they are restricted to the Cartesian system. Then: a = a 1 e 1 + a 2 e 2 + a 3 e 3 = ∑ i = 1 3 a i e i {\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}=\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}} It is standard to use the Einstein notation—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness: a = ∑ i = 1 3 a i e i ≡ a i e i {\displaystyle \mathbf {a} =\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}\equiv a_{i}\mathbf {e} _{i}} An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and not indices. (It is informal to say "i = x, y, z"). === Second-order tensors in three dimensions === A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b. Analogous to vectors, it can be written as a linear combination of the tensor basis ex ⊗ ex ≡ exx, ex ⊗ ey ≡ exy, ..., ez ⊗ ez ≡ ezz (the right-hand side of each identity is only an abbreviation, nothing more): T = ( a x e x + a y e y + a z e z ) ⊗ ( b x e x + b y e y + b z e z ) = a x b x e x ⊗ e x + a x b y e x ⊗ e y + a x b z e x ⊗ e z + a y b x e y ⊗ e x + a y b y e y ⊗ e y + a y b z e y ⊗ e z + a z b x e z ⊗ e x + a z b y e z ⊗ e y + a z b z e z ⊗ e z {\displaystyle {\begin{aligned}\mathbf {T} =\quad &\left(a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}\right)\otimes \left(b_{\text{x}}\mathbf {e} _{\text{x}}+b_{\text{y}}\mathbf {e} _{\text{y}}+b_{\text{z}}\mathbf {e} _{\text{z}}\right)\\[5pt]{}=\quad &a_{\text{x}}b_{\text{x}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{x}}+a_{\text{x}}b_{\text{y}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{y}}+a_{\text{x}}b_{\text{z}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{z}}\\[4pt]{}+{}&a_{\text{y}}b_{\text{x}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{x}}+a_{\text{y}}b_{\text{y}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{y}}+a_{\text{y}}b_{\text{z}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{z}}\\[4pt]{}+{}&a_{\text{z}}b_{\text{x}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{x}}+a_{\text{z}}b_{\text{y}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{y}}+a_{\text{z}}b_{\text{z}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{z}}\end{aligned}}} Representing each basis tensor as a matrix: e x ⊗ e x ≡ e xx = ( 1 0 0 0 0 0 0 0 0 ) , e x ⊗ e y ≡ e xy = ( 0 1 0 0 0 0 0 0 0 ) , e z ⊗ e z ≡ e zz = ( 0 0 0 0 0 0 0 0 1 ) {\displaystyle {\begin{aligned}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{x}}&\equiv \mathbf {e} _{\text{xx}}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}}\,,&\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{y}}&\equiv \mathbf {e} _{\text{xy}}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}}\,,&\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{z}}&\equiv \mathbf {e} _{\text{zz}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}}\end{aligned}}} then T can be represented more systematically as a matrix: T = ( a x b x a x b y a x b z a y b x a y b y a y b z a z b x a z b y a z b z ) {\displaystyle \mathbf {T} ={\begin{pmatrix}a_{\text{x}}b_{\text{x}}&a_{\text{x}}b_{\text{y}}&a_{\text{x}}b_{\text{z}}\\a_{\text{y}}b_{\text{x}}&a_{\text{y}}b_{\text{y}}&a_{\text{y}}b_{\text{z}}\\a_{\text{z}}b_{\text{x}}&a_{\text{z}}b_{\text{y}}&a_{\text{z}}b_{\text{z}}\end{pmatrix}}} See matrix multiplication for the notational correspondence between matrices and the dot and tensor products. More generally, whether or not T is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates Txx, Txy, ..., Tzz: T = T xx e xx + T xy e xy + T xz e xz + T yx e yx + T yy e yy + T yz e yz + T zx e zx + T zy e zy + T zz e zz {\displaystyle {\begin{aligned}\mathbf {T} =\quad &T_{\text{xx}}\mathbf {e} _{\text{xx}}+T_{\text{xy}}\mathbf {e} _{\text{xy}}+T_{\text{xz}}\mathbf {e} _{\text{xz}}\\[4pt]{}+{}&T_{\text{yx}}\mathbf {e} _{\text{yx}}+T_{\text{yy}}\mathbf {e} _{\text{yy}}+T_{\text{yz}}\mathbf {e} _{\text{yz}}\\[4pt]{}+{}&T_{\text{zx}}\mathbf {e} _{\text{zx}}+T_{\text{zy}}\mathbf {e} _{\text{zy}}+T_{\text{zz}}\mathbf {e} _{\text{zz}}\end{aligned}}} while in terms of tensor indices: T = T i j e i j ≡ ∑ i j T i j e i ⊗ e j , {\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,} and in matrix form: T = ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) {\displaystyle \mathbf {T} ={\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}} Second-order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors – they are multilinear functions. A second-order tensor T which takes in a vector u of some magnitude and direction will return a vector v; of a different magnitude and in a different direction to u, in general. The notation used for functions in mathematical analysis leads us to write v − T(u), while the same idea can be expressed in matrix and index notations (including the summation convention), respectively: ( v x v y v z ) = ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) ( u x u y u z ) , v i = T i j u j {\displaystyle {\begin{aligned}{\begin{pmatrix}v_{\text{x}}\\v_{\text{y}}\\v_{\text{z}}\end{pmatrix}}&={\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}u_{\text{x}}\\u_{\text{y}}\\u_{\text{z}}\end{pmatrix}}\,,&v_{i}&=T_{ij}u_{j}\end{aligned}}} By "linear", if u = ρr + σs for two scalars ρ and σ and vectors r and s, then in function and index notations: v = T ( ρ r + σ s ) = ρ T ( r ) + σ T ( s ) v i = T i j ( ρ r j + σ s j ) = ρ T i j r j + σ T i j s j {\displaystyle {\begin{aligned}\mathbf {v} &=&&\mathbf {T} (\rho \mathbf {r} +\sigma \mathbf {s} )&=&&\rho \mathbf {T} (\mathbf {r} )+\sigma \mathbf {T} (\mathbf {s} )\\[1ex]v_{i}&=&&T_{ij}(\rho r_{j}+\sigma s_{j})&=&&\rho T_{ij}r_{j}+\sigma T_{ij}s_{j}\end{aligned}}} and similarly for the matrix notation. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner. Both provide the physical interpretation of directions; vectors have one direction, while second-order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction. The use of second-order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the dot product of two vectors is always a scalar, while the cross product of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot obtain a new vector of any magnitude in any direction. (See also below for more on the dot and cross products). The tensor product of two vectors is a second-order tensor, although this has no obvious directional interpretation by itself. The previous idea can be continued: if T takes in two vectors p and q, it will return a scalar r. In function notation we write r = T(p, q), while in matrix and index notations (including the summation convention) respectively: r = ( p x p y p z ) ( T xx T xy T xz T yx T yy T yz T zx T zy T zz ) ( q x q y q z ) = p i T i j q j {\displaystyle r={\begin{pmatrix}p_{\text{x}}&p_{\text{y}}&p_{\text{z}}\end{pmatrix}}{\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}q_{\text{x}}\\q_{\text{y}}\\q_{\text{z}}\end{pmatrix}}=p_{i}T_{ij}q_{j}} The tensor T is linear in both input vectors. When vectors and tensors are written without reference to components, and indices are not used, sometimes a dot ⋅ is placed where summations over indices (known as tensor contractions) are taken. For the above cases: v = T ⋅ u r = p ⋅ T ⋅ q {\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {T} \cdot \mathbf {u} \\r&=\mathbf {p} \cdot \mathbf {T} \cdot \mathbf {q} \end{aligned}}} motivated by the dot product notation: a ⋅ b ≡ a i b i {\displaystyle \mathbf {a} \cdot \mathbf {b} \equiv a_{i}b_{i}} More generally, a tensor of order m which takes in n vectors (where n is between 0 and m inclusive) will return a tensor of order m − n, see Tensor § As multilinear maps for further generalizations and details. The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary within throughout space, in which case we have vector fields and tensor fields, and can also depend on time. Following are some examples: For the electrical conduction example, the index and matrix notations would be: J i = σ i j E j ≡ ∑ j σ i j E j ( J x J y J z ) = ( σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz ) ( E x E y E z ) {\displaystyle {\begin{aligned}J_{i}&=\sigma _{ij}E_{j}\equiv \sum _{j}\sigma _{ij}E_{j}\\{\begin{pmatrix}J_{\text{x}}\\J_{\text{y}}\\J_{\text{z}}\end{pmatrix}}&={\begin{pmatrix}\sigma _{\text{xx}}&\sigma _{\text{xy}}&\sigma _{\text{xz}}\\\sigma _{\text{yx}}&\sigma _{\text{yy}}&\sigma _{\text{yz}}\\\sigma _{\text{zx}}&\sigma _{\text{zy}}&\sigma _{\text{zz}}\end{pmatrix}}{\begin{pmatrix}E_{\text{x}}\\E_{\text{y}}\\E_{\text{z}}\end{pmatrix}}\end{aligned}}} while for the rotational kinetic energy T: T = 1 2 ω i I i j ω j ≡ 1 2 ∑ i j ω i I i j ω j , = 1 2 ( ω x ω y ω z ) ( I xx I xy I xz I yx I yy I yz I zx I zy I zz ) ( ω x ω y ω z ) . {\displaystyle {\begin{aligned}T&={\frac {1}{2}}\omega _{i}I_{ij}\omega _{j}\equiv {\frac {1}{2}}\sum _{ij}\omega _{i}I_{ij}\omega _{j}\,,\\&={\frac {1}{2}}{\begin{pmatrix}\omega _{\text{x}}&\omega _{\text{y}}&\omega _{\text{z}}\end{pmatrix}}{\begin{pmatrix}I_{\text{xx}}&I_{\text{xy}}&I_{\text{xz}}\\I_{\text{yx}}&I_{\text{yy}}&I_{\text{yz}}\\I_{\text{zx}}&I_{\text{zy}}&I_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}\omega _{\text{x}}\\\omega _{\text{y}}\\\omega _{\text{z}}\end{pmatrix}}\,.\end{aligned}}} See also constitutive equation for more specialized examples. === Vectors and tensors in n dimensions === In n-dimensional Euclidean space over the real numbers, R n {\displaystyle \mathbb {R} ^{n}} , the standard basis is denoted e1, e2, e3, ... en. Each basis vector ei points along the positive xi axis, with the basis being orthonormal. Component j of ei is given by the Kronecker delta: ( e i ) j = δ i j {\displaystyle (\mathbf {e} _{i})_{j}=\delta _{ij}} A vector in R n {\displaystyle \mathbb {R} ^{n}} takes the form: a = a i e i ≡ ∑ i a i e i . {\displaystyle \mathbf {a} =a_{i}\mathbf {e} _{i}\equiv \sum _{i}a_{i}\mathbf {e} _{i}\,.} Similarly for the order-2 tensor above, for each vector a and b in R n {\displaystyle \mathbb {R} ^{n}} : T = a i b j e i j ≡ ∑ i j a i b j e i ⊗ e j , {\displaystyle \mathbf {T} =a_{i}b_{j}\mathbf {e} _{ij}\equiv \sum _{ij}a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,} or more generally: T = T i j e i j ≡ ∑ i j T i j e i ⊗ e j . {\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,.} == Transformations of Cartesian vectors (any number of dimensions) == === Meaning of "invariance" under coordinate transformations === The position vector x in R n {\displaystyle \mathbb {R} ^{n}} is a simple and common example of a vector, and can be represented in any coordinate system. Consider the case of rectangular coordinate systems with orthonormal bases only. It is possible to have a coordinate system with rectangular geometry if the basis vectors are all mutually perpendicular and not normalized, in which case the basis is orthogonal but not orthonormal. However, orthonormal bases are easier to manipulate and are often used in practice. The following results are true for orthonormal bases, not orthogonal ones. In one rectangular coordinate system, x as a contravector has coordinates xi and basis vectors ei, while as a covector it has coordinates xi and basis covectors ei, and we have: x = x i e i , x = x i e i {\displaystyle {\begin{aligned}\mathbf {x} &=x^{i}\mathbf {e} _{i}\,,&\mathbf {x} &=x_{i}\mathbf {e} ^{i}\end{aligned}}} In another rectangular coordinate system, x as a contravector has coordinates xi and basis ei, while as a covector it has coordinates xi and basis ei, and we have: x = x ¯ i e ¯ i , x = x ¯ i e ¯ i {\displaystyle {\begin{aligned}\mathbf {x} &={\bar {x}}^{i}{\bar {\mathbf {e} }}_{i}\,,&\mathbf {x} &={\bar {x}}_{i}{\bar {\mathbf {e} }}^{i}\end{aligned}}} Each new coordinate is a function of all the old ones, and vice versa for the inverse function: x ¯ i = x ¯ i ( x 1 , x 2 , … ) ⇌ x i = x i ( x ¯ 1 , x ¯ 2 , … ) x ¯ i = x ¯ i ( x 1 , x 2 , … ) ⇌ x i = x i ( x ¯ 1 , x ¯ 2 , … ) {\displaystyle {\begin{aligned}{\bar {x}}{}^{i}={\bar {x}}{}^{i}\left(x^{1},x^{2},\ldots \right)\quad &\rightleftharpoons \quad x{}^{i}=x{}^{i}\left({\bar {x}}^{1},{\bar {x}}^{2},\ldots \right)\\{\bar {x}}{}_{i}={\bar {x}}{}_{i}\left(x_{1},x_{2},\ldots \right)\quad &\rightleftharpoons \quad x{}_{i}=x{}_{i}\left({\bar {x}}_{1},{\bar {x}}_{2},\ldots \right)\end{aligned}}} and similarly each new basis vector is a function of all the old ones, and vice versa for the inverse function: e ¯ j = e ¯ j ( e 1 , e 2 , … ) ⇌ e j = e j ( e ¯ 1 , e ¯ 2 , … ) e ¯ j = e ¯ j ( e 1 , e 2 , … ) ⇌ e j = e j ( e ¯ 1 , e ¯ 2 , … ) {\displaystyle {\begin{aligned}{\bar {\mathbf {e} }}{}_{j}={\bar {\mathbf {e} }}{}_{j}\left(\mathbf {e} _{1},\mathbf {e} _{2},\ldots \right)\quad &\rightleftharpoons \quad \mathbf {e} {}_{j}=\mathbf {e} {}_{j}\left({\bar {\mathbf {e} }}_{1},{\bar {\mathbf {e} }}_{2},\ldots \right)\\{\bar {\mathbf {e} }}{}^{j}={\bar {\mathbf {e} }}{}^{j}\left(\mathbf {e} ^{1},\mathbf {e} ^{2},\ldots \right)\quad &\rightleftharpoons \quad \mathbf {e} {}^{j}=\mathbf {e} {}^{j}\left({\bar {\mathbf {e} }}^{1},{\bar {\mathbf {e} }}^{2},\ldots \right)\end{aligned}}} for all i, j. A vector is invariant under any change of basis, so if coordinates transform according to a transformation matrix L, the bases transform according to the matrix inverse L−1, and conversely if the coordinates transform according to inverse L−1, the bases transform according to the matrix L. The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are linearly transformed according to the following rules: where Lij represents the entries of the transformation matrix (row number is i and column number is j) and (L−1)ik denotes the entries of the inverse matrix of the matrix Lik. If L is an orthogonal transformation (orthogonal matrix), the objects transforming by it are defined as Cartesian tensors. This geometrically has the interpretation that a rectangular coordinate system is mapped to another rectangular coordinate system, in which the norm of the vector x is preserved (and distances are preserved). The determinant of L is det(L) = ±1, which corresponds to two types of orthogonal transformation: (+1) for rotations and (−1) for improper rotations (including reflections). There are considerable algebraic simplifications, the matrix transpose is the inverse from the definition of an orthogonal transformation: L T = L − 1 ⇒ ( L − 1 ) i j = ( L T ) i j = ( L ) j i = L j i {\displaystyle {\boldsymbol {\mathsf {L}}}^{\textsf {T}}={\boldsymbol {\mathsf {L}}}^{-1}\Rightarrow \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{i}{}^{j}=\left({\boldsymbol {\mathsf {L}}}^{\textsf {T}}\right)_{i}{}^{j}=({\boldsymbol {\mathsf {L}}})^{j}{}_{i}={\mathsf {L}}^{j}{}_{i}} From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between raising and lowering indices, and in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for exponents. All indices will be lowered in the remainder of this article. One can determine the actual raised and lowered indices by considering which quantities are covectors or contravectors, and the relevant transformation rules. Exactly the same transformation rules apply to any vector a, not only the position vector. If its components ai do not transform according to the rules, a is not a vector. Despite the similarity between the expressions above, for the change of coordinates such as xj = Lijxi, and the action of a tensor on a vector like bi = Tij aj, L is not a tensor, but T is. In the change of coordinates, L is a matrix, used to relate two rectangular coordinate systems with orthonormal bases together. For the tensor relating a vector to a vector, the vectors and tensors throughout the equation all belong to the same coordinate system and basis. === Derivatives and Jacobian matrix elements === The entries of L are partial derivatives of the new or old coordinates with respect to the old or new coordinates, respectively. Differentiating xi with respect to xk: ∂ x ¯ i ∂ x k = ∂ ∂ x k ( x j L j i ) = L j i ∂ x j ∂ x k = δ k j L j i = L k i {\displaystyle {\frac {\partial {\bar {x}}_{i}}{\partial x_{k}}}={\frac {\partial }{\partial x_{k}}}(x_{j}{\mathsf {L}}_{ji})={\mathsf {L}}_{ji}{\frac {\partial x_{j}}{\partial x_{k}}}=\delta _{kj}{\mathsf {L}}_{ji}={\mathsf {L}}_{ki}} so L i j ≡ L i j = ∂ x ¯ j ∂ x i {\displaystyle {{\mathsf {L}}_{i}}^{j}\equiv {\mathsf {L}}_{ij}={\frac {\partial {\bar {x}}_{j}}{\partial x_{i}}}} is an element of the Jacobian matrix. There is a (partially mnemonical) correspondence between index positions attached to L and in the partial derivative: i at the top and j at the bottom, in each case, although for Cartesian tensors the indices can be lowered. Conversely, differentiating xj with respect to xi: ∂ x j ∂ x ¯ k = ∂ ∂ x ¯ k ( x ¯ i ( L − 1 ) i j ) = ∂ x ¯ i ∂ x ¯ k ( L − 1 ) i j = δ k i ( L − 1 ) i j = ( L − 1 ) k j {\displaystyle {\frac {\partial x_{j}}{\partial {\bar {x}}_{k}}}={\frac {\partial }{\partial {\bar {x}}_{k}}}\left({\bar {x}}_{i}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}\right)={\frac {\partial {\bar {x}}_{i}}{\partial {\bar {x}}_{k}}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}=\delta _{ki}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{kj}} so ( L − 1 ) i j ≡ ( L − 1 ) i j = ∂ x j ∂ x ¯ i {\displaystyle \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{i}{}^{j}\equiv \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}={\frac {\partial x_{j}}{\partial {\bar {x}}_{i}}}} is an element of the inverse Jacobian matrix, with a similar index correspondence. Many sources state transformations in terms of the partial derivatives: and the explicit matrix equations in 3d are: x ¯ = L x ( x ¯ 1 x ¯ 2 x ¯ 3 ) = ( ∂ x ¯ 1 ∂ x 1 ∂ x ¯ 1 ∂ x 2 ∂ x ¯ 1 ∂ x 3 ∂ x ¯ 2 ∂ x 1 ∂ x ¯ 2 ∂ x 2 ∂ x ¯ 2 ∂ x 3 ∂ x ¯ 3 ∂ x 1 ∂ x ¯ 3 ∂ x 2 ∂ x ¯ 3 ∂ x 3 ) ( x 1 x 2 x 3 ) {\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&={\boldsymbol {\mathsf {L}}}\mathbf {x} \\{\begin{pmatrix}{\bar {x}}_{1}\\{\bar {x}}_{2}\\{\bar {x}}_{3}\end{pmatrix}}&={\begin{pmatrix}{\frac {\partial {\bar {x}}_{1}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{1}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{1}}{\partial x_{3}}}\\{\frac {\partial {\bar {x}}_{2}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{2}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{2}}{\partial x_{3}}}\\{\frac {\partial {\bar {x}}_{3}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{3}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{3}}{\partial x_{3}}}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\end{aligned}}} similarly for x = L − 1 x ¯ = L T x ¯ {\displaystyle \mathbf {x} ={\boldsymbol {\mathsf {L}}}^{-1}{\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}^{\textsf {T}}{\bar {\mathbf {x} }}} === Projections along coordinate axes === As with all linear transformations, L depends on the basis chosen. For two orthonormal bases e ¯ i ⋅ e ¯ j = e i ⋅ e j = δ i j , \| e i \| = \| e ¯ i \| = 1 , {\displaystyle {\begin{aligned}{\bar {\mathbf {e} }}_{i}\cdot {\bar {\mathbf {e} }}_{j}&=\mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}\,,&\left\|\mathbf {e} _{i}\right\|&=\left\|{\bar {\mathbf {e} }}_{i}\right\|=1\,,\end{aligned}}} projecting x to the x axes: x ¯ i = e ¯ i ⋅ x = e ¯ i ⋅ x j e j = x i L i j , {\displaystyle {\bar {x}}_{i}={\bar {\mathbf {e} }}_{i}\cdot \mathbf {x} ={\bar {\mathbf {e} }}_{i}\cdot x_{j}\mathbf {e} _{j}=x_{i}{\mathsf {L}}_{ij}\,,} projecting x to the x axes: x i = e i ⋅ x = e i ⋅ x ¯ j e ¯ j = x ¯ j ( L − 1 ) j i . {\displaystyle x_{i}=\mathbf {e} _{i}\cdot \mathbf {x} =\mathbf {e} _{i}\cdot {\bar {x}}_{j}{\bar {\mathbf {e} }}_{j}={\bar {x}}_{j}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ji}\,.