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Wikipedia:Method of dominant balance#0
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In mathematics, the method of dominant balance approximates the solution to an equation by solving a simplified form of the equation containing 2 or more of the equation's terms that most influence (dominate) the solution and excluding terms contributing only small modifications to this approximate solution. Following an initial solution, iteration of the procedure may generate additional terms of an asymptotic expansion providing a more accurate solution. An early example of the dominant balance method is the Newton polygon method. Newton developed this method to find an explicit approximation for an algebraic function. Newton expressed the function as proportional to the independent variable raised to a power, retained only the lowest-degree polynomial terms (dominant terms), and solved this simplified reduced equation to obtain an approximate solution. Dominant balance has a broad range of applications, solving differential equations arising in fluid mechanics, plasma physics, turbulence, combustion, nonlinear optics, geophysical fluid dynamics, and neuroscience. == Asymptotic relations == The functions f ( z ) {\textstyle f(z)} and g ( z ) {\displaystyle g(z)} of parameter or independent variable z {\textstyle z} and the quotient f ( z ) / g ( z ) {\textstyle f(z)/g(z)} have limits as z {\textstyle z} approaches the limit L {\textstyle L} . The function f ( z ) {\textstyle f(z)} is much less than g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} , written as f ( z ) ≪ g ( z ) ( z → L ) {\textstyle f(z)\ll g(z)\ (z\to L)} , if the limit of the quotient f ( z ) / g ( z ) {\textstyle f(z)/g(z)} is zero as z {\textstyle z} approaches L {\textstyle L} . The relation f ( z ) {\textstyle f(z)} is lower order than g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} , written using little-o notation f ( z ) = o ( g ( z ) ) ( z → L ) {\textstyle f(z)=o(g(z))\ (z\to L)} , is identical to the f ( z ) {\textstyle f(z)} is much less than g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} relation. The function f ( z ) {\textstyle f(z)} is equivalent to g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} , written as f ( z ) ∼ g ( z ) ( z → L ) {\textstyle f(z)\sim g(z)\ (z\to L)} , if the limit of the quotient f ( z ) / g ( z ) {\textstyle f(z)/g(z)} is 1 as z {\textstyle z} approaches L {\textstyle L} . This result indicates that the zero function, f ( z ) = 0 {\textstyle f(z)=0} for all values of z {\textstyle z} , can never be equivalent to any other function. Asymptotically equivalent functions remain asymptotically equivalent under integration if requirements related to convergence are met. There are more specific requirements for asymptotically equivalent functions to remain asymptotically equivalent under differentiation. == Equation properties == An equation's approximate solution is s ( z ) {\textstyle s(z)} as z {\textstyle z} approaches limit L {\textstyle L} . The equation's terms that may be constants or contain this solution are T 0 ( s ) , T 1 ( s ) , … , T n ( s ) {\textstyle T_{0}(s),T_{1}(s),\ldots ,T_{n}(s)} . If the approximate solution is fully correct, the equation's terms sum to zero in this equation: T 0 ( s ) + T 1 ( s ) + … + T n ( s ) = 0. {\displaystyle T_{0}(s)+T_{1}(s)+\ldots +T_{n}(s)=0.} For distinct integer indices i , j {\textstyle i,j} , this equation is a sum of 2 terms and a remainder R i j ( s ) {\textstyle R_{ij}(s)} expressed as T i ( s ) + T j ( s ) + R i j ( s ) = 0 R i j ( s ) = ∑ k = 0 k ≠ i , k ≠ j n T k ( s ) . {\displaystyle {\begin{aligned}&T_{i}(s)+T_{j}(s)+R_{ij}(s)=0\\&R_{ij}(s)=\sum _{{k=0} \atop {k\neq i,k\neq j}}^{n}T_{k}(s).\end{aligned}}} Balance equation terms T i ( s ) {\textstyle T_{i}(s)} and T j ( s ) {\textstyle T_{j}(s)} means make these terms equal and asymptotically equivalent by finding the function s ( z ) {\textstyle s(z)} that solves the reduced equation T i ( s ) + T j ( s ) = 0 {\textstyle T_{i}(s)+T_{j}(s)=0} with T i ( s ) ≠ 0 {\textstyle T_{i}(s)\neq 0} and T j ( s ) ≠ 0 {\textstyle T_{j}(s)\neq 0} . This solution s ( z ) {\textstyle s(z)} is consistent if terms T i ( s ) {\textstyle T_{i}(s)} and T j ( s ) {\textstyle T_{j}(s)} are dominant; dominant means the remaining equation terms R i j ( s ) {\textstyle R_{ij}(s)} are much less than terms T i ( s ) {\textstyle T_{i}(s)} and T j ( s ) {\textstyle T_{j}(s)} as z {\textstyle z} approaches L {\textstyle L} . A consistent solution that balances two equation terms may generate an accurate approximation to the full equation's solution for z {\textstyle z} values approaching L {\textstyle L} . Approximate solutions arising from balancing different terms of an equation may generate distinct approximate solutions e.g. inner and outer layer solutions. Substituting the scaled function s ( z ) = ( z − L ) p s ~ ( z ) {\textstyle s(z)=(z-L)^{p}{\tilde {s}}(z)} into the equation and taking the limit as z {\textstyle z} approaches L {\textstyle L} may generate simplified reduced equations for distinct exponent values of p {\textstyle p} . These simplified equations are called distinguished limits and identify balanced dominant equation terms. The scale transformation generates the scaled functions. The dominant balance method applies scale transformations to balance equation terms whose factors contain distinct exponents. For example, T i ( s ) {\textstyle T_{i}(s)} contains factor ( z − L ) q {\textstyle (z-L)^{q}} and term T j ( s ) {\textstyle T_{j}(s)} contains factor ( z − L ) r {\textstyle (z-L)^{r}} with q ≠ r {\textstyle q\neq r} . Scaled functions are applied to differential equations when z {\textstyle z} is an equation parameter, not the differential equation´s independent variable. The Kruskal-Newton diagram facilitates identifying the required scaled functions needed for dominant balance of algebraic and differential equations. For differential equation solutions containing an irregular singularity, the leading behavior is the first term of an asymptotic series solution that remains when the independent variable z {\textstyle z} approaches an irregular singularity L {\textstyle L} . The controlling factor is the fastest changing part of the leading behavior. It is advised to "show that the equation for the function obtained by factoring off the dominant balance solution from the exact solution itself has a solution that varies less rapidly than the dominant balance solution." == Algorithm == The input is the set of equation terms and the limit L. The output is the set of approximate solutions. For each pair of distinct equation terms T i ( s ) , T j ( s ) {\textstyle T_{i}(s),T_{j}(s)} the algorithm applies a scale transformation if needed, balances the selected terms by finding a function that solves the reduced equation and then determines if this function is consistent. If the function balances the terms and is consistent, the algorithm adds the function to the set of approximate solutions, otherwise the algorithm rejects the function. The process is repeated for each pair of distinct equation terms. Inputs Set of equation terms { T 0 ( s ) , T 1 ( s ) , … , T n ( s ) } {\textstyle \{T_{0}(s),T_{1}(s),\ldots ,T_{n}(s)\}} and limit L {\textstyle L} Output Set of approximate solutions { s 0 ( z ) , s 1 ( z ) , … } {\textstyle \{s_{0}(z),s_{1}(z),\dots \}} For each pair of distinct equation terms T i ( s ) , T j ( s ) {\textstyle T_{i}(s),T_{j}(s)} do: Apply a scale transformation if needed. Solve the reduced equation: T i ( s ) + T j ( s ) = 0 {\textstyle T_{i}(s)+T_{j}(s)=0} with T i ( s ) ≠ 0 {\textstyle T_{i}(s)\neq 0} and T j ( s ) ≠ 0 {\textstyle T_{j}(s)\neq 0} . Verify consistency: R i j ( s ) ≪ T i ( s ) ( z → L ) {\textstyle R_{ij}(s)\ll T_{i}(s)\ (z\to L)} and R i j ( s ) ≪ T j ( s ) ( z → L ) . {\textstyle R_{ij}(s)\ll T_{j}(s)\ (z\to L).} If function s ( z ) {\textstyle s(z)} is consistent and solves the reduced equation, add this function to the set of approximate solutions, otherwise reject the function. == Improved accuracy == The method may be iterated to generate additional terms of an asymptotic expansion to provide a more accurate solution. Iterative methods such as the Newton-Raphson method may generate a more accurate solution. A perturbation series, using the approximate solution as the first term, may also generate a more accurate solution. == Examples == === Algebraic function === The dominant balance method will find an explicit approximate expression for the multi-valued function s = s ( z ) {\textstyle s=s(z)} defined by the equation 1 − 16 s + z s 5 = 0 {\textstyle 1-16s+zs^{5}=0} as z {\textstyle z} approaches zero. ==== Input ==== The set of equation terms is { 1 , − 16 s , z s 5 } {\textstyle \{1,-16s,zs^{5}\}} and the limit is zero. ==== First term pair ==== Select the terms 1 {\textstyle 1} and − 16 s {\textstyle -16s} . The scale transformation is not required. Solve the reduced equation: 1 − 16 s = 0 , s ( z ) = 1 16 {\displaystyle 1-16s=0,s(z)={\tfrac {1}{16}}} . Verify consistency: z s 5 ≪ 1 ( z → 0 ) , z s 5 ≪ 16 s ( z → 0 ) {\displaystyle zs^{5}\ll 1\ (z\to 0),\ zs^{5}\ll 16s\ (z\to 0)\ } for s ( z ) = 1 16 . {\displaystyle s(z)={\tfrac {1}{16}}.} Add this function to the set of approximate solutions: s 0 ( z ) = 1 16 {\displaystyle s_{0}(z)={\tfrac {1}{16}}} . ==== Second term pair ==== Select the terms − 16 s {\displaystyle -16s} and z s 5 {\displaystyle zs^{5}} . Apply the scale transformation s = z − 1 / 4 s ~ {\displaystyle s=z^{-1/4}{\tilde {s}}} . The transformed equation is z 1 / 4 − 16 s ~ + s ~ 5 = 0 {\displaystyle z^{1/4}-16{\tilde {s}}+{\tilde {s}}^{5}=0} . Solve the reduced equation: − 16 s ~ + s ~ 5 = 0 , s ~ = 2 , − 2 , 2 i , − 2 i {\displaystyle -16{\tilde {s}}+{\tilde {s}}^{5}=0,\ {\tilde {s}}=2,-2,2i,-2i} . Verify consistency: z 1 / 4 ≪ 16 s ~ ( z → 0 ) , z 1 / 4 ≪ s ~ 5 ( z → 0 ) {\displaystyle z^{1/4}\ll 16{\tilde {s}}\ (z\to 0),\ z^{1/4}\ll {\tilde {s}}^{5}\ (z\to 0)\ } for s ~ = 2 , − 2 , 2 i , − 2 i . {\displaystyle {\tilde {s}}=2,-2,2i,-2i.} Add these functions to the set of approximate solutions: s 1 ( z ) = 2 z 1 / 4 , s 2 ( z ) = − 2 z 1 / 4 , s 3 ( z ) = 2 i z 1 / 4 , s 4 ( z ) = − 2 i z 1 / 4 . {\displaystyle s_{1}(z)={\frac {2}{z^{1/4}}},s_{2}(z)={\frac {-2}{z^{1/4}}},s_{3}(z)={\frac {2i}{z^{1/4}}},s_{4}(z)={\frac {-2i}{z^{1/4}}}.} ==== Third term pair ==== Select the terms 1 {\displaystyle 1} and z s 5 {\displaystyle zs^{5}} . Apply the scale transformation s = z − 1 / 5 s ~ {\displaystyle s=z^{-1/5}{\tilde {s}}} . The transformed equation is 1 − 16 z − 1 / 5 s ~ + s ~ 5 = 0. {\displaystyle 1-16z^{-1/5}{\tilde {s}}+{\tilde {s}}^{5}=0.} Solve the reduced equation: 1 + s ~ 5 = 0 , s ~ = ( − 1 ) 1 / 5 . {\displaystyle 1+{\tilde {s}}^{5}=0,\ {\tilde {s}}=(-1)^{1/5}.} The function is not consistent: − 16 z − 1 / 5 s ~ ≫ 1 ( z → 0 ) , z − 1 / 5 s ~ ≫ s ~ 5 ( z → 0 ) {\displaystyle -16z^{-1/5}{\tilde {s}}\gg 1\ (z\to 0),\ z^{-1/5}{\tilde {s}}\gg {\tilde {s}}^{5}\ (z\to 0)\ } for s ~ = ( − 1 ) 1 / 5 . {\displaystyle {\tilde {s}}=(-1)^{1/5}.} Reject this function: s = z − 1 / 5 ( − 1 ) 1 / 5 . {\displaystyle s=z^{-1/5}(-1)^{1/5}.} ==== Output ==== The set of approximate solutions has 5 functions: { 1 16 , 2 z 1 / 4 , − 2 z 1 / 4 , 2 i z 1 / 4 , − 2 i z 1 / 4 } . {\displaystyle \left\{{\frac {1}{16}},{\frac {2}{z^{1/4}}},{\frac {-2}{z^{1/4}}},{\frac {2i}{z^{1/4}}},{\frac {-2i}{z^{1/4}}}\right\}.} ==== Perturbation series solution ==== The approximate solutions are the first terms in the perturbation series solutions. s 0 ( z ) = 1 16 + 1 16777216 z 1 + 5 17592186044416 z 2 + … , s 1 ( z ) = 2 z 1 / 4 − 1 64 − 5 16384 z 1 4 − 5 524288 z 1 2 − … , s 2 ( z ) = − 2 z 1 / 4 − 1 64 + 5 16384 z 1 4 − 5 524288 z 1 2 + … , s 3 ( z ) = 2 i z 1 / 4 − 1 64 + 5 i 16384 z 1 4 + 5 524288 z 1 2 − … s 4 ( z ) = − 2 i z 1 / 4 − 1 64 − 5 i 16384 z 1 4 + 5 524288 z 1 2 + … , {\displaystyle {\begin{aligned}&s_{0}(z)={\frac {1}{16}}+{\frac {1}{16777216}}z^{1}+{\frac {5}{17592186044416}}z^{2}+\ldots ,\\&s_{1}(z)={\frac {2}{z^{1/4}}}-{\frac {1}{64}}-{\frac {5}{16384}}z^{\frac {1}{4}}-{\frac {5}{524288}}z^{\frac {1}{2}}-\ldots ,\\&s_{2}(z)=-{\frac {2}{z^{1/4}}}-{\frac {1}{64}}+{\frac {5}{16384}}z^{\frac {1}{4}}-{\frac {5}{524288}}z^{\frac {1}{2}}+\ldots ,\\&s_{3}(z)={\frac {2i}{z^{1/4}}}-{\frac {1}{64}}+{\frac {5i}{16384}}z^{\frac {1}{4}}+{\frac {5}{524288}}z^{\frac {1}{2}}-\ldots \\&s_{4}(z)=-{\frac {2i}{z^{1/4}}}-{\frac {1}{64}}-{\frac {5i}{16384}}z^{\frac {1}{4}}+{\frac {5}{524288}}z^{\frac {1}{2}}+\ldots ,\\\end{aligned}}} === Differential equation === The differential equation z 3 w ′ ′ − w = 0 {\textstyle z^{3}w^{\prime \prime }-w=0} is known to have a solution with an exponential leading term. The transformation w ( z ) = e s ( z ) {\textstyle w(z)=e^{s(z)}} leads to the differential equation 1 − z 3 ( s ′ ) 2 − z 3 s ′ ′ = 0 {\textstyle 1-z^{3}(s^{\prime })^{2}-z^{3}s^{\prime \prime }=0} . The dominant balance method will find an approximate solution as z {\textstyle z} approaches zero. Scaled functions will not be used because z {\textstyle z} is the differential equation's independent variable, not a differential equation parameter. ==== Input ==== The set of equation terms is { 1 , − z 3 ( s ′ ) 2 , − z 3 s ′ ′ } {\textstyle \{1,-z^{3}(s^{\prime })^{2},-z^{3}s^{\prime \prime }\}} and the limit is zero. ===== First term pair ===== Select 1 {\displaystyle 1} and − z 3 ( s ′ ) 2 {\displaystyle -z^{3}(s^{\prime })^{2}} . The scale transformation is not required. Solve the reduced equation: 1 − z 3 ( s ′ ) 2 = 0 , s ( z ) = ± 2 z − 1 / 2 {\displaystyle 1-z^{3}(s^{\prime })^{2}=0,\ s(z)=\pm 2z^{-1/2}} Verify consistency: z 3 s ′ ′ ≪ 1 ( z → 0 ) , z 3 s ′ ′ ≪ z 3 ( s ′ ) 2 ( z → 0 ) {\displaystyle z^{3}s^{\prime \prime }\ll 1\ (z\to 0),\ z^{3}s^{\prime \prime }\ll z^{3}(s^{\prime })^{2}\ (z\to 0)} for s ( z ) = ± 2 z − 1 / 2 . {\displaystyle s(z)=\pm 2z^{-1/2}.} Add these 2 functions to the set of approximate solutions: s + ( z ) = + 2 z − 1 / 2 , s − ( z ) = − 2 z − 1 / 2 . {\displaystyle s_{+}(z)=+2z^{-1/2},\ s_{-}(z)=-2z^{-1/2}.} ==== Second term pair ==== Select 1 {\displaystyle 1} and − z 3 s ′ ′ {\displaystyle -z^{3}s^{\prime \prime }} The scale transformation is not required. Solve the reduced equation: 1 − z 3 s ′ ′ = 0 , s ( z ) = 1 2 z − 1 {\displaystyle 1-z^{3}s^{\prime \prime }=0,\ s(z)={\tfrac {1}{2}}z^{-1}} The function is not consistent: z 3 ( s ′ ) 2 ≫ 1 ( z → 0 ) , z 3 ( s ′ ) 2 ≫ z 3 s ′ ′ ( z → 0 ) {\displaystyle z^{3}(s^{\prime })^{2}\gg 1\ (z\to 0),\ z^{3}(s^{\prime })^{2}\gg z^{3}s^{\prime \prime }\ (z\to 0)} for s ( z ) = 1 2 z − 1 . {\displaystyle s(z)={\tfrac {1}{2}}z^{-1}.} Reject this function: s ( z ) = 1 2 z − 1 . {\displaystyle s(z)={\tfrac {1}{2}}z^{-1}.} . ==== Third term pair ==== Select − z 3 ( s ′ ) 2 {\displaystyle -z^{3}(s^{\prime })^{2}} and − z 3 s ′ ′ {\displaystyle -z^{3}s^{\prime \prime }} . The scale transformation is not required. Solve the reduced equation: z 3 ( s ′ ) 2 + z 3 s ′ ′ = 0 , s ( z ) = ln z {\displaystyle z^{3}(s^{\prime })^{2}+z^{3}s^{\prime \prime }=0,\ s(z)=\ln z} . The function is not consistent: 1 ≫ z 3 ( s ′ ) 2 ( z → 0 ) {\displaystyle 1\gg z^{3}(s^{\prime })^{2}\ (z\to 0)\ } and 1 ≫ z 3 s ′ ′ ( z → 0 ) {\displaystyle \ 1\gg \ z^{3}s^{\prime \prime }\ (z\to 0)} for s ( z ) = ln z . {\displaystyle s(z)=\ln z.} Reject this function: s ( z ) = ln z . {\displaystyle s(z)=\ln z.} ==== Output ==== The set of approximate solutions has 2 functions: { + 2 z − 1 / 2 , − 2 z − 1 / 2 } . {\displaystyle \left\{+2z^{-1/2},-2z^{-1/2}\right\}.} ==== Find 2-term solutions ==== Using the 1-term solution, a 2-term solution is s 2 ± ( z ) = ± 2 z − 1 / 2 + s ( z ) . {\displaystyle s_{2\pm }(z)=\pm 2z^{-1/2}+s(z).} Substitution of this 2-term solution into the original differential equation generates a new differential equation: 1 − z 3 ( s 2 ± ′ ) 2 − z 3 s 2 ± ′ ′ = 0 ± 1 ∓ 4 3 z s ′ + 2 3 z 5 / 2 ( s ′ ) 2 + 2 3 z 5 / 2 s ′ ′ = 0. {\displaystyle {\begin{aligned}1-z^{3}(s_{2\pm }^{\prime })^{2}-z^{3}s_{2\pm }^{\prime \prime }&=0\\\pm 1\mp {\frac {4}{3}}zs^{\prime }+{\frac {2}{3}}z^{5/2}(s^{\prime })^{2}+{\frac {2}{3}}z^{5/2}s^{\prime \prime }&=0.\end{aligned}}} ==== Input ==== The set of equation terms is { ± 1 , ∓ 4 3 z s ′ , 2 3 z 5 / 2 ( s ′ ) 2 , 2 3 z 5 / 2 s ′ ′ } {\textstyle \{\pm 1,\mp {\frac {4}{3}}zs^{\prime },{\frac {2}{3}}z^{5/2}(s^{\prime })^{2},{\frac {2}{3}}z^{5/2}s^{\prime \prime }\}} and the limit is zero. ===== First term pair ===== 1. Select 1 {\displaystyle 1} and − 4 3 z s ′ {\displaystyle -{\tfrac {4}{3}}zs^{\prime }} . 2. The scale transformation is not required. 3. Solve the reduced equation: 1 − 4 3 z s ′ = 0 , s ( z ) = 3 4 ln z {\displaystyle 1-{\tfrac {4}{3}}zs^{\prime }=0,\ s(z)={\tfrac {3}{4}}\ln z} . 4. Verify consistency: 2 3 z 5 / 2 ( s ′ ) 2 + 2 3 z 5 / 2 s ′ ′ ≪ 1 ( z → 0 ) , for s ( z ) = 3 4 ln z {\displaystyle {\tfrac {2}{3}}z^{5/2}(s^{\prime })^{2}+{\tfrac {2}{3}}z^{5/2}s^{\prime \prime }\ll 1\ (z\to 0),{\text{for}}\ s(z)={\tfrac {3}{4}}\ln z} 2 3 z 5 / 2 ( s ′ ) 2 + 2 3 z 5 / 2 s ′ ′ ≪ 4 3 z s ′ ( z → 0 ) for s ( z ) = 3 4 ln z . {\displaystyle {\tfrac {2}{3}}z^{5/2}(s^{\prime })^{2}+{\tfrac {2}{3}}z^{5/2}s^{\prime \prime }\ll {\tfrac {4}{3}}zs^{\prime }\ (z\to 0)\ {\text{for}}\ s(z)={\tfrac {3}{4}}\ln z.} 5. Add these functions to the set of approximate solutions: s 2 + ( z ) = + 2 z − 1 / 2 + 3 4 ln z {\textstyle s_{2+}(z)=+2z^{-1/2}+{\tfrac {3}{4}}\ln z} s 2 − ( z ) = − 2 z − 1 / 2 + 3 4 ln z {\textstyle s_{2-}(z)=-2z^{-1/2}+{\tfrac {3}{4}}\ln z} . ==== Other term pairs ==== For other term pairs, the functions that solve the reduced equations are not consistent. ==== Output ==== The set of approximate solutions has 2 functions: { + 2 z − 1 / 2 + 3 4 ln z , − 2 z − 1 / 2 + 3 4 ln z } . {\displaystyle \left\{+2z^{-1/2}+{\tfrac {3}{4}}\ln z,-2z^{-1/2}+{\tfrac {3}{4}}\ln z\right\}.} ==== Asymptotic expansion ==== The next iteration generates a 3-term solution s 3 ± ( z ) = ± 2 z − 1 / 2 + 3 4 ln ( z ) + h ( z ) {\textstyle s_{3\pm }(z)=\pm 2z^{-1/2}+{\tfrac {3}{4}}\operatorname {ln} (z)+h(z)} with h ( z ) ≪ 1 ( z → 0 ) {\textstyle h(z)\ll 1\ (z\to 0)} and this means that a power series expansion can represent the remainder of the solution. The dominant balance method generates the leading term to this asymptotic expansion with constant A {\textstyle A} and expansion coefficients determined by substitution into the full differential equation: w ( z ) = A z 3 / 4 e ± 2 z − 1 / 2 ( ∑ n = 0 m a n z n / 2 ) {\displaystyle w(z)=Az^{3/4}e^{\pm 2z^{-1/2}}\left(\sum _{n=0}^{m}\ a_{n}z^{n/2}\right)} a n + 1 = ± ( n − 1 / 2 ) ( n + 3 / 2 ) a n 4 ( n + 1 ) . {\displaystyle a_{n+1}=\pm {\frac {(n-1/2)(n+3/2)a_{n}}{4(n+1)}}.} A partial sum of this non-convergent series generates an approximate solution. The leading term corresponds to the Liouville-Green (LG) or Wentzel–Kramers–Brillouin (WKB) approximation. == Citations == == References == == See also == Asymptotic analysis
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Wikipedia:Method of exhaustion#0
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The method of exhaustion (Latin: methodus exhaustionis) is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too. == History == The idea originated in the late 5th century BC with Antiphon, although it is not entirely clear how well he understood it. The theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. The first use of the term was in 1647 by Gregory of Saint Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen as a precursor to the methods of calculus. The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. An important alternative approach was Cavalieri's principle, also termed the method of indivisibles which eventually evolved into the infinitesimal calculus of Roberval, Torricelli, Wallis, Leibniz, and others. === Euclid === Euclid used the method of exhaustion to prove the following six propositions in the 12th book of his Elements. Proposition 2: The area of circles is proportional to the square of their diameters. Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11: The volume of a cone (or cylinder) of the same height is proportional to the area of the base. Proposition 12: The volume of a cone (or cylinder) that is similar to another is proportional to the cube of the ratio of the diameters of the bases. Proposition 18: The volume of a sphere is proportional to the cube of its diameter. === Archimedes === Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a sequence of polygons with an increasing number of sides and a corresponding increase in area. The quotients formed by the area of these polygons divided by the square of the circle radius can be made arbitrarily close to π as the number of polygon sides becomes large, proving that the area inside the circle of radius r is πr2, π being defined as the ratio of the circumference to the diameter (C/d). He also provided the bounds 3 + 10/71 < π < 3 + 10/70, (giving a range of 1/497) by comparing the perimeters of the circle with the perimeters of the inscribed and circumscribed 96-sided regular polygons. Other results he obtained with the method of exhaustion included The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height (the quadrature of the parabola); The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; The volume of a sphere is 4 times that of a cone having a base of the same radius and height equal to this radius; The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter; The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length; Use of the method of exhaustion also led to the successful evaluation of an infinite geometric series (for the first time); == See also == The Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule Pythagorean Theorem == References ==
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Wikipedia:Method of matched asymptotic expansions#0
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In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, each of which is valid (i.e. accurate) for part of the range of the independent variable, and then combining these different solutions together to give a single approximate solution that is valid for the whole range of values of the independent variable. In the Russian literature, these methods were known under the name of "intermediate asymptotics" and were introduced in the work of Yakov Zeldovich and Grigory Barenblatt. == Method overview == In a large class of singularly perturbed problems, the domain may be divided into two or more subdomains. In one of these, often the largest, the solution is accurately approximated by an asymptotic series found by treating the problem as a regular perturbation (i.e. by setting a relatively small parameter to zero). The other subdomains consist of one or more small regions in which that approximation is inaccurate, generally because the perturbation terms in the problem are not negligible there. These areas are referred to as transition layers in general, and specifically as boundary layers or interior layers depending on whether they occur at the domain boundary (as is the usual case in applications) or inside the domain, respectively. An approximation in the form of an asymptotic series is obtained in the transition layer(s) by treating that part of the domain as a separate perturbation problem. This approximation is called the inner solution, and the other is the outer solution, named for their relationship to the transition layer(s). The outer and inner solutions are then combined through a process called "matching" in such a way that an approximate solution for the whole domain is obtained. == A simple example == Consider the boundary value problem ε y ″ + ( 1 + ε ) y ′ + y = 0 , {\displaystyle \varepsilon y''+(1+\varepsilon )y'+y=0,} where y {\displaystyle y} is a function of independent time variable t {\displaystyle t} , which ranges from 0 to 1, the boundary conditions are y ( 0 ) = 0 {\displaystyle y(0)=0} and y ( 1 ) = 1 {\displaystyle y(1)=1} , and ε {\displaystyle \varepsilon } is a small parameter, such that 0 < ε ≪ 1 {\displaystyle 0<\varepsilon \ll 1} . === Outer solution, valid for t = O(1) === Since ε {\displaystyle \varepsilon } is very small, our first approach is to treat the equation as a regular perturbation problem, i.e. make the approximation ε = 0 {\displaystyle \varepsilon =0} , and hence find the solution to the problem y ′ + y = 0. {\displaystyle y'+y=0.} Alternatively, consider that when y {\displaystyle y} and t {\displaystyle t} are both of size O(1), the four terms on the left hand side of the original equation are respectively of sizes O ( ε ) {\displaystyle O(\varepsilon )} , O(1), O ( ε ) {\displaystyle O(\varepsilon )} and O(1). The leading-order balance on this timescale, valid in the distinguished limit ε → 0 {\displaystyle \varepsilon \to 0} , is therefore given by the second and fourth terms, i.e., y ′ + y = 0. {\displaystyle y'+y=0.} This has solution y = A e − t {\displaystyle y=Ae^{-t}} for some constant A {\displaystyle A} . Applying the boundary condition y ( 0 ) = 0 {\displaystyle y(0)=0} , we would have A = 0 {\displaystyle A=0} ; applying the boundary condition y ( 1 ) = 1 {\displaystyle y(1)=1} , we would have A = e {\displaystyle A=e} . It is therefore impossible to satisfy both boundary conditions, so ε = 0 {\displaystyle \varepsilon =0} is not a valid approximation to make across the whole of the domain (i.e. this is a singular perturbation problem). From this we infer that there must be a boundary layer at one of the endpoints of the domain where ε {\displaystyle \varepsilon } needs to be included. This region will be where ε {\displaystyle \varepsilon } is no longer negligible compared to the independent variable t {\displaystyle t} , i.e. t {\displaystyle t} and ε {\displaystyle \varepsilon } are of comparable size, i.e. the boundary layer is adjacent to t = 0 {\displaystyle t=0} . Therefore, the other boundary condition y ( 1 ) = 1 {\displaystyle y(1)=1} applies in this outer region, so A = e {\displaystyle A=e} , i.e. y O = e 1 − t {\displaystyle y_{\mathrm {O} }=e^{1-t}} is an accurate approximate solution to the original boundary value problem in this outer region. It is the leading-order solution. === Inner solution, valid for t = O(ε) === In the inner region, t {\displaystyle t} and ε {\displaystyle \varepsilon } are both tiny, but of comparable size, so define the new O(1) time variable τ = t / ε {\displaystyle \tau =t/\varepsilon } . Rescale the original boundary value problem by replacing t {\displaystyle t} with τ ε {\displaystyle \tau \varepsilon } , and the problem becomes 1 ε y ″ ( τ ) + ( 1 + ε ) 1 ε y ′ ( τ ) + y ( τ ) = 0 , {\displaystyle {\frac {1}{\varepsilon }}y''(\tau )+\left({1+\varepsilon }\right){\frac {1}{\varepsilon }}y'(\tau )+y(\tau )=0,} which, after multiplying by ε {\displaystyle \varepsilon } and taking ε = 0 {\displaystyle \varepsilon =0} , is y ″ + y ′ = 0. {\displaystyle y''+y'=0.