} Hence the components reduce to direction cosines between the xi and xj axes: L i j = e ¯ i ⋅ e j = cos ⁡ θ i j ( L − 1 ) i j = e i ⋅ e ¯ j = cos ⁡ θ j i {\displaystyle {\begin{aligned}{\mathsf {L}}_{ij}&={\bar {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}=\cos \theta _{ij}\\\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ij}&=\mathbf {e} _{i}\cdot {\bar {\mathbf {e} }}_{j}=\cos \theta _{ji}\end{aligned}}} where θij and θji are the angles between the xi and xj axes. In general, θij is not equal to θji, because for example θ12 and θ21 are two different angles. The transformation of coordinates can be written: and the explicit matrix equations in 3d are: x ¯ = L x ( x ¯ 1 x ¯ 2 x ¯ 3 ) = ( e ¯ 1 ⋅ e 1 e ¯ 1 ⋅ e 2 e ¯ 1 ⋅ e 3 e ¯ 2 ⋅ e 1 e ¯ 2 ⋅ e 2 e ¯ 2 ⋅ e 3 e ¯ 3 ⋅ e 1 e ¯ 3 ⋅ e 2 e ¯ 3 ⋅ e 3 ) ( x 1 x 2 x 3 ) = ( cos ⁡ θ 11 cos ⁡ θ 12 cos ⁡ θ 13 cos ⁡ θ 21 cos ⁡ θ 22 cos ⁡ θ 23 cos ⁡ θ 31 cos ⁡ θ 32 cos ⁡ θ 33 ) ( x 1 x 2 x 3 ) {\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&={\boldsymbol {\mathsf {L}}}\mathbf {x} \\{\begin{pmatrix}{\bar {x}}_{1}\\{\bar {x}}_{2}\\{\bar {x}}_{3}\end{pmatrix}}&={\begin{pmatrix}{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{3}\\{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{3}\\{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{3}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{11}&\cos \theta _{12}&\cos \theta _{13}\\\cos \theta _{21}&\cos \theta _{22}&\cos \theta _{23}\\\cos \theta _{31}&\cos \theta _{32}&\cos \theta _{33}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\end{aligned}}} similarly for x = L − 1 x ¯ = L T x ¯ {\displaystyle \mathbf {x} ={\boldsymbol {\mathsf {L}}}^{-1}{\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}^{\textsf {T}}{\bar {\mathbf {x} }}} The geometric interpretation is the xi components equal to the sum of projecting the xj components onto the xj axes. The numbers ei⋅ej arranged into a matrix would form a symmetric matrix (a matrix equal to its own transpose) due to the symmetry in the dot products, in fact it is the metric tensor g. By contrast ei⋅ej or ei⋅ej do not form symmetric matrices in general, as displayed above. Therefore, while the L matrices are still orthogonal, they are not symmetric. Apart from a rotation about any one axis, in which the xi and xi for some i coincide, the angles are not the same as Euler angles, and so the L matrices are not the same as the rotation matrices. == Transformation of the dot and cross products (three dimensions only) == The dot product and cross product occur very frequently, in applications of vector analysis to physics and engineering, examples include: power transferred P by an object exerting a force F with velocity v along a straight-line path: P = v ⋅ F {\displaystyle P=\mathbf {v} \cdot \mathbf {F} } tangential velocity v at a point x of a rotating rigid body with angular velocity ω: v = ω × x {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} } potential energy U of a magnetic dipole of magnetic moment m in a uniform external magnetic field B: U = − m ⋅ B {\displaystyle U=-\mathbf {m} \cdot \mathbf {B} } angular momentum J for a particle with position vector r and momentum p: J = r × p {\displaystyle \mathbf {J} =\mathbf {r} \times \mathbf {p} } torque τ acting on an electric dipole of electric dipole moment p in a uniform external electric field E: τ = p × E {\displaystyle {\boldsymbol {\tau }}=\mathbf {p} \times \mathbf {E} } induced surface current density jS in a magnetic material of magnetization M on a surface with unit normal n: j S = M × n {\displaystyle \mathbf {j} _{\mathrm {S} }=\mathbf {M} \times \mathbf {n} } How these products transform under orthogonal transformations is illustrated below. === Dot product, Kronecker delta, and metric tensor === The dot product ⋅ of each possible pairing of the basis vectors follows from the basis being orthonormal. For perpendicular pairs we have e x ⋅ e y = e y ⋅ e z = e z ⋅ e x = e y ⋅ e x = e z ⋅ e y = e x ⋅ e z = 0 {\displaystyle {\begin{array}{llll}\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{z}}&=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{x}}&=\\\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{x}}&=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{z}}&=0\end{array}}} while for parallel pairs we have e x ⋅ e x = e y ⋅ e y = e z ⋅ e z = 1. {\displaystyle \mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{z}}=1.} Replacing Cartesian labels by index notation as shown above, these results can be summarized by e i ⋅ e j = δ i j {\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}} where δij are the components of the Kronecker delta. The Cartesian basis can be used to represent δ in this way. In addition, each metric tensor component gij with respect to any basis is the dot product of a pairing of basis vectors: g i j = e i ⋅ e j . {\displaystyle g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}.} For the Cartesian basis the components arranged into a matrix are: g = ( g xx g xy g xz g yx g yy g yz g zx g zy g zz ) = ( e x ⋅ e x e x ⋅ e y e x ⋅ e z e y ⋅ e x e y ⋅ e y e y ⋅ e z e z ⋅ e x e z ⋅ e y e z ⋅ e z ) = ( 1 0 0 0 1 0 0 0 1 ) {\displaystyle \mathbf {g} ={\begin{pmatrix}g_{\text{xx}}&g_{\text{xy}}&g_{\text{xz}}\\g_{\text{yx}}&g_{\text{yy}}&g_{\text{yz}}\\g_{\text{zx}}&g_{\text{zy}}&g_{\text{zz}}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{z}}\\\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{z}}\\\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{z}}\\\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}} so are the simplest possible for the metric tensor, namely the δ: g i j = δ i j {\displaystyle g_{ij}=\delta _{ij}} This is not true for general bases: orthogonal coordinates have diagonal metrics containing various scale factors (i.e. not necessarily 1), while general curvilinear coordinates could also have nonzero entries for off-diagonal components. The dot product of two vectors a and b transforms according to a ⋅ b = a ¯ j b ¯ j = a i L i j b k ( L − 1 ) j k = a i δ i k b k = a i b i {\displaystyle \mathbf {a} \cdot \mathbf {b} ={\bar {a}}_{j}{\bar {b}}_{j}=a_{i}{\mathsf {L}}_{ij}b_{k}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{jk}=a_{i}\delta _{i}{}_{k}b_{k}=a_{i}b_{i}} which is intuitive, since the dot product of two vectors is a single scalar independent of any coordinates. This also applies more generally to any coordinate systems, not just rectangular ones; the dot product in one coordinate system is the same in any other. === Cross product, Levi-Civita symbol, and pseudovectors === For the cross product (×) of two vectors, the results are (almost) the other way round. Again, assuming a right-handed 3d Cartesian coordinate system, cyclic permutations in perpendicular directions yield the next vector in the cyclic collection of vectors: e x × e y = e z e y × e z = e x e z × e x = e y e y × e x = − e z e z × e y = − e x e x × e z = − e y {\displaystyle {\begin{aligned}\mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{z}}&\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{z}}&=\mathbf {e} _{\text{x}}&\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{x}}&=\mathbf {e} _{\text{y}}\\[1ex]\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{x}}&=-\mathbf {e} _{\text{z}}&\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{y}}&=-\mathbf {e} _{\text{x}}&\mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{z}}&=-\mathbf {e} _{\text{y}}\end{aligned}}} while parallel vectors clearly vanish: e x × e x = e y × e y = e z × e z = 0 {\displaystyle \mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{z}}={\boldsymbol {0}}} and replacing Cartesian labels by index notation as above, these can be summarized by: e i × e j = { + e k cyclic permutations: ( i , j , k ) = ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) − e k anticyclic permutations: ( i , j , k ) = ( 2 , 1 , 3 ) , ( 3 , 2 , 1 ) , ( 1 , 3 , 2 ) 0 i = j {\displaystyle \mathbf {e} _{i}\times \mathbf {e} _{j}={\begin{cases}+\mathbf {e} _{k}&{\text{cyclic permutations: }}(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\[2pt]-\mathbf {e} _{k}&{\text{anticyclic permutations: }}(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\[2pt]{\boldsymbol {0}}&i=j\end{cases}}} where i, j, k are indices which take values 1, 2, 3. It follows that: e k ⋅ e i × e j = { + 1 cyclic permutations: ( i , j , k ) = ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) , ( 3 , 1 , 2 ) − 1 anticyclic permutations: ( i , j , k ) = ( 2 , 1 , 3 ) , ( 3 , 2 , 1 ) , ( 1 , 3 , 2 ) 0 i = j or j = k or k = i {\displaystyle {\mathbf {e} _{k}\cdot \mathbf {e} _{i}\times \mathbf {e} _{j}}={\begin{cases}+1&{\text{cyclic permutations: }}(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\[2pt]-1&{\text{anticyclic permutations: }}(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\[2pt]0&i=j{\text{ or }}j=k{\text{ or }}k=i\end{cases}}} These permutation relations and their corresponding values are important, and there is an object coinciding with this property: the Levi-Civita symbol, denoted by ε. The Levi-Civita symbol entries can be represented by the Cartesian basis: ε i j k = e i ⋅ e j × e k {\displaystyle \varepsilon _{ijk}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}\times \mathbf {e} _{k}} which geometrically corresponds to the volume of a cube spanned by the orthonormal basis vectors, with sign indicating orientation (and not a "positive or negative volume"). Here, the orientation is fixed by ε123 = +1, for a right-handed system. A left-handed system would fix ε123 = −1 or equivalently ε321 = +1. The scalar triple product can now be written: c ⋅ a × b = c i e i ⋅ a j e j × b k e k = ε i j k c i a j b k {\displaystyle \mathbf {c} \cdot \mathbf {a} \times \mathbf {b} =c_{i}\mathbf {e} _{i}\cdot a_{j}\mathbf {e} _{j}\times b_{k}\mathbf {e} _{k}=\varepsilon _{ijk}c_{i}a_{j}b_{k}} with the geometric interpretation of volume (of the parallelepiped spanned by a, b, c) and algebraically is a determinant:: 23 c ⋅ a × b = \| c x a x b x c y a y b y c z a z b z \| {\displaystyle \mathbf {c} \cdot \mathbf {a} \times \mathbf {b} ={\begin{vmatrix}c_{\text{x}}&a_{\text{x}}&b_{\text{x}}\\c_{\text{y}}&a_{\text{y}}&b_{\text{y}}\\c_{\text{z}}&a_{\text{z}}&b_{\text{z}}\end{vmatrix}}} This in turn can be used to rewrite the cross product of two vectors as follows: ( a × b ) i = e i ⋅ a × b = ε ℓ j k ( e i ) ℓ a j b k = ε ℓ j k δ i ℓ a j b k = ε i j k a j b k ⇒ a × b = ( a × b ) i e i = ε i j k a j b k e i {\displaystyle {\begin{aligned}(\mathbf {a} \times \mathbf {b} )_{i}={\mathbf {e} _{i}\cdot \mathbf {a} \times \mathbf {b} }&=\varepsilon _{\ell jk}{(\mathbf {e} _{i})}_{\ell }a_{j}b_{k}=\varepsilon _{\ell jk}\delta _{i\ell }a_{j}b_{k}=\varepsilon _{ijk}a_{j}b_{k}\\\Rightarrow \quad {\mathbf {a} \times \mathbf {b} }=(\mathbf {a} \times \mathbf {b} )_{i}\mathbf {e} _{i}&=\varepsilon _{ijk}a_{j}b_{k}\mathbf {e} _{i}\end{aligned}}} Contrary to its appearance, the Levi-Civita symbol is not a tensor, but a pseudotensor, the components transform according to: ε ¯ p q r = det ( L ) ε i j k L i p L j q L k r . {\displaystyle {\bar {\varepsilon }}_{pqr}=\det({\boldsymbol {\mathsf {L}}})\varepsilon _{ijk}{\mathsf {L}}_{ip}{\mathsf {L}}_{jq}{\mathsf {L}}_{kr}\,.} Therefore, the transformation of the cross product of a and b is: ( a ¯ × b ¯ ) i = ε ¯ i j k a ¯ j b ¯ k = det ( L ) ε p q r L p i L q j L r k a m L m j b n L n k = det ( L ) ε p q r L p i L q j ( L − 1 ) j m L r k ( L − 1 ) k n a m b n = det ( L ) ε p q r L p i δ q m δ r n a m b n = det ( L ) L p i ε p q r a q b r = det ( L ) ( a × b ) p L p i {\displaystyle {\begin{aligned}&\left({\bar {\mathbf {a} }}\times {\bar {\mathbf {b} }}\right)_{i}\\[1ex]{}={}&{\bar {\varepsilon }}_{ijk}{\bar {a}}_{j}{\bar {b}}_{k}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}{\mathsf {L}}_{qj}{\mathsf {L}}_{rk}\;\;a_{m}{\mathsf {L}}_{mj}\;\;b_{n}{\mathsf {L}}_{nk}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}\;\;{\mathsf {L}}_{qj}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{jm}\;\;{\mathsf {L}}_{rk}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{kn}\;\;a_{m}\;\;b_{n}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}\;\;\delta _{qm}\;\;\delta _{rn}\;\;a_{m}\;\;b_{n}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;{\mathsf {L}}_{pi}\;\;\varepsilon _{pqr}a_{q}b_{r}\\[1ex]{}={}&\det({\boldsymbol {\mathsf {L}}})\;\;(\mathbf {a} \times \mathbf {b} )_{p}{\mathsf {L}}_{pi}\end{aligned}}} and so a × b transforms as a pseudovector, because of the determinant factor. The tensor index notation applies to any object which has entities that form multidimensional arrays – not everything with indices is a tensor by default. Instead, tensors are defined by how their coordinates and basis elements change under a transformation from one coordinate system to another. Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector. === Applications of the δ tensor and ε pseudotensor === Other identities can be formed from the δ tensor and ε pseudotensor, a notable and very useful identity is one that converts two Levi-Civita symbols adjacently contracted over two indices into an antisymmetrized combination of Kronecker deltas: ε i j k ε p q k = δ i p δ j q − δ i q δ j p {\displaystyle \varepsilon _{ijk}\varepsilon _{pqk}=\delta _{ip}\delta _{jq}-\delta _{iq}\delta _{jp}} The index forms of the dot and cross products, together with this identity, greatly facilitate the manipulation and derivation of other identities in vector calculus and algebra, which in turn are used extensively in physics and engineering. For instance, it is clear the dot and cross products are distributive over vector addition: a ⋅ ( b + c ) = a i ( b i + c i ) = a i b i + a i c i = a ⋅ b + a ⋅ c a × ( b + c ) = e i ε i j k a j ( b k + c k ) = e i ε i j k a j b k + e i ε i j k a j c k = a × b + a × c {\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )&=a_{i}(b_{i}+c_{i})=a_{i}b_{i}+a_{i}c_{i}=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} \\[1ex]\mathbf {a} \times (\mathbf {b} +\mathbf {c} )&=\mathbf {e} _{i}\varepsilon _{ijk}a_{j}(b_{k}+c_{k})=\mathbf {e} _{i}\varepsilon _{ijk}a_{j}b_{k}+\mathbf {e} _{i}\varepsilon _{ijk}a_{j}c_{k}=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} \end{aligned}}} without resort to any geometric constructions – the derivation in each case is a quick line of algebra. Although the procedure is less obvious, the vector triple product can also be derived. Rewriting in index notation: [ a × ( b × c ) ] i = ε i j k a j ( ε k ℓ m b ℓ c m ) = ( ε i j k ε k ℓ m ) a j b ℓ c m {\displaystyle \left[\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\right]_{i}=\varepsilon _{ijk}a_{j}(\varepsilon _{k\ell m}b_{\ell }c_{m})=(\varepsilon _{ijk}\varepsilon _{k\ell m})a_{j}b_{\ell }c_{m}} and because cyclic permutations of indices in the ε symbol does not change its value, cyclically permuting indices in εkℓm to obtain εℓmk allows us to use the above δ-ε identity to convert the ε symbols into δ tensors: [ a × ( b × c ) ] i = ( δ i ℓ δ j m − δ i m δ j ℓ ) a j b ℓ c m = δ i ℓ δ j m a j b ℓ c m − δ i m δ j ℓ a j b ℓ c m = a j b i c j − a j b j c i = [ ( a ⋅ c ) b − ( a ⋅ b ) c ] i {\displaystyle {\begin{aligned}\left[\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\right]_{i}{}={}&\left(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell }\right)a_{j}b_{\ell }c_{m}\\{}={}&\delta _{i\ell }\delta _{jm}a_{j}b_{\ell }c_{m}-\delta _{im}\delta _{j\ell }a_{j}b_{\ell }c_{m}\\{}={}&a_{j}b_{i}c_{j}-a_{j}b_{j}c_{i}\\{}={}&\left[(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} \right]_{i}\end{aligned}}} thusly: a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} } Note this is antisymmetric in b and c, as expected from the left hand side. Similarly, via index notation or even just cyclically relabelling a, b, and c in the previous result and taking the negative: ( a × b ) × c = ( c ⋅ a ) b − ( c ⋅ b ) a {\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} -(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} } and the difference in results show that the cross product is not associative. More complex identities, like quadruple products; ( a × b ) ⋅ ( c × d ) , ( a × b ) × ( c × d ) , … {\displaystyle (\mathbf {a} \times \mathbf {b} )\cdot (\mathbf {c} \times \mathbf {d} ),\quad (\mathbf {a} \times \mathbf {b} )\times (\mathbf {c} \times \mathbf {d} ),\quad \ldots } and so on, can be derived in a similar manner. == Transformations of Cartesian tensors (any number of dimensions) == Tensors are defined as quantities which transform in a certain way under linear transformations of coordinates. === Second order === Let a = aiei and b = biei be two vectors, so that they transform according to aj = aiLij, bj = biLij. Taking the tensor product gives: a ⊗ b = a i e i ⊗ b j e j = a i b j e i ⊗ e j {\displaystyle \mathbf {a} \otimes \mathbf {b} =a_{i}\mathbf {e} _{i}\otimes b_{j}\mathbf {e} _{j}=a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}} then applying the transformation to the components a ¯ p b ¯ q = a i L i p b j L j q = L i p L j q a i b j {\displaystyle {\bar {a}}_{p}{\bar {b}}_{q}=a_{i}{\mathsf {L}}_{i}{}_{p}b_{j}{\mathsf {L}}_{j}{}_{q}={\mathsf {L}}_{i}{}_{p}{\mathsf {L}}_{j}{}_{q}a_{i}b_{j}} and to the bases e ¯ p ⊗ e ¯ q = ( L − 1 ) p i e i ⊗ ( L − 1 ) q j e j = ( L − 1 ) p i ( L − 1 ) q j e i ⊗ e j = L i p − 1 L j q − 1 e i ⊗ e j {\displaystyle {\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}\mathbf {e} _{i}\otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\mathbf {e} _{j}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\mathbf {e} _{i}\otimes \mathbf {e} _{j}={\mathsf {L}}_{ip}^{-1}{\mathsf {L}}_{jq}^{-1}\mathbf {e} _{i}\otimes \mathbf {e} _{j}} gives the transformation law of an order-2 tensor. The tensor a⊗b is invariant under this transformation: a ¯ p b ¯ q e ¯ p ⊗ e ¯ q = L k p L ℓ q a k b ℓ ( L − 1 ) p i ( L − 1 ) q j e i ⊗ e j = L k p ( L − 1 ) p i L ℓ q ( L − 1 ) q j a k b ℓ e i ⊗ e j = δ k i δ ℓ j a k b ℓ e i ⊗ e j = a i b j e i ⊗ e j {\displaystyle {\begin{aligned}{\bar {a}}_{p}{\bar {b}}_{q}{\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}{}={}&{\mathsf {L}}_{kp}{\mathsf {L}}_{\ell q}a_{k}b_{\ell }\,\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\[1ex]{}={}&{\mathsf {L}}_{kp}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{pi}{\mathsf {L}}_{\ell q}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{qj}\,a_{k}b_{\ell }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\[1ex]{}={}&\delta _{k}{}_{i}\delta _{\ell j}\,a_{k}b_{\ell }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\[1ex]{}={}&a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\end{aligned}}} More generally, for any order-2 tensor R = R i j e i ⊗ e j , {\displaystyle \mathbf {R} =R_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,} the components transform according to; R ¯ p q = L i p L j q R i j , {\displaystyle {\bar {R}}_{pq}={\mathsf {L}}_{i}{}_{p}{\mathsf {L}}_{j}{}_{q}R_{ij},} and the basis transforms by: e ¯ p ⊗ e ¯ q = ( L − 1 ) i p e i ⊗ ( L − 1 ) j q e j {\displaystyle {\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{ip}\mathbf {e} _{i}\otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{jq}\mathbf {e} _{j}} If R does not transform according to this rule – whatever quantity R may be – it is not an order-2 tensor. === Any order === More generally, for any order p tensor T = T j 1 j 2 ⋯ j p e j 1 ⊗ e j 2 ⊗ ⋯ e j p {\displaystyle \mathbf {T} =T_{j_{1}j_{2}\cdots j_{p}}\mathbf {e} _{j_{1}}\otimes \mathbf {e} _{j_{2}}\otimes \cdots \mathbf {e} _{j_{p}}} the components transform according to; T ¯ j 1 j 2 ⋯ j p = L i 1 j 1 L i 2 j 2 ⋯ L i p j p T i 1 i 2 ⋯ i p {\displaystyle {\bar {T}}_{j_{1}j_{2}\cdots j_{p}}={\mathsf {L}}_{i_{1}j_{1}}{\mathsf {L}}_{i_{2}j_{2}}\cdots {\mathsf {L}}_{i_{p}j_{p}}T_{i_{1}i_{2}\cdots i_{p}}} and the basis transforms by: e ¯ j 1 ⊗ e ¯ j 2 ⋯ ⊗ e ¯ j p = ( L − 1 ) j 1 i 1 e i 1 ⊗ ( L − 1 ) j 2 i 2 e i 2 ⋯ ⊗ ( L − 1 ) j p i p e i p {\displaystyle {\bar {\mathbf {e} }}_{j_{1}}\otimes {\bar {\mathbf {e} }}_{j_{2}}\cdots \otimes {\bar {\mathbf {e} }}_{j_{p}}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j_{1}i_{1}}\mathbf {e} _{i_{1}}\otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j_{2}i_{2}}\mathbf {e} _{i_{2}}\cdots \otimes \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{j_{p}i_{p}}\mathbf {e} _{i_{p}}} For a pseudotensor S of order p, the components transform according to; S ¯ j 1 j 2 ⋯ j p = det ( L ) L i 1 j 1 L i 2 j 2 ⋯ L i p j p S i 1 i 2 ⋯ i p . {\displaystyle {\bar {S}}_{j_{1}j_{2}\cdots j_{p}}=\det({\boldsymbol {\mathsf {L}}}){\mathsf {L}}_{i_{1}j_{1}}{\mathsf {L}}_{i_{2}j_{2}}\cdots {\mathsf {L}}_{i_{p}j_{p}}S_{i_{1}i_{2}\cdots i_{p}}\,.