} Alternatively, consider that when t {\displaystyle t} has reduced to size O ( ε ) {\displaystyle O(\varepsilon )} , then y {\displaystyle y} is still of size O(1) (using the expression for y O {\displaystyle y_{\mathrm {O} }} ), and so the four terms on the left hand side of the original equation are respectively of sizes O ( ε − 1 ) {\displaystyle O(\varepsilon ^{-1})} , O ( ε − 1 ) {\displaystyle O(\varepsilon ^{-1})} , O(1) and O(1). The leading-order balance on this timescale, valid in the distinguished limit ε → 0 {\displaystyle \varepsilon \to 0} , is therefore given by the first and second terms, i.e. y ″ + y ′ = 0. {\displaystyle y''+y'=0.} This has solution y = B − C e − τ {\displaystyle y=B-Ce^{-\tau }} for some constants B {\displaystyle B} and C {\displaystyle C} . Since y ( 0 ) = 0 {\displaystyle y(0)=0} applies in this inner region, this gives B = C {\displaystyle B=C} , so an accurate approximate solution to the original boundary value problem in this inner region (it is the leading-order solution) is y I = B ( 1 − e − τ ) = B ( 1 − e − t / ε ) . {\displaystyle y_{\mathrm {I} }=B\left({1-e^{-\tau }}\right)=B\left({1-e^{-t/\varepsilon }}\right).} === Matching === We use matching to find the value of the constant B {\displaystyle B} . The idea of matching is that the inner and outer solutions should agree for values of t {\displaystyle t} in an intermediate (or overlap) region, i.e. where ε ≪ t ≪ 1 {\displaystyle \varepsilon \ll t\ll 1} . We need the outer limit of the inner solution to match the inner limit of the outer solution, i.e., lim τ → ∞ y I = lim t → 0 y O , {\displaystyle \lim _{\tau \to \infty }y_{\mathrm {I} }=\lim _{t\to 0}y_{\mathrm {O} },} which gives B = e {\displaystyle B=e} . The above problem is the simplest of the simple problems dealing with matched asymptotic expansions. One can immediately calculate that e 1 − t {\displaystyle e^{1-t}} is the entire asymptotic series for the outer region whereas the O ( ε ) {\displaystyle {\mathcal {O}}(\varepsilon )} correction to the inner solution y I {\displaystyle y_{\mathrm {I} }} is B ( 1 − e − t / ε ) {\textstyle B(1-e^{-t/\varepsilon })} and the constant of integration B {\displaystyle B} must be obtained from inner-outer matching. Notice, the intuitive idea for matching of taking the limits i.e. lim τ → ∞ y I = lim t → 0 y O , {\textstyle \lim _{\tau \to \infty }y_{\mathrm {I} }=\lim _{t\to 0}y_{\mathrm {O} },} doesn't apply at this level. This is simply because the underlined term doesn't converge to a limit. The methods to follow in these types of cases are either to go for a) method of an intermediate variable or using b) the Van-Dyke matching rule. The former method is cumbersome and works always whereas the Van-Dyke matching rule is easy to implement but with limited applicability. A concrete boundary value problem having all the essential ingredients is the following. Consider the boundary value problem ε y ″ − x 2 y ′ − y = 1 , y ( 0 ) = y ( 1 ) = 1 {\displaystyle \varepsilon y''-x^{2}y'-y=1,\quad y(0)=y(1)=1} The conventional outer expansion y O = y 0 + ε y 1 + ⋯ {\displaystyle y_{\mathrm {O} }=y_{0}+\varepsilon y_{1}+\cdots } gives y 0 = α e 1 / x − 1 {\displaystyle y_{0}=\alpha e^{1/x}-1} , where α {\displaystyle \alpha } must be obtained from matching. The problem has boundary layers both on the left and on the right. The left boundary layer near 0 {\displaystyle 0} has a thickness ε 1 / 2 {\displaystyle \varepsilon ^{1/2}} whereas the right boundary layer near 1 {\displaystyle 1} has thickness ε {\displaystyle \varepsilon } . Let us first calculate the solution on the left boundary layer by rescaling X = x / ε 1 / 2 , Y = y {\displaystyle X=x/\varepsilon ^{1/2},\;Y=y} , then the differential equation to satisfy on the left is Y ″ − ε 1 / 2 X 2 Y ′ − Y = 1 , Y ( 0 ) = 1 {\displaystyle Y''-\varepsilon ^{1/2}X^{2}Y'-Y=1,\quad Y(0)=1} and accordingly, we assume an expansion Y l = Y 0 l + ε 1 / 2 Y 1 / 2 l + ⋯ {\displaystyle Y^{l}=Y_{0}^{l}+\varepsilon ^{1/2}Y_{1/2}^{l}+\cdots } . The O ( 1 ) {\displaystyle {\mathcal {O}}(1)} inhomogeneous condition on the left provides us the reason to start the expansion at O ( 1 ) {\displaystyle {\mathcal {O}}(1)} . The leading order solution is Y 0 l = 2 e − X − 1 {\displaystyle Y_{0}^{l}=2e^{-X}-1} . This with 1 − 1 {\displaystyle 1-1} van-Dyke matching gives α = 0 {\displaystyle \alpha =0} . Let us now calculate the solution on the right rescaling X = ( 1 − x ) / ε , Y = y {\displaystyle X=(1-x)/\varepsilon ,\;Y=y} , then the differential equation to satisfy on the right is Y ″ + ( 1 − 2 ε X + ε 2 X 2 ) Y ′ − ε Y = ε , Y ( 1 ) = 1 , {\displaystyle Y''+\left(1-2\varepsilon X+\varepsilon ^{2}X^{2}\right)Y'-\varepsilon Y=\varepsilon ,\quad Y(1)=1,} and accordingly, we assume an expansion Y r = Y 0 r + ε Y 1 r + ⋯ . {\displaystyle Y^{r}=Y_{0}^{r}+\varepsilon Y_{1}^{r}+\cdots .} The O ( 1 ) {\displaystyle {\mathcal {O}}(1)} inhomogeneous condition on the right provides us the reason to start the expansion at O ( 1 ) {\displaystyle {\mathcal {O}}(1)} . The leading order solution is Y 0 r = ( 1 − B ) + B e − X {\displaystyle Y_{0}^{r}=(1-B)+Be^{-X}} . This with 1 − 1 {\displaystyle 1-1} van-Dyke matching gives B = 2 {\displaystyle B=2} . Proceeding in a similar fashion if we calculate the higher order-corrections we get the solutions as Y l = 2 e − X − 1 + ε 1 / 2 e − X ( X 3 3 + X 2 2 + X 2 ) + O ( ε ) , X = x ε 1 / 2 . {\displaystyle Y^{l}=2e^{-X}-1+\varepsilon ^{1/2}e^{-X}\left({\frac {X^{3}}{3}}+{\frac {X^{2}}{2}}+{\frac {X}{2}}\right)+{\mathcal {O}}(\varepsilon ),\quad X={\frac {x}{\varepsilon ^{1/2}}}.} y ≡ − 1. {\displaystyle y\equiv -1.} Y r = 2 e − X − 1 + 2 ε e − X ( X + X 2 ) + O ( ε 2 ) , X = 1 − x ε . {\displaystyle Y^{r}=2e^{-X}-1+2\varepsilon e^{-X}\left(X+X^{2}\right)+{\mathcal {O}}(\varepsilon ^{2}),\quad X={\frac {1-x}{\varepsilon }}.} === Composite solution === To obtain our final, matched, composite solution, valid on the whole domain, one popular method is the uniform method. In this method, we add the inner and outer approximations and subtract their overlapping value, y o v e r l a p {\displaystyle \,y_{\mathrm {overlap} }} , which would otherwise be counted twice. The overlapping value is the outer limit of the inner boundary layer solution, and the inner limit of the outer solution; these limits were above found to equal e {\displaystyle e} . Therefore, the final approximate solution to this boundary value problem is, y ( t ) = y I + y O − y o v e r l a p = e ( 1 − e − t / ε ) + e 1 − t − e = e ( e − t − e − t / ε ) . {\displaystyle y(t)=y_{\mathrm {I} }+y_{\mathrm {O} }-y_{\mathrm {overlap} }=e\left({1-e^{-t/\varepsilon }}\right)+e^{1-t}-e=e\left({e^{-t}-e^{-t/\varepsilon }}\right).} Note that this expression correctly reduces to the expressions for y I {\displaystyle y_{\mathrm {I} }} and y O {\displaystyle y_{\mathrm {O} }} when t {\displaystyle t} is O ( ε ) {\displaystyle O(\varepsilon )} and O(1), respectively. === Accuracy === This final solution satisfies the problem's original differential equation (shown by substituting it and its derivatives into the original equation). Also, the boundary conditions produced by this final solution match the values given in the problem, up to a constant multiple. This implies, due to the uniqueness of the solution, that the matched asymptotic solution is identical to the exact solution up to a constant multiple. This is not necessarily always the case, any remaining terms should go to zero uniformly as ε → 0 {\displaystyle \varepsilon \rightarrow 0} . Not only does our solution successfully approximately solve the problem at hand, it closely approximates the problem's exact solution. It happens that this particular problem is easily found to have exact solution y ( t ) = e − t − e − t / ε e − 1 − e − 1 / ε , {\displaystyle y(t)={\frac {e^{-t}-e^{-t/\varepsilon }}{e^{-1}-e^{-1/\varepsilon }}},} which has the same form as the approximate solution, by the multiplying constant. The approximate solution is the first term in a binomial expansion of the exact solution in powers of e 1 − 1 / ε {\displaystyle e^{1-1/\varepsilon }} . === Location of boundary layer === Conveniently, we can see that the boundary layer, where y ′ {\displaystyle y'} and y ″ {\displaystyle y''} are large, is near t = 0 {\displaystyle t=0} , as we supposed earlier. If we had supposed it to be at the other endpoint and proceeded by making the rescaling τ = ( 1 − t ) / ε {\displaystyle \tau =(1-t)/\varepsilon } , we would have found it impossible to satisfy the resulting matching condition. For many problems, this kind of trial and error is the only way to determine the true location of the boundary layer. == Harder problems == The problem above is a simple example because it is a single equation with only one dependent variable, and there is one boundary layer in the solution. Harder problems may contain several co-dependent variables in a system of several equations, and/or with several boundary and/or interior layers in the solution. It is often desirable to find more terms in the asymptotic expansions of both the outer and the inner solutions. The appropriate form of these expansions is not always clear: while a power-series expansion in ε {\displaystyle \varepsilon } may work, sometimes the appropriate form involves fractional powers of ε {\displaystyle \varepsilon } , functions such as ε log ε {\displaystyle \varepsilon \log \varepsilon } , et cetera. As in the above example, we will obtain outer and inner expansions with some coefficients which must be determined by matching. == Second-order differential equations == === Schrödinger-like second-order differential equations === A method of matched asymptotic expansions - with matching of solutions in the common domain of validity - has been developed and used extensively by Dingle and Müller-Kirsten for the derivation of asymptotic expansions of the solutions and characteristic numbers (band boundaries) of Schrödinger-like second-order differential equations with periodic potentials - in particular for the Mathieu equation (best example), Lamé and ellipsoidal wave equations, oblate and prolate spheroidal wave equations, and equations with anharmonic potentials. === Convection–diffusion equations === Methods of matched asymptotic expansions have been developed to find approximate solutions to the Smoluchowski convection–diffusion equation, which is a singularly perturbed second-order differential equation. The problem has been studied particularly in the context of colloid particles in linear flow fields, where the variable is given by the pair distribution function around a test particle. In the limit of low Péclet number, the convection–diffusion equation also presents a singularity at infinite distance (where normally the far-field boundary condition should be placed) due to the flow field being linear in the interparticle separation. This problem can be circumvented with a spatial Fourier transform as shown by Jan Dhont. A different approach to solving this problem was developed by Alessio Zaccone and coworkers and consists in placing the boundary condition right at the boundary layer distance, upon assuming (in a first-order approximation) a constant value of the pair distribution function in the outer layer due to convection being dominant there. This leads to an approximate theory for the encounter rate of two interacting colloid particles in a linear flow field in good agreement with the full numerical solution. When the Péclet number is significantly larger than one, the singularity at infinite separation no longer occurs and the method of matched asymptotics can be applied to construct the full solution for the pair distribution function across the entire domain. == See also == Asymptotic analysis Multiple-scale analysis Activation energy asymptotics == References ==
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Wikipedia:Method of steepest descent#0
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In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form ∫ C f ( z ) e λ g ( z ) d z , {\displaystyle \int _{C}f(z)e^{\lambda g(z)}\,dz,} where C is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that the following conditions hold: C′ passes through one or more zeros of the derivative g′(z), the imaginary part of g(z) is constant on C′. The method of steepest descent was first published by Debye (1909), who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note by Riemann (1863) about hypergeometric functions. The contour of steepest descent has a minimax property, see Fedoryuk (2001). Siegel (1932) described some other unpublished notes of Riemann, where he used this method to derive the Riemann–Siegel formula. == Basic idea == The method of steepest descent is a method to approximate a complex integral of the form I ( λ ) = ∫ C f ( z ) e λ g ( z ) d z {\displaystyle I(\lambda )=\int _{C}f(z)e^{\lambda g(z)}\,\mathrm {d} z} for large λ → ∞ {\displaystyle \lambda \rightarrow \infty } , where f ( z ) {\displaystyle f(z)} and g ( z ) {\displaystyle g(z)} are analytic functions of z {\displaystyle z} . Because the integrand is analytic, the contour C {\displaystyle C} can be deformed into a new contour C ′ {\displaystyle C'} without changing the integral. In particular, one seeks a new contour on which the imaginary part, denoted ℑ ( ⋅ ) {\displaystyle \Im (\cdot )} , of g ( z ) = ℜ [ g ( z ) ] + i ℑ [ g ( z ) ] {\displaystyle g(z)=\Re [g(z)]+i\,\Im [g(z)]} is constant ( ℜ ( ⋅ ) {\displaystyle \Re (\cdot )} denotes the real part). Then I ( λ ) = e i λ ℑ { g ( z ) } ∫ C ′ f ( z ) e λ ℜ { g ( z ) } d z , {\displaystyle I(\lambda )=e^{i\lambda \Im \{g(z)\}}\int _{C'}f(z)e^{\lambda \Re \{g(z)\}}\,\mathrm {d} z,} and the remaining integral can be approximated with other methods like Laplace's method. == Etymology == The method is called the method of steepest descent because for analytic g ( z ) {\displaystyle g(z)} , constant phase contours are equivalent to steepest descent contours. If g ( z ) = X ( z ) + i Y ( z ) {\displaystyle g(z)=X(z)+iY(z)} is an analytic function of z = x + i y {\displaystyle z=x+iy} , it satisfies the Cauchy–Riemann equations ∂ X ∂ x = ∂ Y ∂ y and ∂ X ∂ y = − ∂ Y ∂ x . {\displaystyle {\frac {\partial X}{\partial x}}={\frac {\partial Y}{\partial y}}\qquad {\text{and}}\qquad {\frac {\partial X}{\partial y}}=-{\frac {\partial Y}{\partial x}}.} Then ∂ X ∂ x ∂ Y ∂ x + ∂ X ∂ y ∂ Y ∂ y = ∇ X ⋅ ∇ Y = 0 , {\displaystyle {\frac {\partial X}{\partial x}}{\frac {\partial Y}{\partial x}}+{\frac {\partial X}{\partial y}}{\frac {\partial Y}{\partial y}}=\nabla X\cdot \nabla Y=0,} so contours of constant phase are also contours of steepest descent. == A simple estimate == Let f, S : Cn → C and C ⊂ Cn. If M = sup x ∈ C ℜ ( S ( x ) ) < ∞ , {\displaystyle M=\sup _{x\in C}\Re (S(x))<\infty ,} where ℜ ( ⋅ ) {\displaystyle \Re (\cdot )} denotes the real part, and there exists a positive real number λ0 such that ∫ C | f ( x ) e λ 0 S ( x ) | d x < ∞ , {\displaystyle \int _{C}\left|f(x)e^{\lambda _{0}S(x)}\right|dx<\infty ,} then the following estimate holds: | ∫ C f ( x ) e λ S ( x ) d x | ⩽ const ⋅ e λ M , ∀ λ ∈ R , λ ⩾ λ 0 . {\displaystyle \left|\int _{C}f(x)e^{\lambda S(x)}dx\right|\leqslant {\text{const}}\cdot e^{\lambda M},\qquad \forall \lambda \in \mathbb {R} ,\quad \lambda \geqslant \lambda _{0}.} Proof of the simple estimate: | ∫ C f ( x ) e λ S ( x ) d x | ⩽ ∫ C | f ( x ) | | e λ S ( x ) | d x ≡ ∫ C | f ( x ) | e λ M | e λ 0 ( S ( x ) − M ) e ( λ − λ 0 ) ( S ( x ) − M ) | d x ⩽ ∫ C | f ( x ) | e λ M | e λ 0 ( S ( x ) − M ) | d x | e ( λ − λ 0 ) ( S ( x ) − M ) | ⩽ 1 = e − λ 0 M ∫ C | f ( x ) e λ 0 S ( x ) | d x ⏟ const ⋅ e λ M . {\displaystyle {\begin{aligned}\left|\int _{C}f(x)e^{\lambda S(x)}dx\right|&\leqslant \int _{C}|f(x)|\left|e^{\lambda S(x)}\right|dx\\&\equiv \int _{C}|f(x)|e^{\lambda M}\left|e^{\lambda _{0}(S(x)-M)}e^{(\lambda -\lambda _{0})(S(x)-M)}\right|dx\\&\leqslant \int _{C}|f(x)|e^{\lambda M}\left|e^{\lambda _{0}(S(x)-M)}\right|dx&&\left|e^{(\lambda -\lambda _{0})(S(x)-M)}\right|\leqslant 1\\&=\underbrace {e^{-\lambda _{0}M}\int _{C}\left|f(x)e^{\lambda _{0}S(x)}\right|dx} _{\text{const}}\cdot e^{\lambda M}.\end{aligned}}} == The case of a single non-degenerate saddle point == === Basic notions and notation === Let x be a complex n-dimensional vector, and S x x ″ ( x ) ≡ ( ∂ 2 S ( x ) ∂ x i ∂ x j ) , 1 ⩽ i , j ⩽ n , {\displaystyle S''_{xx}(x)\equiv \left({\frac {\partial ^{2}S(x)}{\partial x_{i}\partial x_{j}}}\right),\qquad 1\leqslant i,\,j\leqslant n,} denote the Hessian matrix for a function S(x). If φ ( x ) = ( φ 1 ( x ) , φ 2 ( x ) , … , φ k ( x ) ) {\displaystyle {\boldsymbol {\varphi }}(x)=(\varphi _{1}(x),\varphi _{2}(x),\ldots ,\varphi _{k}(x))} is a vector function, then its Jacobian matrix is defined as φ x ′ ( x ) ≡ ( ∂ φ i ( x ) ∂ x j ) , 1 ⩽ i ⩽ k , 1 ⩽ j ⩽ n . {\displaystyle {\boldsymbol {\varphi }}_{x}'(x)\equiv \left({\frac {\partial \varphi _{i}(x)}{\partial x_{j}}}\right),\qquad 1\leqslant i\leqslant k,\quad 1\leqslant j\leqslant n.} A non-degenerate saddle point, z0 ∈ Cn, of a holomorphic function S(z) is a critical point of the function (i.e., ∇S(z0) = 0) where the function's Hessian matrix has a non-vanishing determinant (i.e., det S z z ″ ( z 0 ) ≠ 0 {\displaystyle \det S''_{zz}(z^{0})\neq 0} ). The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point: === Complex Morse lemma === The Morse lemma for real-valued functions generalizes as follows for holomorphic functions: near a non-degenerate saddle point z0 of a holomorphic function S(z), there exist coordinates in terms of which S(z) − S(z0) is exactly quadratic. To make this precise, let S be a holomorphic function with domain W ⊂ Cn, and let z0 in W be a non-degenerate saddle point of S, that is, ∇S(z0) = 0 and det S z z ″ ( z 0 ) ≠ 0 {\displaystyle \det S''_{zz}(z^{0})\neq 0} . Then there exist neighborhoods U ⊂ W of z0 and V ⊂ Cn of w = 0, and a bijective holomorphic function φ : V → U with φ(0) = z0 such that ∀ w ∈ V : S ( φ ( w ) ) = S ( z 0 ) + 1 2 ∑ j = 1 n μ j w j 2 , det φ w ′ ( 0 ) = 1 , {\displaystyle \forall w\in V:\qquad S({\boldsymbol {\varphi }}(w))=S(z^{0})+{\frac {1}{2}}\sum _{j=1}^{n}\mu _{j}w_{j}^{2},\quad \det {\boldsymbol {\varphi }}_{w}'(0)=1,} Here, the μj are the eigenvalues of the matrix S z z ″ ( z 0 ) {\displaystyle S_{zz}''(z^{0})} . === The asymptotic expansion in the case of a single non-degenerate saddle point === Assume f (z) and S(z) are holomorphic functions in an open, bounded, and simply connected set Ωx ⊂ Cn such that the Ix = Ωx ∩ Rn is connected; ℜ ( S ( z ) ) {\displaystyle \Re (S(z))} has a single maximum: max z ∈ I x ℜ ( S ( z ) ) = ℜ ( S ( x 0 ) ) {\displaystyle \max _{z\in I_{x}}\Re (S(z))=\Re (S(x^{0}))} for exactly one point x0 ∈ Ix; x0 is a non-degenerate saddle point (i.e., ∇S(x0) = 0 and det S x x ″ ( x 0 ) ≠ 0 {\displaystyle \det S''_{xx}(x^{0})\neq 0} ). Then, the following asymptotic holds where μj are eigenvalues of the Hessian S x x ″ ( x 0 ) {\displaystyle S''_{xx}(x^{0})} and ( − μ j ) − 1 2 {\displaystyle (-\mu _{j})^{-{\frac {1}{2}}}} are defined with arguments This statement is a special case of more general results presented in Fedoryuk (1987). Equation (8) can also be written as where the branch of det ( − S x x ″ ( x 0 ) ) {\displaystyle {\sqrt {\det \left(-S_{xx}''(x^{0})\right)}}} is selected as follows ( det ( − S x x ″ ( x 0 ) ) ) − 1 2 = exp ( − i Ind ( − S x x ″ ( x 0 ) ) ) ∏ j = 1 n | μ j | − 1 2 , Ind ( − S x x ″ ( x 0 ) ) = 1 2 ∑ j = 1 n arg ( − μ j ) , | arg ( − μ j ) | < π 2 . {\displaystyle {\begin{aligned}\left(\det \left(-S_{xx}''(x^{0})\right)\right)^{-{\frac {1}{2}}}&=\exp \left(-i{\text{ Ind}}\left(-S_{xx}''(x^{0})\right)\right)\prod _{j=1}^{n}\left|\mu _{j}\right|^{-{\frac {1}{2}}},\\{\text{Ind}}\left(-S_{xx}''(x^{0})\right)&={\tfrac {1}{2}}\sum _{j=1}^{n}\arg(-\mu _{j}),&&|\arg(-\mu _{j})|<{\tfrac {\pi }{2}}.\end{aligned}}} Consider important special cases: If S(x) is real valued for real x and x0 in Rn (aka, the multidimensional Laplace method), then Ind ( − S x x ″ ( x 0 ) ) = 0. {\displaystyle {\text{Ind}}\left(-S_{xx}''(x^{0})\right)=0.} If S(x) is purely imaginary for real x (i.e., ℜ ( S ( x ) ) = 0 {\displaystyle \Re (S(x))=0} for all x in Rn) and x0 in Rn (aka, the multidimensional stationary phase method), then Ind ( − S x x ″ ( x 0 ) ) = π 4 sign S x x ″ ( x 0 ) , {\displaystyle {\text{Ind}}\left(-S_{xx}''(x^{0})\right)={\frac {\pi }{4}}{\text{sign }}S_{xx}''(x_{0}),} where sign S x x ″ ( x 0 ) {\displaystyle {\text{sign }}S_{xx}''(x_{0})} denotes the signature of matrix S x x ″ ( x 0 ) {\displaystyle S_{xx}''(x_{0})} , which equals to the number of negative eigenvalues minus the number of positive ones. It is noteworthy that in applications of the stationary phase method to the multidimensional WKB approximation in quantum mechanics (as well as in optics), Ind is related to the Maslov index see, e.g., Chaichian & Demichev (2001) and Schulman (2005). == The case of multiple non-degenerate saddle points == If the function S(x) has multiple isolated non-degenerate saddle points, i.e., ∇ S ( x ( k ) ) = 0 , det S x x ″ ( x ( k ) ) ≠ 0 , x ( k ) ∈ Ω x ( k ) , {\displaystyle \nabla S\left(x^{(k)}\right)=0,\quad \det S''_{xx}\left(x^{(k)}\right)\neq 0,\quad x^{(k)}\in \Omega _{x}^{(k)},} where { Ω x ( k ) } k = 1 K {\displaystyle \left\{\Omega _{x}^{(k)}\right\}_{k=1}^{K}} is an open cover of Ωx, then the calculation of the integral asymptotic is reduced to the case of a single saddle point by employing the partition of unity. The partition of unity allows us to construct a set of continuous functions ρk(x) : Ωx → [0, 1], 1 ≤ k ≤ K, such that ∑ k = 1 K ρ k ( x ) = 1 , ∀ x ∈ Ω x , ρ k ( x ) = 0 ∀ x ∈ Ω x ∖ Ω x ( k ) . {\displaystyle {\begin{aligned}\sum _{k=1}^{K}\rho _{k}(x)&=1,&&\forall x\in \Omega _{x},\\\rho _{k}(x)&=0&&\forall x\in \Omega _{x}\setminus \Omega _{x}^{(k)}.\end{aligned}}} Whence, ∫ I x ⊂ Ω x f ( x ) e λ S ( x ) d x ≡ ∑ k = 1 K ∫ I x ⊂ Ω x ρ k ( x ) f ( x ) e λ S ( x ) d x . {\displaystyle \int _{I_{x}\subset \Omega _{x}}f(x)e^{\lambda S(x)}dx\equiv \sum _{k=1}^{K}\int _{I_{x}\subset \Omega _{x}}\rho _{k}(x)f(x)e^{\lambda S(x)}dx.} Therefore as λ → ∞ we have: ∑ k = 1 K ∫ a neighborhood of x ( k ) f ( x ) e λ S ( x ) d x = ( 2 π λ ) n 2 ∑ k = 1 K e λ S ( x ( k ) ) ( det ( − S x x ″ ( x ( k ) ) ) ) − 1 2 f ( x ( k ) ) , {\displaystyle \sum _{k=1}^{K}\int _{{\text{a neighborhood of }}x^{(k)}}f(x)e^{\lambda S(x)}dx=\left({\frac {2\pi }{\lambda }}\right)^{\frac {n}{2}}\sum _{k=1}^{K}e^{\lambda S\left(x^{(k)}\right)}\left(\det \left(-S_{xx}''\left(x^{(k)}\right)\right)\right)^{-{\frac {1}{2}}}f\left(x^{(k)}\right),} where equation (13) was utilized at the last stage, and the pre-exponential function f (x) at least must be continuous. == The other cases == When ∇S(z0) = 0 and det S z z ″ ( z 0 ) = 0 {\displaystyle \det S''_{zz}(z^{0})=0} , the point z0 ∈ Cn is called a degenerate saddle point of a function S(z). Calculating the asymptotic of ∫ f ( x ) e λ S ( x ) d x , {\displaystyle \int f(x)e^{\lambda S(x)}dx,} when λ → ∞, f (x) is continuous, and S(z) has a degenerate saddle point, is a very rich problem, whose solution heavily relies on the catastrophe theory. Here, the catastrophe theory replaces the Morse lemma, valid only in the non-degenerate case, to transform the function S(z) into one of the multitude of canonical representations. For further details see, e.g., Poston & Stewart (1978) and Fedoryuk (1987). Integrals with degenerate saddle points naturally appear in many applications including optical caustics and the multidimensional WKB approximation in quantum mechanics. The other cases such as, e.g., f (x) and/or S(x) are discontinuous or when an extremum of S(x) lies at the integration region's boundary, require special care (see, e.g., Fedoryuk (1987) and Wong (1989)). == Extensions and generalizations == An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems. Given a contour C in the complex sphere, a function f defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If f and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution. An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour. The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of the Russian mathematician Alexander Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, steepest descent contours solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov). The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics. Another extension is the Method of Chester–Friedman–Ursell for coalescing saddle points and uniform asymptotic extensions. == See also == Pearcey integral Stationary phase approximation Laplace's method == Notes == == References == Chaichian, M.; Demichev, A. (2001), Path Integrals in Physics Volume 1: Stochastic Process and Quantum Mechanics, Taylor & Francis, p. 174, ISBN 075030801X Debye, P. (1909), "Näherungsformeln für die Zylinderfunktionen für große Werte des Arguments und unbeschränkt veränderliche Werte des Index", Mathematische Annalen, 67 (4): 535–558, doi:10.1007/BF01450097, S2CID 122219667 English translation in Debye, Peter J. W. (1954), The collected papers of Peter J. W. Debye, Interscience Publishers, Inc., New York, ISBN 978-0-918024-58-9, MR 0063975 {{citation}}: ISBN / Date incompatibility (help) Deift, P.; Zhou, X. (1993), "A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation", Ann. of Math., vol. 137, no. 2, The Annals of Mathematics, Vol. 137, No. 2, pp. 295–368, arXiv:math/9201261, doi:10.2307/2946540, JSTOR 2946540, S2CID 12699956. Erdelyi, A. (1956), Asymptotic Expansions, Dover. Fedoryuk, M. V. (2001) [1994], "Saddle point method", Encyclopedia of Mathematics, EMS Press. Fedoryuk, M. V. (1987), Asymptotic: Integrals and Series, Nauka, Moscow [in Russian]. Kamvissis, S.; McLaughlin, K. T.-R.; Miller, P. (2003), "Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation", Annals of Mathematics Studies, vol. 154, Princeton University Press. Riemann, B. (1863), Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita (Unpublished note, reproduced in Riemann's collected papers.) Siegel, C. L. (1932), "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, 2: 45–80 Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966. Translated in Barkan, Eric; Sklar, David (2018), "On Riemanns Nachlass for Analytic Number Theory: A translation of Siegel's Uber", arXiv:1810.05198 [math.HO]. Poston, T.; Stewart, I. (1978), Catastrophe Theory and Its Applications, Pitman. Schulman, L. S. (2005), "Ch. 17: The Phase of the Semiclassical Amplitude", Techniques and Applications of Path Integration, Dover, ISBN 0486445283 Wong, R. (1989), Asymptotic approximations of integrals, Academic Press.