} == Pseudovectors as antisymmetric second order tensors == The antisymmetric nature of the cross product can be recast into a tensorial form as follows. Let c be a vector, a be a pseudovector, b be another vector, and T be a second order tensor such that: c = a × b = T ⋅ b {\displaystyle \mathbf {c} =\mathbf {a} \times \mathbf {b} =\mathbf {T} \cdot \mathbf {b} } As the cross product is linear in a and b, the components of T can be found by inspection, and they are: T = ( 0 − a z a y a z 0 − a x − a y a x 0 ) {\displaystyle \mathbf {T} ={\begin{pmatrix}0&-a_{\text{z}}&a_{\text{y}}\\a_{\text{z}}&0&-a_{\text{x}}\\-a_{\text{y}}&a_{\text{x}}&0\\\end{pmatrix}}} so the pseudovector a can be written as an antisymmetric tensor. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by v = ω × x, this can be rewritten as v = Ω ⋅ x where Ω is the tensor corresponding to the pseudovector ω: Ω = ( 0 − ω z ω y ω z 0 − ω x − ω y ω x 0 ) {\displaystyle {\boldsymbol {\Omega }}={\begin{pmatrix}0&-\omega _{\text{z}}&\omega _{\text{y}}\\\omega _{\text{z}}&0&-\omega _{\text{x}}\\-\omega _{\text{y}}&\omega _{\text{x}}&0\\\end{pmatrix}}} For an example in electromagnetism, while the electric field E is a vector field, the magnetic field B is a pseudovector field. These fields are defined from the Lorentz force for a particle of electric charge q traveling at velocity v: F = q ( E + v × B ) = q ( E − B × v ) {\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )=q(\mathbf {E} -\mathbf {B} \times \mathbf {v} )} and considering the second term containing the cross product of a pseudovector B and velocity vector v, it can be written in matrix form, with F, E, and v as column vectors and B as an antisymmetric matrix: ( F x F y F z ) = q ( E x E y E z ) − q ( 0 − B z B y B z 0 − B x − B y B x 0 ) ( v x v y v z ) {\displaystyle {\begin{pmatrix}F_{\text{x}}\\F_{\text{y}}\\F_{\text{z}}\\\end{pmatrix}}=q{\begin{pmatrix}E_{\text{x}}\\E_{\text{y}}\\E_{\text{z}}\\\end{pmatrix}}-q{\begin{pmatrix}0&-B_{\text{z}}&B_{\text{y}}\\B_{\text{z}}&0&-B_{\text{x}}\\-B_{\text{y}}&B_{\text{x}}&0\\\end{pmatrix}}{\begin{pmatrix}v_{\text{x}}\\v_{\text{y}}\\v_{\text{z}}\\\end{pmatrix}}} If a pseudovector is explicitly given by a cross product of two vectors (as opposed to entering the cross product with another vector), then such pseudovectors can also be written as antisymmetric tensors of second order, with each entry a component of the cross product. The angular momentum of a classical pointlike particle orbiting about an axis, defined by J = x × p, is another example of a pseudovector, with corresponding antisymmetric tensor: J = ( 0 − J z J y J z 0 − J x − J y J x 0 ) = ( 0 − ( x p y − y p x ) ( z p x − x p z ) ( x p y − y p x ) 0 − ( y p z − z p y ) − ( z p x − x p z ) ( y p z − z p y ) 0 ) {\displaystyle \mathbf {J} ={\begin{pmatrix}0&-J_{\text{z}}&J_{\text{y}}\\J_{\text{z}}&0&-J_{\text{x}}\\-J_{\text{y}}&J_{\text{x}}&0\\\end{pmatrix}}={\begin{pmatrix}0&-(xp_{\text{y}}-yp_{\text{x}})&(zp_{\text{x}}-xp_{\text{z}})\\(xp_{\text{y}}-yp_{\text{x}})&0&-(yp_{\text{z}}-zp_{\text{y}})\\-(zp_{\text{x}}-xp_{\text{z}})&(yp_{\text{z}}-zp_{\text{y}})&0\\\end{pmatrix}}} Although Cartesian tensors do not occur in the theory of relativity; the tensor form of orbital angular momentum J enters the spacelike part of the relativistic angular momentum tensor, and the above tensor form of the magnetic field B enters the spacelike part of the electromagnetic tensor. == Vector and tensor calculus == The following formulae are only so simple in Cartesian coordinates – in general curvilinear coordinates there are factors of the metric and its determinant – see tensors in curvilinear coordinates for more general analysis. === Vector calculus === Following are the differential operators of vector calculus. Throughout, let Φ(r, t) be a scalar field, and A ( r , t ) = A x ( r , t ) e x + A y ( r , t ) e y + A z ( r , t ) e z B ( r , t ) = B x ( r , t ) e x + B y ( r , t ) e y + B z ( r , t ) e z {\displaystyle {\begin{aligned}\mathbf {A} (\mathbf {r} ,t)&=A_{\text{x}}(\mathbf {r} ,t)\mathbf {e} _{\text{x}}+A_{\text{y}}(\mathbf {r} ,t)\mathbf {e} _{\text{y}}+A_{\text{z}}(\mathbf {r} ,t)\mathbf {e} _{\text{z}}\\[1ex]\mathbf {B} (\mathbf {r} ,t)&=B_{\text{x}}(\mathbf {r} ,t)\mathbf {e} _{\text{x}}+B_{\text{y}}(\mathbf {r} ,t)\mathbf {e} _{\text{y}}+B_{\text{z}}(\mathbf {r} ,t)\mathbf {e} _{\text{z}}\end{aligned}}} be vector fields, in which all scalar and vector fields are functions of the position vector r and time t. The gradient operator in Cartesian coordinates is given by: ∇ = e x ∂ ∂ x + e y ∂ ∂ y + e z ∂ ∂ z {\displaystyle \nabla =\mathbf {e} _{\text{x}}{\frac {\partial }{\partial x}}+\mathbf {e} _{\text{y}}{\frac {\partial }{\partial y}}+\mathbf {e} _{\text{z}}{\frac {\partial }{\partial z}}} and in index notation, this is usually abbreviated in various ways: ∇ i ≡ ∂ i ≡ ∂ ∂ x i {\displaystyle \nabla _{i}\equiv \partial _{i}\equiv {\frac {\partial }{\partial x_{i}}}} This operator acts on a scalar field Φ to obtain the vector field directed in the maximum rate of increase of Φ: ( ∇ Φ ) i = ∇ i Φ {\displaystyle \left(\nabla \Phi \right)_{i}=\nabla _{i}\Phi } The index notation for the dot and cross products carries over to the differential operators of vector calculus.: 197 The directional derivative of a scalar field Φ is the rate of change of Φ along some direction vector a (not necessarily a unit vector), formed out of the components of a and the gradient: a ⋅ ( ∇ Φ ) = a j ( ∇ Φ ) j {\displaystyle \mathbf {a} \cdot (\nabla \Phi )=a_{j}(\nabla \Phi )_{j}} The divergence of a vector field A is: ∇ ⋅ A = ∇ i A i {\displaystyle \nabla \cdot \mathbf {A} =\nabla _{i}A_{i}} Note the interchange of the components of the gradient and vector field yields a different differential operator A ⋅ ∇ = A i ∇ i {\displaystyle \mathbf {A} \cdot \nabla =A_{i}\nabla _{i}} which could act on scalar or vector fields. In fact, if A is replaced by the velocity field u(r, t) of a fluid, this is a term in the material derivative (with many other names) of continuum mechanics, with another term being the partial time derivative: D D t = ∂ ∂ t + u ⋅ ∇ {\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla } which usually acts on the velocity field leading to the non-linearity in the Navier-Stokes equations. As for the curl of a vector field A, this can be defined as a pseudovector field by means of the ε symbol: ( ∇ × A ) i = ε i j k ∇ j A k {\displaystyle \left(\nabla \times \mathbf {A} \right)_{i}=\varepsilon _{ijk}\nabla _{j}A_{k}} which is only valid in three dimensions, or an antisymmetric tensor field of second order via antisymmetrization of indices, indicated by delimiting the antisymmetrized indices by square brackets (see Ricci calculus): ( ∇ × A ) i j = ∇ i A j − ∇ j A i = 2 ∇ [ i A j ] {\displaystyle \left(\nabla \times \mathbf {A} \right)_{ij}=\nabla _{i}A_{j}-\nabla _{j}A_{i}=2\nabla _{[i}A_{j]}} which is valid in any number of dimensions. In each case, the order of the gradient and vector field components should not be interchanged as this would result in a different differential operator: ε i j k A j ∇ k = A i ∇ j − A j ∇ i = 2 A [ i ∇ j ] {\displaystyle \varepsilon _{ijk}A_{j}\nabla _{k}=A_{i}\nabla _{j}-A_{j}\nabla _{i}=2A_{[i}\nabla _{j]}} which could act on scalar or vector fields. Finally, the Laplacian operator is defined in two ways, the divergence of the gradient of a scalar field Φ: ∇ ⋅ ( ∇ Φ ) = ∇ i ( ∇ i Φ ) {\displaystyle \nabla \cdot (\nabla \Phi )=\nabla _{i}(\nabla _{i}\Phi )} or the square of the gradient operator, which acts on a scalar field Φ or a vector field A: ( ∇ ⋅ ∇ ) Φ = ( ∇ i ∇ i ) Φ ( ∇ ⋅ ∇ ) A = ( ∇ i ∇ i ) A {\displaystyle {\begin{aligned}(\nabla \cdot \nabla )\Phi &=(\nabla _{i}\nabla _{i})\Phi \\(\nabla \cdot \nabla )\mathbf {A} &=(\nabla _{i}\nabla _{i})\mathbf {A} \end{aligned}}} In physics and engineering, the gradient, divergence, curl, and Laplacian operator arise inevitably in fluid mechanics, Newtonian gravitation, electromagnetism, heat conduction, and even quantum mechanics. Vector calculus identities can be derived in a similar way to those of vector dot and cross products and combinations. For example, in three dimensions, the curl of a cross product of two vector fields A and B: [ ∇ × ( A × B ) ] i = ε i j k ∇ j ( ε k ℓ m A ℓ B m ) = ( ε i j k ε ℓ m k ) ∇ j ( A ℓ B m ) = ( δ i ℓ δ j m − δ i m δ j ℓ ) ( B m ∇ j A ℓ + A ℓ ∇ j B m ) = ( B j ∇ j A i + A i ∇ j B j ) − ( B i ∇ j A j + A j ∇ j B i ) = ( B j ∇ j ) A i + A i ( ∇ j B j ) − B i ( ∇ j A j ) − ( A j ∇ j ) B i = [ ( B ⋅ ∇ ) A + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) − ( A ⋅ ∇ ) B ] i {\displaystyle {\begin{aligned}&\left[\nabla \times (\mathbf {A} \times \mathbf {B} )\right]_{i}\\{}={}&\varepsilon _{ijk}\nabla _{j}(\varepsilon _{k\ell m}A_{\ell }B_{m})\\{}={}&(\varepsilon _{ijk}\varepsilon _{\ell mk})\nabla _{j}(A_{\ell }B_{m})\\{}={}&(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })(B_{m}\nabla _{j}A_{\ell }+A_{\ell }\nabla _{j}B_{m})\\{}={}&(B_{j}\nabla _{j}A_{i}+A_{i}\nabla _{j}B_{j})-(B_{i}\nabla _{j}A_{j}+A_{j}\nabla _{j}B_{i})\\{}={}&(B_{j}\nabla _{j})A_{i}+A_{i}(\nabla _{j}B_{j})-B_{i}(\nabla _{j}A_{j})-(A_{j}\nabla _{j})B_{i}\\{}={}&\left[(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )-(\mathbf {A} \cdot \nabla )\mathbf {B} \right]_{i}\\\end{aligned}}} where the product rule was used, and throughout the differential operator was not interchanged with A or B. Thus: ∇ × ( A × B ) = ( B ⋅ ∇ ) A + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) − ( A ⋅ ∇ ) B {\displaystyle \nabla \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )-(\mathbf {A} \cdot \nabla )\mathbf {B} } === Tensor calculus === One can continue the operations on tensors of higher order. Let T = T(r, t) denote a second order tensor field, again dependent on the position vector r and time t. For instance, the gradient of a vector field in two equivalent notations ("dyadic" and "tensor", respectively) is: ( ∇ A ) i j ≡ ( ∇ ⊗ A ) i j = ∇ i A j {\displaystyle (\nabla \mathbf {A} )_{ij}\equiv (\nabla \otimes \mathbf {A} )_{ij}=\nabla _{i}A_{j}} which is a tensor field of second order. The divergence of a tensor is: ( ∇ ⋅ T ) j = ∇ i T i j {\displaystyle (\nabla \cdot \mathbf {T} )_{j}=\nabla _{i}T_{ij}} which is a vector field. This arises in continuum mechanics in Cauchy's laws of motion – the divergence of the Cauchy stress tensor σ is a vector field, related to body forces acting on the fluid. == Difference from the standard tensor calculus == Cartesian tensors are as in tensor algebra, but Euclidean structure of and restriction of the basis brings some simplifications compared to the general theory. The general tensor algebra consists of general mixed tensors of type (p, q): T = T j 1 j 2 ⋯ j q i 1 i 2 ⋯ i p e i 1 i 2 ⋯ i p j 1 j 2 ⋯ j q {\displaystyle \mathbf {T} =T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}\mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}} with basis elements: e i 1 i 2 ⋯ i p j 1 j 2 ⋯ j q = e i 1 ⊗ e i 2 ⊗ ⋯ e i p ⊗ e j 1 ⊗ e j 2 ⊗ ⋯ e j q {\displaystyle \mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}=\mathbf {e} _{i_{1}}\otimes \mathbf {e} _{i_{2}}\otimes \cdots \mathbf {e} _{i_{p}}\otimes \mathbf {e} ^{j_{1}}\otimes \mathbf {e} ^{j_{2}}\otimes \cdots \mathbf {e} ^{j_{q}}} the components transform according to: T ¯ ℓ 1 ℓ 2 ⋯ ℓ q k 1 k 2 ⋯ k p = L i 1 k 1 L i 2 k 2 ⋯ L i p k p ( L − 1 ) ℓ 1 j 1 ( L − 1 ) ℓ 2 j 2 ⋯ ( L − 1 ) ℓ q j q T j 1 j 2 ⋯ j q i 1 i 2 ⋯ i p {\displaystyle {\bar {T}}_{\ell _{1}\ell _{2}\cdots \ell _{q}}^{k_{1}k_{2}\cdots k_{p}}={\mathsf {L}}_{i_{1}}{}^{k_{1}}{\mathsf {L}}_{i_{2}}{}^{k_{2}}\cdots {\mathsf {L}}_{i_{p}}{}^{k_{p}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{\ell _{1}}{}^{j_{1}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{\ell _{2}}{}^{j_{2}}\cdots \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{\ell _{q}}{}^{j_{q}}T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}} as for the bases: e ¯ k 1 k 2 ⋯ k p ℓ 1 ℓ 2 ⋯ ℓ q = ( L − 1 ) k 1 i 1 ( L − 1 ) k 2 i 2 ⋯ ( L − 1 ) k p i p L j 1 ℓ 1 L j 2 ℓ 2 ⋯ L j q ℓ q e i 1 i 2 ⋯ i p j 1 j 2 ⋯ j q {\displaystyle {\bar {\mathbf {e} }}_{k_{1}k_{2}\cdots k_{p}}^{\ell _{1}\ell _{2}\cdots \ell _{q}}=\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{k_{1}}{}^{i_{1}}\left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{k_{2}}{}^{i_{2}}\cdots \left({\boldsymbol {\mathsf {L}}}^{-1}\right)_{k_{p}}{}^{i_{p}}{\mathsf {L}}_{j_{1}}{}^{\ell _{1}}{\mathsf {L}}_{j_{2}}{}^{\ell _{2}}\cdots {\mathsf {L}}_{j_{q}}{}^{\ell _{q}}\mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}} For Cartesian tensors, only the order p + q of the tensor matters in a Euclidean space with an orthonormal basis, and all p + q indices can be lowered. A Cartesian basis does not exist unless the vector space has a positive-definite metric, and thus cannot be used in relativistic contexts. == History == Dyadic tensors were historically the first approach to formulating second-order tensors, similarly triadic tensors for third-order tensors, and so on. Cartesian tensors use tensor index notation, in which the variance may be glossed over and is often ignored, since the components remain unchanged by raising and lowering indices. == See also == Tensor algebra Tensor calculus Tensors in curvilinear coordinates Rotation group == References == === General references === D. C. Kay (1988). Tensor Calculus. Schaum's Outlines. McGraw Hill. pp. 18–19, 31–32. ISBN 0-07-033484-6. M. R. Spiegel; S. Lipcshutz; D. Spellman (2009). Vector analysis. Schaum's Outlines (2nd ed.). McGraw Hill. p. 227. ISBN 978-0-07-161545-7. J.R. Tyldesley (1975). An introduction to tensor analysis for engineers and applied scientists. Longman. pp. 5–13. ISBN 0-582-44355-5. === Further reading and applications === S. Lipcshutz; M. Lipson (2009). Linear Algebra. Schaum's Outlines (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1. Pei Chi Chou (1992). Elasticity: Tensor, Dyadic, and Engineering Approaches. Courier Dover Publications. ISBN 048-666-958-0. T. W. Körner (2012). Vectors, Pure and Applied: A General Introduction to Linear Algebra. Cambridge University Press. p. 216. ISBN 978-11070-3356-6. R. Torretti (1996). Relativity and Geometry. Courier Dover Publications. p. 103. ISBN 0-4866-90466. J. J. L. Synge; A. Schild (1978). Tensor Calculus. Courier Dover Publications. p. 128. ISBN 0-4861-4139-X. C. A. Balafoutis; R. V. Patel (1991). Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach. The Kluwer International Series in Engineering and Computer Science: Robotics: vision, manipulation and sensors. Vol. 131. Springer. ISBN 0792-391-454. S. G. Tzafestas (1992). Robotic systems: advanced techniques and applications. Springer. ISBN 0-792-317-491. T. Dass; S. K. Sharma (1998). Mathematical Methods In Classical And Quantum Physics. Universities Press. p. 144. ISBN 817-371-0899. G. F. J. Temple (2004). Cartesian Tensors: An Introduction. Dover Books on Mathematics Series. Dover. ISBN 0-4864-3908-9. H. Jeffreys (1961). Cartesian Tensors. Cambridge University Press. ISBN 9780521054232. {{cite book}}: ISBN / Date incompatibility (help) == External links == Cartesian Tensors V. N. Kaliakin, Brief Review of Tensors, University of Delaware R. E. Hunt, Cartesian Tensors, University of Cambridge