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Wikipedia:Metric differential#0
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In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions. == Discussion == Rademacher's theorem states that a Lipschitz map f : Rn → Rm is differentiable almost everywhere in Rn; in other words, for almost every x, f is approximately linear in any sufficiently small range of x. If f is a function from a Euclidean space Rn that takes values instead in a metric space X, it doesn't immediately make sense to talk about differentiability since X has no linear structure a priori. Even if you assume that X is a Banach space and ask whether a Fréchet derivative exists almost everywhere, this does not hold. For example, consider the function f : [0,1] → L1([0,1]), mapping the unit interval into the space of integrable functions, defined by f(x) = χ[0,x], this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ x ≤ y≤ 1, then | f ( x ) − f ( y ) | = ∫ 0 1 | χ [ 0 , x ] ( t ) − χ [ 0 , y ] ( t ) | d t = ∫ x y d t = | x − y | , {\displaystyle |f(x)-f(y)|=\int _{0}^{1}|\chi _{[0,x]}(t)-\chi _{[0,y]}(t)|\,dt=\int _{x}^{y}\,dt=|x-y|,} but one can verify that limh→0(f(x + h) − f(x))/h does not converge to an L1 function for any x in [0,1], so it is not differentiable anywhere. However, if you look at Rademacher's theorem as a statement about how a Lipschitz function stabilizes as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of f instead of its linear properties. == Definition and existence of the metric differential == A substitute for a derivative of f:Rn → X is the metric differential of f at a point z in Rn which is a function on Rn defined by the limit M D ( f , z ) ( x ) = lim r → 0 d X ( f ( z + r x ) , f ( z ) ) r {\displaystyle MD(f,z)(x)=\lim _{r\rightarrow 0}{\frac {d_{X}(f(z+rx),f(z))}{r}}} whenever the limit exists (here d X denotes the metric on X). A theorem due to Bernd Kirchheim states that a Rademacher theorem in terms of metric differentials holds: for almost every z in Rn, MD(f, z) is a seminorm and d X ( f ( x ) , f ( y ) ) − M D ( f , z ) ( x − y ) = o ( | x − z | + | y − z | ) . {\displaystyle d_{X}(f(x),f(y))-MD(f,z)(x-y)=o(|x-z|+|y-z|).} The little-o notation employed here means that, at values very close to z, the function f is approximately an isometry from Rn with respect to the seminorm MD(f, z) into the metric space X. == References ==
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Wikipedia:Micha Perles#0
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Micha Asher Perles is an Israeli mathematician working in geometry, a professor emeritus at the Hebrew University. He earned his Ph.D. in 1964 from the Hebrew University, under the supervision of Branko Grünbaum. His contributions include: The Perles configuration, a set of nine points in the Euclidean plane whose collinearities can be realized only by using irrational numbers as coordinates. Perles used this configuration to prove the existence of irrational polytopes in higher dimensions. The Perles–Sauer–Shelah lemma, a result in extremal set theory whose proof was credited to Perles by Saharon Shelah. The pumping lemma for context-free languages, a widely used method for proving that a language is not context-free that Perles discovered with Yehoshua Bar-Hillel and Eli Shamir. Notable students of Perles include Noga Alon, Gil Kalai, and Nati Linial. == References == == External links == Micha Asher Perles' home page Micha Perles at DBLP Bibliography Server Micha A. Perles' online publications at arXiv Micha Perles at the Mathematics Genealogy Project
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Wikipedia:Michael A. B. Deakin#0
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Michael Andrew Bernard Deakin (12 August 1939 – 5 August 2014) was an Australian mathematician and mathematics educator. He was known for his work as a writer and editor of Function, a mathematics magazine aimed at high school students, and as a biographer of ancient Greek mathematician Hypatia. He won the B. H. Neumann award of the Australian Mathematics Trust in 2003 for his "rich and varied commitment to mathematics enrichment". == Education and career == Deakin was born 12 August 1939. He grew up in Tasmania, and moved to Melbourne late in his high school education, taking a second matriculation year studying Latin at St Patrick's College, East Melbourne before entering the University of Melbourne in 1957. He completed a bachelor's degree with second-class honours in mathematics at Melbourne in 1961. He went on to earn a master's degree there in 1963, with a thesis on integral equations supervised by Russell Love. Deakin moved to the University of Chicago in 1963 for graduate study, and completed his Ph.D. in 1966, under the supervision of mathematical biophysicist Herbert Landahl. He became a lecturer at Monash University in Melbourne in 1967, but then in 1970 moved to Papua New Guinea to become reader-in-charge in the mathematics department of the Institute of Higher Technical Education. He returned to Monash as a senior reader in 1973. He earned a master's degree in education in 1975 from the University of Exeter, and remained at Monash for the rest of his career. He died on 5 August 2014, survived by his widow, Rayda, and the children of his first marriage. == Function == In 1976 a group of mathematicians at Monash University led by department chair Gordon Preston recognized the need for a journal focused on "mathematics as mathematicians themselves would recognise it, but addressed to secondary students". A secondary but explicit goal was to encourage young women in mathematics, as at that time their under-representation was already recognized. Later, over beers with friends from other disciplines, Deakin found the name for the new journal, Function. The journal was published from 1977 to 2004, and Deakin became a founding member of its editorial board, its most frequent contributor, and, for much of its existence, its editor-in-chief. == Hypatia == Deakin published the first of his several articles on Hypatia in 1992 in Function. In 2007, he published the book Hypatia of Alexandria: Mathematician and Martyr (Prometheus Books). Aimed at a popular audience, the book is "at least in part, a response to Maria Dzielska's Hypatia of Alexandria", which had focused on the historical and literary legacy of Hypatia at the expense of her mathematics, and which Deakin had previously reviewed for the American Mathematical Monthly. In his book, Deakin organizes the work of other scholars on Hypatia's mathematics, particularly relying on the earlier work of Wilbur Knorr, rather than formulating new theories on this aspect of Hypatia's work. He argues that, unlike some other neoplatonists of her time, Hypatia was not opposed to Christianity, and that the Christian mob that killed her likely misunderstood her philosophical position. However, his book has been criticized for treating Hypatia more as an idealized icon, or a caricature of a female mathematician, than as a realistic person of her times. == See also == Wine/water paradox == References == == External links == Function magazine
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Wikipedia:Michael Barber (academic)#0
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Michael Newton Barber (born 30 April 1947) is a mathematician, physicist and academic. He was Vice-Chancellor of Flinders University in South Australia from 2008 until 2014. == Career == Barber studied at the University of New South Wales, where he received the University Medal in applied mathematics and graduated with first class honours. He received a PhD from Cornell University in the USA in theoretical physics in 1972. He is best known for the scaling theory of finite size effects at phase transitions, which he introduced together with Michael Fisher. Barber worked at the Australian National University and University of New South Wales as an academic in the 1970s and 1980s. in 1990, he was appointed Dean of the Faculty of Science at the ANU. He assumed his first office-bearing position as Pro Vice-Chancellor (Research) at the University of Western Australia in 1994 and held the position until 2002. That year, Barber was appointed Executive Director, Science Planning at the CSIRO. In 2006, he was appointed Group Executive, Information, Manufacturing and Minerals. He held this position until his appointment as vice-chancellor at Flinders University in 2008—a position he held until December 2014. During his time as vice-chancellor he worked "to enhance the University’s contribution to the ‘new South Australia’ with its strong defence and resources sectors." In 2008 he stated: "The University is already more engaged with the defence and resource industries than many people recognise but we need to further strengthen the science and technology base at Flinders so that the University is better positioned to grasp more of the opportunities in the new South Australia. We need to work through this in a planned and deliberate way to find appropriate strategies to address it." In 2012, Barber was a member of the South Australian State Advisory Council of the Committee for Economic Development of Australia (CEDA). That same year, Barber's salary for his position as vice-chancellor of Flinders University became a controversial topic. The Advertiser revealed that he would receive between $710,000 and $719,999 in 2012. In 2010 his salary had been $545,000. A spokesperson from Flinders University stated that his remuneration was "in line" with industry standards. Throughout his career, Barber has acted as an advisor on science and research matters to government and industry in Australia and overseas. == Recognition == Barber was elected a Fellow of the Australian Academy of Science in 1992. He served as the organisation's Secretary, Science Policy from 2001 until 2005. In 2001 he was awarded the Centenary Medal "for service to Australian society through university administration and scientific research". In 2009 he was elected a Fellow of the Society for Industrial and Applied Mathematics (SIAM). In 2018 Barber was appointed an Officer of the Order of Australia (AO) for "distinguished service to higher education administration, and in the field of mathematical physics, particularly statistical mechanics, as an academic and researcher, and through contributions to science policy reform". == Personal life == Barber's father was noted Australian botanist and geneticist Horace Barber . == References == == External links == Biographical entry, Encyclopedia of Australian Science SIAM fellows: class of 2009
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Wikipedia:Michael Barr (mathematician)#0
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Michael Barr (born January 22, 1937) is an American mathematician who is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University. == Early life and education == He was born in Philadelphia, Pennsylvania, and graduated from the 202nd class of Central High School in June 1954. He graduated from the University of Pennsylvania in February 1959 and received a PhD from the same school in June 1962. == Career == Barr studied mathematics at the University of Pennsylvania, graduating with a bachelor's degree in 1959 and a doctorate in 1962 under David Kent Harrison (Cohomology of Commutative Algebras). He was then an instructor at Columbia University and from 1964 Assistant Professor and later Associate Professor at the University of Illinois Urbana-Champaign. In 1968 he became Associate Professor and in 1972 Professor at McGill University. In 1967 and 1975/76 he was a visiting scientist at ETH Zurich and in 1970/71 at the University of Fribourg and in 1989/90 a visiting professor at the University of Pennsylvania. In 1970 he was an invited speaker at the International Congress of Mathematicians in Nice (Non-abelian full embedding: outline). His earlier work was in homological algebra, but his principal research area for a number of years has been category theory. He is well known to theoretical computer scientists for his book Category Theory for Computing Science (1990) with Charles Wells, as well as for the development of *-autonomous categories and Chu spaces which have found various applications in computer science. His monograph *-autonomous categories (1979), and his books Toposes, Triples, and Theories (1985), also coauthored with Wells, and Acyclic Models (2002), are aimed at more specialized audiences. In 2011 Michael Barr and his wife Marcia published an English translation of Grothendieck's fundamental Tôhoku paper. Barr is on the editorial boards of Mathematical Structures in Computer Science and the electronic journal Homology, Homotopy and Applications, and is editor of the electronic journal Theory and Applications of Categories. == References == == External links == Toposes, Triples and Theories, updated edition of text first published in 1985. Category Theory for Computing Science, updated 3rd edition of the book. Some aspects of homological Algebra, translation of Grothendieck's Tôhoku paper http://www.tac.mta.ca/tac (Theory and Applications of Categories) https://web.archive.org/web/20080704125156/http://www.math.rutgers.edu/hha/geninfo.html (Homology, Homotopy and Applications) Michael Barr at the Mathematics Genealogy Project
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Wikipedia:Michael Benedicks#0
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Michael Benedicks, born 1949, is a Professor of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, Sweden. He received his Ph.D. from the Royal Institute of Technology in 1980. His doctoral advisor was Professor Harold S. Shapiro. He was a visiting scholar at the Institute for Advanced Study in the summer of 1989. He became a Professor of Mathematics at the Royal Institute of Technology in 1991. His research interests include dynamic systems. For example, he has studied Hénon maps together with Professor Lennart Carleson. He became a member of the Royal Swedish Academy of Sciences in 2007. == References == == External links == Prof. Benedicks's website
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Wikipedia:Michael Dinneen#0
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Michael J. Dinneen is an American-New Zealand mathematician and computer scientist working as a senior lecturer at the University of Auckland, New Zealand. He is deputy director of the Center for Discrete Mathematics and Theoretical Computer Science. He does research in combinatorial optimization, distributed computing, and graph theory. Dinneen was educated at the University of Idaho (BS), Washington State University, and University of Victoria (MS and PhD). He worked at the Los Alamos National Laboratory before moving to New Zealand and the University of Auckland. == Selected bibliography == Michael J. Dinneen, Georgy Gimel'farb, and Mark C. Wilson. Introduction to Algorithms, Data Structures and Formal Languages. Pearson (Education New Zealand), 2004. ISBN 1-877258-79-2 (pages 253). Cristian S. Calude, Michael J. Dinneen, and Chi-Kou Shu. Computing a glimpse of randomness. "Experimental Mathematics", 11(2):369-378, 2002. http://www.cs.auckland.ac.nz/~cristian/Calude361_370.pdf Joshua J. Arulanandham, Cristian S. Calude, and Michael J. Dinneen. A fast natural algorithm for searching. "Theoretical Computer Science", 320(1):3-13, 2004. http://authors.elsevier.com/sd/article/S0304397504001914 Michael J. Dinneen, Bakhadyr Khoussainov, André Nies (eds.). Computation, Physics and Beyond - International Workshop on Theoretical Computer Science, WTCS 2012, Dedicated to Cristian S. Calude on the Occasion of His 60th Birthday, Auckland, New Zealand, February 21–24, 2012, Revised Selected and Invited Papers. Lecture Notes in Computer Science 7160, Springer 2012. https://www.springer.com/gp/book/9783642276538 == References == == External links == Michael J. Dinneen Home Page List of publications of Michael J. Dinneen CDMTCS at the University of Auckland List of publications of Michael J. Dinneen at DBLP
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Wikipedia:Michael Eastwood#0
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Kyle Eastwood (born May 19, 1968) is an American jazz bassist and film composer. He studied film at the University of Southern California for two years before embarking on a music career. After becoming a session player in the early 1990s and leading his own quartet, he released his first solo album, From There to Here, in 1998. His album The View From Here was released in 2013 by Jazz Village. In addition to his solo albums, Eastwood has composed music for nine of his father Clint Eastwood's films. Eastwood plays fretted and fretless electric bass guitar and double bass. == Early life == Kyle Clinton Eastwood was born May 19, 1968, the son of Margaret Neville Eastwood (née Johnson) (born 1931) and actor-director Clint Eastwood. He has a sister, Alison, who was born in 1972. He also has six known paternal half-siblings: Laurie (b. 1954), Kimber (b. 1964), Scott (b. 1986), Kathryn (b. 1988), Francesca (b. 1993) and Morgan (b. 1996). == Career == === Music === Eastwood comes from a musical family, as noted in an October 27, 2006, article from The Independent newspaper: When I told my father, film actor/director Clint Eastwood, I wanted to be a musician, he was happy about it. Music has always been important to my family. My parents gave me my taste in music and my love of jazz from an early age. My father plays piano, my mother used to play, and my mother's mother was a music teacher at Northwestern University in Illinois. Music was prominent in the Eastwood home. According to his biography with Hopper Management, Eastwood grew up listening to records by jazz legends such as Miles Davis, Dave Brubeck, Thelonious Monk, and the Stan Kenton Big Band with his parents, who were both jazz lovers. Eastwood attended the Monterey Jazz Festival numerous times with his parents. "One advantage of having a famous father was I got to go backstage," Eastwood explained in an interview conducted by stepmother Dina Ruiz Eastwood. "I met a lot of artists, greats like Dizzy Gillespie and Sarah Vaughan. Looking back on that, I can see how much the musicians I met there influenced my career." Eastwood began playing bass guitar in high school, learning R&B, Motown, and reggae tunes by ear. After studying with French bassist Bunny Brunel, he began playing gigs in New York City and Los Angeles, forming the Kyle Eastwood Quartet which contributed to Eastwood After Hours: Live at Carnegie Hall (1996), a concert in honor of Clint Eastwood and his dedication to jazz. Clint Eastwood has always been supportive of, and interested in, Kyle's work, as Eastwood told The Independent: "As far as my father is concerned, as long as I was serious about my music career, he was supportive of me." Two years later, in 1998, Sony released his first album, From There to Here, a collection of jazz standards and original compositions. After signing with the UK's Candid Records in 2004, Eastwood moved to Dave Koz's label, Rendezvous, which released his albums Paris Blue (2005), and Now (2006). In addition to his solo albums, Eastwood has also contributed music to nine of his father's films: The Rookie (1990), Mystic River (2002), Million Dollar Baby (2004), Flags of Our Fathers (2006), Letters from Iwo Jima (2006), Changeling (2008), Gran Torino (2008), Invictus (2009) and J. Edgar (2011). He was nominated with music partner Michael Stevens for a 2006 Chicago Film Critics Association Award for Original Score (Letters from Iwo Jima). In 2014 Eastwood and Matt McGuire contributed to the score of the documentary Homme Less about homeless photographer Mark Reay. === Other work === Kyle Eastwood provided the voice of "Daddy" in "Daddy and Son" (2007) and the voice of 1980s-era DJ Andy Wright for the computer game The Movies (2005). He had a supporting role in the 1982 Clint Eastwood film Honkytonk Man. == Personal life == Eastwood has a daughter, Graylen (b. March 28, 1994) with Laura Gomez. They married in May 1995 and filed for divorce in 2005. In 2006, he denied real estate agent Sam Kelley's claim that the two had an eight-year homosexual affair. Eastwood married Cynthia Ramirez in September 2014 at his father's Mission Ranch Hotel in Carmel, California. == Discography == === Studio albums === === Compilation albums === === Soundtracks === == Filmography == === Composer/performer/arranger === 1990 The Rookie as composer, "Red Zone" with Michael Stevens 1991 Regarding Henry as uncredited performer 2003 Mystic River as composer, "Cosmo", "Black Emerald Blues" with Michael Stevens 2004 Million Dollar Baby as composer, "Boxing Baby", "Solferino", "Blue Diner" with Michael Stevens 2006 Letters from Iwo Jima as composer, with Michael Stevens 2006 Flags of Our Fathers as arranger 2007 Rails & Ties as music by 2008 Changeling as arrangements 2008 Gran Torino as composer, with Michael Stevens 2009 Invictus as composer, with Michael Stevens 2011 J. Edgar as composer, "Red Sails in the Sunset", "I Only Have Eyes for You" === Actor === 1976 The Outlaw Josey Wales as Josey's Son (uncredited) 1980 Bronco Billy as Orphan (uncredited) 1982 Honkytonk Man as Whit Stovall 1990 The Rookie as Band Member at Ackerman's House Party (uncredited) 1995 The Bridges of Madison County as James Rivers Band 2007 Summer Hours as James 2011 J. Edgar as Member of The "Stork Club Band" == References == == External links == Official website Kyle Eastwood at IMDb Profile at All About Jazz Kyle Eastwood at Allmusic
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Wikipedia:Michael Fekete#0
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Michael (Mihály) Fekete (Hebrew: מיכאל פקטה; 19 July 1886 – 13 May 1957) was a Hungarian-Israeli mathematician. == Biography == Michael Fekete was born in Zenta, Austria-Hungary (today Senta, Serbia). He received his PhD in 1909 from the University of Budapest (later renamed Eötvös Loránd University). He studied under Lipót Fejér. After completing his PhD he switched to University of Göttingen, which was considered a mathematics hub. In 1914, he returned to the University of Budapest, where he attained the title of Privatdozent. Fekete also worked as a private math tutor. Among his students was János Neumann. In 1922, Fekete published a paper together with Neumann on extremal polynomials, which was Neumann's first scientific paper. Fekete dedicated the majority of his scientific work to the transfinite diameter. In 1928 Fekete immigrated to Mandate Palestine and was among the first instructors of the Institute of Mathematics at the Hebrew University of Jerusalem. In 1929 he was promoted to professor, and eventually headed the institute, succeeding Edmund Landau and Adolf Abraham Halevi Fraenkel. He later became the dean of Natural Sciences, and between the years 1946–1948 he was Hebrew University Provost. Among his students were Aryeh Dvoretzky and Michael Bahir Maschler. == Awards and recognition == In 1955, Fekete was awarded the Israel Prize for exact sciences. == See also == Fekete problem Fekete polynomial Fekete–Szegő inequality Fekete's lemma Fekete constant == References == == Literature == Joseph, Anthony; Melnikov, Anna; Rentschler, Rudolf (2003). Studies in Memory of Issai Schur. New York: Springfield. ISBN 978-1-4612-0045-1. Hersh, Reuben (2015). Peter Lax, Mathematician. American Mathematical Soc. p. 168. ISBN 978-1-4704-1708-6. == External links ==
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Wikipedia:Michael Frame#0
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Michael Frame is an American mathematician and retired Yale professor. He is a co-author, along with Amelia Urry, of Fractal Worlds: Grown, Built, and Imagined. At Yale, he was a colleague of Benoit Mandelbrot and helped Mandelbrot develop a curriculum within the mathematics department. == Early years and education == Michael Frame was born in 1951 and grew up in St. Albans, West Virginia. After leaving his physics major because the lab requirement was "something in biophysics with killing frogs," Frame, a vegetarian, received a bachelor's degree in mathematics at Union College as a first-generation college student. In 1978, he completed a PhD in mathematics at Tulane University. == Teaching career == Michael Frame came to work at Yale University at the invitation of his colleague Benoit Mandelbrot. At Yale, Frame called himself "the stupidest guy in the department...the dimmest bulb in the pack here," and focused on his teaching contributions. He received the McCredie Prize for best use of technology in teaching at Yale College, the Dylan Hixon '88 Prize for teaching excellence in the natural sciences, and the Yale Phi Beta Kappa chapter's DeVane medal for undergraduate teaching. == Work with Mandelbrot == Benoit Mandelbrot includes a section on Michael Frame in his posthumously published autobiography The Fractalist: Memoir of a Scientific Maverick. In the section, called "Michael Frame, Friend and Colleague," he calls Frame an "indispensable" professor. In 1997, Mandelbrot and Frame held a meeting of teachers of fractal geometry. According to Mandelbrot, as far as he knew, this was the "first scientific meeting totally dedicated to the teaching of fractals. This eventually culminated in the 2002 publication of the book Fractals, Graphics, and Mathematics Education, which was co-authored by Frame and Mandelbrot. == References ==
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Wikipedia:Michael Lin (mathematician)#0
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Michael Lin (Hebrew: מיכאל לין; born June 8, 1942) is an Israeli mathematician, who has published scientific articles in the field of probability concentrating on Markov chains and ergodic theory. He serves as professor emeritus at the Department of Mathematics in Ben-Gurion University of the Negev (BGU). Additionally, he is a member of the academic board and serves as the academic coordinator at Achva Academic College. == Biography == Michael Lin was born in Israel. He holds a Bachelor of Science in Mathematics and Physics from The Hebrew University of Jerusalem (1963), Master of Science in Mathematics (1967) and a PhD in Mathematics also from The Hebrew University of Jerusalem (1971). In 1971 he was appointed as an assistant professor in Ohio State University. In 1976 he returned to Israel and became a senior lecturer in the Department of Mathematics at Ben-Gurion University of the Negev. Only 4 years later, at 1979, he became an associate professor and in 1984 he became a full professor. In 2011, Professor Lin retired and nowadays he serves as professor emeritus. During his career at Ben-Gurion University of the Negev he acted as: Computer Science Coordinator, Department of Mathematics and Computer Science, BGU. Member of BGU Computer Policy committee. Chairman and Computer Science Coordinator, Department of Mathematics and Computer Science, BGU. Senate representative to Executive Committee of Board of Trustees of BGU. Senate representative to the BGU Executive Committee's subcommittee for student affairs. President, Israel Mathematical Union. Head of the Ethical Code Committee of BGU. In 2004 Professor Lin also acted as a member of the committee electing the recipients of the Israel Prize in mathematics. == Research and publications == Professor Lin's published work focuses on two main areas of research in the field of probability: Ergodic theory and Markov chain. More specifically, he researched in several areas: mean and individual Ergodic theory, Central limit theorem and functional analysis. == References ==
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Wikipedia:Michael Makkai#0
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Michael Makkai (Hungarian: Makkai Mihály; 24 June 1939 in Budapest, Hungary) is a Canadian mathematician of Hungarian origin, specializing in mathematical logic. He works in model theory, category theory, algebraic logic, type theory and the theory of topoi. == Career == === Academic biography === Makkai was awarded his PhD from the Eötvös Loránd University, Budapest, in 1966, having been supervised by Rózsa Péter and Andrzej Mostowski. He then worked at the Mathematical Institute of the Hungarian Academy of Sciences. Between 1974 and 2010, he was professor of mathematics at McGill University, retiring in 2010. He is also an external member of the Hungarian Academy of Sciences (1995). === Work === With Leo Harrington and Saharon Shelah he proved the Vaught conjecture for ω-stable theories. With Robert Paré he further developed the theory of Accessible Categories. Makkai has an Erdős number of 1, having published "Some Remarks on Set Theory, X" with Paul Erdős in 1966. == Selected publications == M. Makkai, G. E. Reyes: First Order Categorical Logic, Lecture Notes in Mathematics, 611, Springer, 1977, viii+301 pp. doi:10.1007/BFb0066201 L. Harrington, M. Makkai, S. Shelah: A proof of Vaught's conjecture for ω-stable theories, Israel Journal of Mathematics, 49(1984), 259–280. doi:10.1007/BF02760651 Michael Makkai, Robert Paré: Accessible categories: the foundations of categorical model theory. Contemporary Mathematics, 104. American Mathematical Society, Providence, RI, 1989. viii+176 pp. ISBN 0-8218-5111-X, doi:10.1090/conm/104 M. Makkai: Duality and Definability in First Order Logic, Memoirs of the American Mathematical Society, 503, 1993, ISSN 0065-9266. doi:10.1090/memo/0503 == References == == External links == Makkai's homepage at the Hungarian Academy of Sciences Makkai's homepage at McGill University
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Wikipedia:Michael Maschler#0
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Michael Bahir Maschler (Hebrew: מיכאל בהיר משלר; July 22, 1927 – July 20, 2008) was an Israeli mathematician well known for his contributions to the field of game theory. He was a professor in the Einstein Institute of Mathematics and the Center for the Study of Rationality at the Hebrew University of Jerusalem. In 2012, the Israeli Chapter of the Game Theory Society founded the Maschler Prize, an annual prize awarded to an outstanding research student in game theory and related topics in Israel. == Biography == Michael B. Maschler was born in Jerusalem on July 22, 1927. == Selected publications == For a complete list of English and Hebrew publications, see Michael Maschler: In Memoriam, above. "The Bargaining Set for Cooperative Games", with R.J. Aumann, 1964, in Advances in Game Theory "The Core of a Cooperative Game", with M. Davis, 1965, Naval Research Logistics Quarterly "Game-Theoretic Aspects of Gradual Disarmament", with R.J. Aumann, 1966, Mathematica "Some Thoughts on the Minimax Principle" with R.J. Aumann, 1972, Management Science "An Advantage of the Bargaining Set over the Core", 1976, JET "Geometric Properties of the Kernel, Nucleolus and Related Solution Concepts", with B. Peleg and L.S. Shapley, 1979, Mathematics of Operations Research "Superadditive Solution for the Nash bargaining Game", with M. Perles, 1981, IJGT "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud", with R.J. Aumann, 1985, JET "The Consistent Shapley Value for Hyperplane Games", with G. Owen, 1989, IJGT "The Consistent Shapley Value for Games without Side Payments", with G. Owen, 1992, in Selten, editor, Rational Interaction "The Bargaining Set, Kernel and Nucleolus", 1992, in Aumann and Hart, editors, Handbook of Game Theory Repeated Games with Incomplete Information, MIT Press, Cambridge, 1995, with R.J. Aumann Insights into Game Theory: An Alternative Mathematical Experience, Cambridge University Press, forthcoming, with Ein-Ya Gura. == References == == Sources == Aumann, Robert J.; Gura, Ein-Ya; Hart, Sergiu (2008). "Michael Maschler: In Memoriam" (PDF). Hebrew University of Jerusalem.
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Wikipedia:Michael O. Rabin#0
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Michael Oser Rabin (Hebrew: מִיכָאֵל עוזר רַבִּין; born September 1, 1931) is an Israeli mathematician, computer scientist, and recipient of the Turing Award. == Biography == === Early life and education === Rabin was born in 1931 in Breslau, Germany (today Wrocław, in Poland), the son of a rabbi. In 1935, he emigrated with his family to Mandatory Palestine. As a young boy, he was very interested in mathematics and his father sent him to the best high school in Haifa, where he studied under mathematician Elisha Netanyahu, who was then a high school teacher. Rabin graduated from the Hebrew Reali School in Haifa in 1948, and was drafted into the army during the 1948 Arab–Israeli War. The mathematician Abraham Fraenkel, who was a professor of mathematics in Jerusalem, intervened with the army command, and Rabin was discharged to study at the university in 1949. Afterwards, he received an M.Sc from Hebrew University of Jerusalem. He began graduate studies at the University of Pennsylvania before receiving a Ph.D. from Princeton University in 1956. === Career === Rabin became Associate Professor of Mathematics at the University of California, Berkeley (1961–62) and MIT (1962-63). Before moving to Harvard University as Gordon McKay Professor of Computer Science in 1981, he was a professor at the Hebrew University. In the late 1950s, he was invited for a summer to do research for IBM at the Lamb Estate in Westchester County, New York with other promising mathematicians and scientists. It was there that he and Dana Scott wrote the paper "Finite Automata and Their Decision Problems". Soon, using nondeterministic automata, they were able to re-prove Kleene's result that finite state machines exactly accept regular languages. As to the origins of what was to become computational complexity theory, the next summer Rabin returned to the Lamb Estate. John McCarthy posed a puzzle to him about spies, guards, and passwords, which Rabin studied and soon after he wrote an article, "Degree of Difficulty of Computing a Function and Hierarchy of Recursive Sets." Nondeterministic machines have become a key concept in computational complexity theory, particularly with the description of the complexity classes P and NP. Rabin then returned to Jerusalem, researching logic, and working on the foundations of what would later be known as computer science. He was an associate professor and the head of the Institute of Mathematics at the Hebrew University at 29 years old, and a full professor by 33. Rabin recalls, "There was absolutely no appreciation of the work on the issues of computing. Mathematicians did not recognize the emerging new field". In 1960, he was invited by Edward F. Moore to work at Bell Labs, where Rabin introduced probabilistic automata that employ coin tosses in order to decide which state transitions to take. He showed examples of regular languages that required a very large number of states, but for which you get an exponential reduction of the number of states with probabilistic automata. In 1966 (published in conference proceedings in 1967), Rabin introduced the notion of polynomial time (introduced independently and very shortly before by Cobham and Edmonds). In 1969, Rabin introduced infinite-tree automata and proved that the monadic second-order theory of n successors (S2S when n = 2) is decidable. A key component of the proof implicitly showed determinacy of parity games, which lie in the third level of the Borel hierarchy. In 1975, Rabin finished his tenure as Rector of the Hebrew University of Jerusalem and went to the Massachusetts Institute of Technology in the USA as a visiting professor. While there, Rabin invented the Miller–Rabin primality test, a randomized algorithm that can determine very quickly (but with a tiny probability of error) whether a number is prime. Rabin's method was based on previous work of Gary Miller that solved the problem deterministically with the assumption that the generalized Riemann hypothesis is true, but Rabin's version of the test made no such assumption. Fast primality testing is key in the successful implementation of most public-key cryptography, and in 2003 Miller, Rabin, Robert M. Solovay, and Volker Strassen were given the Paris Kanellakis Award for their work on primality testing. In 1976 he was invited by Joseph Traub to meet at Carnegie Mellon University and presented the primality test, which Traub called "revolutionary". In 1979, Rabin invented the Rabin cryptosystem, the first asymmetric cryptosystem whose security was proved equivalent to the intractability of integer factorization. In 1981, Rabin reinvented a weak variant of the technique of oblivious transfer invented by Wiesner under the name of multiplexing, allowing a sender to transmit a message to a receiver where the receiver has some probability between 0 and 1 of learning the message, with the sender being unaware whether the receiver was able to do so. In 1987, Rabin, together with Richard Karp, created one of the most well-known efficient string search algorithms, the Rabin–Karp string search algorithm, known for its rolling hash. Rabin's more recent research has concentrated on computer security. He is currently the Thomas J. Watson Sr. Professor of Computer Science, Emeritus at Harvard University and Professor of Computer Science (Emeritus) at Hebrew University. During the spring semester of 2007, he was a visiting professor at Columbia University teaching Introduction to Cryptography. == Awards and honours == Rabin is a foreign member of the United States National Academy of Sciences, a member of the American Philosophical Society, a member of the American Academy of Arts and Sciences, a member of the French Academy of Sciences, and a foreign member of the Royal Society. In 1976, the Turing Award was awarded jointly to Rabin and Dana Scott for a paper written in 1959, the citation for which states that the award was granted: For their joint paper "Finite Automata and Their Decision Problems," which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) [sic] classic paper has been a continuous source of inspiration for subsequent work in this field. In 1995, Rabin was awarded the Israel Prize, in computer sciences. In 2010, Rabin was awarded the Tel Aviv University Dan David Prize ("Future" category), jointly with Leonard Kleinrock and Gordon E. Moore, for Computers and Telecommunications. Rabin was awarded an Honorary Doctor of Science from Harvard University in 2017. == Personal life == Rabin has a daughter, computer scientist Tal Rabin. == See also == Oblivious transfer Rabin automaton Rabin fingerprint Hyper-encryption List of Israel Prize recipients == References == == External links == Short Description in an Information Science Hall of Fame at University of Pittsburgh Oblivious transfer Quotes from some of Professor Rabin's classes Website for one of Rabin's courses Description of Rabin's research by Richard J. Lipton
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Wikipedia:Michael Stifel#0
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John Michael Stipe (; born January 4, 1960) is an American singer, songwriter and artist, best known as the lead singer and lyricist of the alternative rock band R.E.M. Stipe was born in Metro Atlanta in January 1960. Due to his father's military commission, his family moved constantly, with Stipe spending part of his childhood in West Germany before finishing high school in suburban St Louis. Stipe attended the University of Georgia in Athens, where he became involved in the local college rock and jangle pop scene. He formed R.E.M. after meeting his bandmates at the university and soon dropped out to pursue music with them. The band issued its debut single, "Radio Free Europe," and subsequently signed to I.R.S. Records, meeting wide acclaim and soon great commercial success. Possessing a distinctive voice, Stipe has been noted for the "mumbling" style of his early career. Since the mid-1980s, Stipe has sung in "wailing, keening, arching vocal figures" that R.E.M. biographer David Buckley compared to Celtic folk artists and Muslim muezzin. He was in charge of R.E.M.'s visual aspect, often selecting album artwork and directing many of the band's music videos. Outside the music industry, he owns and runs two film production studios, C-00 and Single Cell Pictures. As a member of R.E.M., Stipe was inducted into the Rock and Roll Hall of Fame in 2007. As a singer-songwriter, Stipe influenced a wide range of artists, including Kurt Cobain of Nirvana and Thom Yorke of Radiohead. Bono of U2 has described his voice as "extraordinary", and Yorke told The Guardian that Stipe is his favorite lyricist, saying "I loved the way he would take an emotion and then take a step back from it and in doing so make it so much more powerful". == Early life and education == Stipe was born on January 4, 1960, in Decatur, Georgia, to Marianne and John Stipe. He was a military brat; his father was a serviceman in the United States Army, having served in Korea as a helicopter pilot. The elder Stipe's career resulted in frequent relocations for his family. His younger sister, Lynda Stipe, was born in 1962 and became the vocalist of Hetch Hetchy. Stipe and his family moved to various locales during his childhood, including West Germany, Texas, Illinois, and Alabama. In 1978, he graduated from high school in Collinsville, Illinois, in suburban St. Louis. His senior photo is pictured in the album art work of Eponymous. Stipe also worked at the local Waffle House. Previous generations of his family were Methodist ministers. At age 14, Stipe was turned on to punk rock by an article in Creem magazine by Lisa Robinson on the CBGB scene. The article featured a photo of Patti Smith, whom Stipe came to idolize. He remembers buying her debut album, Horses, the day it came out. "Since then, I never looked back." == Career == === Boat Of === In the early 1980s, Stipe played in the group Boat Of with Tom Smith, who would later found the groups Peach of Immortality and To Live and Shave in LA. Also in Boat Of were Carol Levy and Mike Green. === R.E.M. === While attending the University of Georgia in Athens, Stipe frequented the Wuxtry record shop, where he met store clerk Peter Buck in 1980. "He was a striking-looking guy and he also bought weird records, which not everyone in the store did," Buck recalled. The two became friends; they eventually decided to form a band and started writing music together, although at the time Stipe was also in a local group named Gangster. Buck and Stipe were soon joined by Bill Berry and Mike Mills, and named themselves R.E.M., a name Stipe selected at random from a dictionary. Stipe was the youngest member of the band. All four members of R.E.M. dropped out of school in 1980 to focus on the new band. Stipe was the last to do so. The band issued its debut single, "Radio Free Europe," on Hib-Tone; it was a college radio success. The band signed to I.R.S. Records for the release of the Chronic Town EP one year later. In 1983, R.E.M. released its debut album, Murmur, which was acclaimed by critics. Stipe's vocals and lyrics received particular attention from listeners. Murmur went on to win the Rolling Stone Critics Poll Album of the Year over Michael Jackson's Thriller. Their second album, Reckoning, followed in 1984. In 1985, R.E.M. traveled to England to record their third album, Fables of the Reconstruction, a difficult process that brought the band to the verge of a break up. After the album was released, relationships in the band remained tense. Gaining weight and acting eccentrically (such as by shaving his hair into a monk's tonsure), Stipe later identified himself as suffering from depression and exhaustion during this period, saying "I was well on my way to losing my mind." They toured in Canada and throughout Europe that year; Stipe had bleached his hair blond during this time. Bill Berry left R.E.M. in 1997, and the other members continued as a three-piece. R.E.M. disbanded amicably in 2011. Stipe confirmed in 2021 that they had no plans to reunite. === Projects === In September 1983, a few months after the release of R.E.M.'s debut album, Stipe participated in a low-budget, forty-five-minute Super-8 film called Just Like a Movie, shot in Athens by New York Rocker magazine photographer Laura Levine, who was a friend of the band. Those with acting roles in the film included Levine, Stipe, his sister Lynda, Matthew Sweet (who formed a short-lived duo, Community Trolls, with Michael Stipe), and R.E.M.'s Bill Berry. The film remains unreleased. In the period between 1990 and 1992, Stipe was involved with the band Chickasaw Mudd Puppies. He co-produced and featured on their two albums: White Dirt (1990) and 8 Track Stomp (1991). Stipe had planned a collaboration with friend Kurt Cobain, lead singer of Nirvana, in 1994; this was partly an attempt to lure Cobain away from his home and his drug addiction. However, they did not manage to compose or record anything before Cobain's death. Stipe was chosen as the godfather of Cobain and Courtney Love's daughter, Frances Bean Cobain. R.E.M. recorded the song "Let Me In" from the 1994 album Monster in tribute to Cobain. In 2023, Stipe would officiate the younger Cobain's wedding to Riley Hawk. Stipe was once very close to fellow alternative rock singer Natalie Merchant and has recorded a few songs with her, including one titled "Photograph," which appeared on a pro-choice benefit album titled Born to Choose, and they appeared live with Peter Gabriel singing Gabriel's single "Red Rain" at the 1996 VH1 Honors and a few other times. Stipe and Tori Amos became friends in the mid-1990s and recorded a duet in 1994 called "It Might Hurt a Bit" for the Don Juan DeMarco motion picture soundtrack. Both Stipe and Amos decided not to release it. In 1998, Stipe published a collection called Two Times Intro: On the Road with Patti Smith. In 2006, Stipe released an EP that comprised six different cover versions of Joseph Arthur's "In The Sun" for the Hurricane Katrina disaster relief fund. One version, recorded in a collaboration with Coldplay's Chris Martin, reached number one on the Canadian Singles Chart. Also in 2006, Stipe appeared on the song "Broken Promise" on the Placebo release Meds. Continuing his non-R.E.M. work in 2006, Stipe sang the song "L'Hôtel" on the tribute album to Serge Gainsbourg titled Monsieur Gainsbourg Revisited and appeared on the song "Dancing on the Lip of a Volcano" on the New York Dolls album One Day It Will Please Us to Remember Even This. He recorded a song with Miguel Bosé on the album Papito, "Lo que ves es lo que hay." Stipe collaborated with Lacoste in 2008 to release his own "holiday collector edition" brand of polo shirt. The design depicts a concert audience from the view of the performer on stage. He appeared with Chris Martin of Coldplay live at Madison Square Garden and online to perform "Losing My Religion" in the 12-12-12 concert raising money for relief from Hurricane Sandy. A new recording from Stipe and featuring Courtney Love was revealed in 2013. The song, "Rio Grande," is taken from Johnny Depp's pirate-themed album, Son of Rogue's Gallery. Stipe also created the soundtrack for The Cold Lands (2013), a film by Stipe's friend director Tom Gilroy. Stipe inducted the American grunge band Nirvana into the Rock and Roll Hall of Fame on April 10, 2014. He debuted his first solo composition at Moogfest in 2017. In June 2017, it was revealed that Stipe had returned to recording, acting as producer and co-writer for Fischerspooner's single "Have Fun Tonight", the lead single from their album Sir. Stipe would go on to produce and co-write the entire Sir album, released on February 16, 2018. Stipe released the solo song "Future, If Future" on March 24, 2018, followed by "Your Capricious Soul" on October 5, 2019. "Drive to the Ocean" was released for his 60th birthday on January 4, 2020. Photography has long been a passion for Stipe and he has been carrying a camera with him since his teenage years when he photographed shows featuring Ramones, The Runaways and Queen. In 2018, Stipe released a book of his photography entitled Volume 1, which featured 35 photographs of such celebrities as River Phoenix and Kurt Cobain. A second volume with Douglas Coupland, Our Interference Times: A Visual Record, was released in 2019. In 2019, Stipe collaborated with Aaron Dessner and Justin Vernon's band Big Red Machine on the single "No Time For Love Like Now." The song was finished and released in 2020 during the COVID-19 pandemic. Stipe began recording his first solo album at Electric Lady Studios in New York City in 2023, writing and producing "synth-infused, poppy" songs with longtime collaborator Andy LeMaster. === Film and television work === In early 1987, Stipe and Jim McKay co-founded C-00 Films, a mixed-media company that was "designed to channel its founder's creative talents towards the creation and promotion of alternative film works." Stipe and his producing partner, Sandy Stern, have served as executive producers on films including Being John Malkovich, Velvet Goldmine, and Man on the Moon. He was also credited as a producer of the 2004 film Saved! In 1998, he worked on Single Cell Pictures, a film production company that released several arthouse/indie movies. Stipe has made a number of acting appearances on film and on television. He appeared in an episode of The Adventures of Pete & Pete as an ice cream man named Captain Scrummy. Stipe has appeared as himself with R.E.M. on Sesame Street, playing a reworked version of "Shiny Happy People" titled "Furry Happy Monsters", and appeared in an episode of The Simpsons titled "Homer the Moe", in which R.E.M. was tricked into playing a show in Homer Simpson's garage. He also appeared as a guest on the Cartoon Network talk show spoof Space Ghost Coast to Coast in the episode "Hungry". Stipe made several short appearances on The Colbert Report. Stipe voiced Schnitzel the Reindeer in the 1999 movie Olive, the Other Reindeer and appeared in the 1996 film Color of a Brisk and Leaping Day. === Political activism === In March 2006, Stipe, along with other musicians, held a protest concert against the Iraq War. In March 2018, Stipe joined the "March for Our Lives" rallies to advocate gun control after the Marjory Stoneman Douglas High School shooting. He also released a teaser of his new song in the rally. In a 2021 interview for Jacobin, Stipe described himself as a democratic socialist, and said that he was a member of the Democratic Party so he could vote in Democratic primaries. He endorsed Bernie Sanders' 2016 and 2020 presidential campaigns. Stipe expressed solidarity with the people of the Gaza Strip during the Gaza war. He signed an October 2023 open letter of artists to President Joe Biden urging a ceasefire in Gaza. == Personal life == Stipe is vegetarian and co-owned a vegetarian restaurant, Guaranteed, in Athens, Georgia. Though many people think he also owned The Grit, he was the landlord of the building and not the restaurant owner. He lives with his long-term partner, photographer Thomas Dozol, in New York and Berlin. In 1983, Stipe met Natalie Merchant of the band 10,000 Maniacs; the two started a friendship, and eventually had a romantic relationship for a period of time. With the success of the albums Out of Time (1991) and Automatic for the People (1992), R.E.M. became mainstream music stars. Around 1992, rumors that Stipe had contracted HIV began to circulate. He responded with the following: Not that I can tell. I wore a hat that said "White House Stop AIDS." I'm skinny. I've always been skinny, except in 1985 when I looked like Marlon Brando, the last time I shaved my head. I was really sick then. Eating potatoes. I think AIDS hysteria would obviously and naturally extend to people who are media figures and anybody of indecipherable or unpronounced sexuality. Anybody who looks gaunt, for whatever reason. Anybody who is associated, for whatever reason – whether it's a hat, or the way I carry myself – as being queer-friendly. In 1994, with questions remaining, Stipe described himself as "an equal opportunity lech," and said he did not define himself as gay, straight or bisexual, but that he was attracted to, and had relationships with, both men and women. In 1995, he appeared on the cover of Out magazine. Stipe described himself as a "queer artist" in Time in 2001 and revealed that he had been in a relationship with "an amazing man" for three years at that point. Stipe reiterated this in a 2004 interview with Butt magazine. When asked if he ever declares himself as gay, Stipe stated, "I don't. I think there's a line drawn between gay and queer, and for me, queer describes something that's more inclusive of the grey areas." In 1999, author Douglas A. Martin published a novel, Outline of My Lover, in which the narrator has a six-year romantic relationship with the unnamed lead singer of a successful Athens, Georgia-based rock band; the book was widely speculated, and later confirmed by its author, to have been a roman à clef based on a real relationship between Martin and Stipe. The two had previously collaborated on two books, both in 1998: The Haiku Year (for which the two had both contributed haiku) and Martin's book of poetry Servicing the Salamander (for which Stipe took the cover photograph). == Musical style == Stipe has a baritone vocal range. His role in the songwriting process for R.E.M. was to write lyrics and devise melodies. While each member was given an equal vote in the songwriting process, Peter Buck has conceded that Stipe, as the band's lyricist, could rarely be persuaded to follow an idea he did not favor. Stipe sings in "wailing, keening, arching vocal figures" that R.E.M. biographer David Buckley compared to Celtic folk artists and Muslim muezzin. Stipe often harmonizes with Mills in songs; in the chorus for "Stand", Mills and Stipe alternate singing lyrics, creating a dialogue. Early articles about the band focused on Stipe's singing style (described as "mumbling" by The Washington Post), which often rendered his lyrics indecipherable. Stipe commented in 1984, "It's just the way I sing. If I tried to control it, it would be pretty false." Stipe has earned recognition from the music industry for his unique voice. Bono remarked in 2003 that Stipe has an "extraordinary voice," adding "I often tell him I think he's a crooner, and he doesn't like that very much. But it is sort of one part some sort of Bing Crosby '50s laid-back crooner, and one part Dolly Parton." In 2023, Rolling Stone ranked Stipe at number 152 on its list of the 200 Greatest Singers of All Time. Stipe insisted that many of his early lyrics were "nonsense," saying in a 1994 online chat, "You all know there aren't words, per se, to a lot of the early stuff. I can't even remember them." In truth, many early R.E.M. songs had definite lyrics that Stipe wrote with care. Stipe explained in 1984 that when he started writing lyrics they were like "simple pictures," but after a year he grew tired of the approach and "started experimenting with lyrics that didn't make exact linear sense, and it's just gone from there." In the mid-1980s, as Stipe's pronunciation while singing became clearer, the band decided that its lyrics should convey ideas on a more literal level. Mills explained, "After you've made three records and you've written several songs and they've gotten better and better lyrically the next step would be to have somebody question you and say, are you saying anything? And Michael had the confidence at that point to say yes...." After what Stipe has referred to as "The Dark Ages of American Politics" [The Reagan/Bush Years], R.E.M. incorporated more politically oriented concerns into his lyrics on Document and Green. "Our political activism and the content of the songs was just a reaction to where we were, and what we were surrounded by, which was just abject horror," Stipe said later. "In 1987 and '88 there was nothing to do but be active." While Stipe continued to write songs with political subject matter like "Ignoreland" and "Final Straw," later albums have focused on other topics. Automatic for the People dealt with "mortality and dying. Pretty turgid stuff," according to Stipe; Monster, meanwhile, critiqued love and mass culture, and Reveal dipped into mysticism. == Discography == == Books == Michael Stipe: Volume 1. Damiani, 2018. ISBN 9788862085915. Contains 35 photographs. Our Interference Times: A Visual Record. With Douglas Coupland. Damiani, 2019. ISBN 978-8862086783. Michael Stipe: Michael Stipe. Damiani, 2021. ISBN 9788862087384. Even the Birds Gave Pause. Damiani. 2023. ISBN 9788862088145. == References == == General references == Buckley, David. R.E.M.: Fiction: An Alternative Biography. Virgin, 2002. ISBN 1-85227-927-3. Jovanovic, Rob (2006). Michael Stipe: The Biography. Portrait. ISBN 0-7499-5098-6. Platt, John, ed. The R.E.M. Companion: Two Decades of Commentary. Schirmer, 1998. ISBN 0-02-864935-4. == External links == Official website Tumblr blog Michael Stipe at AllMusic Michael Stipe discography at Discogs Michael Stipe at IMDb
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Wikipedia:Michael Strabo#0
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Michael Strabo (born November 6, 1975) is a Danish financier. He is the founder and managing director of Strabo Investments Limited, a Malta incorporated capital markets and corporate finance focused firm. He has publicly advocated for companies to implement shareholder value enhancing strategies, actively pushing Danske Bank A/S to implement a share buyback program and arguing the Danske Bank Board of Directors ought to consider a sale of the company in order to unlock shareholder value. Strabo graduated from University College London (UCL) with a B.Sc. joint honours degree in both mathematics and statistics. He holds the Chartered Financial Analyst (CFA) designation from the CFA Institute. == References ==
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Wikipedia:Michael Vaughan-Lee#0
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Michael Rogers Vaughan-Lee is a mathematician and retired academic. He was Professor of Mathematics at the University of Oxford from 1996 to 2010 and a tutor at Christ Church, Oxford, between 1971 and 2010. == Career == Vaughan-Lee completed his Doctor of Philosophy (DPhil) degree at the University of Oxford in 1968 and then taught at Vanderbilt University for two years as an assistant professor. In 1970, he was appointed to a lectureship at the University of Queensland, but resigned the following year and returned to the United Kingdom to become a tutor in mathematics at Christ Church, Oxford, where he remained until he retired in 2010. In 1996, he was awarded the title of Professor of Mathematics by the University of Oxford; since retirement in 2010, he has been an emeritus professor. == Research == Vaughan-Lee specialises in group theory, especially the restricted Burnside problem. He has also made contributions relating to Engel Lie algebras, computational algebra, and other areas. === Selected publications === "Lie rings of groups of prime exponent", Journal of the Australian Mathematical Society, vol. 49 (1990), pp. 386–398. The Restricted Burnside Problem (Oxford University Press, 1st ed., 1990; 2nd ed., 1993). (with E. I. Zel'manov) "Upper bounds in the restricted Burnside problem", Journal of Algebra, vol. 162 (1993), pp. 107–145. "An algorithm for computing graded algebras", Journal of Symbolic Computation, vol. 16 (1993), pp. 345–354. "The nilpotency class of finite groups of exponent p", Transactions of the American Mathematical Society, vol. 346 (1994), pp. 617–640. (with E. I. Zel'manov) "Upper bounds in the restricted Burnside problem II", International Journal of Algebra and Computation, vol. 6 (1996), pp. 735–744. "Engel-4 groups of exponent 5", Proceedings of the London Mathematical Society, vol. 74 (1997), pp. 306–334. "Superalgebras and dimensions of algebras", International Journal of Algebra and Computation, vol. 8 (1998), pp. 97–125. (with M. F. Newman) "Engel-4 groups of exponent 5. II. Orders", Proceedings of the London Mathematical Society, vol. 79 (1999), pp. 283–317. (with E. I. Zel'manov) "Bounds in the restricted Burnside problem", Journal of the Australian Mathematical Society, vol. 67 (1999), pp. 261–271. (with Daniel Groves) "Finite groups of bounded exponent", Bulletin of the London Mathematical Society, vol. 35 (2003), pp. 37–40. "Simple Lie Algebras of Low Dimension Over GF (2)", LMS Journal of Computation and Mathematics, vol. 9 (2006), pp. pp. 174–192. "On 4-Engel Groups", LMS Journal of Computation and Mathematics, vol. 10 (2007), pp. 341–353. == References ==
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Wikipedia:Michał Falkener#0