Wikipedia:Cassini and Catalan identities#0	Cassini's identity (sometimes called Simson's identity) and Catalan's identity are mathematical identities for the Fibonacci numbers. Cassini's identity, a special case of Catalan's identity, states that for the nth Fibonacci number, F n − 1 F n + 1 − F n 2 = ( − 1 ) n . {\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.} Note here F 0 {\displaystyle F_{0}} is taken to be 0, and F 1 {\displaystyle F_{1}} is taken to be 1. Catalan's identity generalizes this: F n 2 − F n − r F n + r = ( − 1 ) n − r F r 2 . {\displaystyle F_{n}^{2}-F_{n-r}F_{n+r}=(-1)^{n-r}F_{r}^{2}.} Vajda's identity generalizes this: F n + i F n + j − F n F n + i + j = ( − 1 ) n F i F j . {\displaystyle F_{n+i}F_{n+j}-F_{n}F_{n+i+j}=(-1)^{n}F_{i}F_{j}.} == History == Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). However Johannes Kepler presumably knew the identity already in 1608. Catalan's identity is named after Eugène Catalan (1814–1894). It can be found in one of his private research notes, entitled "Sur la série de Lamé" and dated October 1879. However, the identity did not appear in print until December 1886 as part of his collected works (Catalan 1886). This explains why some give 1879 and others 1886 as the date for Catalan's identity (Tuenter 2022, p. 314). The Hungarian-British mathematician Steven Vajda (1901–95) published a book on Fibonacci numbers (Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, 1989) which contains the identity carrying his name. However, the identity had been published earlier in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly, and in 1901 by Alberto Tagiuri in Periodico di Matematica. == Proof of Cassini identity == === Proof by matrix theory === A quick proof of Cassini's identity may be given (Knuth 1997, p. 81) by recognising the left side of the equation as a determinant of a 2×2 matrix of Fibonacci numbers. The result is almost immediate when the matrix is seen to be the nth power of a matrix with determinant −1: F n − 1 F n + 1 − F n 2 = det [ F n + 1 F n F n F n − 1 ] = det [ 1 1 1 0 ] n = ( det [ 1 1 1 0 ] ) n = ( − 1 ) n . {\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=\det \left[{\begin{matrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{matrix}}\right]=\det \left[{\begin{matrix}1&1\\1&0\end{matrix}}\right]^{n}=\left(\det \left[{\begin{matrix}1&1\\1&0\end{matrix}}\right]\right)^{n}=(-1)^{n}.} === Proof by induction === Consider the induction statement: F n − 1 F n + 1 − F n 2 = ( − 1 ) n {\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}} The base case n = 1 {\displaystyle n=1} is true. Assume the statement is true for n {\displaystyle n} . Then: F n − 1 F n + 1 − F n 2 + F n F n + 1 − F n F n + 1 = ( − 1 ) n {\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}+F_{n}F_{n+1}-F_{n}F_{n+1}=(-1)^{n}} F n − 1 F n + 1 + F n F n + 1 − F n 2 − F n F n + 1 = ( − 1 ) n {\displaystyle F_{n-1}F_{n+1}+F_{n}F_{n+1}-F_{n}^{2}-F_{n}F_{n+1}=(-1)^{n}} F n + 1 ( F n − 1 + F n ) − F n ( F n + F n + 1 ) = ( − 1 ) n {\displaystyle F_{n+1}(F_{n-1}+F_{n})-F_{n}(F_{n}+F_{n+1})=(-1)^{n}} F n + 1 2 − F n F n + 2 = ( − 1 ) n {\displaystyle F_{n+1}^{2}-F_{n}F_{n+2}=(-1)^{n}} F n F n + 2 − F n + 1 2 = ( − 1 ) n + 1 {\displaystyle F_{n}F_{n+2}-F_{n+1}^{2}=(-1)^{n+1}} so the statement is true for all integers n > 0 {\displaystyle n>0} . == Proof of Catalan identity == We use Binet's formula, that F n = ϕ n − ψ n 5 {\displaystyle F_{n}={\frac {\phi ^{n}-\psi ^{n}}{\sqrt {5}}}} , where ϕ = 1 + 5 2 {\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}} and ψ = 1 − 5 2 {\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}} . Hence, ϕ + ψ = 1 {\displaystyle \phi +\psi =1} and ϕ ψ = − 1 {\displaystyle \phi \psi =-1} . So, 5 ( F n 2 − F n − r F n + r ) {\displaystyle 5(F_{n}^{2}-F_{n-r}F_{n+r})} = ( ϕ n − ψ n ) 2 − ( ϕ n − r − ψ n − r ) ( ϕ n + r − ψ n + r ) {\displaystyle =(\phi ^{n}-\psi ^{n})^{2}-(\phi ^{n-r}-\psi ^{n-r})(\phi ^{n+r}-\psi ^{n+r})} = ( ϕ 2 n − 2 ϕ n ψ n + ψ 2 n ) − ( ϕ 2 n − ϕ n ψ n ( ϕ − r ψ r + ϕ r ψ − r ) + ψ 2 n ) {\displaystyle =(\phi ^{2n}-2\phi ^{n}\psi ^{n}+\psi ^{2n})-(\phi ^{2n}-\phi ^{n}\psi ^{n}(\phi ^{-r}\psi ^{r}+\phi ^{r}\psi ^{-r})+\psi ^{2n})} = − 2 ϕ n ψ n + ϕ n ψ n ( ϕ − r ψ r + ϕ r ψ − r ) {\displaystyle =-2\phi ^{n}\psi ^{n}+\phi ^{n}\psi ^{n}(\phi ^{-r}\psi ^{r}+\phi ^{r}\psi ^{-r})} Using ϕ ψ = − 1 {\displaystyle \phi \psi =-1} , = − ( − 1 ) n 2 + ( − 1 ) n ( ϕ − r ψ r + ϕ r ψ − r ) {\displaystyle =-(-1)^{n}2+(-1)^{n}(\phi ^{-r}\psi ^{r}+\phi ^{r}\psi ^{-r})} and again as ϕ = − 1 ψ {\displaystyle \phi ={\frac {-1}{\psi }}} , = − ( − 1 ) n 2 + ( − 1 ) n − r ( ψ 2 r + ϕ 2 r ) {\displaystyle =-(-1)^{n}2+(-1)^{n-r}(\psi ^{2r}+\phi ^{2r})} The Lucas number L n {\displaystyle L_{n}} is defined as L n = ϕ n + ψ n {\displaystyle L_{n}=\phi ^{n}+\psi ^{n}} , so = − ( − 1 ) n 2 + ( − 1 ) n − r L 2 r {\displaystyle =-(-1)^{n}2+(-1)^{n-r}L_{2r}} Because L 2 n = 5 F n 2 + 2 ( − 1 ) n {\displaystyle L_{2n}=5F_{n}^{2}+2(-1)^{n}} = − ( − 1 ) n 2 + ( − 1 ) n − r ( 5 F r 2 + 2 ( − 1 ) r ) {\displaystyle =-(-1)^{n}2+(-1)^{n-r}(5F_{r}^{2}+2(-1)^{r})} = − ( − 1 ) n 2 + ( − 1 ) n − r 2 ( − 1 ) r + ( − 1 ) n − r 5 F r 2 {\displaystyle =-(-1)^{n}2+(-1)^{n-r}2(-1)^{r}+(-1)^{n-r}5F_{r}^{2}} = − ( − 1 ) n 2 + ( − 1 ) n 2 + ( − 1 ) n − r 5 F r 2 {\displaystyle =-(-1)^{n}2+(-1)^{n}2+(-1)^{n-r}5F_{r}^{2}} = ( − 1 ) n − r 5 F r 2 {\displaystyle =(-1)^{n-r}5F_{r}^{2}} Cancelling the 5 {\displaystyle 5} 's gives the result. == Notes == == References == Catalan, Eugène-Charles (December 1886). "CLXXXIX. — Sur la série de Lamé". Mémoires de la Société Royale des Sciences de Liège. Deuxième Série. 13: 319–321. Knuth, Donald Ervin (1997), The Art of Computer Programming, Volume 1: Fundamental Algorithms, The Art of Computer Programming, vol. 1 (3rd ed.), Reading, Mass: Addison-Wesley, ISBN 0-201-89683-4 Simson, R. (1753). "An Explication of an Obscure Passage in Albert Girard's Commentary upon Simon Stevin's Works". Philosophical Transactions of the Royal Society of London. 48: 368–376. doi:10.1098/rstl.1753.0056. Tuenter, Hans J. H. (November 2022). "Fibonacci Summation Identities arising from Catalan's Identity". The Fibonacci Quarterly. 60 (4): 312–319. doi:10.1080/00150517.2022.12427460. MR 4539699. Zbl 1512.11025. Werman, M.; Zeilberger, D. (1986). "A bijective proof of Cassini's Fibonacci identity". Discrete Mathematics. 58 (1): 109. doi:10.1016/0012-365X(86)90194-9. MR 0820846. == External links == Proof of Cassini's identity Proof of Catalan's Identity Cassini formula for Fibonacci numbers Fibonacci and Phi Formulae
Wikipedia:Casus irreducibilis#0	Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced to the computation of square and cube roots. Cardano's formula for solution in radicals of a cubic equation was discovered at this time. It applies in the casus irreducibilis, but, in this case, requires the computation of the square root of a negative number, which involves knowledge of complex numbers, unknown at the time. The casus irreducibilis occurs when the three solutions are real and distinct, or, equivalently, when the discriminant is positive. It is only in 1843 that Pierre Wantzel proved that there cannot exist any solution in real radicals in the casus irreducibilis. == The three cases of the discriminant == Let a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} be a cubic equation with a ≠ 0 {\displaystyle a\neq 0} . Then the discriminant is given by D := ( ( x 1 − x 2 ) ( x 1 − x 3 ) ( x 2 − x 3 ) ) 2 = 18 a b c d − 4 a c 3 − 27 a 2 d 2 + b 2 c 2 − 4 b 3 d . {\displaystyle D:={\bigl (}(x_{1}-x_{2})(x_{1}-x_{3})(x_{2}-x_{3}){\bigr )}^{2}=18abcd-4ac^{3}-27a^{2}d^{2}+b^{2}c^{2}-4b^{3}d~.} It appears in the algebraic solution and is the square of the product Δ := ∏ j < k ( x j − x k ) = ( x 1 − x 2 ) ( x 1 − x 3 ) ( x 2 − x 3 ) ( = ± D ) {\displaystyle \Delta :=\prod _{j<k}(x_{j}-x_{k})=(x_{1}-x_{2})(x_{1}-x_{3})(x_{2}-x_{3})\qquad \qquad {\bigl (}\!=\pm {\sqrt {D}}{\bigr )}} of the ( 3 2 ) = 3 {\displaystyle {\tbinom {3}{2}}=3} differences of the 3 roots x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} . If D < 0, then the polynomial has one real root and two complex non-real roots. Δ ∈ i R × {\displaystyle \Delta \in i\mathbb {R} ^{\times }} is purely imaginary.Although there are cubic polynomials with negative discriminant which are irreducible in the modern sense, casus irreducibilis does not apply. If D = 0, then Δ = 0 {\displaystyle \Delta =0} and two of the roots are equal. These two roots are also a root of the derivative of the polyomial. So, they are also a root of the greatest common divisor of the polynomial and its derivative, which can be computed with the Euclidean algorithm for polynomials. It follows that the three roots are real, and if the coefficients are rational numbers, the same is true for the roots. That is, all the roots are expressible without radicals. If D > 0, then Δ ∈ R × {\displaystyle \Delta \in \mathbb {R} ^{\times }} is non-zero and real, and there are three distinct real roots which are expressed by Cardano's formula as sums of two complex conjugates cube roots.Because complex numbers were not known in the 16th century, this case has been termed casus irreducibilis, because the computation of the roots could not, at that time, be reduced to the computation of square and cube roots. == Formal statement and proof == More generally, suppose that F is a formally real field, and that p(x) ∈ F[x] is a cubic polynomial, irreducible over F, but having three real roots (roots in the real closure of F). Then casus irreducibilis states that it is impossible to express a solution of p(x) = 0 by radicals with real radicands. To prove this, note that the discriminant D is positive. Form the field extension F(√D) = F(∆). Since this is F or a quadratic extension of F (depending in whether or not D is a square in F), p(x) remains irreducible in it. Consequently, the Galois group of p(x) over F(√D) is the cyclic group C3. Suppose that p(x) = 0 can be solved by real radicals. Then p(x) can be split by a tower of cyclic extensions F ⊂ F ( D ) ⊂ F ( D , α 1 p 1 ) ⊂ ⋯ ⊂ K ⊂ K ( α 3 ) {\displaystyle F\subset F({\sqrt {D}})\subset F({\sqrt {D}},{\sqrt[{p_{1}}]{\alpha _{1}}})\subset \cdots \subset K\subset K({\sqrt[{3}]{\alpha }})} At the final step of the tower, p(x) is irreducible in the penultimate field K, but splits in K(3√α) for some α. But this is a cyclic field extension, and so must contain a conjugate of 3√α and therefore a primitive 3rd root of unity. However, there are no primitive 3rd roots of unity in a real closed field, since the primitive 3rd roots of unity are the roots of the quadratic equation ⁠ x 2 + x + 1 = 0 {\displaystyle x^{2}+x+1=0} ⁠ which has a negative discriminant. == Solution in non-real radicals == === Cardano's solution === The equation ax3 + bx2 + cx + d = 0 can be depressed to a monic trinomial by dividing by a {\displaystyle a} and substituting x = t − ⁠b/3a⁠ (the Tschirnhaus transformation), giving the equation t3 + pt + q = 0 where p = 3 a c − b 2 3 a 2 {\displaystyle p={\frac {3ac-b^{2}}{3a^{2}}}} q = 2 b 3 − 9 a b c + 27 a 2 d 27 a 3 . {\displaystyle q={\frac {2b^{3}-9abc+27a^{2}d}{27a^{3}}}.} Then regardless of the number of real roots, by Cardano's solution the three roots are given by t k = ω k − q 2 + q 2 4 + p 3 27 3 + ω k 2 − q 2 − q 2 4 + p 3 27 3 {\displaystyle t_{k}=\omega _{k}{\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+\omega _{k}^{2}{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}} where ω k {\displaystyle \omega _{k}} (k=1, 2, 3) is a cube root of 1 ( ω 1 = 1 {\displaystyle \omega _{1}=1} , ω 2 = − 1 2 + 3 2 i {\displaystyle \omega _{2}=-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i} , and ω 3 = − 1 2 − 3 2 i {\displaystyle \omega _{3}=-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i} , where i is the imaginary unit). Here if the radicands under the cube roots are non-real, the cube roots expressed by radicals are defined to be any pair of complex conjugate cube roots, while if they are real these cube roots are defined to be the real cube roots. Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if ⁠q2/4⁠ + ⁠p3/27⁠ < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the cube root of a complex number (the cube root cannot itself be placed in the form α + βi with specifically given expressions in real radicals for α and β, since doing so would require independently solving the original cubic). Even in the reducible case in which one of three real roots is rational and hence can be factored out by polynomial long division, Cardano's formula (unnecessarily in this case) expresses that root (and the others) in terms of non-real radicals. === Example === The cubic equation 2 x 3 − 9 x 2 − 6 x + 3 = 0 {\displaystyle 2x^{3}-9x^{2}-6x+3=0} is irreducible, because if it could be factored there would be a linear factor giving a rational solution, while none of the possible roots given by the rational root test are actually roots. Since its discriminant is positive, it has three real roots, so it is an example of casus irreducibilis. These roots can be expressed as t k = 3 − ω k 39 − 26 i 3 − ω k 2 39 + 26 i 3 2 {\displaystyle t_{k}={\frac {3-\omega _{k}{\sqrt[{3}]{39-26i}}-\omega _{k}^{2}{\sqrt[{3}]{39+26i}}}{2}}} for k ∈ { 1 , 2 , 3 } {\displaystyle k\in \left\{1,2,3\right\}} . The solutions are in radicals and involve the cube roots of complex conjugate numbers. == Trigonometric solution in terms of real quantities == While casus irreducibilis cannot be solved in radicals in terms of real quantities, it can be solved trigonometrically in terms of real quantities. Specifically, the depressed monic cubic equation t 3 + p t + q = 0 {\displaystyle t^{3}+pt+q=0} is solved by t k = 2 − p 3 cos ⁡ [ 1 3 arccos ⁡ ( 3 q 2 p − 3 p ) − k 2 π 3 ] for k = 0 , 1 , 2 . {\displaystyle t_{k}=2{\sqrt {-{\frac {p}{3}}}}\cos \left[{\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)-k{\frac {2\pi }{3}}\right]\quad {\text{for}}\quad k=0,1,2\,.} These solutions are in terms of real quantities if and only if q 2 4 + p 3 27 < 0 {\displaystyle {q^{2} \over 4}+{p^{3} \over 27}<0} — i.e., if and only if there are three real roots. The formula involves starting with an angle whose cosine is known, trisecting the angle by multiplying it by 1/3, and taking the cosine of the resulting angle and adjusting for scale. Although cosine and its inverse function (arccosine) are transcendental functions, this solution is algebraic in the sense that cos ⁡ [ arccos ⁡ ( x ) / 3 ] {\displaystyle \cos \left[\arccos \left(x\right)/3\right]} is an algebraic function, equivalent to angle trisection. == Relation to angle trisection == The distinction between the reducible and irreducible cubic cases with three real roots is related to the issue of whether or not an angle is trisectible by the classical means of compass and unmarked straightedge. For any angle θ, one-third of this angle has a cosine that is one of the three solutions to 4 x 3 − 3 x − cos ⁡ ( θ ) = 0. {\displaystyle 4x^{3}-3x-\cos(\theta )=0.} Likewise, θ⁄3 has a sine that is one of the three real solutions to 4 y 3 − 3 y + sin ⁡ ( θ ) = 0. {\displaystyle 4y^{3}-3y+\sin(\theta )=0.} In either case, if the rational root test reveals a rational solution, x or y minus that root can be factored out of the polynomial on the left side, leaving a quadratic that can be solved for the remaining two roots in terms of a square root; then all of these roots are classically constructible since they are expressible in no higher than square roots, so in particular cos(θ⁄3) or sin(θ⁄3) is constructible and so is the associated angle θ⁄3. On the other hand, if the rational root test shows that there is no rational root, then casus irreducibilis applies, cos(θ⁄3) or sin(θ⁄3) is not constructible, the angle θ⁄3 is not constructible, and the angle θ is not classically trisectible. As an example, while a 180° angle can be trisected into three 60° angles, a 60° angle cannot be trisected with only compass and straightedge. Using triple-angle formulae one can see that cos ⁠π/3⁠ = 4x3 − 3x where x = cos(20°). Rearranging gives 8x3 − 6x − 1 = 0, which fails the rational root test as none of the rational numbers suggested by the theorem is actually a root. Therefore, the minimal polynomial of cos(20°) has degree 3, whereas the degree of the minimal polynomial of any constructible number must be a power of two. Expressing cos(20°) in radicals results in cos ⁡ ( π 9 ) = 1 − i 3 3 + 1 + i 3 3 2 2 3 {\displaystyle \cos \left({\frac {\pi }{9}}\right)={\frac {{\sqrt[{3}]{1-i{\sqrt {3}}}}+{\sqrt[{3}]{1+i{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} which involves taking the cube root of complex numbers. Note the similarity to eiπ/3 = ⁠1+i√3/2⁠ and e−iπ/3 = ⁠1−i√3/2⁠. The connection between rational roots and trisectability can also be extended to some cases where the sine and cosine of the given angle is irrational. Consider as an example the case where the given angle θ is a vertex angle of a regular pentagon, a polygon that can be constructed classically. For this angle 5θ/3 is 180°, and standard trigonometric identities then give cos ⁡ ( θ ) + cos ⁡ ( θ / 3 ) = 2 cos ⁡ ( θ / 3 ) cos ⁡ ( 2 θ / 3 ) = − 2 cos ⁡ ( θ / 3 ) cos ⁡ ( θ ) {\displaystyle \cos(\theta )+\cos(\theta /3)=2\cos(\theta /3)\cos(2\theta /3)=-2\cos(\theta /3)\cos(\theta )} thus cos ⁡ ( θ / 3 ) = − cos ⁡ ( θ ) / ( 1 + 2 cos ⁡ ( θ ) ) . {\displaystyle \cos(\theta /3)=-\cos(\theta )/(1+2\cos(\theta )).} The cosine of the trisected angle is rendered as a rational expression in terms of the cosine of the given angle, so the vertex angle of a regular pentagon can be trisected (mechanically, by simply drawing a diagonal). == Generalization == Casus irreducibilis can be generalized to higher degree polynomials as follows. Let p ∈ F[x] be an irreducible polynomial which splits in a formally real extension R of F (i.e., p has only real roots). Assume that p has a root in K ⊆ R {\displaystyle K\subseteq R} which is an extension of F by radicals. Then the degree of p is a power of 2, and its splitting field is an iterated quadratic extension of F.: 571–572 Thus for any irreducible polynomial whose degree is not a power of 2 and which has all roots real, no root can be expressed purely in terms of real radicals, i.e. it is a casus irreducibilis in the (16th century) sense of this article. Moreover, if the polynomial degree is a power of 2 and the roots are all real, then if there is a root that can be expressed in real radicals it can be expressed in terms of square roots and no higher-degree roots, as can the other roots, and so the roots are classically constructible. Casus irreducibilis for quintic polynomials is discussed by Dummit.: 17 === Relation to angle pentasection (quintisection) and higher === The distinction between the reducible and irreducible quintic cases with five real roots is related to the issue of whether or not an angle with rational cosine or rational sine is pentasectible (able to be split into five equal parts) by the classical means of compass and unmarked straightedge. For any angle θ, one-fifth of this angle has a cosine that is one of the five real roots of the equation 16 x 5 − 20 x 3 + 5 x − cos ⁡ ( θ ) = 0. {\displaystyle 16x^{5}-20x^{3}+5x-\cos(\theta )=0.} Likewise, ⁠θ/5⁠ has a sine that is one of the five real roots of the equation 16 y 5 − 20 y 3 + 5 y − sin ⁡ ( θ ) = 0. {\displaystyle 16y^{5}-20y^{3}+5y-\sin(\theta )=0.} In either case, if the rational root test yields a rational root x1, then the quintic is reducible since it can be written as a factor (x—x1) times a quartic polynomial. But if the test shows that there is no rational root, then the polynomial may be irreducible, in which case casus irreducibilis applies, cos(θ⁄5) and sin(θ⁄5) are not constructible, the angle θ⁄5 is not constructible, and the angle θ is not classically pentasectible. An example of this is when one attempts to construct a 25-gon (icosipentagon) with compass and straightedge. While a pentagon is relatively easy to construct, a 25-gon requires an angle pentasector as the minimal polynomial for cos(14.4°) has degree 10: cos ⁡ ( 2 π 5 ) = 5 − 1 4 16 x 5 − 20 x 3 + 5 x + 1 − 5 4 = 0 x = cos ⁡ ( 2 π 25 ) 4 ( 16 x 5 − 20 x 3 + 5 x + 1 − 5 4 ) ( 16 x 5 − 20 x 3 + 5 x + 1 + 5 4 ) = 0 4 ( 16 x 5 − 20 x 3 + 5 x ) 2 + 2 ( 16 x 5 − 20 x 3 + 5 x ) − 1 = 0 1024 x 10 − 2560 x 8 + 2240 x 6 + 32 x 5 − 800 x 4 − 40 x 3 + 100 x 2 + 10 x − 1 = 0. {\displaystyle {\begin{aligned}\cos \left({\frac {2\pi }{5}}\right)&={\frac {{\sqrt {5}}-1}{4}}\\16x^{5}-20x^{3}+5x+{\frac {1-{\sqrt {5}}}{4}}&=0\qquad \qquad x=\cos \left({\frac {2\pi }{25}}\right)\\4\left(16x^{5}-20x^{3}+5x+{\frac {1-{\sqrt {5}}}{4}}\right)\left(16x^{5}-20x^{3}+5x+{\frac {1+{\sqrt {5}}}{4}}\right)&=0\\4\left(16x^{5}-20x^{3}+5x\right)^{2}+2\left(16x^{5}-20x^{3}+5x\right)-1&=0\\1024x^{10}-2560x^{8}+2240x^{6}+32x^{5}-800x^{4}-40x^{3}+100x^{2}+10x-1&=0.\end{aligned}}} Thus, e 2 π i / 5 = − 1 + 5 4 + 10 + 2 5 4 i e − 2 π i / 5 = − 1 + 5 4 − 10 + 2 5 4 i cos ⁡ ( 2 π 25 ) = − 1 + 5 − i 10 + 2 5 5 + − 1 + 5 + i 10 + 2 5 5 2 4 5 . {\displaystyle {\begin{aligned}e^{2\pi i/5}&={\frac {-1+{\sqrt {5}}}{4}}+{\frac {\sqrt {10+2{\sqrt {5}}}}{4}}i\\e^{-2\pi i/5}&={\frac {-1+{\sqrt {5}}}{4}}-{\frac {\sqrt {10+2{\sqrt {5}}}}{4}}i\\\cos \left({\frac {2\pi }{25}}\right)&={\frac {{\sqrt[{5}]{-1+{\sqrt {5}}-i{\sqrt {10+2{\sqrt {5}}}}}}+{\sqrt[{5}]{-1+{\sqrt {5}}+i{\sqrt {10+2{\sqrt {5}}}}}}}{2{\sqrt[{5}]{4}}}}.\end{aligned}}} == Notes == == References == Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9. See in particular Section 1.3 Cubic Equations over the Real Numbers (pp. 15–22) and Section 8.6 The Casus Irreducibilis (pp. 220–227). van der Waerden, Bartel Leendert (2003), Modern Algebra I, F. Blum, J.R. Schulenberg, Springer, ISBN 978-0-387-40624-4 == External links == casus irreducibilis at PlanetMath.