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Michael Falkener, Michael of Wroclaw or Michael de Wratislava (Polish: Michał Wrocławczyk; Latin: Michael Vratislaviensis; ca. 1450 or 1460 in Wrocław – 1534) was a Silesian Scholastic philosopher, astronomer, astrologer, mathematician, theologian, philologist, and professor of the Kraków Academy. == Life == Michał Falkener was born in Silesia to a family of wealthy German burghers. In Latin—the language favored by medieval European scholars, and used in his works—he is sometimes referred to as "Vratislaviensis" or "Wratislaviensis" ("the Wrocławian") in addition to "Michaelis de Vratislauia" ("Michael of Wrocław"). In Polish he is, respectively, "Wrocławczyk" and "Michał z Wrocławia" ("Michael of Wrocław"). In German, the place identifier is "of Breslau"—"von Breslau" or "aus Breslau." Falkener entered the arts faculty of the Kraków Academy in 1478, earning his bachelor's in 1481 and defending his master's thesis in 1488. Later he lectured there on astronomy, astrology, mathematics, physics, logic, grammar, and rhetoric, as well as scholastic and Aristotelian philosophy. His students included Nicolaus Copernicus. In 1495 he entered the Collegium Minus, and in 1501 the Collegium Maius. In 1512 he joined the theological faculty, where he earned a doctorate in 1517. He twice (1499 and 1504) served as rector of the faculty of arts. For several years he headed the Bursa Niemiecka, succeeding John of Głogów. Falkener was a Thomist but an incomplete one since, in addition to Peripatetic-Thomist proofs for the existence of God, he also accepted St. Anselm's proofs. In addition to more medieval pursuits, Falkener was interested in humanism: he knew and taught on classical and humanist authors, appreciating their linguistic and artistic abilities in particular. He published and edited important introductions to and commentaries on song collections and religious texts. Falkener's first printed astrological predictions were published for the years 1494–95; 1506 saw the first edition of his Introductorium astronomiae Cracoviensis elucidans almanach. He bequeathed his personal library to the Kraków Academy. == See also == History of philosophy in Poland List of Poles Gesamtkatalog der Wiegendrucke == Works == Iudicium Cracoviense. Leipzig, 1494. [1] [2] Introductorium astronomie Cracoviense elucidans Almanach. Kraków, 1506; Kraków, 1507; Kraków, 1513; Kraków, 1517 [3] Introductorium Dyalecticae quod congestum logicum appellatur. Kraków, 1509; Nuremberg, 1511 [4]; Kraków, 1515; Argentoratum/Strasbourg, 1515 [5]. Expositio hymnorumque interpretatio pro iuniorum eruditione. Kraków: J. Haller, 1516. (Collection of psalms, hymns and chants with a literary critical introduction and philosophico-theological commentaries.) Epithoma figurarum in libros physicorum et De anima Arystotelis. Kraków, J. Haller, 1518. Epithoma conclusionum theologicalium: pro introductione in quator libros sententiarum magistri Petri Lombardi. Kraków: J. Haller, 1521. Prosarum dilucidatio ac earundem interpretatio... pro studiorum eruditione. Kraków: F. Ungler, 1530. (Collection of church texts, both rhymed and unrhymed, with commentary.) === Works as editor === Computus novus totius fere astronomiae fundamentum pulcherrimum continens. Kraków: drukarnia J. Haller, 1517; Kraków: J. Haller, 1504; Lipsk: J. Tanner, 1504; Kraków, 1508; Kraków 1514 (2 edycje); Kraków, 1518 (2 editions); Kraków, 1524; and others. (Collection of astronomical lessons and mnemonic verses.) == Literature == Ludwik Nowak, Michael Falkener de Vratislavia, Congestum logicum, Introductonium dialecticae, published by Akademia Teologii Katolickiej (Academy of Catholic Theology), 1990. Bibliografia Literatury Polskiej – Nowy Korbut, vol. 2 Piśmiennictwo Staropolskie, Państwowy Instytut Wydawniczy, Warszawa 1964, p. 154–155. == Notes == == References == Władysław Tatarkiewicz, Historia filozofii (History of Philosophy), volume one, Warsaw, Państwowe Wydawnictwo Naukowe, 1978. == External links == Works by Michał Falkener in digital library Polona
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Wikipedia:Michał Misiurewicz#0
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In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. This term makes less sense for maps in greater generality that have more than one free critical point because some critical points might be periodic and others not. These points are named after the Polish-American mathematician Michał Misiurewicz, who was the first to study them. == Mathematical notation == A parameter c {\displaystyle c} is a Misiurewicz point M k , n {\displaystyle M_{k,n}} if it satisfies the equations: f c ( k ) ( z c r ) = f c ( k + n ) ( z c r ) {\displaystyle f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})} and: f c ( k − 1 ) ( z c r ) ≠ f c ( k + n − 1 ) ( z c r ) {\displaystyle f_{c}^{(k-1)}(z_{cr})\neq f_{c}^{(k+n-1)}(z_{cr})} so: M k , n = c : f c ( k ) ( z c r ) = f c ( k + n ) ( z c r ) {\displaystyle M_{k,n}=c:f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})} where: z c r {\displaystyle z_{cr}} is a critical point of f c {\displaystyle f_{c}} , k {\displaystyle k} and n {\displaystyle n} are positive integers, f c ( k ) {\displaystyle f_{c}^{(k)}} denotes the k {\displaystyle k} -th iterate of f c {\displaystyle f_{c}} . == Name == The term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent; that is, in which there exists a neighbourhood for every critical point that is not visited by the orbit of this critical point. This meaning is firmly established in the context of the dynamics of iterated interval maps. Only in very special cases does a quadratic polynomial have a strictly periodic and unique critical point. In this restricted sense, the term is used in complex dynamics; a more appropriate one would be Misiurewicz–Thurston points (after William Thurston, who investigated post-critically finite rational maps). == Quadratic maps == A complex quadratic polynomial has only one critical point. By a suitable conjugation any quadratic polynomial can be transformed into a map of the form P c ( z ) = z 2 + c {\displaystyle P_{c}(z)=z^{2}+c} which has a single critical point at z = 0 {\displaystyle z=0} . The Misiurewicz points of this family of maps are roots of the equations: P c ( k ) ( 0 ) = P c ( k + n ) ( 0 ) , {\displaystyle P_{c}^{(k)}(0)=P_{c}^{(k+n)}(0),} Subject to the condition that the critical point is not periodic, where: k is the pre-period n is the period P c ( n ) = P c ( P c ( n − 1 ) ) {\displaystyle P_{c}^{(n)}=P_{c}(P_{c}^{(n-1)})} denotes the n-fold composition of P c ( z ) = z 2 + c {\displaystyle P_{c}(z)=z^{2}+c} with itself i.e. the nth iteration of P c {\displaystyle P_{c}} . For example, the Misiurewicz points with k= 2 and n= 1, denoted by M2,1, are roots of: P c ( 2 ) ( 0 ) = P c ( 3 ) ( 0 ) ⇒ c 2 + c = ( c 2 + c ) 2 + c ⇒ c 4 + 2 c 3 = 0. {\displaystyle {\begin{aligned}&P_{c}^{(2)}(0)=P_{c}^{(3)}(0)\\\Rightarrow {}&c^{2}+c=(c^{2}+c)^{2}+c\\\Rightarrow {}&c^{4}+2c^{3}=0.\end{aligned}}} The root c= 0 is not a Misiurewicz point because the critical point is a fixed point when c= 0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point M2,1 at c = −2. === Properties of Misiurewicz points of complex quadratic mapping === Misiurewicz points belong to, and are dense in, the boundary of the Mandelbrot set. If c {\displaystyle c} is a Misiurewicz point, then the associated filled Julia set is equal to the Julia set and means the filled Julia set has no interior. If c {\displaystyle c} is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling (in particular the cycle that the critical orbit falls onto). The Mandelbrot set and Julia set J c {\displaystyle J_{c}} are locally asymptotically self-similar around Misiurewicz points. ==== Types ==== Misiurewicz points in the context of the Mandelbrot set can be classified based on several criteria. One such criterion is the number of external rays that converge on such a point. Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more external arguments (or angles). Non-branch points have exactly two external rays (these correspond to points lying on arcs within the Mandelbrot set). These non-branch points are generally more subtle and challenging to identify in visual representations. End points, or branch tips, have only one external ray converging on them. Another criterion for classifying Misiurewicz points is their appearance within a plot of a subset of the Mandelbrot set. Misiurewicz points can be found at the centers of spirals as well as at points where two or more branches meet. According to the Branch Theorem of the Mandelbrot set, all branch points of the Mandelbrot set are Misiurewicz points. Most Misiurewicz parameters within the Mandelbrot set exhibit a "center of a spiral". This occurs due to the behavior at a Misiurewicz parameter where the critical value jumps onto a repelling periodic cycle after a finite number of iterations. At each point during the cycle, the Julia set exhibits asymptotic self-similarity through complex multiplication by the derivative of this cycle. If the derivative is non-real, it implies that the Julia set near the periodic cycle has a spiral structure. Consequently, a similar spiral structure occurs in the Julia set near the critical value, and by Tan Lei's theorem, also in the Mandelbrot set near any Misiurewicz parameter for which the repelling orbit has a non-real multiplier. The visibility of the spiral shape depends on the value of this multiplier. The number of arms in the spiral corresponds to the number of branches at the Misiurewicz parameter, which in turn equals the number of branches at the critical value in the Julia set. Even the principal Misiurewicz point M 3 , 1 {\displaystyle M_{3,1}} in the 1/3-limb, located at the end of the parameter rays at angles 9/56, 11/56, and 15/56, is asymptotically a spiral with infinitely many turns, although this is difficult to discern without magnification. ==== External arguments ==== External arguments of Misiurewicz points, measured in turns are: Rational numbers Proper fractions with an even denominator Dyadic fractions with denominator = 2 b {\displaystyle =2^{b}} and finite (terminating) expansion: 1 2 10 = 0.5 10 = 0.1 2 {\displaystyle {\frac {1}{2}}_{10}=0.5_{10}=0.1_{2}} Fractions with a denominator = a ⋅ 2 b ; a , b ∈ N ; 2 ∤ b {\displaystyle =a\cdot 2^{b};a,b\in \mathbb {N} ;2\nmid b} and repeating expansion: 1 6 10 = 1 2 × 3 10 = 0.16666... 10 = 0.0 ( 01 ) . . . 2 . {\displaystyle {\frac {1}{6}}_{10}={\frac {1}{2\times 3}}_{10}=0.16666..._{10}=0.0(01)..._{2}.} The subscript number in each of these expressions is the base of the numeral system being used. === Examples of Misiurewicz points of complex quadratic mapping === ==== End points ==== Point c = M 2 , 2 = i {\displaystyle c=M_{2,2}=i} is considered an end point as it is a tip of a filament, and the landing point of the external ray for the angle 1/6. Its critical orbit is { 0 , i , i − 1 , − i , i − 1 , − i . . . } {\displaystyle \{0,i,i-1,-i,i-1,-i...\}} . Point c = M 2 , 1 = − 2 {\displaystyle c=M_{2,1}=-2} is considered an end point as it is the endpoint of the main antenna of the Mandelbrot set. and the landing point of only one external ray (parameter ray) of angle 1/2. It is also considered an end point because its critical orbit is { 0 , − 2 , 2 , 2 , 2 , . . . } {\displaystyle \{0,-2,2,2,2,...\}} , following the Symbolic sequence = C L R R R ... with a pre-period of 2 and period of 1. ==== Branch points ==== Point c = − 0.10109636384562... + i 0.95628651080914... = M 3 , 1 {\displaystyle c=-0.10109636384562...+i\,0.95628651080914...=M_{3,1}} is considered a branch point because it is a principal Misiurewicz point of the 1/3 limb and has 3 external rays: 9/56, 11/56 and 15/56. ==== Other points ==== These are points which are not-branch and not-end points. Point c = − 0.77568377 + i 0.13646737 {\displaystyle c=-0.77568377+i\,0.13646737} is near a Misiurewicz point M 23 , 2 {\displaystyle M_{23,2}} . This can be seen because it is a center of a two-arms spiral, the landing point of 2 external rays with angles: 8388611 25165824 {\displaystyle {\frac {8388611}{25165824}}} and 8388613 25165824 {\displaystyle {\frac {8388613}{25165824}}} where the denominator is 3 ∗ 2 23 {\displaystyle 3*2^{23}} , and has a preperiodic point with pre-period k = 23 {\displaystyle k=23} and period n = 2 {\displaystyle n=2} . Point c = − 1.54368901269109 {\displaystyle c=-1.54368901269109} is near a Misiurewicz point M 3 , 1 {\displaystyle M_{3,1}} , as it is the landing point for pair of rays: 5 12 {\displaystyle {\frac {5}{12}}} , 7 12 {\displaystyle {\frac {7}{12}}} and has pre-period k = 3 {\displaystyle k=3} and period n = 1 {\displaystyle n=1} . == See also == Arithmetic dynamics Feigenbaum point Dendrite (mathematics) == References == == Further reading == == External links == Preperiodic (Misiurewicz) points in the Mandelbrot set by Evgeny Demidov M & J-sets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel Misiurewicz Point of the logistic map by J. C. Sprott
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Wikipedia:Michel Bierlaire#0
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Michel Bierlaire (born 1967 in Namur, Belgium) is a Belgian-Swiss applied mathematician specialized in transportation modeling and optimization. He is a professor at EPFL (École Polytechnique Fédérale de Lausanne) and the head of the Transport and Mobility Laboratory. == Career == Bierlaire received a PhD in mathematics from University of Namur in 1996 for his thesis on "Mathematical models for transportation demand analysis" that was supervised by Philippe Toint. He then joined as a research associate the Intelligent Transportation Systems Program at the Massachusetts Institute of Technology where he worked on the design and development of DynaMIT, a real-time software simulation tool designed to "effectively support the operation of Advanced Traveler Information Systems (ATIS) and Advanced Traffic Management Systems (ATMS)." In 1998, he joined EPFL first as a senior scientist (Maître d'enseignement et de recherche) at the Operations Research Group at the Institute of Mathematics. In 2006, he was made associate professor at the EPFL's School of Architecture, Civil and Environmental Engineering and became the founding director of the Transport and Mobility Laboratory. Since 2012, he has been a full professor at the EPFL. At the EPFL, he created in 2010 the Doctoral Program in Civil and Environmental Engineering, that he chaired until 2017. In 2012, Bierlaire founded hEART, the European Association for Research in Transportation that he chaired from 2012 to 2015. == Research == Bierlaire's research targets at developing mathematical models replicating the complexity of mobility behavior of individuals and goods for all modes of transportation. He aims to develop solutions to transportation problems that also include the implications of mobility on land use, economics, and the environment, among others. His work focuses on modelling travel behaviours by employing choice and activity-based models; on developing operations research models based on vehicle routing, scheduling, and timetabling; and on the fusion of those models. His further interests encompass intelligent transportation systems and the reproduction of pedestrian flow patterns. He creates and tests mathematical models and algorithms for applications in operations research that include continuous and discrete optimization, queuing theory, graphs, and simulation. Apart from implementations in transportation demand analysis, his work also finds active use in other domains such as marketing and image analysis. His multidisciplinary research draws next to mathematics also on computer vision, image analysis, hospital management and marketing. === Biogeme === Bierlaire is the lead developer of Biogeme, an open source project that performs the maximum likelihood estimation of parametric discrete choice models. It is working using pandas, a Python data analysis library. == Teaching == Bierlaire has developed several online courses, one discrete choice models, and three on optimization. Together with Moshe Ben-Akiva at MIT, Daniel McFadden and Joan Walker, both at University of California, Berkeley, he is offering a course on "Discrete Choice Analysis: Predicting Individual Behaviour and Market Demand" that is designed for professionals from academia and industry. == Distinctions == On invitation from the Association of European Operational Research Societies, Bierlaire initiated the EURO Journal on Transportation and Logistics, whose editor in chief he was between 2011 and 2019. Since 2012, he has been an associate editor of the journal Operations Research. He was an associate editor of the Journal of Choice Modelling since its conception in 2007 until 2017. == Selected works == Bierlaire, Michel (2015). Optimization: principles and algorithms. Lausanne, Switzerland: EPFL Press. ISBN 978-1482203455. Fosgerau, Mogens; McFadden, Daniel; Bierlaire, Michel (2013). "Choice probability generating functions" (PDF). Journal of Choice Modelling. 8: 1–18. doi:10.1016/j.jocm.2013.05.002. Bierlaire, Michel; Chen, Jingmin; Newman, Jeffrey (2013). "A probabilistic map matching method for smartphone GPS data". Transportation Research Part C: Emerging Technologies. 26: 78–98. doi:10.1016/j.trc.2012.08.001. Farooq, Bilal; Bierlaire, Michel; Hurtubia, Ricardo; Flötteröd, Gunnar (2013). "Simulation based population synthesis". Transportation Research Part B: Methodological. 58: 243–263. doi:10.1016/j.trb.2013.09.012. Osorio, Carolina; Bierlaire, Michel (2013). "A Simulation-Based Optimization Framework for Urban Transportation Problems". Operations Research. 61 (6): 1333–1345. doi:10.1287/opre.2013.1226. hdl:1721.1/89831. S2CID 8250849. Glerum, Aurélie; Stankovikj, Lidija; Thémans, Michaël; Bierlaire, Michel (2014). "Forecasting the Demand for Electric Vehicles: Accounting for Attitudes and Perceptions". Transportation Science. 48 (4): 483–499. doi:10.1287/trsc.2013.0487. Umang, Nitish; Bierlaire, Michel; Vacca, Ilaria (2013). "Exact and heuristic methods to solve the berth allocation problem in bulk ports". Transportation Research Part E: Logistics and Transportation Review. 54: 14–31. doi:10.1016/j.tre.2013.03.003. S. Sharif Azadeh, B. Atasoy, M. E. Ben-Akiva, M. Bierlaire, and M. Y. Maknoon. "Choice-driven dial-a-ride problem for demand responsive mobility service." Transportation Research Part B: Methodological 161 (2022): 128-149. Paneque, M. P., Bierlaire, M., Gendron, B., & Sharif Azadeh, S. (2021). Integrating advanced discrete choice models in mixed integer linear optimization. Transportation Research Part B: Methodological, 146, 26-49. == References == == External links == Michel Bierlaire publications indexed by Google Scholar Website of the Transport and Mobility Laboratory q Website of the European Association for Research in Transportation
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Wikipedia:Michel Deza#0
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Michel Marie Deza (27 April 1939 – 23 November 2016) was a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory. He was the retired director of research at the French National Centre for Scientific Research (CNRS), the vice president of the European Academy of Sciences, a research professor at the Japan Advanced Institute of Science and Technology, and one of the three founding editors-in-chief of the European Journal of Combinatorics. Deza graduated from Moscow University in 1961, after which he worked at the Soviet Academy of Sciences until emigrating to France in 1972. In France, he worked at CNRS from 1973 until his 2005 retirement. He has written eight books and about 280 academic papers with 75 different co-authors, including four papers with Paul Erdős, giving him an Erdős number of 1. The papers from a conference on combinatorics, geometry and computer science, held in Luminy, France in May 2007, have been collected as a special issue of the European Journal of Combinatorics in honor of Deza's 70th birthday. == Selected papers == Deza, M. (1974), "Solution d'un problème de Erdös-Lovász", Journal of Combinatorial Theory, Series B, 16 (2): 166–167, doi:10.1016/0095-8956(74)90059-8, MR 0337635. This paper solved a conjecture of Paul Erdős and László Lovász (in [1], p. 406) that a sufficiently large family of k-subsets of any n-element universe, in which the intersection of every pair of k-subsets has exactly t elements, has a common t-element set shared by all the members of the family. Manoussakis writes that Deza is sorry not to have kept and framed the US$100 check from Erdős for the prize for solving the problem, and that this result inspired Deza to pursue a lifestyle of mathematics and travel similar to that of Erdős. Deza, M.; Frankl, P.; Singhi, N. M. (1983), "On functions of strength t", Combinatorica, 3 (3–4): 331–339, doi:10.1007/BF02579189, MR 0729786, S2CID 46336677. This paper considers functions ƒ from subsets of some n-element universe to integers, with the property that, when A is a small set, the sum of the function values of the supersets of A is zero. The strength of the function is the maximum value t such that all sets A of t or fewer elements have this property. If a family of sets F has the property that it contains all the sets that have nonzero values for some function ƒ of strength at most t, F is t-dependent; the t-dependent families form the dependent sets of a matroid, which Deza and his co-authors investigate. Deza, M.; Laurent, M. (1992), "Facets for the cut cone I", Mathematical Programming, 56 (1–3): 121–160, doi:10.1007/BF01580897, MR 1183645, S2CID 18981099. This paper in polyhedral combinatorics describes some of the facets of a polytope that encodes cuts in a complete graph. As the maximum cut problem is NP-complete, but could be solved by linear programming given a complete description of this polytope's facets, such a complete description is unlikely. Deza, A.; Deza, M.; Fukuda, K. (1996), "On skeletons, diameters and volumes of metric polyhedra", Combinatorics and Computer Science (PDF), Lecture Notes in Computer Science, vol. 1120, Springer-Verlag, pp. 112–128, doi:10.1007/3-540-61576-8_78, ISBN 978-3-540-61576-7, MR 1448925. This paper with his son Antoine Deza, a fellow of the Fields Institute who holds a Canada Research Chair in Combinatorial Optimization at McMaster University, combines Michel Deza's interests in polyhedral combinatorics and metric spaces; it describes the metric polytope, whose points represent symmetric distance matrices satisfying the triangle inequality. For metric spaces with seven points, for instance, this polytope has 21 dimensions (the 21 pairwise distances between the points) and 275,840 vertices. Chepoi, V.; Deza, M.; Grishukhin, V. (1997), "Clin d'oeil on L1-embeddable planar graphs", Discrete Applied Mathematics, 80 (1): 3–19, doi:10.1016/S0166-218X(97)00066-8, MR 1489057. Much of Deza's work concerns isometric embeddings of graphs (with their shortest path metric) and metric spaces into vector spaces with the L1 distance; this paper is one of many in this line of research. An earlier result of Deza showed that every L1 metric with rational distances could be scaled by an integer and embedded into a hypercube; this paper shows that for the metrics coming from planar graphs (including many graphs arising in chemical graph theory), the scale factor can always be taken to be 2. == Books == Deza, M.; Laurent, M. (1997), Geometry of cuts and metrics, Algorithms and Combinatorics, vol. 15, Springer, doi:10.1007/978-3-642-04295-9, ISBN 3-540-61611-X, MR 1460488. As MathSciNet reviewer Alexander Barvinok writes, this book describes "many interesting connections ... among polyhedral combinatorics, local Banach geometry, optimization, graph theory, geometry of numbers, and probability". Deza, M.; Grishukhin, V.; Shtogrin, M. (2004), Scale-isometric polytopal graphs in hypercubes and cubic lattices, Imperial College Press, doi:10.1142/9781860945489, ISBN 1-86094-421-3, MR 2051396, archived from the original on 2012-02-25, retrieved 2009-05-20. A sequel to Geometry of cuts and metrics, this book concentrates more specifically on L1 metrics. Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0-444-52087-2. Reviewed in Newsletter of the European Mathematical Society 64 (June 2007), p. 57. This book is organized as a list of distances of many types, each with a brief description. Deza, M.; Dutour Sikirić, M. (2008), Geometry of chemical graphs: polycycles and two-faced maps, Encyclopedia of Mathematics and its Applications, vol. 119, Cambridge University Press, doi:10.1017/CBO9780511721311, ISBN 978-0-521-87307-9, MR 2429120. This book describes the graph-theoretic and geometric properties of fullerenes and their generalizations, planar graphs in which all faces are cycles with only two possible lengths. Deza, M.; Deza, E. (2009), Encyclopedia of Distances, Springer-Verlag, ISBN 978-3-642-00233-5, Deza, E.; Deza, M. (2011), Figurate Numbers, World Scientific, ISBN 978-981-4355-48-3. Deza, M.; Deza, E. (2013), Encyclopedia of Distances, 2nd revised edition, Springer-Verlag, ISBN 978-3-642-30957-1. Deza, M.; Deza, E. (2014), Encyclopedia of Distances, 3rd revised edition, Springer-Verlag, ISBN 978-3-662-44341-5. Deza, M.; Deza, E. (2016), Encyclopedia of Distances, 4th revised edition, Springer-Verlag, ISBN 978-3-662-52844-0. Deza, M.; Dutour Sikirić, M.; Shtogrin, M. (2015), Geometric Structure of Chemistry-relevant Graphs, Springer, ISBN 978-81-322-2448-8. Deza, E.; Deza, M.; Dutour Sikirić, M. (2016), Generalizations of Finite Metrics and Cuts, World Scientific, ISBN 978-98-147-4039-5. == Poetry in Russian == Deza, M. (1983), 59--62, Sintaksis, Paris (http://dc.lib.unc.edu/cdm/item/collection/rbr/?id=30912). Deza, M. (2014), Poems and interviews, Probel-2000, Moscow, ISBN 978-5-98604-442-2 (https://web.archive.org/web/20161026002230/http://www.liga.ens.fr/~deza/InRussian/DEZA-M.pdf). Deza, M. (2016), 75--77, Probel-2000, Moscow, ISBN 978-5-98604-555-9 (https://web.archive.org/web/20161022031836/http://www.liga.ens.fr/~deza/InRussian/DEZA-M2.pdf). == References == == Further reading == Agudo, Pierre (January 24, 1998), "Le mathématicien a besoin d'être aimé", l'Humanité (in French) == External links == Deza's web page as of August 17, 2016 on Wayback Machine Archived copy of Deza's web page, with note of demise
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Wikipedia:Michel Kervaire#0
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Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra. He introduced the Kervaire semi-characteristic. He was the first to show the existence of topological n-manifolds with no differentiable structure (using the Kervaire invariant), and (with John Milnor) computed the number of exotic spheres in dimensions greater than four, known as Kervaire–Milnor groups. He is also well known for fundamental contributions to high-dimensional knot theory. The solution of the Kervaire invariant problem was announced by Michael Hopkins in Edinburgh on 21 April 2009. == Education == He was the son of André Kervaire (a French industrialist) and Nelly Derancourt. After completing high school in France, Kervaire pursued his studies at ETH Zurich (1947–1952), receiving a Ph.D. in 1955. His thesis, entitled Courbure intégrale généralisée et homotopie, was written under the direction of Heinz Hopf and Beno Eckmann. == Career == Kervaire was a professor at New York University's Courant Institute from 1959 to 1971, and then at the University of Geneva from 1971 to 1997, when he retired. He received an honorary doctorate from the University of Neuchâtel in 1986; he was also an honorary member of the Swiss Mathematical Society. == See also == Homology sphere Kervaire manifold Plus construction == Selected publications == Kervaire, Michel (1960), "A manifold which does not admit any differentiable structure", Commentarii Mathematici Helvetici, 34: 257–270, doi:10.1007/BF02565940, MR 0139172, S2CID 120977898 Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF). Annals of Mathematics. 77 (3). Princeton University Press: 504–537. doi:10.2307/1970128. JSTOR 1970128. MR 0148075. This paper describes the structure of the group of smooth structures on an n-sphere for n > 4. Kervaire, Michel (1965), "Les nœuds de dimensions supérieures", Bulletin de la Société Mathématique de France, 93: 225–271, doi:10.24033/bsmf.1624, MR 0189052 Kervaire, Michel (1969), "Smooth homology spheres and their fundamental groups", Transactions of the American Mathematical Society, 144: 67–72, doi:10.2307/1995269, JSTOR 1995269, MR 0253347 Kervaire, Michel A.; Eliahou, Shalom (1990), "Minimal resolutions of some monomial ideals", Journal of Algebra, 129 (1): 1–25, doi:10.1016/0021-8693(90)90237-I, MR 1037391 == Notes == == References == Eliahou, Shalom; de la Harpe, Pierre; Hausmann, Jean-Claude; Weber, Claude (2008), "Michel Kervaire 1927–2007" (PDF), Notices of the American Mathematical Society, 55 (8): 960–961, ISSN 0002-9920, MR 2441527 == External links == Michel Kervaire at the Mathematics Genealogy Project Michel Kervaire in German, French and Italian in the online Historical Dictionary of Switzerland. Michel Kervaire's work in surgery and knot theory (Slides of lectures given by Andrew Ranicki at the Kervaire Memorial Symposium, Geneva, February 2009)
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Wikipedia:Michel Mandjes#0
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Michael Robertus Hendrikus "Michel" Mandjes (born 14 February 1970 in Zaandam) is a Dutch mathematician, known for several contributions to queueing theory and applied probability theory. His research interests include queueing models for telecommunications, traffic management and analysis, and network economics. He holds a full-professorship (Applied Probability and Queueing Theory) at the University of Amsterdam (Korteweg-de Vries Institute for Mathematics). From September 2004 he is advisor of the "Queueing and Performance Analysis" theme at EURANDOM, Eindhoven. He is author of the book "Large deviations for Gaussian queues", and is associate editor of the journals Stochastic Models and Queuing Systems. He contributed to the book Queues and Lévy fluctuation theory, published in 2015. == Books == "Large deviations for Gaussian queues" (2007) == References == == External links == Homepage
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Wikipedia:Michel Van den Bergh#0
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Michel Van den Bergh (born 25 July 1960) is a Belgian mathematician and professor at the Vrije Universiteit Brussel and does research at Hasselt University. His research interest is on the fundamental relationship between algebra and geometry. In 2003, he was awarded the Francqui Prize on Exact Sciences. Van den Bergh obtained his Ph.D. in mathematics from the University of Antwerp in 1985, with thesis Algebraic Elements in Finite Dimensional Division Algebras written under the direction of Fred Van Oystaeyen and Jan Maria Hendrik Van Geel. == References == == External links == Michel Van den Bergh publications indexed by Google Scholar Publications of Michel Van den Bergh
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Wikipedia:Michel Waldschmidt#0
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Michel Waldschmidt (born June 17, 1946 at Nancy, France) is a French mathematician, specializing in number theory, especially transcendental numbers. == Career == Waldschmidt was educated at Lycée Henri Poincaré and the University of Nancy until 1968. In 1972 he defended his thesis, titled Indépendance algébrique de nombres transcendants (Algebraic independence of transcendental numbers) and directed by Jean Fresnel, the University of Bordeaux. He was a research associate of the CNRS at the University of Bordeaux in 1971–1972. He was then a lecturer at Paris-Sud 11 University in 1972–1973. He became a lecturer at the University of Paris VI (Pierre et Marie Curie) and then became a professor there in 1973. He is a member of the Institut de mathématiques de Jussieu. Waldschmidt was also a visiting professor at places including the École normale supérieure. From 2001 to 2004 he was president of the Mathematical Society of France. He is a member of several mathematical societies, including the EMS, the AMS and Ramanujan Mathematical Society. == Awards and honors == Waldschmidt was awarded the Albert Châtelet Prize in 1974, the CNRS Silver Medal in 1978, the Marquet Prize of Academy of Sciences in 1980 and the Special Award of the Hardy–Ramanujan Society in 1986. In 2021, he was awarded the Bertrand Russell Prize by the American Mathematical Society. == Selected publications == Diophantine approximation on linear algebraic groups. Springer, 2000 ISBN 978-3-540-66785-8 Nombres transcendants, Lecture Notes in Mathematics, vol. 402, 1974, Springer ISBN 978-3-540-06874-7 Nombres transcendants et groupes algébriques, Astérisque, vol. 69/70, 1979, 2e tirage 1987 Transcendence Methods, Queens Papers in Pure and Applied Mathematics, 1979 With J.-M. Luck, P. Moussa, C. Itzykson (eds.), From Number Theory to Physics, 1995 == References == == External links == Official website Homepage to Jussieu Michel Waldschmidt at the Mathematics Genealogy Project Biography at the Wayback Machine (archived February 23, 2008) on frenchsciencetoday.org search on author Michel Waldschmidt from Google Scholar
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Wikipedia:Michela Procesi#0
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Michela Procesi (born 1973) is an Italian mathematician specializing in Hamiltonian partial differential equations such as the nonlinear Schrödinger equation or wave equation. The Degasperis–Procesi equation is named for her. She is a professor of mathematics at Roma Tre University. == Education and career == Procesi was born in 1973 in Rome, the daughter of mathematician Claudio Procesi. She earned a laurea in physics at the Sapienza University of Rome in 1998, and continued at la Sapienza for a PhD in mathematics in 2002. Her dissertation, Estimates on Hamiltonian splittings: tree techniques in the theory of homoclinic splitting and Arnold diffusion for a-priori stable systems, was supervised by Luigi Chierchia. She became a postdoctoral researcher at the International School for Advanced Studies in Trieste and, with the support of the Istituto Nazionale di Alta Matematica Francesco Severi, at Roma Tre University. After continued work as a researcher at the University of Naples Federico II and la Sapienza, she obtained a position as an associate professor at Roma Tre University in 2015. She has been a full professor there since 2019. == Recognition == Procesi was an invited speaker at the 2022 (virtual) International Congress of Mathematicians. == References == == External links == Home page Michela Procesi publications indexed by Google Scholar
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Wikipedia:Michela Redivo-Zaglia#0
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Michela Redivo-Zaglia is an Italian numerical analyst known for her works on numerical linear algebra and on extrapolation-based acceleration of numerical methods. She is an associate professor in the department of mathematics at the University of Padua. == Education and career == Redivo-Zaglia earned a degree in mathematics at the University of Padua in 1975, and completed her Ph.D. in 1992 at the University of Lille in France. Her dissertation, Extrapolation, Méthodes de Lanczos et Polynômes Orthogonaux: Théorie et Conception de Logiciels was supervised by Claude Brezinski. She worked at the University of Padua, in the department of electronics and computer science, from 1984 to 1998, when she became an associate professor in 1998 at the University of Calabria. She subsequently returned to Padua as an associate professor. == Books == Redivo-Zaglia's books include: Extrapolation Methods: Theory and Practice (with Claude Brezinski, North-Holland, 1991) Méthodes Numériques Directes de l’Algèbre Matricielle (Direct Numerical Methods for Matrix Algebra, with Claude Brezinski, Ellipses, 2004) Méthodes Numériques Itératives: Algèbre Linéaire et Non Linéaire (Iterative Numerical Methods: Linear and Nonlinear Algebra, with Claude Brezinski, Ellipses, 2006) Extrapolation and Rational Approximation: The Works of the Main Contributors (with Claude Brezinski, Springer, 2020) She is also the author of four textbooks on computer science and numerical analysis in Italian. == Recognition == In 2019, a workshop on numerical analysis was held at the University of Porto, dedicated to Redivo-Zaglia and her advisor Claude Brezinski, "due to their important contributions to this field of research". == References == == External links == Home page Michela Redivo-Zaglia publications indexed by Google Scholar
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Wikipedia:Michela Varagnolo#0
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Michela Varagnolo is a mathematician whose research topics have included representation theory, Hecke algebra, Schur–Weyl duality, Yangians, and quantum affine algebras. She earned a doctorate in 1993 at the University of Pisa, under the supervision of Corrado de Concini, and is maître de conférences in the department of mathematics at CY Cergy Paris University, affiliated there with the research laboratory on analysis, geometry, and modeling. Varagnolo was an invited speaker at the 2014 International Congress of Mathematicians. In 2019, with Éric Vasserot, she won the Prix de l'État of the French Academy of Sciences for their work on the geometric representation theory of Hecke algebras and quantum groups. == References ==
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Wikipedia:Michelangelo Ricci#0
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Michelangelo Ricci (1619–1682) was an Italian mathematician and a Cardinal of the Roman Catholic Church. == Biography == Michelangelo Ricci was born on 30 January 1619 in Rome, then capital of the Papal States, to a family of low social standing that originated in Bergamo. He studied theology and law in Rome, where he was a contemporary of René-François de Sluse. He also studied mathematics under Benedetto Castelli who himself had been a student of Galileo Galilei. He was a friend of Evangelista Torricelli, kept close links with contemporary scientific culture, and played an important role in the development of the Galilean school. Like de Sluze, he spent his entire career in the Roman Catholic Church and served the pope in various roles on several occasions. A trained theologian, he acted as consultant to various Congregations of the Roman Curia. Having suffered from epilepsy since his birth, he was (according to canon law of the time) disqualified from ordination. Nonetheless, he was created a Cardinal-Deacon in the Consistory of 1 September 1681 by Pope Innocent XI, with the title 'Cardinal-Deacon of Santa Maria in Aquiro'. His position in the church was very useful for protecting his friends and fellow scientists in their controversies with the opposing scholastic school. He played a significant part in the theoretical debates and experiments that led up to Torricelli's discovery of atmospheric pressure and invention of the mercury barometer. In particular he followed the experiments in this field by Gasparo Berti, in Rome. There is an unpublished manuscript by Ricci, devoted to algebra, in the library of the Mathematical Institute of Genoa. It shows that by 1640 he was familiar with the 'New Algebra' of François Viète. In this book he provides a critique of the solutions given by the geometer Marino Ghetaldi of Ragusa in his De Resolutione et Compositione Matematica to the problems posed by Apollonius of Perga. His published mathematical work is summarised in a treatise of nineteen pages, Exercitatio geometrica, de maximis et minimis (1666) in which he studies the maxima of functions of the form x m ( a − x ) n {\displaystyle x^{m}(a-x)^{n}} and tangents to curves with equation y m = k x n {\displaystyle y^{m}=kx^{n}} , using methods that are an early form of induction. This treatise was much admired by his contemporaries and has recently been republished as an appendix to Mercator's 'Logarithmo-Technia' (1688). He also studied spirals (1644) and cycloids (1674) and recognised that the study of tangents and the calculation of areas are reciprocal operations. Ricci is also known for his correspondence with Torricelli, Vincenzo Viviani, René de Sluze and Cardinal Leopoldo de' Medici, founder of the Accademia del Cimento. These letters give his thoughts on paraboloids and hyperboloids when cut by parallel planes, on the surface of a ring, and on the properties of the vacuum. It was Ricci who welcomed Marin Mersenne, when he came to Italy to present the work of René Descartes. In optics he studied the magnifying effect of lenses. With Giovanni Battista Baliani he discussed the Galilean revolution. Ricci was also a close associate of Antonio Nardi, another mathematician in Rome. He collaborated with Nardi on his Scene, an unpublished manuscript that circulated among Torricelli and others, in which Ricci included some of the material that would later appear in his 1666 Geometrica Exercitatio. Following the death of Torricelli and the disappearance of Bonaventura Cavalieri, he was requested to collate and publish his correspondence with these two men. However he declined the invitation, leading to Torricelli's work being forgotten for some time. He endeavoured to defend Francesco Redi against Cardinal Leopold when Redi published his Esperienze Intorno alla Generazione degl'Insetti, arguing against the spontaneous generation of insects. Ricci was also the backer of Abbot Francesco Nazzari in the publication of the first review of Italian literature, the Giornale de' Letterati (1668-1683). He died in Rome, age 63, on 12 May 1682. == Works == Michaelis Angeli Riccii Geometrica exercitatio, Romae: apud Nicolaum Angelum Tinassium, 1666 Decretum sacrae Congregationis indulgentijs, sacrisque reliquijs praepositae, Romae et Pataui: typis reuerendae Camerae Apostolicae, 1678 Decretum Aloysius card. Homodeus, Romae, et Pataui, Romae et Pataui: typis reuerendae Camerae Apostolicae, 1678 Decretum sacrae Congregationis Indulgentiarum, Romae: typis reuerendae Camerae Apostolicae, 1679 Logarithmotechnia Nicolaus Mercator. Beigebunden Exercitatio geometrica, Hildesheim; New York: Olms, 1975 == References == == Sources == «RICCI MICHELANGELO, Cardinale». In: Gaetano Moroni, Dizionario di erudizione storico-ecclesiastica, Vol. LVII, Venezia: Tipografia Emiliana, 1852, p. 177 Vitarum Italorum doctrina excellentium qui saeculo XVIII floruerunt decas I-VI. Auctore Angelo Fabronio, Romae: typis S. Michaelis apud Junchium: prostant venales apud Natalem Barbiellini in foro Pasquini, 1769, Vol. II, p. 200 == External links == (in English) Biography on the University of Saint-Andrews site (in English) Biography on the History of Science Museum, Florence, site (in Italian) Biography on a site devoted to Francesco Redi (in English) The Cardinals of the Holy Roman Church, Biography
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Wikipedia:Michele Benzi#0
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Michele Benzi (born 1962 in Bologna) is an Italian mathematician who works as a full professor in the Scuola Normale Superiore in Pisa. He is known for his contributions to numerical linear algebra and its applications, especially to the solution of sparse linear systems and the study of preconditioners. == Previous career == He worked as assistant professor at the University of Bologna from 1993 to 1996, then at Cerfacs in Toulouse from 1996 to 1997, and then at the Los Alamos National Laboratory for three years. He transferred to the Emory University in Atlanta in 2000, where he held the endowed chair of Samuel Candler Dobbs professor starting from 2012 to 2018. Subsequently, he moved back to Italy to the Scuola Normale Superiore in Pisa as a full professor. == Awards == Benzi was named a SIAM Fellow in 2012, and a fellow of the American Mathematical Society in 2018 "for his contributions in numerical linear algebra, exposition, and service to the profession". He was elected as a member of Academia Europaea in 2019. As of 2022, he is the editor in chief of the SIAM Journal on Matrix Analysis and Applications, and he is editor of approximately 20 journals. He won a SIAM Outstanding Paper Prize in 2001. == Selected publications == Numerical solution of saddle point problems. M Benzi, GH Golub, J Liesen. Acta Numerica 14, 1-137 Preconditioning techniques for large linear systems: a survey. M Benzi. Journal of Computational Physics 182 (2), 418-477 A preconditioner for generalized saddle point problems. M Benzi, GH Golub. SIAM Journal on Matrix Analysis and Applications 26 (1), 20-41 The physics of communicability in complex networks. E Estrada, N Hatano, M Benzi. Physics Reports 514 (3), 89-119 == References ==
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Wikipedia:Michiel Hazewinkel#0
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Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. == Biography == Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. He received his BA in mathematics and physics in 1963, his MA in mathematics with a minor in philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as assistant professor at the University of Amsterdam in 1969. In 1970 he became associate professor at the Erasmus University Rotterdam, where in 1972 he was appointed professor of mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker (1975), M. van de Vel (1975), Jo Ritzen (1977), and Gerard van der Hoek (1980). From 1973 to 1975 he was also Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student. From 1982 to 1985 he was appointed part-time professor extraordinarius in mathematics at the Erasmus Universiteit Rotterdam, and part-time head of the Department of Pure Mathematics at the Centre for Mathematics and Computer (CWI) in Amsterdam. In 1985 he was also appointed professor extraordinarius in mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven (1986), Huib-Jan Imbens (1989), J. Scholma (1990) and F. Wainschtein (1992). At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became professor of mathematics and head of the Department of Algebra, Analysis and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977; for Acta Applicandae Mathematicae since its foundation in 1983; and associate editor for Chaos, Solitons & Fractals since 1991. He was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977; Mathematics and Geophysics for Reidel Publishing in 1981; Encyclopedia of Mathematics for Kluwer Academic Publishers from 1987 to 1994; and Handbook of Algebra in 6 volumes for Elsevier Science Publishers from 1995 to 2009. Hazewinkel was member of 15 professional societies in the field of mathematics, and participated in numerous administrative tasks in institutes, program committee, steering committee, consortiums, councils and boards. In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. == Publications == Hazewinkel has authored and edited several books, and numerous articles. Books, selection : 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems. With Rudolf E. Kálmán. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel (1976). "Moduli and canonical forms for linear dynamical systems II: The topological case" (PDF). Mathematical Systems Theory. 10 (1): 363–385. doi:10.1007/BF01683285. S2CID 25234632. Archived from the original (PDF) on 12 December 2013. Hazewinkel, Michiel; Marcus, Steven I. (1982). "On Lie algebras and finite dimensional filtering" (PDF). Stochastics. 7 (1–2): 29–62. doi:10.1080/17442508208833212. S2CID 119818672. Archived from the original (PDF) on 12 December 2013. Hazewinkel, M.; Marcus, S. I.; Sussmann, H. J. (1983). "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem" (PDF). Systems & Control Letters. 3 (6): 331–340. doi:10.1016/0167-6911(83)90074-9. S2CID 122766978. Archived from the original (PDF) on 2013-12-12. Retrieved 2013-09-10. Hazewinkel, Michiel (2001). "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164 (2): 283–300. doi:10.1006/aima.2001.2017. == References == == External links == Homepage
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Wikipedia:Michèle Moons#0
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Michèle Moons (1951-1998) was a Belgian scientist, leading researches on Celestial Mechanics for the department of mathematics of Facultés Universitaires Notre Dame de la Paix in Namur (Belgium). She developed an analytical theory of the liberation of the moon in the early 1980s, that is widely used by several centers analyzing the moon's motion. She has also worked on the effects of resonant motion in the minor planet belt. For nearly ten years, she was assistant editor of the journal Celestial Mechanics and Dynamical Astronomy. Her name has been given to the main-belt asteroid 7805 Moons, discovered in 1960 by Cornelis Johannes van Houten, Ingrid van Houten-Groeneveld and Tom Gehrels at Palomar Observatory. == References == == External links == 7805 Moons
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Wikipedia:Miggy Biller#0
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Margherita Joan (Miggy) Biller is a British mathematics teacher, the head of mathematics at York College. She was named an MBE in the 2016 New Year Honours "for services to mathematics in further education". Biller taught mathematics at St Peter's School, York before moving to York College in 1988. At York College, she taught mathematics prodigy Daniel Lightwing, after whom the main character of the film X+Y was modelled. Her husband, Peter Biller, is a historian at the University of York. == References ==
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Wikipedia:Miguel Walsh#0
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Miguel Nicolás Walsh is an Argentine mathematician working in number theory and ergodic theory. He is a professor at the University of Buenos Aires. == Education and career == Walsh has previously held a Clay Research Fellowship and was a fellow of Merton College at the University of Oxford. He is a professor of mathematics at the University of Buenos Aires. He was also Member of the Mathematical Sciences Research Institute at Berkeley, Senior Fellow of the Institute for Pure and Applied Mathematics at UCLA and von Neumann Fellow of the Institute for Advanced Study at Princeton. Walsh was born in Buenos Aires, Argentina. He obtained his undergraduate degree in 2010 from the University of Buenos Aires and his PhD, also from the same institution, in 2012. == Recognition == He received the MCA Prize in 2013. In 2014, he was awarded the ICTP Ramanujan Prize for his contributions to mathematics. He is the youngest recipient to date of both awards. In June 2017 Walsh was invited to present his research at the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil. In 2021, he was selected as Plenary Speaker of the Mathematical Congress of the Americas. In 2024, he was awarded the inaugural IMSA Prize of the Institute of the Mathematical Sciences of America during the Mathematical Waves Conference in Miami. That same year, he received the UMALCA Award of the Mathematical Union of Latin America and the Caribbean. He is one of two 2024 recipients of the Salem Prize, given "for contributions to ergodic theory, analytic number theory, and the development of the polynomial method, including a convergence theorem for nonconventional ergodic averages, bounds on the local Fourier uniformity of multiplicative functions, and bounds on rational points on varieties". == Selected publications == Walsh, Miguel N. (2020-11-01). "The polynomial method over varieties". Inventiones Mathematicae. 222 (2): 469–512. arXiv:1811.07865. Bibcode:2020InMat.222..469W. doi:10.1007/s00222-020-00975-6. ISSN 1432-1297. S2CID 119650372. Walsh, Miguel N. (2014-04-29). "The algebraicity of ill-distributed sets". Geometric and Functional Analysis. 24 (3): 959–967. arXiv:1307.0259. doi:10.1007/s00039-014-0286-3. ISSN 1016-443X. S2CID 119695020. Walsh, Miguel N. (2012-01-01). "Norm convergence of nilpotent ergodic averages". Annals of Mathematics. 175 (3): 1667–1688. arXiv:1109.2922. doi:10.4007/annals.2012.175.3.15. S2CID 55768070. Walsh, Miguel N. (2012-07-15). "The inverse sieve problem in high dimensions". Duke Mathematical Journal. 161 (10): 2001–2022. arXiv:1105.1551. doi:10.1215/00127094-1645788. hdl:11336/68296. ISSN 0012-7094. S2CID 119317224. == References == == External links == Walsh's homepage at the University of Oxford lanacion.com – El matemático argentino Miguel Walsh, de 26 años, ganó el premio Ramanujan (in Spanish) mincyt – Interview with Miguel Walsh ICTP – 2014 Ramanujan Prize Announced
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Wikipedia:Miguel da Silva#0
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Miguel da Silva (c. 1480 – 5 June 1556) was a Portuguese nobleman, the second son of Diogo da Silva, 1st Count of Portalegre and of his wife Maria de Ayala, a Castilian noblewomen. He was ambassador of the king of Portugal to several popes, and papal ambassador to the Emperor and others. Sometimes referred to through antonomasia as the Cardinal of Viseu (Portuguese: Cardeal de Viseu), he was Bishop of Viseu (Portugal), and Apostolic Administrator of the diocese of Massa Maritima (Tuscany). He was a cardinal of the Holy Roman Church from 1539 to 1556, and served as governor of several papal states. == Education and career == Silva was educated at the University of Paris, then in Siena, then Bologna, and finally in Rome. After his stay in Rome, he travelled to Venice, and from there he returned to Portugal, visiting several European principalities along the way. On his return to Portugal in 1502, he was appointed escrivão da puridade, or keeper of the royal seal, to the eldest son of King Manuel I, who succeeded him as John III of Portugal. He was appointed by King Manuel I of Portugal as ambassador to Rome in 1514. He served in that post during the reigns of popes Leo X, Adrian VI and Clement VII. Both Leo X and Clement VII wanted to make him a Cardinal, but were opposed by the Portuguese Crown. He was recalled to Lisbon in 1525 where he served as member of the Royal Council. Pope Clement VII appointed him Bishop of Viseu on 21 November 1526. He resigned the see on 22 April 1547, in favor of Cardinal Alessandro Farnese, the grandson of Pope Paul III. Pope Paul III finally elevated Miguel da Silva to the cardinalate on 19 December 1539, though the appointment was kept secret (in pectore) for the time being. Falling out of favour with King John III of Portugal, D. Miguel da Silva ran away to Rome in 1540, where he was warmly welcomed to the Curia by Paul III. His status as a Cardinal was revealed in 1541, and on 6 February 1542 he was assigned the titular church of Ss. XII Apostolorum. King John III of Portugal promptly condemned him on a charge of treason and revoked his Portuguese nationality. On 30 August 1542, Silva was named Legate to the Emperor Charles V. On 9 January 1545, he was appointed Legate of the Marches of Ancona, and on 19 March 1545 was also named governor of Fermo. He served as papal legate to Venice and Bologna. On 20 May 1549, he was named Apostolic Administrator of the diocese of Massa Marittima by Pope Paul III. Silva took part in the papal conclave following the death of Paul III, which began on 29 November 1549 and concluded on 7 February 1550 with the election of Cardinal Giovanni Maria Ciocchi del Monte, who took the name Julius III. He also took part in the conclave following the death of Julius III, which began on 5 April 1555 and ended on 9 April 1555, with the election of Cardinal Marcello Cervini, who took the name Marcellus II. He died three weeks later, on 30 April. A conclave followed immediately, opening on 15 May 1555 and concluding on 23 May with the election of Giampetro Carafa (Paul IV). Silva died in Rome on 5 June 1556, and was buried in the church of S. Maria in Trastevere, which had been his titular church since 11 December 1553. Greatly praised for his classical culture and command of ancient languages, he was a personal friend of the painter Raffaello Sanzio. Baldassare Castiglione dedicated his masterpiece Il Cortegiano to Silva. == References == == Sources == Cardella, Lorenzo (1793). Memorie storiche de' cardinali della santa Romana chiesa (in Italian). Vol. Tomo quattro (4). Roma: Pagliarini. pp. 233–236. Eubel, Conradus (ed.); Gulik, Guilelmus (1923). Hierarchia catholica (in Latin). Vol. Tomus 3 (second ed.). Münster: Libreria Regensbergiana. {{cite book}}: |first1= has generic name (help) Deswarte, Sylvie, "La Rome de D. Miguel da Silva (1515-1525)," O Humanismo Português. Primeiro Simpósio Nacional, 21-25 de Outubro de 1985. Lisboa: Il Centenario da Academia das Ciencias de Lisboa, 1988, pp. 177–307. Deswarte, Sylvie, Il "perfetto cortegiano," D. Miguel da Silva. Roma: Bulzoni Editore, 1989. Paiva, J.P. Os Bispos de Portugal e do Império 1495-1777. Coimbra, Universidade de Coimbra, 2006
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Wikipedia:Mihaela Ignatova#0
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Mihaela Ignatova is a Bulgarian mathematician who won the 2020 Sadosky Prize of the Association for Women in Mathematics for her research in mathematical analysis, and in particular in partial differential equations and fluid dynamics. == Education == In 2004, Ignatova earned both a bachelor's degree from Sofia University and a master's degree from the University of Nantes. She earned a second master's degree from Sofia University in 2006, working under the supervision of mathematician Emil Horozov. She then completed PhD studies from University of Southern California in 2011 under the supervision of Igor Kukavica. == Career == After working as a visiting assistant professor at the University of California, Riverside, a postdoctoral researcher at Stanford University, and an instructor at Princeton University, she moved to Temple University as an assistant professor in 2018. == References == == External links == Home page
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Wikipedia:Mihailo Petrović Alas#0