Wikipedia:Category of matrices#0	In mathematics, the category of matrices, often denoted M a t {\displaystyle \mathbf {Mat} } , is the category whose objects are natural numbers and whose morphisms are matrices, with composition given by matrix multiplication. == Construction == Let A {\displaystyle A} be an n × m {\displaystyle n\times m} real matrix, i.e. a matrix with n {\displaystyle n} rows and m {\displaystyle m} columns. Given a p × q {\displaystyle p\times q} matrix B {\displaystyle B} , we can form the matrix multiplication B A {\displaystyle BA} or B ∘ A {\displaystyle B\circ A} only when q = n {\displaystyle q=n} , and in that case the resulting matrix is of dimension p × m {\displaystyle p\times m} . In other words, we can only multiply matrices A {\displaystyle A} and B {\displaystyle B} when the number of rows of A {\displaystyle A} matches the number of columns of B {\displaystyle B} . One can keep track of this fact by declaring an n × m {\displaystyle n\times m} matrix to be of type m → n {\displaystyle m\to n} , and similarly a p × q {\displaystyle p\times q} matrix to be of type q → p {\displaystyle q\to p} . This way, when q = n {\displaystyle q=n} the two arrows have matching source and target, m → n → p {\displaystyle m\to n\to p} , and can hence be composed to an arrow of type m → p {\displaystyle m\to p} . This is precisely captured by the mathematical concept of a category, where the arrows, or morphisms, are the matrices, and they can be composed only when their domain and codomain are compatible (similar to what happens with functions). In detail, the category M a t R {\displaystyle \mathbf {Mat} _{\mathbb {R} }} is constructed as follows: It has natural numbers as objects; Given numbers m {\displaystyle m} and n {\displaystyle n} , a morphism m → n {\displaystyle m\to n} is an n × m {\displaystyle n\times m} matrix, i.e. a matrix with n {\displaystyle n} rows and m {\displaystyle m} columns; The identity morphism at each object n {\displaystyle n} is given by the n × n {\displaystyle n\times n} identity matrix; The composition of morphisms A : m → n {\displaystyle A:m\to n} and B : n → p {\displaystyle B:n\to p} (i.e. of matrices n × m {\displaystyle n\times m} and p × n {\displaystyle p\times n} ) is given by matrix multiplication. More generally, one can define the category M a t F {\displaystyle \mathbf {Mat} _{\mathbb {F} }} of matrices over a fixed field F {\displaystyle \mathbb {F} } , such as the one of complex numbers. == Properties == The category of matrices M a t R {\displaystyle \mathbf {Mat} _{\mathbb {R} }} is equivalent to the category of finite-dimensional real vector spaces and linear maps. This is witnessed by the functor mapping the number n {\displaystyle n} to the vector space R n {\displaystyle \mathbb {R} ^{n}} , and an n × m {\displaystyle n\times m} matrix to the corresponding linear map R m → R n {\displaystyle \mathbb {R} ^{m}\to \mathbb {R} ^{n}} . A possible interpretation of this fact is that, as mathematical theories, abstract finite-dimensional vector spaces and concrete matrices have the same expressive power. More generally, the category of matrices M a t F {\displaystyle \mathbf {Mat} _{\mathbb {F} }} is equivalent to the category of finite-dimensional vector spaces over the field F {\displaystyle \mathbb {F} } and F {\displaystyle \mathbb {F} } -linear maps. A linear row operation on a n × m {\displaystyle n\times m} matrix A {\displaystyle A} can be equivalently obtained by applying the same operation to the n × n {\displaystyle n\times n} identity matrix, and then multiplying the resulting n × n {\displaystyle n\times n} matrix with A {\displaystyle A} . In particular, elementary row operations correspond to elementary matrices. This fact can be seen as an instance of the Yoneda lemma for the category of matrices. The transpose operation makes the category of matrices a dagger category. The same can be said about the conjugate transpose in the case of complex numbers. == Particular subcategories == For every fixed n {\displaystyle n} , the morphisms n → n {\displaystyle n\to n} of M a t R {\displaystyle \mathbf {Mat} _{\mathbb {R} }} are the n × n {\displaystyle n\times n} matrices, and form a monoid, canonically isomorphic to the monoid of linear endomorphisms of R n {\displaystyle \mathbb {R} ^{n}} . In particular, the invertible n × n {\displaystyle n\times n} matrices form a group. The same can be said for a generic field F {\displaystyle \mathbb {F} } . A stochastic matrix is a real matrix of nonnegative entries, such that the sum of each column is one. Stochastic matrices include the identity and are closed under composition, and so they form a subcategory of M a t R {\displaystyle \mathbf {Mat} _{\mathbb {R} }} . == Citations == == References == == External links == The Yoneda lemma in the category of matrices, tutorial video.
Wikipedia:Category of modules#0	In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that). Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. == Properties == The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules over some ring. Projective limits and inductive limits exist in the categories of left and right modules. Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category. == Objects == A monoid object of the category of modules over a commutative ring R is exactly an associative algebra over R. A compact object in R-Mod is exactly a finitely presented module. == Category of vector spaces == The category K-Vect (some authors use VectK) has all vector spaces over a field K as objects, and K-linear maps as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod (some authors use ModR), the category of left R-modules. Much of linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond exactly to the cardinal numbers, and that K-Vect is equivalent to the subcategory of K-Vect which has as its objects the vector spaces Kn, where n is any cardinal number. == Generalizations == The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives). == See also == Algebraic K-theory (the important invariant of the category of modules.) Category of rings Derived category Module spectrum Category of graded vector spaces Category of representations Change of rings Morita equivalence stable module category == References == === Bibliography === Bourbaki. "Algèbre linéaire". Algèbre. Dummit, David; Foote, Richard. Abstract Algebra. Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001. == External links == Mod at the nLab
Wikipedia:Caterina Consani#0	Caterina (Katia) Consani (born 1963) is an Italian mathematician specializing in arithmetic geometry. She is a professor of mathematics at Johns Hopkins University. == Contributions == Consani is the namesake of the Consani–Scholten quintic, a quintic threefold that she described with Jasper Scholten in 2001,[Q3] and of the Connes–Consani plane connection, a relationship between the field with one element and certain group actions on projective spaces investigated by Consani with Alain Connes.[AC] She is also known for her work with Matilde Marcolli on Arakelov theory and noncommutative geometry.[NG] == Education and career == Consani was born January 9, 1963, in Chiavari. She earned a laurea in mathematics in 1986 at the University of Genoa, a doctorate (dottorato di ricerca) in 1992 from the University of Genoa and the University of Turin, and a second doctorate in 1996 from the University of Chicago. Her first doctoral dissertation was Teoria dell’ intersezione e K-teoria su varietà singolari, supervised by Claudio Pedrini, and her second dissertation was Double Complexes and Euler L-factors on Degenerations of Algebraic Varieties, supervised by Spencer Bloch. She was a C. L. E. Moore instructor at the Massachusetts Institute of Technology from 1996 to 1999, overlapping with a research visit in 1998 to the University of Cambridge. After additional postdoctoral research at the Institute for Advanced Study, she became an assistant professor at the University of Toronto in 2000, and moved to Johns Hopkins in 2005. == Recognition == Consani was elected as a Fellow of the American Mathematical Society in the 2024 class of fellows. == Selected publications == == References ==
Wikipedia:Catherine Bandle#0	Catherine Bandle (born 22 March 1943) is a Swiss mathematician known for her research on differential equations, including semilinear elliptic equations and reaction-diffusion equations, and for her book on isoperimetric inequalities. She is a professor emerita of mathematics at the University of Basel. == Education and career == Bandle completed her doctorate (Dr. math.) at ETH Zurich in 1971. Her dissertation, Konstruktion isoperimetrischer Ungleichungen der mathematischen Physik aus solchen der Geometrie, concerned isoperimetric inequalities and was jointly supervised by Joseph Hersch and Alfred Huber. Like Alice Roth before her, she received the ETH Silver Medal for her dissertation, and she continued at ETH Zurich for a habilitation in 1974. She was the first woman mathematician and one of the earliest women to earn a habilitation at ETH Zurich. She became a professor at the University of Basel in 1975 and remained there until her retirement in 2003. She has studied destabilized elliptic equations with her friend and collaborator Maria Assunta Pozio. == Book == Bandle is the author of the book Isoperimetric Inequalities and Applications (Pitman, 1980). It was only the second book to study the applications of isoperimetric inequalities in mathematical physics, after the 1951 book Isoperimetric Inequalities in Mathematical Physics by George Pólya and Gábor Szegő, and at the time of its publication was considerably more up-to-date on recent developments in the subject. In 2023, Bandle published the book Shape Optimization - Variations of Domains and Applications together with Alfred Wagner from RWTH Aachen University. == References ==
Wikipedia:Catherine Doléans-Dade#0	In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation d Y t = Y t − d X t , Y 0 = 1 , {\displaystyle dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,} where Y − {\displaystyle Y_{-}} denotes the process of left limits, i.e., Y t − = lim s ↑ t Y s {\displaystyle Y_{t-}=\lim _{s\uparrow t}Y_{s}} . The concept is named after Catherine Doléans-Dade. Stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since X {\displaystyle X} measures the cumulative percentage change in Y {\displaystyle Y} . == Notation and terminology == Process Y {\displaystyle Y} obtained above is commonly denoted by E ( X ) {\displaystyle {\mathcal {E}}(X)} . The terminology "stochastic exponential" arises from the similarity of E ( X ) = Y {\displaystyle {\mathcal {E}}(X)=Y} to the natural exponential of X {\displaystyle X} : If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation d Y t / d t = Y t d X t / d t {\displaystyle dY_{t}/\mathrm {d} t=Y_{t}dX_{t}/dt} , whose solution is Y = exp ⁡ ( X − X 0 ) {\displaystyle Y=\exp(X-X_{0})} . == General formula and special cases == Without any assumptions on the semimartingale X {\displaystyle X} , one has E ( X ) t = exp ⁡ ( X t − X 0 − 1 2 [ X ] t c ) ∏ s ≤ t ( 1 + Δ X s ) exp ⁡ ( − Δ X s ) , t ≥ 0 , {\displaystyle {\mathcal {E}}(X)_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}^{c}{\Bigr )}\prod _{s\leq t}(1+\Delta X_{s})\exp(-\Delta X_{s}),\qquad t\geq 0,} where [ X ] c {\displaystyle [X]^{c}} is the continuous part of quadratic variation of X {\displaystyle X} and the product extends over the (countably many) jumps of X up to time t. If X {\displaystyle X} is continuous, then E ( X ) = exp ⁡ ( X − X 0 − 1 2 [ X ] ) . {\displaystyle {\mathcal {E}}(X)=\exp {\Bigl (}X-X_{0}-{\frac {1}{2}}[X]{\Bigr )}.} In particular, if X {\displaystyle X} is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion. If X {\displaystyle X} is continuous and of finite variation, then E ( X ) = exp ⁡ ( X − X 0 ) . {\displaystyle {\mathcal {E}}(X)=\exp(X-X_{0}).} Here X {\displaystyle X} need not be differentiable with respect to time; for example, X {\displaystyle X} can be the Cantor function. == Properties == Stochastic exponential cannot go to zero continuously, it can only jump to zero. Hence, the stochastic exponential of a continuous semimartingale is always strictly positive. Once E ( X ) {\displaystyle {\mathcal {E}}(X)} has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when Δ X = − 1 {\displaystyle \Delta X=-1} . Unlike the natural exponential exp ⁡ ( X t ) {\displaystyle \exp(X_{t})} , which depends only of the value of X {\displaystyle X} at time t {\displaystyle t} , the stochastic exponential E ( X ) t {\displaystyle {\mathcal {E}}(X)_{t}} depends not only on X t {\displaystyle X_{t}} but on the whole history of X {\displaystyle X} in the time interval [ 0 , t ] {\displaystyle [0,t]} . For this reason one must write E ( X ) t {\displaystyle {\mathcal {E}}(X)_{t}} and not E ( X t ) {\displaystyle {\mathcal {E}}(X_{t})} . Natural exponential of a semimartingale can always be written as a stochastic exponential of another semimartingale but not the other way around. Stochastic exponential of a local martingale is again a local martingale. All the formulae and properties above apply also to stochastic exponential of a complex-valued X {\displaystyle X} . This has application in the theory of conformal martingales and in the calculation of characteristic functions. == Useful identities == Yor's formula: for any two semimartingales U {\displaystyle U} and V {\displaystyle V} one has E ( U ) E ( V ) = E ( U + V + [ U , V ] ) {\displaystyle {\mathcal {E}}(U){\mathcal {E}}(V)={\mathcal {E}}(U+V+[U,V])} == Applications == Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential E ( X ) {\displaystyle {\mathcal {E}}(X)} of a continuous local martingale X {\displaystyle X} is a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's condition. == Derivation of the explicit formula for continuous semimartingales == For any continuous semimartingale X, take for granted that Y {\displaystyle Y} is continuous and strictly positive. Then applying Itō's formula with ƒ(Y) = log(Y) gives log ⁡ ( Y t ) − log ⁡ ( Y 0 ) = ∫ 0 t 1 Y u d Y u − ∫ 0 t 1 2 Y u 2 d [ Y ] u = X t − X 0 − 1 2 [ X ] t . {\displaystyle {\begin{aligned}\log(Y_{t})-\log(Y_{0})&=\int _{0}^{t}{\frac {1}{Y_{u}}}\,dY_{u}-\int _{0}^{t}{\frac {1}{2Y_{u}^{2}}}\,d[Y]_{u}=X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}.\end{aligned}}} Exponentiating with Y 0 = 1 {\displaystyle Y_{0}=1} gives the solution Y t = exp ⁡ ( X t − X 0 − 1 2 [ X ] t ) , t ≥ 0. {\displaystyle Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}{\Bigr )},\qquad t\geq 0.} This differs from what might be expected by comparison with the case where X has finite variation due to the existence of the quadratic variation term [X] in the solution. == See also == Stochastic logarithm == References == Jacod, J.; Shiryaev, A. N. (2003), Limit Theorems for Stochastic Processes (2nd ed.), Springer, pp. 58–61, ISBN 3-540-43932-3 Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4
Wikipedia:Catherine Greenhill#0	Catherine Greenhill is an Australian mathematician known for her research on random graphs, combinatorial enumeration and Markov chains. She is a professor of mathematics in the School of Mathematics and Statistics at the University of New South Wales, and an editor-in-chief of the Electronic Journal of Combinatorics. == Education and career == Greenhill did her undergraduate studies at the University of Queensland, and remained there for a master's degree, working with Anne Penfold Street there. She earned her Ph.D. in 1996 at the University of Oxford, under the supervision of Peter M. Neumann. Her dissertation was From Multisets to Matrix Groups: Some Algorithms Related to the Exterior Square. After postdoctoral research with Martin Dyer at the University of Leeds and Nick Wormald at the University of Melbourne, Greenhill joined the University of New South Wales in 2003. She was promoted to associate professor in 2014, becoming the first female mathematician to earn such a promotion at UNSW. == Recognition == Greenhill was the 2010 winner of the Hall Medal of the Institute of Combinatorics and its Applications. She was president of the Combinatorial Mathematics Society of Australasia for 2011–2013. In 2015 the Australian Academy of Science awarded her their Christopher Heyde Medal for distinguished research in the mathematical sciences. She was elected a Fellow of the Australian Academy of Science in 2022. == References == == External links == Catherine Greenhill publications indexed by Google Scholar