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Mihailo Petrović Alas (Serbian Cyrillic: Михаило Петровић Алас; 6 May 1868 – 8 June 1943), was a Serbian mathematician and inventor. He was also a distinguished professor at Belgrade University, an academic, fisherman, philosopher, writer, publicist, musician, businessman, traveler and volunteer in the Balkan Wars, the First and Second World Wars. He was a student of Henri Poincaré, Paul Painlevé, Charles Hermite and Émile Picard. Petrović contributed significantly to the study of differential equations and phenomenology, founded engineering mathematics in Serbia, and invented one of the first prototypes of a hydraulic analog computer. == Biography == Petrović was born on 6 May 1868, in Belgrade, as the first child of Nikodim, a professor of theology, and Milica (née Lazarević). He finished the First Belgrade Gymnasium in 1885, and afterwards enrolled at the natural science-mathematical section of the Faculty of Philosophy in Belgrade. At the time when he finished his studies in Serbia in 1889, several Serbian mathematicians who had acquired their doctorate degrees abroad, like Dr. Dimitrije Nešić (at Vienna and Karlsruhe Institute of Technology), Dr. Dimitrije Danić (at Jena, 1885) and Bogdan Gavrilović (at Budapest, 1887) were beginning to make a name for themselves. Subsequently, in September 1889, he too went abroad, to Paris to receive further education, and to prepare for the entrance exam to the École Normale Supérieure. He got a degree in mathematical sciences from Sorbonne University in 1891. He worked on preparing his doctoral dissertation, and on 21 June 1894 he defended his PhD degree at the Sorbonne, and received a title Docteur des sciences mathematiques (doctor of mathematical sciences). His doctorate was in the field of differential equations. In 1894, Petrović became a professor of mathematics at the Belgrade's Grande école (which later became the University of Belgrade). In those days, he was one of the greatest experts for differential equations. He held lectures until his retirement in 1938. In 1897, he became an associate member of the Serbian Royal Academy and associate member of the Yugoslav Academy of Sciences and Arts in Zagreb. He became a full member of the Serbian Royal Academy in 1899, when he was only 31. In 1882, he became a fisherman apprentice, and in 1895 he took an exam to become a master fisherman. Mihailo Petrović got the nickname "Alas" (river fisher) because of his passion for fishery. He was not only an aficionado, but expert as well. He participated in legislative talks regarding the fishery convention with Romania, and in talks with Austria-Hungary about the protection of the fishery on Sava, Drina and Danube rivers. Alas published expert papers and reports on the fish-fauna found in the Macedonian lakes, such as Skadar Lake and Ohrid Lake. He played violin, and in 1896, founded a musical society named Suz. Mihailo Petrović Alas also constructed a hydrointegrator, and won the gold medal at the World Exposition in Paris 1900. When in 1905 the Grande école was transformed into the University of Belgrade Petrović was among first eight regular professors, who elected other professors. He patented a total of 10 inventions, published 300 scientific works and a number of books and journals from his sea expeditions. These expeditions included trips to Azores, Newfoundland and Labrador, Suez Canal, Madagascar, Cape Verde, Canary Islands, Greenland, Iceland, Bermuda Triangle, Caribbean and others. Petrovć also visited both the North and South poles, researched the culture of Eskimos and took part in whale hunting expeditions. He received numerous awards and acknowledgments and was a member of several foreign science academies (Prague, Bucharest, Warsaw, Kraków) and scientific societies. In 1927, when Jovan Cvijić died, members of the Serbian academy proposed Mihailo Petrović as the new president of academy, but the authorities did not accept this proposal. Probable reason for this was the fact that Mihailo Petrović Alas was first a private tutor and mentor and later a close friend of the prince Đorđe P. Karađorđević, the king's brother, who was arrested in 1925, and held in house arrest. In 1931, members of the academy unanimously proposed Alas for the president of the academy, but authorities again dismissed this proposal. Mathematician and physicist Bogdan Gavrilović, a fellow professor, was nominated instead. In 1939, he became an honorary doctor at the University of Belgrade. In the same year, he received the order of Saint Sava, first class. He also founded the Belgrade School of Mathematics, which produced a number of mathematicians who continued Alas's work. All doctoral dissertations defended on the Belgrade University since 1912 until the Second World War were under his mentorship. Alas participated in the Balkan Wars and in the First World War as an officer, and after the war he served as a reserve officer. He practised cryptography, and his cipher systems were used by the Yugoslav army until World War II. When the Second World War broke out in Yugoslavia, he was again called into the army and the Germans captured him. After a while, he was released because of illness. On 8 June 1943, professor Petrović died in his home in Kosančićev Venac Street in Belgrade. Ninth Belgrade Gymnasium "Mihailo Petrović Alas" and Primary School in John's Street is a high school in Belgrade, Serbia named after him. Alas and fellow scientist Milutin Milanković were close friends for several decades. Due to his scientific work and results, Mihailo Petrović Alas is among the greatest Serbian mathematicians as well as one of the 100 most prominent Serbs. In the Association for Culture, Art and International Cooperation "Adligat" in Belgrade there is an extensive fund of documents from the legacy of Mihailo Petrović Alas, including his childhood photos, letters from his grandfather who educated him, diplomas, notes, a whole bundle of published and unpublished manuscripts, as well as numerous exam reports signed by him, among which is the report on the defense of the graduation exam jointly signed by Mihailo Petrović and Milutin Milanković. == Awards and memberships == Source: Member of Serbian academy of sciences and arts Member of Yugoslav academy Member of Academy of Sciences of the Czech Republic Member of academy, Bucharest Member of academy, Warsaw Member of academy, Kraków Member of various societies, Prague Member of various societies in Paris Member of various societies in Berlin Member of various societies in France Member of society of Italian mathematicians, Palermo Member of society of German mathematicians, Leipzig Member of Shevchenko Scientific Society, Lviv Member of scientific expedition for explorationof the South Pole Member of Rotary Club, Belgrade Order of Miloš the Great Order of St. Sava, 1st degree Order of St. Sava, 2nd degree Order of St. Sava, 3rd degree Order of the Romanian crown, 3rd degree Honorary brevet from London's society of mathematicians Honorary president of Yugoslav Alliance of students of mathematics Honorary doctor of science, University of Belgrade Dean of Faculty of Philosophy, Belgrade == Selected works == == See also == Mika Alas's House, where he lived, worked, and died, is a designated historic site. Bogdan Gavrilović Jovan Karamata == References == == Notes == Trifunović, Dragan (1991). Bard srpske matematike Mihailo Petrović Alas, Prilog intelektualnoj biografiji. Belgrade: Zavod za udžbenike i nastavna sredstva. == Further reading == Notice sur les travaux scientifiques de Mishel Petrovitch. Paris: Press. 1922. Publications mathematiques de l'Universite de Belgrade. Belgrade. 1938.{{cite book}}: CS1 maint: location missing publisher (link) Matematički vesnik. Belgrade. 1939.{{cite book}}: CS1 maint: location missing publisher (link) Milanković, Milutin (1946). Mika Alas. Belgrade.{{cite book}}: CS1 maint: location missing publisher (link) Zbornik radova Matematičkog instituta, SANU. Belgrade. 1953.{{cite book}}: CS1 maint: location missing publisher (link) Mihailo Petrović – čovek, filozof, matematičar. Belgrade: Matematička biblioteka. 1968. Spomenica Mihaila Petrovića. Belgrade. 1968.{{cite book}}: CS1 maint: location missing publisher (link) Trifunović, Dragan (1969). Letopis života i rada Mihaila Petrovića. Belgrade.{{cite book}}: CS1 maint: location missing publisher (link) Trifunović, Dragan (1976). Proučavanje modelovanja u delu Mihaila Petrovića. Belgrade.{{cite book}}: CS1 maint: location missing publisher (link) Todorčević, V.; Šegan-Radonjić, M. (15 September 2019). "Mihailo Petrović Alas: Mathematician and Master Fisherman". The Mathematical Intelligencer. 41 (3): 44–50. doi:10.1007/s00283-019-09901-y. == External links == Mihailo Petrović Alas at the Mathematics Genealogy Project Mihailo Petrovic The First Century of the International Commission on Mathematical Instruction, Petrovic Brilliant mind of mathematician, globetrotter and fisherman – Mihajlo Petrovich Alas Mihailo Petrović Alas: Life, Work, Times (2019)
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Wikipedia:Mihalj Šilobod Bolšić#0
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Mihalj Šilobod Bolšić (1 November 1724 – 4 April 1787) was a Croatian Roman Catholic priest, mathematician, writer, and musical theorist primarily known for writing the first Croatian arithmetics textbook Arithmetika Horvatzka (published in Zagreb, 1758). == Biography == Mihalj was born in Podgrađe Podokićko on 1 November 1724 as the son of Andrija Šilobod and Margareta Gunarić (Guunarich). He was baptised one day later at the local church as evidenced by an extract from the register of baptisms for Mihalj Šilobod, located at the parish in Podgrađe Podokićko. Andrija Šilobod was then a prominent senior member of the Karlovac Military Generalate of the Slunj regiment. From 1735 to 1739, Andrija participated in military operations against the Turks, as was required by the Habsburg Kingdom of Croatia. Mihalj's brother, Ivan (Johan) Šilobod, was also a soldier in the Slunj infantry. Andrija and Ivan were awarded nobility by Queen Maria Theresa of Austria in 1758 for their military services in the Croatian Military Frontier in 1741. Mihalj was initially schooled at the Jesuit school in Samobor, Zagreb. Later, he studied philosophy at the University of Vienna and continued further education in theology at the University of Bologna. Once he finished university in 1747, he went to work as a chaplain in the towns of Tuhlje and Ivanec, in what is now Hrvatsko Zagorje. He pioneered literacy efforts for the low-income Croatian community in a number of parishes, making him a standout even in that very deprived region. In 1751, Šilobod was appointed pastor in Martinska Ves near Sisak. In 1783, he renovated the dilapidated local parish church at Sv. Nedelja, which was formerly in the care of the Erdödy family. These works included raising the walls, naves and sanctuaries, adding a new sacristy, painting the main altar painted, and arranging a tomb for the pastors in which he is also buried. The big clock on the church tower (horologium) drew special attention and praise from the monarchy, which he created and placed himself. Additionally, he commissioned the construction of the Sveti Rok brick choir, which is still in use today. In the same period, while he served in Sv. Nedjelja, he particularly advocated for the construction of the first public school in Sveti Martin pod Okićem in 1761, the neighboring parish of where he was born. Mihalj's father Andreas bought from the land for this school from the Catholic Church and, with the assistance of the Zrinski family, transferred the ownership to his son Mihalj. Mihalj drew out the plans for the school and raised enough money to pay for the building's stone foundations and half of the bottom story during his lifetime; however, after his passing, his successors were unable to gather enough money to continue building. Had Mihael lived long enough to see the project through, the school would have been the first post-war public institution of learning in rural Croatia. Nevertheless, the uncompleted structure did not go to waste and in January 1867, it housed its first class of pupils. The elementary school (OŠ Mihaela Šiloboda) still retains Mihalj's namesake to this day. He died in Sveta Nedelja on 4 April 1787 and was buried in one of the tombs in Sv. Nedjelja. In his life, Mihalj was pious and never used the acquired nobility in his title or professional career. In fact, his father Andrija claimed that the priest signed all of his documents with the simple name "M. Šilobod" out of a desire for humility and to avoid the privileges and trappings of nobility. In Arithmetika Horvatzka, he uses both his birth and baptismal surname, Šilobod, and his second ("different") last name, Bolšić, to honour the feminine lineage of his family tree, which originated with his grandmother and had a profound impact on his life's social orientation. Many people from Hrvatsko Zagorje and Sisak Posavina, as well as Catholic Church leaders and a high-ranking military delegation from the Slunj Regiment, were documented as having travelled to Sv. Nedelja for the burial. == Major works == Šilobod wrote mainly in Latin and Croatian. In addition to his main works, he also wrote various poems and songs, primary in Latin. His major publications include: Zbirka crkvenih pjesama ("Collection of Church Songs"; 1757) – An anthology of traditional songs and hymns from rural Croatia. Arithmetika Horvatszka koju na obchinszku vszega országos haszen, y, z-potrebochu vnogemi izebranemi Példami obilnò iztolnachil, y na szvetlo dal. (1758.) Fundamentum cantus Gregoriani seu chroralis pro Captu Tyronis discipuli, ex probatis authoribus collectum, et brevi, ac facili dialogica methodo in lucem expositum opera, ac studio. ("The basis of the singing of Gregorian melodies, or the chroralis definite for the disciples saw it, from the classical authors, and deposited in a short time, exposed to the light of the work of the Method in the rich and of easy dialogica, and a studio."). (1760.) Cabala de lesu Lotto. ("Cabala about the game of lotto and various fortunes") (1768.) Cabala, to je na vszakoiachka pitanya kratki ter vendar prikladni odgovori vu horvaczkom jeziku. (Unknown Year) – A small practical lexicon or a short answer to all students' questions, especially concerning questions about Croatian language, and some church works. === Cithara Octochord Collection (1751) === Šilobod was the editor of the third edition of the Cithara Octochord Collection (1751), according to musicological study. This edition represents a form of restoration intervention in the Zagreb church's liturgical-musical programme. CO's greatest creative worth and innovation are hymnic forms with innovative text and music content, in which Šilobod undoubtedly played an active role as a poet, musician, and composer. === Zbirka crkvenih pjesama (1757) === After moving into the rural countryside and immersing himself in the local culture he began compiling traditional songs and hymns that had been passed down orally until then. He gathered a treasury of songs, and published them in Zagreb under Zbirka Crkvenih Pjesama ("Collection of Church Songs;" 1757), a book with almost 300 pages of unlicensed sheet music. === Arithmetika Horvatzka (1758) === His best known work, Arithmetika Horvatzka (1758), written in the vernacular Kajkavian dialect of Croatian language, established a complete system of arithmetic terminology in Croatian, and vividly used examples from everyday life in Croatia to present mathematical operations. Šilobod drew inspiration for his arithmetic handbook from a variety of sources, including the classic arithmetic handbook by Italian author Giuseppe Maria Figatelli. It was published 1758 in Zagreb, divided in four parts and signed as Mihalj Šilobod-Bolšić, as appropriate to the Kajkavian dialect. Public education in the Habsburg Monarchy was not institutionalised until the middle of the 18th century. The teaching service was sometimes done by municipally assigned teachers, and sometimes by parish priests. Students were taught to read and write, but it wasn't until the middle of the 18th century that they were also taught to count. Calculus textbooks were only available in foreign languages at the time, and the subject had to be taught in Croatian. The textbook's creation was prompted by a desire to uplift the material conditions of the local people through education. Mihalj identified that there was a severe lack of accounting skills in local organisers who struggled to handle even the most basic financial and business calculations. As the contemporary Croatian mathematician Mate Zoricic describes: "This is the main evil and the cause of poverty and misery in the people, and while foreigners, knowing the calculation, come to Dalmatia and get rich in a short time, sometimes it is not enough for our people even to nutrition." Thus, the book was written for a wide range of people who were literate but could not read Latin, such as business people, hosts, and anyone else who might need an account. Because of this, it is full of examples from real-world situations. In addition, it consistently demonstrates how one currency may be exchanged for another by pointing out the conversion of currencies and assigning activities in which Groschen and Kreuzer occur together. The general structure of Arithmetika Horvatzka adapted the structure and presentation style of similarly utilitarian books from the same time period: The first section of Šilobod's manual covers the four basic arithmetic operations (addition, subtraction, multiplication, and division), the second section covers all operations with fractions, the third section covers the simple and complex rule of three, and the fourth section covers real-world business applications (such as accounts, debts, profits and losses, and mix calculus). This fourth section also includes some additional in-depth explanations, puzzles and riddles, presumably aimed at students who have previously demonstrated an adequate grasp of the material presented in sections one through three. One of the puzzles, for example, states: Two fathers and two sons once hunted three rabbits, and each of them liked one rabbit. How can that be? Explanation: He had to be father, son and son's son, i.e. grandson. So two fathers and two sons each have a rabbit."Silobod hoped that including riddles in his book would make the material more engaging and readable for the general public." Tables of various uses are presented on the final 36 pages of the book that are not numbered. Knowledge of fundamental accounting and all accounts that emerge in real life, especially in commerce, were regarded as essential learning. The book's content, explanations and exercises serve their intended function by including the computation of mixes and other calculations that are part of everyday economic mathematics. Since the guidebook was the first to be written in Croatian language, this is the first appearance of any Croatian mathematical terms. Although it was clear that Šilobod had made use of words that were in dictionaries, this was clearly insufficient for his purposes. On the other hand, dictionaries lacked entries for the more advanced mathematical terminology since they were not commonly used. Because of this, Šilobod realised he was up against an enormous challenge: his handbook would be the first documented record of the preexisting mathematical language in the Kaikavian region. For this reason, he would have had to modify existing dictionary terms, especially that found in J. Belloszténcz's lexicon. Later, he would have had to create the terminology he need, drawing on the origins of existing words with overlapping meanings from the Kaikavian dialect of Croatian. Evidently, he had made up some names by adapting Latin terminology to Kaikavian use. === Fundamentum cantus Gregoriani seu chroralis (1760) === Fundamentum cantus Gregoriani seu chroralis is considered to be his second most famous publication. It delves into musical-Gregorian melodic principles. It is divided into six dialogues in which they discuss: the birth of music or singing (dialogue one); drawing, notes, and clefs (dialogue two); solmization (dialogue three); intervals (dialogue four); genuine and plagal tones (dialogue five); and intonation (dialogue six). For over a century, this handbook was used as a liber textus in the Zagreb seminary. As of today, this work is still studied in the Accademia Nazionale di Santa Cecilia in Rome and is considered a great theoretical guide to choral singing even after a century has passed. == Legacy == Because it was the first appearance of the systematically used Croatian mathematical vocabulary, Šilobod's Arithmatika Horvatzka has had enormous cultural value in Croatia. In addition, the manual played a crucial part in the education in Croatian people at a time when there was a pressing need to increase students' understanding of accounting's function in business and economics. Even a century after his death, the famous phrase that Arithmetika printed at the time had a huge resonance among the people "Ak ne buš učil matematiku išel buš k Šilobodu da te poduči!" ("If you didn't teach mathematics, you went to Šilobod to teach you!") – is still prevalent. His name has entered several other idioms, especially among Kajkavians. For example, it has long been customary to encourage anybody who calculates incorrectly to pray to Šilobod, and when someone wishes to brag about successfully counting or multiplying, they may playfully add, "It would be the same with Šilobod." Hrvatska pošta issued a commemorative stamp in 2008, celebrating 250 years since the publishing of Arithmatika Horvatzka. Irena Mišurac Zorica concluded in her analysis of the work that Mihalj Šilobod Bolšić demonstrated a very high pedagogical and methodical maturity, creating a standard for further development of mathematics curriculum and education. In addition to the stamp, a scientific symposium was organised in Sveta Nedjelja to honour Mihalj Šilobod Bolšić. It was the first scientific conference honouring the priest and teacher who is revered in Kajkavian culture. The Samobor Museum produced a book in 2009 that included the writings of the seven people who had taken part in the scientific meeting: Ph.D. Stjepan Razum, prof. Domagoj Sremi, prof. PhD Alojz Jembriha, MSc sc. Irena Mišurac Zorica, PhD Zvonimir Jakobovi, MSc sc. Marijane Bori, and PhD Katarina Koprek. These writings help provide light on the life and accomplishments of Mihalj, an important figure in the spread of mathematics education in the 18th century. To celebrate the event, the Museum issued a reproduction of the entire Arithmatika Horvatzka in separate volumes, and the collection includes a transcription of Šilobod's prologue to the work in its appendix. == Gallery == == See also == Šilobod Sveta Nedelja, Zagreb County Samobor Croatian literature Catholic Church in Croatia Timeline of Croatian history List of Catholic clergy scientists List of Catholic authors List of Catholic priests List of people from Croatia List of Croatian-language poets List of noble families of Croatia List of important publications in mathematics == References == == External links == Arithmetika Horvatszka full text available via archive.org. Fundamentum cantus gregoriani seu choralis full text available via archive.org. Cithara octochorda full text available via archive.org. Mijo Šilobod Bolšić on Hrvatske enciklopedije
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Wikipedia:Mikael Rørdam#0
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Mikael Rørdam (born 7 January 1959, Copenhagen) is a Danish mathematician, specializing in the theory of operator algebras and its applications. == Education and career == Rørdam graduated with master's degree from the University of Copenhagen in 1984. He received his PhD from the University of Pennsylvania with thesis The theory of unitary rank and regular approximation under the supervision of Richard Kadison. In the spring of 1988 Rørdam was a postdoc at the University of Toronto. At Odense University he was an Adjunkt (assistant professor) from 1988 to 1991 and a Lektor (associate professor) from 1991 to 1997. He was a Lektor at the University of Copenhagen from 1998 to 2002 and full professor from 2002 to 2007 at the University of Southern Denmark. Since 2008 he is a full professor at the University of Copenhagen. Rørdam was elected a member of the Royal Danish Academy of Sciences and Letters in 2004. He was an invited speaker of the International Congress of Mathematicians in 2006 in Madrid. He was a member of the board of the Mittag-Leffler Institute from 2010 to 2016. He was a plenary speaker at the International Workshop on Operator Theory and its Applications (IWOTA) in 2018 in Shanghai. == Selected publications == === Articles === Rørdam, Mikael (1991). "On the structure of simple C*-algebras tensored with a UHF-algebra". Journal of Functional Analysis. 100 (1): 1–17. doi:10.1016/0022-1236(91)90098-P. ISSN 0022-1236. Rørdam, Mikael (1992). "On the structure of simple C*-algebras tensored with a UHF-algebra, II". Journal of Functional Analysis. 107 (2): 255–269. doi:10.1016/0022-1236(92)90106-S. Blackadar, Bruce; Rørdam, Mikael (1992). "Extending states on preordered semigroups and the existence of quasitraces on C*-algebras". Journal of Algebra. 152 (1): 240–247. doi:10.1016/0021-8693(92)90098-7. ISSN 0021-8693. Rørdam, M. (1995). "Classification of Certain Infinite Simple C*-Algebras". Journal of Functional Analysis. 131 (2): 415–458. doi:10.1006/jfan.1995.1095. Kirchberg, Eberhard; Rørdam, Mikael (2000). "Non-simple purely infinite C*-algebras". American Journal of Mathematics. 122 (3): 637–666. doi:10.1353/ajm.2000.0021. S2CID 18204703. Kirchberg, Eberhard; Rørdam, Mikael (2002). "Infinite Non-simple C*-Algebras: Absorbing the Cuntz Algebra O∞". Advances in Mathematics. 167 (2): 195–264. doi:10.1006/aima.2001.2041. Rørdam, M. (2002). "Classification of Nuclear, Simple C*-algebras". Classification of Nuclear C*-Algebras. Entropy in Operator Algebras. Encyclopaedia of Mathematical Sciences. Vol. 126. pp. 1–145. doi:10.1007/978-3-662-04825-2_1. ISBN 978-3-642-07605-3. Rørdam, Mikael (2003). "A simple C*-algebra with a finite and an infinite projection". Acta Mathematica. 191 (1): 109–142. arXiv:math/0204339. doi:10.1007/BF02392697. S2CID 11212596. Rørdam, Mikael (2004). "The stable and the real rank of z {\displaystyle z} -absorbing C*-algebras". International Journal of Mathematics. 15 (10): 1065–1084. doi:10.1142/S0129167X04002661. 2004 Rørdam, Mikael; Winter, Wilhelm (2010). "The Jiang–Su algebra revisited". Journal für die Reine und Angewandte Mathematik. 2010 (642). arXiv:0801.2259. doi:10.1515/crelle.2010.039. S2CID 14639268. Rørdam, Mikael (2019). "Fixed-points in the cone of traces on a C*-algebra". Transactions of the American Mathematical Society. 371 (12): 8879–8906. doi:10.1090/tran/7797. ISSN 0002-9947. === Books === Rørdam, M.; Larsen, Flemming; Laustsen, N. J. (2000-07-20). An Introduction to K-Theory for C*-Algebras. ISBN 9780521789448. Rørdam, M.; Størmer, Erling (2001-11-20). Classification of Nuclear C*-Algebras. Entropy in Operator Algebras. ISBN 9783540423058. == References ==
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Wikipedia:Mike Steel (mathematician)#0
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Michael Anthony Steel (born May 1960) is a New Zealand mathematician and statistician, a Distinguished Professor of mathematics and statistics and the Director of the Biomathematics Research Centre at the University of Canterbury in Christchurch, New Zealand. He is known for his research on modelling and reconstructing evolutionary trees. == Biography == Steel studied at the University of Canterbury, earning a bachelor's degree in 1982, a masters in 1983, and a degree in journalism in 1985. He then moved to Massey University, where he received his Ph.D. in 1989, supervised by Michael D. Hendy and David Penny. He joined the Canterbury faculty in 1994. == Awards and honours == Steel won the Hamilton Memorial Prize of the Royal Society of New Zealand in 1994; this prize is given annually to a New Zealand mathematician for work done within five years of a Ph.D. In 1999 he won the research award of the New Zealand Mathematical Society "for his fundamental contributions to the mathematical understanding of phylogeny, demonstrating a capacity for hard creative work in combinatorics and statistics and an excellent understanding of the biological implications of his results." He became a fellow of the Royal Society of New Zealand in 2003. In 2018, Steel was elected as a Fellow of the International Society for Computational Biology, for his outstanding contributions to the fields of computational biology and bioinformatics. == Selected publications == Lockhart, Peter J., Michael A. Steel, Michael D. Hendy, and David Penny. "Recovering evolutionary trees under a more realistic model of sequence evolution." Molecular biology and evolution 11, no. 4 (1994): 605–612. Esser, Christian, Nahal Ahmadinejad, Christian Wiegand, Carmen Rotte, Federico Sebastiani, Gabriel Gelius-Dietrich, Katrin Henze et al. "A genome phylogeny for mitochondria among α-proteobacteria and a predominantly eubacterial ancestry of yeast nuclear genes." Molecular Biology and Evolution 21, no. 9 (2004): 1643–1660. Erdős, Péter L., Michael A. Steel, László A. Székely, and Tandy J. Warnow. "A few logs suffice to build (almost) all trees (I)." Random Structures & Algorithms 14, no. 2 (1999): 153–184. Erdös, Péter L., Michael A. Steel, LászlóA Székely, and Tandy J. Warnow. "A few logs suffice to build (almost) all trees: part II." Theoretical Computer Science 221, no. 1-2 (1999): 77–118. == References == == External links == Home page Citations on Google scholar