Wikipedia:Catherine Sulem#0	Catherine Sulem (born 1955) is a mathematician and violinist at the University of Toronto. She has completed a monograph "Nonlinear Schrodinger Equation: Self-Focusing Instability and Wave Collapse" together with her brother Pierre-Louis Sulem, which appears in applied Mathematical Sciences. == Awards and honours == Sulem is the winner of the fourth Krieger–Nelson Prize, for "important breakthroughs in understanding of many nonlinear phenomena associated with the focusing nonlinear Schrödinger equation and the water wave problem". She is also a fellow of the American Mathematical Society. In 2015, she was elected a Fellow of the Royal Society of Canada. In 2018 the Canadian Mathematical Society listed her in their inaugural class of fellows. In 2019 she gave the AWM-SIAM Sonia Kovalevsky Lecture, entitled The Dynamics of Ocean Waves, at the 7th ICIAM in Valencia. This lecture is awarded jointly by Association of Women in Mathematics and SIAM. In 2020, Sulem was awarded the CRM-Fields-PIMS Prize, the premier Canadian research prize in the mathematical sciences. She was elected to the 2023 Class of SIAM Fellows. == Selected publications == Books Sulem, Catherine; Sulem, Pierre-Louis (1999), The nonlinear Schrödinger equation: Self-focusing and wave collapse, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, ISBN 0-387-98611-1, MR 1696311. Research articles Sulem, Catherine; Sulem, Pierre-Louis; Frisch, Hélène (1983), "Tracing complex singularities with spectral methods", Journal of Computational Physics, 50 (1): 138–161, Bibcode:1983JCoPh..50..138S, doi:10.1016/0021-9991(83)90045-1, MR 0702063. Sulem, P.-L.; Sulem, C.; Bardos, C. (1986), "On the continuous limit for a system of classical spins", Communications in Mathematical Physics, 107 (3): 431–454, Bibcode:1986CMaPh.107..431S, doi:10.1007/bf01220998, MR 0866199, S2CID 122322088. Craig, W.; Sulem, C. (1993), "Numerical simulation of gravity waves", Journal of Computational Physics, 108 (1): 73–83, Bibcode:1993JCoPh.108...73C, doi:10.1006/jcph.1993.1164, MR 1239970. Buslaev, Vladimir S.; Sulem, Catherine (2003), "On asymptotic stability of solitary waves for nonlinear Schrödinger equations", Annales de l'Institut Henri Poincaré, 20 (3): 419–475, doi:10.1016/S0294-1449(02)00018-5, MR 1972870. Craig, W.; Guyenne, P.; Hammack, J.; Henderson, D.; Sulem, C. (2006), "Solitary water wave interactions", Physics of Fluids, 18 (5): 057106–057106–25, Bibcode:2006PhFl...18e7106C, doi:10.1063/1.2205916, MR 2259317. == References == == External links == Professor of Mathematics, University of Toronto List of Publications
Wikipedia:Cathleen Synge Morawetz#0	Cathleen Synge Morawetz (May 5, 1923 – August 8, 2017) was a Canadian mathematician who spent much of her career in the United States. Morawetz's research was mainly in the study of the partial differential equations governing fluid flow, particularly those of mixed type occurring in transonic flow. She was professor emerita at the Courant Institute of Mathematical Sciences at the New York University, where she had also served as director from 1984 to 1988. She was president of the American Mathematical Society from 1995 to 1996. She was awarded the National Medal of Science in 1998. == Childhood == Morawetz's father, John Lighton Synge, nephew of John Millington Synge, was an Irish mathematician, specializing in the geometry of general relativity. Her mother also studied mathematics for a time. Her uncle was Edward Hutchinson Synge who is credited as the inventor of the Near-field scanning optical microscope and very large astronomical telescopes, based on multiple mirrors. Her childhood was split between Ireland and Canada. Both her parents were supportive of her interest in mathematics and science, and it was a woman mathematician, Cecilia Krieger, who had been a family friend for many years and later encouraged Morawetz to pursue a PhD in mathematics. Morawetz said her father was influential in stimulating her interest in mathematics, but he wondered whether her studying mathematics would be wise (suggesting they might fight like the Bernoulli brothers). == Education == A 1945 graduate of the University of Toronto, she married Herbert Morawetz, a chemist, on October 28, 1945. She received her master's degree in 1946 at the Massachusetts Institute of Technology. Morawetz got a job at New York University where she edited Supersonic Flow and Shock Waves by Richard Courant and Kurt Otto Friedrichs. She earned her Ph.D. in 1951 at New York University, with a thesis on the stability of a spherical implosion, under the supervision of Kurt Otto Friedrichs. Her thesis was entitled Contracting Spherical Shocks Treated by a Perturbation Method. == Career == After earning her doctorate, she spent a year as a research associate at MIT before returning to work as a research associate at the Courant Institute of Mathematical Sciences at NYU, for five more years. During this time she had no teaching requirements and could focus purely on research. She published work on a variety of topics in applied mathematics including viscosity, compressible fluids and transonic flows. Even if an aircraft remains subsonic, the air flowing around the wing can reach supersonic velocity. The mix of air at supersonic and subsonic velocity creates shock waves that can slow the airplane. Turning to the mathematics of transonic flow, she showed that specially designed shockless airfoils could not, in fact, prevent shocks. Shocks developed in response to even small perturbations, such as a gust of wind or an imperfection in a wing. This discovery opened up the problem of developing a theory for a flow with shocks. Subsequently, the shocks she predicted mathematically now have been observed in experiments as air flows around the wing of a plane. In 1957 she became an assistant professor at Courant. At this point she began to work more closely with her colleagues publishing important joint papers with Peter Lax and Ralph Phillips on the decay of solutions to the wave equation around a star shaped obstacle. She continued with important solo work on the wave equation and transonic flow around a profile until she was promoted to full professor by 1965. At this point her research expanded to a variety of problems including papers on the Tricomi equation the nonrelativistic wave equation including questions of decay and scattering. Her first doctoral student, Lesley Sibner, was graduated in 1964. In the 1970s she worked on questions of scattering theory and the nonlinear wave equation. She proved what is now known as the Morawetz Inequality. She died on August 8, 2017, in New York City. == Honors == In 1980, Morawetz won a Lester R. Ford Award. In 1981, she became the first woman to deliver the Gibbs Lecture of The American Mathematical Society, and in 1982 presented an Invited Address at a meeting of the Society for Industrial and Applied Mathematics. She received honorary degrees from Eastern Michigan University in 1980, Brown University, and Smith College in 1982, and Princeton in 1990. In 1983 and in 1988, she was selected as a Noether Lecturer. She was elected to the American Academy of Arts and Sciences in 1984. She was named Outstanding Woman Scientist for 1993 by the Association for Women in Science. In 1995, she became the second woman elected to the office of president of the American Mathematical Society. In 1996, she was awarded an honorary ScD degree by Trinity College Dublin, where her father JL Synge had been a student and later a faculty member. That same year, she was elected to the American Philosophical Society. In 1998 she was awarded the National Medal of Science; she was the first woman to receive the medal for work in mathematics. In 2004, she received the Leroy P. Steele Prize for Lifetime Achievement. In 2006, she won the George David Birkhoff Prize in Applied Mathematics. In 2012, she became a fellow of the American Mathematical Society. Morawetz was a member of the National Academy of Sciences and was the first woman to belong to the Applied Mathematics Section of that organization. == Publications == Morawetz, Cathleen (10 July 1956). "Note on a Maximum Principle and a Uniqueness Theorem for an Elliptic-Hyperbolic Equation". Proceedings of the Royal Society of London, Series A. 236 (1204): 141–144. Bibcode:1956RSPSA.236..141M. doi:10.1098/rspa.1956.0119. JSTOR 99873. S2CID 120556787. —— (1956). "On the non-existence of continuous transonic flows past profiles I". Communications on Pure and Applied Mathematics. 9 (1): 45–68. doi:10.1002/cpa.3160090104. —— (10 September 1968). "Time Decay for the Nonlinear Klein-Gordon Equation". Proceedings of the Royal Society of London, Series A. 306 (1486): 291–296. Bibcode:1968RSPSA.306..291M. doi:10.1098/rspa.1968.0151. JSTOR 2416107. S2CID 123634895. —— (1972). "On the Modes of Decay for the Wave Equation in the Exterior of a Reflecting Body". Proceedings of the Royal Irish Academy, Section A. 72: 113–120. JSTOR 20488719. —— (1979). "Nonlinear conservation equations". The American Mathematical Monthly. 86 (4): 284–287. doi:10.2307/2320747. JSTOR 2320747. —— (1978). "Geometrical Optics and the Singing of Whales". The American Mathematical Monthly. 85 (7): 548–554. doi:10.2307/2320862. JSTOR 2320862. —— (1982). "The mathematical approach to the sonic barrier". Bulletin of the American Mathematical Society. New Series. 6 (2): 127–145. doi:10.1090/s0273-0979-1982-14965-5. MR 0640941. Morawetz, Cathleen; Kriegsmann, Gregory A. (August 1983). "The Calculations of an Inverse Potential Problem". SIAM Journal on Applied Mathematics. 43 (4): 844–854. doi:10.1137/0143055. JSTOR 2101366. Bayliss, Alvin; Li, Yanyan; Morawetz, Cathleen (April 1989). "Scattering by a Potential Using Hyperbolic Methods". Mathematics of Computation. 52 (186): 321–338. doi:10.2307/2008470. JSTOR 2008470. —— (November 1992). "Giants". The American Mathematical Monthly. 99 (9): 819–828. doi:10.2307/2324117. JSTOR 2324117. == Personal life == Morawetz lived in Greenwich Village with her husband Herbert Morawetz, a polymer chemist. They had four children, eight grandchildren, and three great grandchildren. Their children are Pegeen Rubinstein, John, Lida Jeck, and Nancy Morawetz (a professor at New York University School of Law who manages its Immigrant Rights Clinic). Upon being honored by the National Organization for Women for successfully combining career and family, Morawetz quipped, "Maybe I became a mathematician because I was so crummy at housework." She said her current non-mathematical interests are "grandchildren and sailing." == See also == List of second-generation Mathematicians == References == This article incorporates material from Cathleen Morawetz on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Gina Bari Kolata (12 October 1979). "Cathleen Morawetz: The Mathematics of Waves". Science. 206 (4415): 206–207. Bibcode:1979Sci...206..206B. doi:10.1126/science.206.4415.206. JSTOR 1749436. PMID 17801781. == External links == Sormani, Christina (August 2018). "The Mathematics of Cathleen Synge Morawetz" (PDF). Notices of the American Mathematical Society. 65 (7): 764–778. doi:10.1090/noti1706. "Science Lives: Cathleen Morawetz". YouTube. Simons Foundation. April 20, 2015; interview by Marsha Berger and Margaret H. Wright; April 28, 2010; May 9, 2012{{cite web}}: CS1 maint: postscript (link) "Emmy Noether Lecture: Variations on Conservation Laws - Cathleen Morawetz [ICM 1998]". YouTube. 21 May 2018.
Wikipedia:Cauchy formula for repeated integration#0	The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives. == Scalar case == Let f be a continuous function on the real line. Then the nth repeated integral of f with base-point a, f ( − n ) ( x ) = ∫ a x ∫ a σ 1 ⋯ ∫ a σ n − 1 f ( σ n ) d σ n ⋯ d σ 2 d σ 1 , {\displaystyle f^{(-n)}(x)=\int _{a}^{x}\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n-1}}f(\sigma _{n})\,\mathrm {d} \sigma _{n}\cdots \,\mathrm {d} \sigma _{2}\,\mathrm {d} \sigma _{1},} is given by single integration f ( − n ) ( x ) = 1 ( n − 1 ) ! ∫ a x ( x − t ) n − 1 f ( t ) d t . {\displaystyle f^{(-n)}(x)={\frac {1}{(n-1)!}}\int _{a}^{x}\left(x-t\right)^{n-1}f(t)\,\mathrm {d} t.} === Proof === A proof is given by induction. The base case with n = 1 is trivial, since it is equivalent to f ( − 1 ) ( x ) = 1 0 ! ∫ a x ( x − t ) 0 f ( t ) d t = ∫ a x f ( t ) d t . {\displaystyle f^{(-1)}(x)={\frac {1}{0!}}\int _{a}^{x}{(x-t)^{0}}f(t)\,\mathrm {d} t=\int _{a}^{x}f(t)\,\mathrm {d} t.} Now, suppose this is true for n, and let us prove it for n + 1. Firstly, using the Leibniz integral rule, note that d d x [ 1 n ! ∫ a x ( x − t ) n f ( t ) d t ] = 1 ( n − 1 ) ! ∫ a x ( x − t ) n − 1 f ( t ) d t . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[{\frac {1}{n!}}\int _{a}^{x}(x-t)^{n}f(t)\,\mathrm {d} t\right]={\frac {1}{(n-1)!}}\int _{a}^{x}(x-t)^{n-1}f(t)\,\mathrm {d} t.} Then, applying the induction hypothesis, f − ( n + 1 ) ( x ) = ∫ a x ∫ a σ 1 ⋯ ∫ a σ n f ( σ n + 1 ) d σ n + 1 ⋯ d σ 2 d σ 1 = ∫ a x [ ∫ a σ 1 ⋯ ∫ a σ n f ( σ n + 1 ) d σ n + 1 ⋯ d σ 2 ] d σ 1 . {\displaystyle {\begin{aligned}f^{-(n+1)}(x)&=\int _{a}^{x}\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n}}f(\sigma _{n+1})\,\mathrm {d} \sigma _{n+1}\cdots \,\mathrm {d} \sigma _{2}\,\mathrm {d} \sigma _{1}\\&=\int _{a}^{x}\left[\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n}}f(\sigma _{n+1})\,\mathrm {d} \sigma _{n+1}\cdots \,\mathrm {d} \sigma _{2}\right]\,\mathrm {d} \sigma _{1}.\end{aligned}}} Note that the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is σ 1 {\displaystyle \sigma _{1}} . Thus, comparing with the case for n = n and replacing x , σ 1 , ⋯ , σ n {\displaystyle x,\sigma _{1},\cdots ,\sigma _{n}} of the formula at induction step n = n with σ 1 , σ 2 , ⋯ , σ n + 1 {\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n+1}} respectively leads to ∫ a σ 1 ⋯ ∫ a σ n f ( σ n + 1 ) d σ n + 1 ⋯ d σ 2 = 1 ( n − 1 ) ! ∫ a σ 1 ( σ 1 − t ) n − 1 f ( t ) d t . {\displaystyle \int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n}}f(\sigma _{n+1})\,\mathrm {d} \sigma _{n+1}\cdots \,\mathrm {d} \sigma _{2}={\frac {1}{(n-1)!}}\int _{a}^{\sigma _{1}}(\sigma _{1}-t)^{n-1}f(t)\,\mathrm {d} t.} Putting this expression inside the square bracket results in = ∫ a x 1 ( n − 1 ) ! ∫ a σ 1 ( σ 1 − t ) n − 1 f ( t ) d t d σ 1 = ∫ a x d d σ 1 [ 1 n ! ∫ a σ 1 ( σ 1 − t ) n f ( t ) d t ] d σ 1 = 1 n ! ∫ a x ( x − t ) n f ( t ) d t . {\displaystyle {\begin{aligned}&=\int _{a}^{x}{\frac {1}{(n-1)!}}\int _{a}^{\sigma _{1}}(\sigma _{1}-t)^{n-1}f(t)\,\mathrm {d} t\,\mathrm {d} \sigma _{1}\\&=\int _{a}^{x}{\frac {\mathrm {d} }{\mathrm {d} \sigma _{1}}}\left[{\frac {1}{n!}}\int _{a}^{\sigma _{1}}(\sigma _{1}-t)^{n}f(t)\,\mathrm {d} t\right]\,\mathrm {d} \sigma _{1}\\&={\frac {1}{n!}}\int _{a}^{x}(x-t)^{n}f(t)\,\mathrm {d} t.\end{aligned}}} It has been shown that this statement holds true for the base case n = 1 {\displaystyle n=1} . If the statement is true for n = k {\displaystyle n=k} , then it has been shown that the statement holds true for n = k + 1 {\displaystyle n=k+1} . Thus this statement has been proven true for all positive integers. This completes the proof. == Generalizations and applications == The Cauchy formula is generalized to non-integer parameters by the Riemann–Liouville integral, where n ∈ Z ≥ 0 {\displaystyle n\in \mathbb {Z} _{\geq 0}} is replaced by α ∈ C , ℜ ( α ) > 0 {\displaystyle \alpha \in \mathbb {C} ,\ \Re (\alpha )>0} , and the factorial is replaced by the gamma function. The two formulas agree when α ∈ Z ≥ 0 {\displaystyle \alpha \in \mathbb {Z} _{\geq 0}} . Both the Cauchy formula and the Riemann–Liouville integral are generalized to arbitrary dimensions by the Riesz potential. In fractional calculus, these formulae can be used to construct a differintegral, allowing one to differentiate or integrate a fractional number of times. Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result. == References == Augustin-Louis Cauchy: Trente-Cinquième Leçon. In: Résumé des leçons données à l’Ecole royale polytechnique sur le calcul infinitésimal. Imprimerie Royale, Paris 1823. Reprint: Œuvres complètes II(4), Gauthier-Villars, Paris, pp. 5–261. Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2 == External links == Alan Beardon (2000). "Fractional calculus II". University of Cambridge. Maurice Mischler (2023). "About some repeated integrals and associated polynomials".
Wikipedia:Cauchy index#0	In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of r(x) = p(x)/q(x) over the real line is the difference between the number of roots of f(z) located in the right half-plane and those located in the left half-plane. The complex polynomial f(z) is such that f(iy) = q(y) + ip(y). We must also assume that p has degree less than the degree of q. == Definition == The Cauchy index was first defined for a pole s of the rational function r by Augustin-Louis Cauchy in 1837 using one-sided limits as: I s r = { + 1 , if lim x ↑ s r ( x ) = − ∞ ∧ lim x ↓ s r ( x ) = + ∞ , − 1 , if lim x ↑ s r ( x ) = + ∞ ∧ lim x ↓ s r ( x ) = − ∞ , 0 , otherwise. {\displaystyle I_{s}r={\begin{cases}+1,&{\text{if }}\displaystyle \lim _{x\uparrow s}r(x)=-\infty \;\land \;\lim _{x\downarrow s}r(x)=+\infty ,\\-1,&{\text{if }}\displaystyle \lim _{x\uparrow s}r(x)=+\infty \;\land \;\lim _{x\downarrow s}r(x)=-\infty ,\\0,&{\text{otherwise.}}\end{cases}}} A generalization over the compact interval [a,b] is direct (when neither a nor b are poles of r(x)): it is the sum of the Cauchy indices I s {\displaystyle I_{s}} of r for each s located in the interval. We usually denote it by I a b r {\displaystyle I_{a}^{b}r} . We can then generalize to intervals of type [ − ∞ , + ∞ ] {\displaystyle [-\infty ,+\infty ]} since the number of poles of r is a finite number (by taking the limit of the Cauchy index over [a,b] for a and b going to infinity). == Examples == Consider the rational function: r ( x ) = 4 x 3 − 3 x 16 x 5 − 20 x 3 + 5 x = p ( x ) q ( x ) . {\displaystyle r(x)={\frac {4x^{3}-3x}{16x^{5}-20x^{3}+5x}}={\frac {p(x)}{q(x)}}.} We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5. Therefore, r(x) has poles x 1 = 0.9511 {\displaystyle x_{1}=0.9511} , x 2 = 0.5878 {\displaystyle x_{2}=0.5878} , x 3 = 0 {\displaystyle x_{3}=0} , x 4 = − 0.5878 {\displaystyle x_{4}=-0.5878} and x 5 = − 0.9511 {\displaystyle x_{5}=-0.9511} , i.e. x j = cos ⁡ ( ( 2 i − 1 ) π / 2 n ) {\displaystyle x_{j}=\cos((2i-1)\pi /2n)} for j = 1 , . . . , 5 {\displaystyle j=1,...,5} . We can see on the picture that I x 1 r = I x 2 r = 1 {\displaystyle I_{x_{1}}r=I_{x_{2}}r=1} and I x 4 r = I x 5 r = − 1 {\displaystyle I_{x_{4}}r=I_{x_{5}}r=-1} . For the pole in zero, we have I x 3 r = 0 {\displaystyle I_{x_{3}}r=0} since the left and right limits are equal (which is because p(x) also has a root in zero). We conclude that I − 1 1 r = 0 = I − ∞ + ∞ r {\displaystyle I_{-1}^{1}r=0=I_{-\infty }^{+\infty }r} since q(x) has only five roots, all in [−1,1]. We cannot use here the Routh–Hurwitz theorem as each complex polynomial with f(iy) = q(y) + ip(y) has a zero on the imaginary line (namely at the origin). == References == == External links == The Cauchy Index
Wikipedia:Cauchy principal value#0	In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain. == Formulation == Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules: For a singularity at a finite number b lim ε → 0 + [ ∫ a b − ε f ( x ) d x + ∫ b + ε c f ( x ) d x ] {\displaystyle \lim _{\;\varepsilon \to 0^{+}\;}\,\,\left[\,\int _{a}^{b-\varepsilon }f(x)\,\mathrm {d} x~+~\int _{b+\varepsilon }^{c}f(x)\,\mathrm {d} x\,\right]} with a < b < c {\displaystyle a<b<c} and where b is the difficult point, at which the behavior of the function f is such that ∫ a b f ( x ) d x = ± ∞ {\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=\pm \infty \quad } for any a < b {\displaystyle a<b} and ∫ b c f ( x ) d x = ∓ ∞ {\displaystyle \int _{b}^{c}f(x)\,\mathrm {d} x=\mp \infty \quad } for any c > b {\displaystyle c>b} . (See plus or minus for the precise use of notations ± and ∓.) For a singularity at infinity ( ∞ {\displaystyle \infty } ) lim a → ∞ ∫ − a a f ( x ) d x {\displaystyle \lim _{a\to \infty }\,\int _{-a}^{a}f(x)\,\mathrm {d} x} where ∫ − ∞ 0 f ( x ) d x = ± ∞ {\displaystyle \int _{-\infty }^{0}f(x)\,\mathrm {d} x=\pm \infty } and ∫ 0 ∞ f ( x ) d x = ∓ ∞ . {\displaystyle \int _{0}^{\infty }f(x)\,\mathrm {d} x=\mp \infty .} In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity. This is usually done by a limit of the form lim η → 0 + lim ε → 0 + [ ∫ b − 1 η b − ε f ( x ) d x + ∫ b + ε b + 1 η f ( x ) d x ] . {\displaystyle \lim _{\;\eta \to 0^{+}}\,\lim _{\;\varepsilon \to 0^{+}}\,\left[\,\int _{b-{\frac {1}{\eta }}}^{b-\varepsilon }f(x)\,\mathrm {d} x\,~+~\int _{b+\varepsilon }^{b+{\frac {1}{\eta }}}f(x)\,\mathrm {d} x\,\right].} In those cases where the integral may be split into two independent, finite limits, lim ε → 0 + \| ∫ a b − ε f ( x ) d x \| < ∞ {\displaystyle \lim _{\;\varepsilon \to 0^{+}\;}\,\left\|\,\int _{a}^{b-\varepsilon }f(x)\,\mathrm {d} x\,\right\|\;<\;\infty } and lim η → 0 + \| ∫ b + η c f ( x ) d x \| < ∞ , {\displaystyle \lim _{\;\eta \to 0^{+}}\;\left\|\,\int _{b+\eta }^{c}f(x)\,\mathrm {d} x\,\right\|\;<\;\infty ,} then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value". The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f ( z ) : z = x + i y , {\displaystyle f(z):z=x+i\,y\;,} with x , y ∈ R , {\displaystyle x,y\in \mathbb {R} \;,} with a pole on a contour C. Define C ( ε ) {\displaystyle C(\varepsilon )} to be that same contour, where the portion inside the disk of radius ε around the pole has been removed. Provided the function f ( z ) {\displaystyle f(z)} is integrable over C ( ε ) {\displaystyle C(\varepsilon )} no matter how small ε becomes, then the Cauchy principal value is the limit: p . v . ⁡ ∫ C f ( z ) d z = lim ε → 0 + ∫ C ( ε ) f ( z ) d z . {\displaystyle \operatorname {p.\!v.} \int _{C}f(z)\,\mathrm {d} z=\lim _{\varepsilon \to 0^{+}}\int _{C(\varepsilon )}f(z)\,\mathrm {d} z.} In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral. If the function f ( z ) {\displaystyle f(z)} is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over C with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals. Principal value integrals play a central role in the discussion of Hilbert transforms. == Distribution theory == Let C c ∞ ( R ) {\displaystyle {C_{c}^{\infty }}(\mathbb {R} )} be the set of bump functions, i.e., the space of smooth functions with compact support on the real line R {\displaystyle \mathbb {R} } . Then the map p . v . ⁡ ( 1 x ) : C c ∞ ( R ) → C {\displaystyle \operatorname {p.\!v.} \left({\frac {1}{x}}\right)\,:\,{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} } defined via the Cauchy principal value as [ p . v . ⁡ ( 1 x ) ] ( u ) = lim ε → 0 + ∫ R ∖ [ − ε , ε ] u ( x ) x d x = lim ε → 0 + ∫ ε + ∞ u ( x ) − u ( − x ) x d x for u ∈ C c ∞ ( R ) {\displaystyle \left[\operatorname {p.\!v.} \left({\frac {1}{x}}\right)\right](u)=\lim _{\varepsilon \to 0^{+}}\int _{\mathbb {R} \setminus [-\varepsilon ,\varepsilon ]}{\frac {u(x)}{x}}\,\mathrm {d} x=\lim _{\varepsilon \to 0^{+}}\int _{\varepsilon }^{+\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\quad {\text{for }}u\in {C_{c}^{\infty }}(\mathbb {R} )} is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the sign function and the Heaviside step function. === Well-definedness as a distribution === To prove the existence of the limit lim ε → 0 + ∫ ε + ∞ u ( x ) − u ( − x ) x d x {\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{\varepsilon }^{+\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x} for a Schwartz function u ( x ) {\displaystyle u(x)} , first observe that u ( x ) − u ( − x ) x {\displaystyle {\frac {u(x)-u(-x)}{x}}} is continuous on [ 0 , ∞ ) , {\displaystyle [0,\infty ),} as lim x ↘ 0 [ u ( x ) − u ( − x ) ] = 0 {\displaystyle \lim _{\,x\searrow 0\,}\;{\Bigl [}u(x)-u(-x){\Bigr ]}~=~0~} and hence lim x ↘ 0 u ( x ) − u ( − x ) x = lim x ↘ 0 u ′ ( x ) + u ′ ( − x ) 1 = 2 u ′ ( 0 ) , {\displaystyle \lim _{x\searrow 0}\,{\frac {u(x)-u(-x)}{x}}~=~\lim _{\,x\searrow 0\,}\,{\frac {u'(x)+u'(-x)}{1}}~=~2u'(0)~,} since u ′ ( x ) {\displaystyle u'(x)} is continuous and L'Hopital's rule applies. Therefore, ∫ 0 1 u ( x ) − u ( − x ) x d x {\displaystyle \int _{0}^{1}\,{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x} exists and by applying the mean value theorem to u ( x ) − u ( − x ) , {\displaystyle u(x)-u(-x),} we get: \| ∫ 0 1 u ( x ) − u ( − x ) x d x \| ≤ ∫ 0 1 \| u ( x ) − u ( − x ) \| x d x ≤ ∫ 0 1 2 x x sup x ∈ R \| u ′ ( x ) \| d x ≤ 2 sup x ∈ R \| u ′ ( x ) \| . {\displaystyle \left\|\,\int _{0}^{1}\,{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\,\right\|\;\leq \;\int _{0}^{1}{\frac {{\bigl \|}u(x)-u(-x){\bigr \|}}{x}}\,\mathrm {d} x\;\leq \;\int _{0}^{1}\,{\frac {\,2x\,}{x}}\,\sup _{x\in \mathbb {R} }\,{\Bigl \|}u'(x){\Bigr \|}\,\mathrm {d} x\;\leq \;2\,\sup _{x\in \mathbb {R} }\,{\Bigl \|}u'(x){\Bigr \|}~.} And furthermore: \| ∫ 1 ∞ u ( x ) − u ( − x ) x d x \| ≤ 2 sup x ∈ R \| x ⋅ u ( x ) \| ⋅ ∫ 1 ∞ d x x 2 = 2 sup x ∈ R \| x ⋅ u ( x ) \| , {\displaystyle \left\|\,\int _{1}^{\infty }{\frac {\;u(x)-u(-x)\;}{x}}\,\mathrm {d} x\,\right\|\;\leq \;2\,\sup _{x\in \mathbb {R} }\,{\Bigl \|}x\cdot u(x){\Bigr \|}~\cdot \;\int _{1}^{\infty }{\frac {\mathrm {d} x}{\,x^{2}\,}}\;=\;2\,\sup _{x\in \mathbb {R} }\,{\Bigl \|}x\cdot u(x){\Bigr \|}~,} we note that the map p . v . ( 1 x ) : C c ∞ ( R ) → C {\displaystyle \operatorname {p.v.} \;\left({\frac {1}{\,x\,}}\right)\,:\,{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} } is bounded by the usual seminorms for Schwartz functions u {\displaystyle u} . Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution. Note that the proof needs u {\displaystyle u} merely to be continuously differentiable in a neighbourhood of 0 and x u {\displaystyle x\,u} to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as u {\displaystyle u} integrable with compact support and differentiable at 0. === More general definitions === The principal value is the inverse distribution of the function x {\displaystyle x} and is almost the only distribution with this property: x f = 1 ⇔ ∃ K : f = p . v . ⁡ ( 1 x ) + K δ , {\displaystyle xf=1\quad \Leftrightarrow \quad \exists K:\;\;f=\operatorname {p.\!v.} \left({\frac {1}{x}}\right)+K\delta ,} where K {\displaystyle K} is a constant and δ {\displaystyle \delta } the Dirac distribution. In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . If K {\displaystyle K} has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by [ p . v . ⁡ ( K ) ] ( f ) = lim ε → 0 ∫ R n ∖ B ε ( 0 ) f ( x ) K ( x ) d x . {\displaystyle [\operatorname {p.\!v.} (K)](f)=\lim _{\varepsilon \to 0}\int _{\mathbb {R} ^{n}\setminus B_{\varepsilon }(0)}f(x)K(x)\,\mathrm {d} x.} Such a limit may not be well defined, or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if K {\displaystyle K} is a continuous homogeneous function of degree − n {\displaystyle -n} whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms. == Examples == Consider the values of two limits: lim a → 0 + ( ∫ − 1 − a d x x + ∫ a 1 d x x ) = 0 , {\displaystyle \lim _{a\to 0+}\left(\int _{-1}^{-a}{\frac {\mathrm {d} x}{x}}+\int _{a}^{1}{\frac {\mathrm {d} x}{x}}\right)=0,} This is the Cauchy principal value of the otherwise ill-defined expression ∫ − 1 1 d x x , (which gives − ∞ + ∞ ) . {\displaystyle \int _{-1}^{1}{\frac {\mathrm {d} x}{x}},{\text{ (which gives }}{-\infty }+\infty {\text{)}}.} Also: lim a → 0 + ( ∫ − 1 − 2 a d x x + ∫ a 1 d x x ) = ln ⁡ 2. {\displaystyle \lim _{a\to 0+}\left(\int _{-1}^{-2a}{\frac {\mathrm {d} x}{x}}+\int _{a}^{1}{\frac {\mathrm {d} x}{x}}\right)=\ln 2.} Similarly, we have lim a → ∞ ∫ − a a 2 x d x x 2 + 1 = 0 , {\displaystyle \lim _{a\to \infty }\int _{-a}^{a}{\frac {2x\,\mathrm {d} x}{x^{2}+1}}=0,} This is the principal value of the otherwise ill-defined expression ∫ − ∞ ∞ 2 x d x x 2 + 1 (which gives − ∞ + ∞ ) . {\displaystyle \int _{-\infty }^{\infty }{\frac {2x\,\mathrm {d} x}{x^{2}+1}}{\text{ (which gives }}{-\infty }+\infty {\text{)}}.} but lim a → ∞ ∫ − 2 a a 2 x d x x 2 + 1 = − ln ⁡ 4. {\displaystyle \lim _{a\to \infty }\int _{-2a}^{a}{\frac {2x\,\mathrm {d} x}{x^{2}+1}}=-\ln 4.} == Notation == Different authors use different notations for the Cauchy principal value of a function f {\displaystyle f} , among others: P V ∫ f ( x ) d x , {\displaystyle PV\int f(x)\,\mathrm {d} x,} p . v . ∫ f ( x ) d x , {\displaystyle \mathrm {p.v.} \int f(x)\,\mathrm {d} x,} ∫ L ∗ f ( z ) d z , {\displaystyle \int _{L}^{*}f(z)\,\mathrm {d} z,} − ∫ f ( x ) d x , {\displaystyle -\!\!\!\!\!\!\int f(x)\,\mathrm {d} x,} as well as P , {\displaystyle P,} P.V., P , {\displaystyle {\mathcal {P}},} P v , {\displaystyle P_{v},} ( C P V ) , {\displaystyle (CPV),} C , {\displaystyle {\mathcal {C}},} and V.P. == See also == Hadamard finite part integral Hilbert transform Sokhotski–Plemelj theorem == References ==
Wikipedia:Cauchy sequence#0	In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences. It is not sufficient for each term to become arbitrarily close to the preceding term. For instance, in the sequence of square roots of natural numbers: a n = n , {\displaystyle a_{n}={\sqrt {n}},} the consecutive terms become arbitrarily close to each other – their differences a n + 1 − a n = n + 1 − n = 1 n + 1 + n < 1 2 n {\displaystyle a_{n+1}-a_{n}={\sqrt {n+1}}-{\sqrt {n}}={\frac {1}{{\sqrt {n+1}}+{\sqrt {n}}}}<{\frac {1}{2{\sqrt {n}}}}} tend to zero as the index n grows. However, with growing values of n, the terms a n {\displaystyle a_{n}} become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that a m − a n > d . {\displaystyle a_{m}-a_{n}>d.} As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets. == In real numbers == A sequence x 1 , x 2 , x 3 , … {\displaystyle x_{1},x_{2},x_{3},\ldots } of real numbers is called a Cauchy sequence if for every positive real number ε , {\displaystyle \varepsilon ,} there is a positive integer N such that for all natural numbers m , n > N , {\displaystyle m,n>N,} \| x m − x n \| < ε , {\displaystyle \|x_{m}-x_{n}\|<\varepsilon ,} where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring x m − x n {\displaystyle x_{m}-x_{n}} to be infinitesimal for every pair of infinite m, n. For any real number r, the sequence of truncated decimal expansions of r forms a Cauchy sequence. For example, when r = π , {\displaystyle r=\pi ,} this sequence is (3, 3.1, 3.14, 3.141, ...). The mth and nth terms differ by at most 10 1 − m {\displaystyle 10^{1-m}} when m < n, and as m grows this becomes smaller than any fixed positive number ε . {\displaystyle \varepsilon .} === Modulus of Cauchy convergence === If ( x 1 , x 2 , x 3 , . . . ) {\displaystyle (x_{1},x_{2},x_{3},...)} is a sequence in the set X , {\displaystyle X,} then a modulus of Cauchy convergence for the sequence is a function α {\displaystyle \alpha } from the set of natural numbers to itself, such that for all natural numbers k {\displaystyle k} and natural numbers m , n > α ( k ) , {\displaystyle m,n>\alpha (k),} \| x m − x n \| < 1 / k . {\displaystyle \|x_{m}-x_{n}\|<1/k.} Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the well-ordering property of the natural numbers (let α ( k ) {\displaystyle \alpha (k)} be the smallest possible N {\displaystyle N} in the definition of Cauchy sequence, taking ε {\displaystyle \varepsilon } to be 1 / k {\displaystyle 1/k} ). The existence of a modulus also follows from the principle of countable choice. Regular Cauchy sequences are sequences with a given modulus of Cauchy convergence (usually α ( k ) = k {\displaystyle \alpha (k)=k} or α ( k ) = 2 k {\displaystyle \alpha (k)=2^{k}} ). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice. Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice. Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by Bishop (2012) and by Bridges (1997) in constructive mathematics textbooks. == In a metric space == Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space X. To do so, the absolute value \| x m − x n \| {\displaystyle \left\|x_{m}-x_{n}\right\|} is replaced by the distance d ( x m , x n ) {\displaystyle d\left(x_{m},x_{n}\right)} (where d denotes a metric) between x m {\displaystyle x_{m}} and x n . {\displaystyle x_{n}.} Formally, given a metric space ( X , d ) , {\displaystyle (X,d),} a sequence of elements of X {\displaystyle X} x 1 , x 2 , x 3 , … {\displaystyle x_{1},x_{2},x_{3},\ldots } is Cauchy, if for every positive real number ε > 0 {\displaystyle \varepsilon >0} there is a positive integer N {\displaystyle N} such that for all positive integers m , n > N , {\displaystyle m,n>N,} the distance d ( x m , x n ) < ε . {\displaystyle d\left(x_{m},x_{n}\right)<\varepsilon .} Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in X. Nonetheless, such a limit does not always exist within X: the property of a space that every Cauchy sequence converges in the space is called completeness, and is detailed below. == Completeness == A metric space (X, d) in which every Cauchy sequence converges to an element of X is called complete. === Examples === The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number. A rather different type of example is afforded by a metric space X which has the discrete metric (where any two distinct points are at distance 1 from each other). Any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term. === Non-example: rational numbers === The rational numbers Q {\displaystyle \mathbb {Q} } are not complete (for the usual distance): There are sequences of rationals that converge (in R {\displaystyle \mathbb {R} } ) to irrational numbers; these are Cauchy sequences having no limit in Q . {\displaystyle \mathbb {Q} .} In fact, if a real number x is irrational, then the sequence (xn), whose n-th term is the truncation to n decimal places of the decimal expansion of x, gives a Cauchy sequence of rational numbers with irrational limit x. Irrational numbers certainly exist in R , {\displaystyle \mathbb {R} ,} for example: The sequence defined by x 0 = 1 , x n + 1 = x n + 2 / x n 2 {\displaystyle x_{0}=1,x_{n+1}={\frac {x_{n}+2/x_{n}}{2}}} consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational square root of 2, see Babylonian method of computing square root. The sequence x n = F n / F n − 1 {\displaystyle x_{n}=F_{n}/F_{n-1}} of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit ϕ {\displaystyle \phi } satisfying ϕ 2 = ϕ + 1 , {\displaystyle \phi ^{2}=\phi +1,} and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number φ = ( 1 + 5 ) / 2 , {\displaystyle \varphi =(1+{\sqrt {5}})/2,} the Golden ratio, which is irrational. The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational for any rational value of x ≠ 0 , {\displaystyle x\neq 0,} but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series. === Non-example: open interval === The open interval X = ( 0 , 2 ) {\displaystyle X=(0,2)} in the set of real numbers with an ordinary distance in R {\displaystyle \mathbb {R} } is not a complete space: there is a sequence x n = 1 / n {\displaystyle x_{n}=1/n} in it, which is Cauchy (for arbitrarily small distance bound d > 0 {\displaystyle d>0} all terms x n {\displaystyle x_{n}} of n > 1 / d {\displaystyle n>1/d} fit in the ( 0 , d ) {\displaystyle (0,d)} interval), however does not converge in X {\displaystyle X} — its 'limit', number 0, does not belong to the space X . {\displaystyle X.} === Other properties === Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ε > 0 , {\displaystyle \varepsilon >0,} beyond some fixed point, every term of the sequence is within distance ε / 2 {\displaystyle \varepsilon /2} of s, so any two terms of the sequence are within distance ε {\displaystyle \varepsilon } of each other. In any metric space, a Cauchy sequence x n {\displaystyle x_{n}} is bounded (since for some N, all terms of the sequence from the N-th onwards are within distance 1 of each other, and if M is the largest distance between x N {\displaystyle x_{N}} and any terms up to the N-th, then no term of the sequence has distance greater than M + 1 {\displaystyle M+1} from x N {\displaystyle x_{N}} ). In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are within distance r/2 of each other, so every term of the original sequence is within distance r of s. These last two properties, together with the Bolzano–Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers (or, more generally, of elements of any complete normed linear space, or Banach space). Such a series ∑ n = 1 ∞ x n {\textstyle \sum _{n=1}^{\infty }x_{n}} is considered to be convergent if and only if the sequence of partial sums ( s m ) {\displaystyle (s_{m})} is convergent, where s m = ∑ n = 1 m x n . {\textstyle s_{m}=\sum _{n=1}^{m}x_{n}.} It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p > q , {\displaystyle p>q,} s p − s q = ∑ n = q + 1 p x n . {\displaystyle s_{p}-s_{q}=\sum _{n=q+1}^{p}x_{n}.} If f : M → N {\displaystyle f:M\to N} is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then ( f ( x n ) ) {\displaystyle (f(x_{n}))} is a Cauchy sequence in N. If ( x n ) {\displaystyle (x_{n})} and ( y n ) {\displaystyle (y_{n})} are two Cauchy sequences in the rational, real or complex numbers, then the sum ( x n + y n ) {\displaystyle (x_{n}+y_{n})} and the product ( x n y n ) {\displaystyle (x_{n}y_{n})} are also Cauchy sequences. == Generalizations == === In topological vector spaces === There is also a concept of Cauchy sequence for a topological vector space X {\displaystyle X} : Pick a local base B {\displaystyle B} for X {\displaystyle X} about 0; then ( x k {\displaystyle x_{k}} ) is a Cauchy sequence if for each member V ∈ B , {\displaystyle V\in B,} there is some number N {\displaystyle N} such that whenever n , m > N , x n − x m {\displaystyle n,m>N,x_{n}-x_{m}} is an element of V . {\displaystyle V.} If the topology of X {\displaystyle X} is compatible with a translation-invariant metric d , {\displaystyle d,} the two definitions agree. === In topological groups === Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence ( x k ) {\displaystyle (x_{k})} in a topological group G {\displaystyle G} is a Cauchy sequence if for every open neighbourhood U {\displaystyle U} of the identity in G {\displaystyle G} there exists some number N {\displaystyle N} such that whenever m , n > N {\displaystyle m,n>N} it follows that x n x m − 1 ∈ U . {\displaystyle x_{n}x_{m}^{-1}\in U.} As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in G . {\displaystyle G.} As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in G {\displaystyle G} that ( x k ) {\displaystyle (x_{k})} and ( y k ) {\displaystyle (y_{k})} are equivalent if for every open neighbourhood U {\displaystyle U} of the identity in G {\displaystyle G} there exists some number N {\displaystyle N} such that whenever m , n > N {\displaystyle m,n>N} it follows that x n y m − 1 ∈ U . {\displaystyle x_{n}y_{m}^{-1}\in U.} This relation is an equivalence relation: It is reflexive since the sequences are Cauchy sequences. It is symmetric since y n x m − 1 = ( x m y n − 1 ) − 1 ∈ U − 1 {\displaystyle y_{n}x_{m}^{-1}=(x_{m}y_{n}^{-1})^{-1}\in U^{-1}} which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since x n z l − 1 = x n y m − 1 y m z l − 1 ∈ U ′ U ″ {\displaystyle x_{n}z_{l}^{-1}=x_{n}y_{m}^{-1}y_{m}z_{l}^{-1}\in U'U''} where U ′ {\displaystyle U'} and U ″ {\displaystyle U''} are open neighbourhoods of the identity such that U ′ U ″ ⊆ U {\displaystyle U'U''\subseteq U} ; such pairs exist by the continuity of the group operation. === In groups === There is also a concept of Cauchy sequence in a group G {\displaystyle G} : Let H = ( H r ) {\displaystyle H=(H_{r})} be a decreasing sequence of normal subgroups of G {\displaystyle G} of finite index. Then a sequence ( x n ) {\displaystyle (x_{n})} in G {\displaystyle G} is said to be Cauchy (with respect to H {\displaystyle H} ) if and only if for any r {\displaystyle r} there is N {\displaystyle N} such that for all m , n > N , x n x m − 1 ∈ H r . {\displaystyle m,n>N,x_{n}x_{m}^{-1}\in H_{r}.} Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on G , {\displaystyle G,} namely that for which H {\displaystyle H} is a local base. The set C {\displaystyle C} of such Cauchy sequences forms a group (for the componentwise product), and the set C 0 {\displaystyle C_{0}} of null sequences (sequences such that ∀ r , ∃ N , ∀ n > N , x n ∈ H r {\displaystyle \forall r,\exists N,\forall n>N,x_{n}\in H_{r}} ) is a normal subgroup of C . {\displaystyle C.} The factor group C / C 0 {\displaystyle C/C_{0}} is called the completion of G {\displaystyle G} with respect to H . {\displaystyle H.} One can then show that this completion is isomorphic to the inverse limit of the sequence ( G / H r ) . {\displaystyle (G/H_{r}).} An example of this construction familiar in number theory and algebraic geometry is the construction of the p {\displaystyle p} -adic completion of the integers with respect to a prime p . {\displaystyle p.} In this case, G {\displaystyle G} is the integers under addition, and H r {\displaystyle H_{r}} is the additive subgroup consisting of integer multiples of p r . {\displaystyle p_{r}.} If H {\displaystyle H} is a cofinal sequence (that is, any normal subgroup of finite index contains some H r {\displaystyle H_{r}} ), then this completion is canonical in the sense that it is isomorphic to the inverse limit of ( G / H ) H , {\displaystyle (G/H)_{H},} where H {\displaystyle H} varies over all normal subgroups of finite index. For further details, see Ch. I.10 in Lang's "Algebra". === In a hyperreal continuum === A real sequence ⟨ u n : n ∈ N ⟩ {\displaystyle \langle u_{n}:n\in \mathbb {N} \rangle } has a natural hyperreal extension, defined for hypernatural values H of the index n in addition to the usual natural n. The sequence is Cauchy if and only if for every infinite H and K, the values u H {\displaystyle u_{H}} and u K {\displaystyle u_{K}} are infinitely close, or adequal, that is, s t ( u H − u K ) = 0 {\displaystyle \mathrm {st} (u_{H}-u_{K})=0} where "st" is the standard part function. === Cauchy completion of categories === Krause (2020) introduced a notion of Cauchy completion of a category. Applied to Q {\displaystyle \mathbb {Q} } (the category whose objects are rational numbers, and there is a morphism from x to y if and only if x ≤ y {\displaystyle x\leq y} ), this Cauchy completion yields R ∪ { ∞ } {\displaystyle \mathbb {R} \cup \left\{\infty \right\}} (again interpreted as a category using its natural ordering). == See also == Modes of convergence (annotated index) – Annotated index of various modes of convergence Dedekind cut – Method of construction of the real numbers == References == == Further reading == Bishop, Errett Albert (2012). Foundations of Constructive Analysis. Ishi Press. ISBN 9784871877145. Bourbaki, Nicolas (1972). Commutative Algebra (English translation ed.). Addison-Wesley / Hermann. ISBN 0-201-00644-8. Bridges, Douglas Sutherland (1997). Foundations of Constructive Analysis. Springer. ISBN 978-0-387-98239-7. Krause, Henning (2020). "Completing perfect complexes: With appendices by Tobias Barthel and Bernhard Keller". Mathematische Zeitschrift. 296 (3–4): 1387–1427. arXiv:1805.10751. doi:10.1007/s00209-020-02490-z. Lang, Serge (1992). Algebra (3d ed.). Reading, Mass.: Addison Wesley Publishing Company. ISBN 978-0-201-55540-0. Zbl 0848.13001. Spivak, Michael (1994). Calculus (3rd ed.). Berkeley, CA: Publish or Perish. ISBN 0-914098-89-6. Archived from the original on 2007-05-17. Retrieved 2007-05-26. Troelstra, A. S.; van Dalen, D. (1988). Constructivism in Mathematics: An Introduction. (for uses in constructive mathematics) == External links == "Cauchy sequence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Wikipedia:Cauchy's functional equation#0	Cauchy's functional equation is the functional equation: f ( x + y ) = f ( x ) + f ( y ) . {\displaystyle f(x+y)=f(x)+f(y).\ } A function f {\displaystyle f} that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely f : x ↦ c x {\displaystyle f\colon x\mapsto cx} for any rational constant c . {\displaystyle c.} Over the real numbers, the family of linear maps f : x ↦ c x , {\displaystyle f:x\mapsto cx,} now with c {\displaystyle c} an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological solutions. For example, an additive function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is linear if: f {\displaystyle f} is continuous (Cauchy, 1821). In fact, it suffices for f {\displaystyle f} to be continuous at one point (Darboux, 1875). f ( x ) ≥ 0 {\displaystyle f(x)\geq 0} or f ( x ) ≤ 0 {\displaystyle f(x)\leq 0} for all x ≥ 0 {\displaystyle x\geq 0} . f {\displaystyle f} is monotonic on any interval. f {\displaystyle f} is bounded above or below on any interval. f {\displaystyle f} is Lebesgue measurable. f ( x n + 1 ) = x n f ( x ) {\displaystyle f\left(x^{n+1}\right)=x^{n}f(x)} for all real x {\displaystyle x} and some positive integer n {\displaystyle n} . The graph of f {\displaystyle f} is not dense in R 2 {\displaystyle \mathbb {R} ^{2}} . On the other hand, if no further conditions are imposed on f , {\displaystyle f,} then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. Such functions are sometimes called Hamel functions. The fifth problem on Hilbert's list is a generalisation of this equation. Functions where there exists a real number c {\displaystyle c} such that f ( c x ) ≠ c f ( x ) {\displaystyle f(cx)\neq cf(x)} are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of Hilbert's third problem from 3D to higher dimensions. This equation is sometimes referred to as Cauchy's additive functional equation to distinguish it from the other functional equations introduced by Cauchy in 1821, the exponential functional equation f ( x + y ) = f ( x ) f ( y ) , {\displaystyle f(x+y)=f(x)f(y),} the logarithmic functional equation f ( x y ) = f ( x ) + f ( y ) , {\displaystyle f(xy)=f(x)+f(y),} and the multiplicative functional equation f ( x y ) = f ( x ) f ( y ) . {\displaystyle f(xy)=f(x)f(y).} == Solutions over the rational numbers == A simple argument, involving only elementary algebra, demonstrates that the set of additive maps f : V → W {\displaystyle f\colon V\to W} , where V , W {\displaystyle V,W} are vector spaces over an extension field of Q {\displaystyle \mathbb {Q} } , is identical to the set of Q {\displaystyle \mathbb {Q} } -linear maps from V {\displaystyle V} to W {\displaystyle W} . Theorem: Let f : V → W {\displaystyle f\colon V\to W} be an additive function. Then f {\displaystyle f} is Q {\displaystyle \mathbb {Q} } -linear. Proof: We want to prove that any solution f : V → W {\displaystyle f\colon V\to W} to Cauchy’s functional equation, f ( x + y ) = f ( x ) + f ( y ) {\displaystyle f(x+y)=f(x)+f(y)} , satisfies f ( q v ) = q f ( v ) {\displaystyle f(qv)=qf(v)} for any q ∈ Q {\displaystyle q\in \mathbb {Q} } and v ∈ V {\displaystyle v\in V} . Let v ∈ V {\displaystyle v\in V} . First note f ( 0 ) = f ( 0 + 0 ) = f ( 0 ) + f ( 0 ) {\displaystyle f(0)=f(0+0)=f(0)+f(0)} , hence f ( 0 ) = 0 {\displaystyle f(0)=0} , and therewith 0 = f ( 0 ) = f ( v + ( − v ) ) = f ( v ) + f ( − v ) {\displaystyle 0=f(0)=f(v+(-v))=f(v)+f(-v)} from which follows f ( − v ) = − f ( v ) {\displaystyle f(-v)=-f(v)} . Via induction, f ( m v ) = m f ( v ) {\displaystyle f(mv)=mf(v)} is proved for any m ∈ N ∪ { 0 } {\displaystyle m\in \mathbb {N} \cup \{0\}} . For any negative integer m ∈ Z {\displaystyle m\in \mathbb {Z} } we know − m ∈ N {\displaystyle -m\in \mathbb {N} } , therefore f ( m v ) = f ( ( − m ) ( − v ) ) = ( − m ) f ( − v ) = ( − m ) ( − f ( v ) ) = m f ( v ) {\displaystyle f(mv)=f((-m)(-v))=(-m)f(-v)=(-m)(-f(v))=mf(v)} . Thus far we have proved f ( m v ) = m f ( v ) {\displaystyle f(mv)=mf(v)} for any m ∈ Z {\displaystyle m\in \mathbb {Z} } . Let n ∈ N {\displaystyle n\in \mathbb {N} } , then f ( v ) = f ( n n − 1 v ) = n f ( n − 1 v ) {\displaystyle f(v)=f(nn^{-1}v)=nf(n^{-1}v)} and hence f ( n − 1 v ) = n − 1 f ( v ) . {\displaystyle f(n^{-1}v)=n^{-1}f(v).} Finally, any q ∈ Q {\displaystyle q\in \mathbb {Q} } has a representation q = m n {\displaystyle q={\frac {m}{n}}} with m ∈ Z {\displaystyle m\in \mathbb {Z} } and n ∈ N {\displaystyle n\in \mathbb {N} } , so, putting things together, f ( q v ) = f ( m n v ) = f ( 1 n ( m v ) ) = 1 n f ( m v ) = 1 n m f ( v ) = q f ( v ) {\displaystyle f(qv)=f\left({\frac {m}{n}}\,v\right)=f\left({\frac {1}{n}}\,(mv)\right)={\frac {1}{n}}\,f(mv)={\frac {1}{n}}\,m\,f(v)=qf(v)} , q.e.d. == Properties of nonlinear solutions over the real numbers == We prove below that any other solutions must be highly pathological functions. In particular, it is shown that any other solution must have the property that its graph { ( x , f ( x ) ) \| x ∈ R } {\displaystyle \{(x,f(x))\vert x\in \mathbb {R} \}} is dense in R 2 , {\displaystyle \mathbb {R} ^{2},} that is, that any disk in the plane (however small) contains a point from the graph. From this it is easy to prove the various conditions given in the introductory paragraph. == Existence of nonlinear solutions over the real numbers == The linearity proof given above also applies to f : α Q → R , {\displaystyle f\colon \alpha \mathbb {Q} \to \mathbb {R} ,} where α Q {\displaystyle \alpha \mathbb {Q} } is a scaled copy of the rationals. This shows that only linear solutions are permitted when the domain of f {\displaystyle f} is restricted to such sets. Thus, in general, we have f ( α q ) = f ( α ) q {\displaystyle f(\alpha q)=f(\alpha )q} for all α ∈ R {\displaystyle \alpha \in \mathbb {R} } and q ∈ Q . {\displaystyle q\in \mathbb {Q} .} However, as we will demonstrate below, highly pathological solutions can be found for functions f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } based on these linear solutions, by viewing the reals as a vector space over the field of rational numbers. Note, however, that this method is nonconstructive, relying as it does on the existence of a (Hamel) basis for any vector space, a statement proved using Zorn's lemma. (In fact, the existence of a basis for every vector space is logically equivalent to the axiom of choice.) There exist models such as the Solovay model where all sets of reals are measurable which are consistent with ZF + DC, and therein all solutions are linear. To show that solutions other than the ones defined by f ( x ) = f ( 1 ) x {\displaystyle f(x)=f(1)x} exist, we first note that because every vector space has a basis, there is a basis for R {\displaystyle \mathbb {R} } over the field Q , {\displaystyle \mathbb {Q} ,} i.e. a set B ⊂ R {\displaystyle {\mathcal {B}}\subset \mathbb {R} } with the property that any x ∈ R {\displaystyle x\in \mathbb {R} } can be expressed uniquely as x = ∑ i ∈ I λ i x i , {\textstyle x=\sum _{i\in I}{\lambda _{i}x_{i}},} where { x i } i ∈ I {\displaystyle \{x_{i}\}_{i\in I}} is a finite subset of B , {\displaystyle {\mathcal {B}},} and each λ i {\displaystyle \lambda _{i}} is in Q . {\displaystyle \mathbb {Q} .} We note that because no explicit basis for R {\displaystyle \mathbb {R} } over Q {\displaystyle \mathbb {Q} } can be written down, the pathological solutions defined below likewise cannot be expressed explicitly. As argued above, the restriction of f {\displaystyle f} to x i Q {\displaystyle x_{i}\mathbb {Q} } must be a linear map for each x i ∈ B . {\displaystyle x_{i}\in {\mathcal {B}}.} Moreover, because x i q ↦ f ( x i ) q {\displaystyle x_{i}q\mapsto f(x_{i})q} for q ∈ Q , {\displaystyle q\in \mathbb {Q} ,} it is clear that f ( x i ) x i {\displaystyle f(x_{i}) \over x_{i}} is the constant of proportionality. In other words, f : x i Q → R {\displaystyle f\colon x_{i}\mathbb {Q} \to \mathbb {R} } is the map ξ ↦ [ f ( x i ) / x i ] ξ . {\displaystyle \xi \mapsto [f(x_{i})/x_{i}]\xi .} Since any x ∈ R {\displaystyle x\in \mathbb {R} } can be expressed as a unique (finite) linear combination of the x i {\displaystyle x_{i}} s, and f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is additive, f ( x ) {\displaystyle f(x)} is well-defined for all x ∈ R {\displaystyle x\in \mathbb {R} } and is given by: f ( x ) = f ( ∑ i ∈ I λ i x i ) = ∑ i ∈ I f ( x i λ i ) = ∑ i ∈ I f ( x i ) λ i . {\displaystyle f(x)=f{\Big (}\sum _{i\in I}\lambda _{i}x_{i}{\Big )}=\sum _{i\in I}f(x_{i}\lambda _{i})=\sum _{i\in I}f(x_{i})\lambda _{i}.} It is easy to check that f {\displaystyle f} is a solution to Cauchy's functional equation given a definition of f {\displaystyle f} on the basis elements, f : B → R . {\displaystyle f\colon {\mathcal {B}}\to \mathbb {R} .} Moreover, it is clear that every solution is of this form. In particular, the solutions of the functional equation are linear if and only if f ( x i ) x i {\displaystyle f(x_{i}) \over x_{i}} is constant over all x i ∈ B . {\displaystyle x_{i}\in {\mathcal {B}}.} Thus, in a sense, despite the inability to exhibit a nonlinear solution, "most" (in the sense of cardinality) solutions to the Cauchy functional equation are actually nonlinear and pathological. == See also == Antilinear map – Conjugate homogeneous additive map Homogeneous function – Function with a multiplicative scaling behaviour Minkowski functional – Function made from a set Semilinear map – homomorphism between modules, paired with the associated homomorphism between the respective base ringsPages displaying wikidata descriptions as a fallback == References == Kuczma, Marek (2009). An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality. Basel: Birkhäuser. ISBN 9783764387495. Hamel, Georg (1905). "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: f(x+y) = f(x) + f(y)". Mathematische Annalen. == External links == Solution to the Cauchy Equation Rutgers University The Hunt for Addi(c)tive Monster Martin Sleziak; et al. (2013). "Overview of basic facts about Cauchy functional equation". StackExchange. Retrieved 20 December 2015.
Wikipedia:Cauchy–Euler operator#0	In mathematics a Cauchy–Euler operator is a differential operator of the form p ( x ) ⋅ d d x {\displaystyle p(x)\cdot {d \over dx}} for a polynomial p. It is named after Augustin-Louis Cauchy and Leonhard Euler. The simplest example is that in which p(x) = x, which has eigenvalues n = 0, 1, 2, 3, ... and corresponding eigenfunctions xn. == See also == Cauchy–Euler equation Sturm–Liouville theory == References ==
Wikipedia:Cauchy–Kovalevskaya theorem#0	In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full result by Sofya Kovalevskaya (1874). == First order Cauchy–Kovalevskaya theorem == This theorem is about the existence of solutions to a system of m differential equations in n dimensions when the coefficients are analytic functions. The theorem and its proof are valid for analytic functions of either real or complex variables. Let K denote the field of real or of complex numbers, and let V = Km and W = Kn. Let A1, ..., An−1 be analytic functions defined on some neighbourhood of (0, 0) in W × V and taking values in the m × m matrices, and let b be an analytic function with values in V defined on the same neighbourhood. Then there is a neighbourhood of 0 in W on which the quasilinear Cauchy problem ∂ x n f = A 1 ( x , f ) ∂ x 1 f + ⋯ + A n − 1 ( x , f ) ∂ x n − 1 f + b ( x , f ) {\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)} with initial condition f ( x ) = 0 {\displaystyle f(x)=0} on the hypersurface x n = 0 {\displaystyle x_{n}=0} has a unique analytic solution ƒ : W → V near 0. Lewy's example shows that the theorem is not more generally valid for all smooth functions. The theorem can also be stated in abstract (real or complex) vector spaces. Let V and W be finite-dimensional real or complex vector spaces, with n = dim W. Let A1, ..., An−1 be analytic functions with values in End (V) and b an analytic function with values in V, defined on some neighbourhood of (0, 0) in W × V. In this case, the same result holds. == Proof by analytic majorization == Both sides of the partial differential equation can be expanded as formal power series and give recurrence relations for the coefficients of the formal power series for f that uniquely determine the coefficients. The Taylor series coefficients of the Ai's and b are majorized in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the Ai's and b has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must converge where the scalar solution converges. == Higher-order Cauchy–Kovalevskaya theorem == If F and fj are analytic functions near 0, then the non-linear Cauchy problem ∂ t k h = F ( x , t , ∂ t j ∂ x α h ) , where j < k and \| α \| + j ≤ k , {\displaystyle \partial _{t}^{k}h=F\left(x,t,\partial _{t}^{j}\,\partial _{x}^{\alpha }h\right),{\text{ where }}j<k{\text{ and }}\|\alpha \|+j\leq k,} with initial conditions ∂ t j h ( x , 0 ) = f j ( x ) , 0 ≤ j < k , {\displaystyle \partial _{t}^{j}h(x,0)=f_{j}(x),\qquad 0\leq j<k,} has a unique analytic solution near 0. This follows from the first order problem by considering the derivatives of h appearing on the right hand side as components of a vector-valued function. === Example === The heat equation ∂ t h = ∂ x 2 h {\displaystyle \partial _{t}h=\partial _{x}^{2}h} with the condition h ( 0 , x ) = 1 1 + x 2 for t = 0 {\displaystyle h(0,x)={1 \over 1+x^{2}}{\text{ for }}t=0} has a unique formal power series solution (expanded around (0, 0)). However this formal power series does not converge for any non-zero values of t, so there are no analytic solutions in a neighborhood of the origin. This shows that the condition \|α\| + j ≤ k above cannot be dropped. (This example is due to Kowalevski.) == Cauchy–Kovalevskaya–Kashiwara theorem == There is a wide generalization of the Cauchy–Kovalevskaya theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kovalevskaya–Kashiwara theorem, due to Masaki Kashiwara (1983). This theorem involves a cohomological formulation, presented in the language of D-modules. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a derived functor E x t 1 {\displaystyle Ext^{1}} . === Example === Let n ≤ m {\displaystyle n\leq m} . Set Y = { x 1 = ⋯ = x n } {\displaystyle Y=\{x_{1}=\cdots =x_{n}\}} . The system ∂ x i f = g i , i = 1 , … , n , {\displaystyle \partial _{x_{i}}f=g_{i},i=1,\ldots ,n,} has a solution f ∈ C { x 1 , … , x m } {\displaystyle f\in \mathbb {C} \{x_{1},\ldots ,x_{m}\}} if and only if the compatibility conditions ∂ x i g j = ∂ x j g i {\displaystyle \partial _{x_{i}}g_{j}=\partial _{x_{j}}g_{i}} are verified. In order to have a unique solution we must include an initial condition f \| Y = h {\displaystyle f\|_{Y}=h} , where h ∈ C { x n + 1 , … , x m } {\displaystyle h\in \mathbb {C} \{x_{n+1},\ldots ,x_{m}\}} . == References == Cauchy, Augustin (1842), "Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles", Comptes rendus, 15 Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–58. Folland, Gerald B. (1995), Introduction to Partial Differential Equations, Princeton University Press, ISBN 0-691-04361-2 Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035 (linear case) Kashiwara, M. (1983), Systems of microdifferential equations, Progress in Mathematics, vol. 34, Birkhäuser, ISBN 0817631380 von Kowalevsky, Sophie (1875), "Zur Theorie der partiellen Differentialgleichung", Journal für die reine und angewandte Mathematik, 80: 1–32 (German spelling of her surname used at that time.) Nakhushev, A.M. (2001) [1994], "Cauchy–Kovalevskaya theorem", Encyclopedia of Mathematics, EMS Press == External links == PlanetMath

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