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Wikipedia:Mikhael Gromov (mathematician)#0
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Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". == Biography == Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father Leonid Gromov was Russian-Slavic and his mother Lea was of Jewish heritage. Both were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him. When Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a master's degree in 1965, a doctorate in 1969 and defended his postdoctoral thesis in 1973. His thesis advisor was Vladimir Rokhlin. Gromov married in 1967. In 1970, he was invited to give a presentation at the International Congress of Mathematicians in Nice, France. However, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings. Disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel. He changed his last name to that of his mother. He received a coded letter saying that, if he could get out of the Soviet Union, he could go to Stony Brook, where a position had been arranged for him. When the request was granted in 1974, he moved directly to New York and worked at Stony Brook. In 1981 he left Stony Brook University to join the faculty of University of Paris VI and in 1982 he became a permanent professor at the Institut des Hautes Études Scientifiques where he remains today. At the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences in New York since 1996. He adopted French citizenship in 1992. == Work == Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.[G00] He is also interested in mathematical biology, the structure of the brain and the thinking process, and the way scientific ideas evolve. Motivated by Nash and Kuiper's isometric embedding theorems and the results on immersions by Morris Hirsch and Stephen Smale, Gromov introduced the h-principle in various formulations. Modeled upon the special case of the Hirsch–Smale theory, he introduced and developed the general theory of microflexible sheaves, proving that they satisfy an h-principle on open manifolds.[G69] As a consequence (among other results) he was able to establish the existence of positively curved and negatively curved Riemannian metrics on any open manifold whatsoever. His result is in counterpoint to the well-known topological restrictions (such as the Cheeger–Gromoll soul theorem or Cartan–Hadamard theorem) on geodesically complete Riemannian manifolds of positive or negative curvature. After this initial work, he developed further h-principles partly in collaboration with Yakov Eliashberg, including work building upon Nash and Kuiper's theorem and the Nash–Moser implicit function theorem. There are many applications of his results, including topological conditions for the existence of exact Lagrangian immersions and similar objects in symplectic and contact geometry. His well-known book Partial Differential Relations collects most of his work on these problems.[G86] Later, he applied his methods to complex geometry, proving certain instances of the Oka principle on deformation of continuous maps to holomorphic maps.[G89] His work initiated a renewed study of the Oka–Grauert theory, which had been introduced in the 1950s. Gromov and Vitali Milman gave a formulation of the concentration of measure phenomena.[GM83] They defined a "Lévy family" as a sequence of normalized metric measure spaces in which any asymptotically nonvanishing sequence of sets can be metrically thickened to include almost every point. This closely mimics the phenomena of the law of large numbers, and in fact the law of large numbers can be put into the framework of Lévy families. Gromov and Milman developed the basic theory of Lévy families and identified a number of examples, most importantly coming from sequences of Riemannian manifolds in which the lower bound of the Ricci curvature or the first eigenvalue of the Laplace–Beltrami operator diverge to infinity. They also highlighted a feature of Lévy families in which any sequence of continuous functions must be asymptotically almost constant. These considerations have been taken further by other authors, such as Michel Talagrand. Since the seminal 1964 publication of James Eells and Joseph Sampson on harmonic maps, various rigidity phenomena had been deduced from the combination of an existence theorem for harmonic mappings together with a vanishing theorem asserting that (certain) harmonic mappings must be totally geodesic or holomorphic. Gromov had the insight that the extension of this program to the setting of mappings into metric spaces would imply new results on discrete groups, following Margulis superrigidity. Richard Schoen carried out the analytical work to extend the harmonic map theory to the metric space setting; this was subsequently done more systematically by Nicholas Korevaar and Schoen, establishing extensions of most of the standard Sobolev space theory. A sample application of Gromov and Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are arithmetic.[GS92] === Riemannian geometry === In 1978, Gromov introduced the notion of almost flat manifolds.[G78] The famous quarter-pinched sphere theorem in Riemannian geometry says that if a complete Riemannian manifold has sectional curvatures which are all sufficiently close to a given positive constant, then M must be finitely covered by a sphere. In contrast, it can be seen by scaling that every closed Riemannian manifold has Riemannian metrics whose sectional curvatures are arbitrarily close to zero. Gromov showed that if the scaling possibility is broken by only considering Riemannian manifolds of a fixed diameter, then a closed manifold admitting such a Riemannian metric, with sectional curvatures sufficiently close to zero, must be finitely covered by a nilmanifold. The proof works by replaying the proofs of the Bieberbach theorem and Margulis lemma. Gromov's proof was given a careful exposition by Peter Buser and Hermann Karcher. In 1979, Richard Schoen and Shing-Tung Yau showed that the class of smooth manifolds which admit Riemannian metrics of positive scalar curvature is topologically rich. In particular, they showed that this class is closed under the operation of connected sum and of surgery in codimension at least three. Their proof used elementary methods of partial differential equations, in particular to do with the Green's function. Gromov and Blaine Lawson gave another proof of Schoen and Yau's results, making use of elementary geometric constructions.[GL80b] They also showed how purely topological results such as Stephen Smale's h-cobordism theorem could then be applied to draw conclusions such as the fact that every closed and simply-connected smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature. They further introduced the new class of enlargeable manifolds, distinguished by a condition in homotopy theory.[GL80a] They showed that Riemannian metrics of positive scalar curvature cannot exist on such manifolds. A particular consequence is that the torus cannot support any Riemannian metric of positive scalar curvature, which had been a major conjecture previously resolved by Schoen and Yau in low dimensions. In 1981, Gromov identified topological restrictions, based upon Betti numbers, on manifolds which admit Riemannian metrics of nonnegative sectional curvature.[G81a] The principal idea of his work was to combine Karsten Grove and Katsuhiro Shiohama's Morse theory for the Riemannian distance function, with control of the distance function obtained from the Toponogov comparison theorem, together with the Bishop–Gromov inequality on volume of geodesic balls. This resulted in topologically controlled covers of the manifold by geodesic balls, to which spectral sequence arguments could be applied to control the topology of the underlying manifold. The topology of lower bounds on sectional curvature is still not fully understood, and Gromov's work remains as a primary result. As an application of Hodge theory, Peter Li and Yau were able to apply their gradient estimates to find similar Betti number estimates which are weaker than Gromov's but allow the manifold to have convex boundary. In Jeff Cheeger's fundamental compactness theory for Riemannian manifolds, a key step in constructing coordinates on the limiting space is an injectivity radius estimate for closed manifolds. Cheeger, Gromov, and Michael Taylor localized Cheeger's estimate, showing how to use Bishop−Gromov volume comparison to control the injectivity radius in absolute terms by curvature bounds and volumes of geodesic balls.[CGT82] Their estimate has been used in a number of places where the construction of coordinates is an important problem. A particularly well-known instance of this is to show that Grigori Perelman's "noncollapsing theorem" for Ricci flow, which controls volume, is sufficient to allow applications of Richard Hamilton's compactness theory. Cheeger, Gromov, and Taylor applied their injectivity radius estimate to prove Gaussian control of the heat kernel, although these estimates were later improved by Li and Yau as an application of their gradient estimates. Gromov made foundational contributions to systolic geometry. Systolic geometry studies the relationship between size invariants (such as volume or diameter) of a manifold M and its topologically non-trivial submanifolds (such as non-contractible curves). In his 1983 paper "Filling Riemannian manifolds"[G83] Gromov proved that every essential manifold M {\displaystyle M} with a Riemannian metric contains a closed non-contractible geodesic of length at most C ( n ) Vol ( M ) 1 / n {\displaystyle C(n)\operatorname {Vol} (M)^{1/n}} . === Gromov−Hausdorff convergence and geometric group theory === In 1981, Gromov introduced the Gromov–Hausdorff metric, which endows the set of all metric spaces with the structure of a metric space.[G81b] More generally, one can define the Gromov-Hausdorff distance between two metric spaces, relative to the choice of a point in each space. Although this does not give a metric on the space of all metric spaces, it is sufficient in order to define "Gromov-Hausdorff convergence" of a sequence of pointed metric spaces to a limit. Gromov formulated an important compactness theorem in this setting, giving a condition under which a sequence of pointed and "proper" metric spaces must have a subsequence which converges. This was later reformulated by Gromov and others into the more flexible notion of an ultralimit.[G93] Gromov's compactness theorem had a deep impact on the field of geometric group theory. He applied it to understand the asymptotic geometry of the word metric of a group of polynomial growth, by taking the limit of well-chosen rescalings of the metric. By tracking the limits of isometries of the word metric, he was able to show that the limiting metric space has unexpected continuities, and in particular that its isometry group is a Lie group.[G81b] As a consequence he was able to settle the Milnor-Wolf conjecture as posed in the 1960s, which asserts that any such group is virtually nilpotent. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces.[G93] Important developments on this topic were given by Bruce Kleiner, Bernhard Leeb, and Pierre Pansu, among others. Another consequence is Gromov's compactness theorem, stating that the set of compact Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric.[G81b] The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, a class of metric spaces studied in detail by Burago, Gromov and Perelman in 1992.[BGP92] Along with Eliyahu Rips, Gromov introduced the notion of hyperbolic groups.[G87] === Symplectic geometry === Gromov's theory of pseudoholomorphic curves is one of the foundations of the modern study of symplectic geometry.[G85] Although he was not the first to consider pseudo-holomorphic curves, he uncovered a "bubbling" phenomena paralleling Karen Uhlenbeck's earlier work on Yang–Mills connections, and Uhlenbeck and Jonathan Sack's work on harmonic maps. In the time since Sacks, Uhlenbeck, and Gromov's work, such bubbling phenomena has been found in a number of other geometric contexts. The corresponding compactness theorem encoding the bubbling allowed Gromov to arrive at a number of analytically deep conclusions on existence of pseudo-holomorphic curves. A particularly famous result of Gromov's, arrived at as a consequence of the existence theory and the monotonicity formula for minimal surfaces, is the "non-squeezing theorem," which provided a striking qualitative feature of symplectic geometry. Following ideas of Edward Witten, Gromov's work is also fundamental for Gromov-Witten theory, which is a widely studied topic reaching into string theory, algebraic geometry, and symplectic geometry. From a different perspective, Gromov's work was also inspirational for much of Andreas Floer's work. Yakov Eliashberg and Gromov developed some of the basic theory for symplectic notions of convexity.[EG91] They introduce various specific notions of convexity, all of which are concerned with the existence of one-parameter families of diffeomorphisms which contract the symplectic form. They show that convexity is an appropriate context for an h-principle to hold for the problem of constructing certain symplectomorphisms. They also introduced analogous notions in contact geometry; the existence of convex contact structures was later studied by Emmanuel Giroux. == Prizes and honors == === Prizes === Prize of the Mathematical Society of Moscow (1971) Oswald Veblen Prize in Geometry (AMS) (1981) Prix Elie Cartan de l'Academie des Sciences de Paris (1984) Prix de l'Union des Assurances de Paris (1989) Wolf Prize in Mathematics (1993) Leroy P. Steele Prize for Seminal Contribution to Research (AMS) (1997) Lobachevsky Medal (1997) Balzan Prize for Mathematics (1999) Kyoto Prize in Mathematical Sciences (2002) Nemmers Prize in Mathematics (2004) Bolyai Prize in 2005 Abel Prize in 2009 "for his revolutionary contributions to geometry" === Honors === Invited speaker to International Congress of Mathematicians: 1970 (Nice), 1978 (Helsinki), 1983 (Warsaw), 1986 (Berkeley) Foreign member of the National Academy of Sciences (1989), the American Academy of Arts and Sciences (1989), the Norwegian Academy of Science and Letters, the Royal Society (2011), and the National Academy of Sciences of Ukraine (2023). Member of the French Academy of Sciences (1997) Delivered the 2007 Pál Turán Memorial Lectures. == See also == == Publications == Books Major articles == Notes == == References == == External links == Media related to Mikhail Leonidovich Gromov at Wikimedia Commons Personal page at Institut des Hautes Études Scientifiques Personal page at NYU Mikhail Gromov at the Mathematics Genealogy Project Anatoly Vershik, "Gromov's Geometry"
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Wikipedia:Mikhail Gennadiyevich Dmitriyev#0
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Dmitriyev or Dmitriev (Russian: Дми́триев) is a common Russian surname that is derived from the male given name Dmitry and literally means Dmitry's. It may refer to: Aleksandr Dmitriyev (conductor) (born 1935), Russian conductor Alexey Dmitriev (born 1985), Russian ice hockey player Andrei Dmitriev (born 1979), Russian political dissident, publicist. Andrei Dmitriev (born 1956), Russian writer Artur Dmitriev (born 1968), Russian Olympic champion in figure skating Dmitri Dmitrijev (born 1982), Estonian politician Dmitriy Dmitriyev (born 1983), Russian professional football player Georgy Dmitriyev (1942–2016), a Russian composer Igor Dmitriev (1927–2008), Russian actor Ivan Dmitriev (1760–1837), Russian poet Ivan Dmitriev (canoeist) (born 1998), Russian canoeist Matvey Dmitriev-Mamonov (1790–1863), Russian poet, public and military figure Maxim Dmitriyev (1913–1990), Soviet army officer and Hero of the Soviet Union Mikhail Gennadiyevich Dmitriyev (born 1947), Soviet and Russian mathematician Nikolai Dmitriev (1898–1954), Soviet linguist Nikolai Dmitriev-Orenburgsky (1838–1898), Russian painter V. Dmitriev, soloist with the Alexandrov Ensemble Vladimir Dmitriyev (1900–1948), Soviet theater designer and painter Vladimir Karpovich Dmitriev (1868–1913), Russian economist, mathematician and statistician Yury Dmitriyev (1911–2006), Soviet theater and art critic Yury Dmitriyev (born 1946), Soviet cyclist Yury A. Dmitriev (born 1956), rights activist and Gulag historian
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Wikipedia:Mikhail Kadets#0
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Mikhail Iosiphovich Kadets (Russian: Михаил Иосифович Кадец, Ukrainian: Михайло Йосипович Кадець, sometimes transliterated as Kadec, 30 November 1923 – 7 March 2011) was a Soviet-born Jewish mathematician working in analysis and the theory of Banach spaces. == Life and work == Kadets was born in Kyiv. In 1943, he was drafted into the army. After demobilisation in 1946, he studied at Kharkov University, graduating in 1950. After several years in Makeevka he returned to Kharkov in 1957, where he spent the remainder of his life working at various institutes. He defended his PhD in 1955 (under the supervision of Boris Levin), and his doctoral dissertation in 1963. He was awarded the State Prize of Ukraine in 2005. After reading the Ukrainian translation of Banach's monograph Théorie des Opérations Linéaires, he became interested in the theory of Banach spaces. In 1966, Kadets solved in the affirmative the Banach–Fréchet problem, asking whether every two separable infinite-dimensional Banach spaces are homeomorphic. He developed the method of equivalent norms, which has found numerous applications. For example, he showed that every separable Banach space admits an equivalent Fréchet differentiable norm if and only if the dual space is separable. Together with Aleksander Pełczyński, he obtained important results on the topological structure of Lp spaces. Kadets also made several contributions to the theory of finite-dimensional normed spaces. Together with M. G. Snobar (1971), he showed that every n {\displaystyle n} -dimensional subspace of a Banach space is the image of a projection of norm at most n . {\displaystyle {\sqrt {n}}.} Together with V. I. Gurarii and V. I. Matsaev, he found the exact order of magnitude of the Banach–Mazur distance between the n {\displaystyle n} -dimensional spaces ℓ p n {\displaystyle \ell _{p}^{n}} and ℓ q n . {\displaystyle \ell _{q}^{n}.} In harmonic analysis, Kadets proved (1964) what is now called the Kadets 1 / 4 {\displaystyle 1/4} theorem, which states that, if | λ n − n | ≤ C < 1 / 4 {\displaystyle |\lambda _{n}-n|\leq C<1/4} for all integers n , {\displaystyle n,} then the sequence ( exp ( i λ n x ) ) n ∈ Z {\displaystyle (\exp(i\lambda _{n}x))_{n\in \mathbb {Z} }} is a Riesz basis in L 2 [ − π , π ] {\displaystyle L_{2}[-\pi ,\pi ]} Kadets was the founder of the Kharkov school of Banach spaces. Together with his son Vladimir Kadets, he authored two books about series in Banach spaces. == Notes == == External links == Kadets memorial website Mikhail Kadets at the Mathematics Genealogy Project
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Wikipedia:Mikhail Lobanov#0
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Mikhail Sergeyevich Lobanov (Russian: Михаил Сергеевич Лобанов; born 24 February 1984) is a Russian mathematician, left-wing politician, trade union activist, and former associate professor at Moscow State University. == Biography == Mikhail Lobanov was born into a family of engineers: his father worked as a builder, and his mother was a garment manufacturing technologist. He graduated from Moscow State University with a degree in discrete mathematics in 2006. He was a Candidate of Physical and Mathematical Sciences since 2009 (the topic of his PhD thesis "On the relationship between algebraic immunity and nonlinearity of Boolean functions"). Since his student years, he participated in grassroots activism and the trade union movement (among other things, he personally supported the strike of workers at a cement plant in the city of Mikhaylov, Ryazan Oblast). A number of intra-university campaigns aimed at protecting the interests of students and teachers of Moscow State University are associated with his name. In 2007, he organized a film club with his comrades at MSU, supported a group of dissatisfied students of the Faculty of Sociology (OD-Group). In 2009, he was among the founders and leaders of the "Initiative Group of students, graduate students and employees of Moscow State University", which arose from a successful campaign against the university administration's attempt to tighten the rules of admission to dormitories. Co-founder (in 2013) of the independent trade union "University Solidarity", member of the trade union committee at Moscow State University and the central council of the trade union association. During the 2012 Russian protests, he was one of the organizers of scientific and educational columns at opposition marches. He is also known as a participant in urban protection and environmental actions (including the protection of historical buildings and green areas from construction business) in the Ramenka district, where the main complex of MSU is located. He, like several other representatives of the scientific and pedagogical community, criticized and protested against the Law on Educational Activities. For his public activities, he was under pressure from the administration; in 2013 and 2018, they attempted to dismiss him from the university, but thanks to the solidarity campaign, he was reinstated as a teacher of the Faculty of Mechanics and Mathematics of MSU. Lobanov himself supported political prisoners, including fellow mathematicians Dmitry Bogatov and Azat Miftakhov, as well as members of the editorial board of the student magazine "DOXA". In the municipal elections in Moscow in 2022, he supported such candidates as Konstantin Konkov, Arseniy Lytar, Denis Zhilin, Vladlena Mokrousova, Nikita Kozlov. On 23 June 2023, the Russian government declared him a foreign agent, and the following July, he was fired from his post as docent of mechanics and mathematics at the Moscow State University. He announced that he was planning to leave Russia. == 2021 legislative election == Despite not being a member of the Communist Party of the Russian Federation, Lobanov received that party's endorsement, as well as support from Smart Voting, as a candidate for the State Duma of the Russian Federation in the 197th Kuntsevo single-mandate constituency. His main competitor was the candidate from the Moscow mayor's office, the TV presenter Yevgeny Popov, who was nominated by United Russia party and endorsed by Mayor Sergey Sobyanin. According to the results of voting in precincts, Lobanov comfortably led by a margin of more than 10,000 votes, however, after the publication of the results of remote electronic voting (DEG), he lost to Popov. According to official data, he received 72,805 votes (31.65%), the best indicator among all opposition candidates in Moscow districts. Mikhail Lobanov did not recognize the results of the elections, urging other opposition candidates to join the fight for annulment of the results. Lobanov took part in the CPRF rally on Pushkinskaya Square on the night of September 20. On September 23, in advance of a planned rally for September 25, police visited Lobanov as well as other protest leaders. == Arrest == Mikhail Lobanov was arrested in the end of December 2022 and held in administrative detention for two weeks. == Political views == Lobanov professes democratic socialism as his core ideology; among ideologically close non-Russian politicians, he names United States Senator Bernie Sanders and former leader of the British Labour Party Jeremy Corbyn. The key points of his electoral program in 2021, using the leitmotif of "fighting blatant economic and political inequality," were raising the minimum wage, increasing the progressive scale of taxation, increasing spending on education and science, canceling pension reform and lowering the retirement age, and protecting the environment. Lobanov's campaign was supported by independent trade unions (Confederation of Labour of Russia) and a number of left-wing organizations (Russian Socialist Movement, Marxist Tendency, Union of Democratic Socialists and others). Mikhail Lobanov opposes the Russian invasion of Ukraine. On 7 June 2022 he was arrested by the police for anti-war banner «No War». On 24 June 2022, he was detained for fifteen days by Russian police and fined 40 000 rubles for having made statements on social media opposing the 2022 Russian invasion of Ukraine. == Electoral history == == References == == External links == Budraitskis, Ilya. "Russia Has a New Socialist Movement: An interview with Mikhail Lobanov". Jacobin. Retrieved 3 October 2021. Sidorov, Dmitry. "A democratic socialist running for Russian parliament. What could go wrong?". openDemocracy. Retrieved 3 October 2021. Сайт кандидата в депутаты Государственной Думы Михаила Лобанова Математик и политик. Как доцент МГУ конкурирует на западе и юго-западе Москвы с «журналистом ВГТРК» // Новая газета
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Wikipedia:Mikhail Menshikov#0
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Mikhail Vasilyevich Menshikov (Russian: Михаи́л Васи́льевич Ме́ньшиков; born 17 January 1948) is a Russian-British mathematician with publications in areas ranging from probability to combinatorics. He currently holds the post of Professor in the University of Durham. Menshikov has made a substantial contribution to percolation theory and the theory of random walks. Menshikov was born in Moscow and went to school in Kharkov, Ukrainian SSR, Soviet Union. He studied at Moscow State University earning all his degrees up to Candidate of Sciences (1976) and Doctor of Sciences (1988). After briefly working in Zhukovsky, Menshikov worked in Moscow State University for many years. His career then took him to the University of São Paulo before becoming a professor at the University of Durham, where he currently lives. == External links == University webpage Mikhail Menshikov at the Mathematics Genealogy Project
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Wikipedia:Mikhail Ostrogradsky#0
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Mikhail Vasilyevich Ostrogradsky (Russian: Михаи́л Васи́льевич Острогра́дский; 24 September 1801 – 1 January 1862), also known as Mykhailo Vasyliovych Ostrohradskyi (Ukrainian: Миха́йло Васи́льович Острогра́дський), was a Russian Imperial mathematician, mechanician, and physicist of Zaporozhian Cossacks ancestry. Ostrogradsky was a student of Timofei Osipovsky and is considered to be a disciple of Leonhard Euler, who was known as one of the leading mathematicians of Imperial Russia. == Life == Ostrogradsky was born on 24 September 1801 in the village of Pashennaya (at the time in the Poltava Governorate, Russian Empire, today in Kremenchuk Raion, Poltava Oblast, Ukraine). From 1816 to 1820, he studied under Timofei Osipovsky (1765–1832) and graduated from the Imperial University of Kharkov. When Osipovsky was suspended on religious grounds in 1820, Ostrogradsky refused to be examined and he never received his Ph.D. degree. From 1822 to 1826, he studied at the Sorbonne and at the Collège de France in Paris, France. In 1828, he returned to the Russian Empire and settled in Saint Petersburg, where he was elected a member of the Academy of Sciences. He also became a professor of the main military engineering school of the Russian Empire. Ostrogradsky died in Poltava in 1862, aged 60. The Kremenchuk Mykhailo Ostrohradskyi National University in Kremenchuk, Poltava oblast, as well as Ostrogradsky street in Poltava, are named after him. == Work == He worked mainly in the mathematical fields of calculus of variations, integration of algebraic functions, number theory, algebra, geometry, probability theory and in the fields of applied mathematics, mathematical physics and classical mechanics. In the latter, his key contributions are in the motion of an elastic body and the development of methods for integration of the equations of dynamics and fluid power, following up on the works of Euler, Joseph Louis Lagrange, Siméon Denis Poisson and Augustin Louis Cauchy. In Russia, his work in these fields was continued by Nikolay Dmitrievich Brashman (1796–1866), August Yulevich Davidov (1823–1885) and especially by Nikolai Yegorovich Zhukovsky (1847–1921). Ostrogradsky did not appreciate the work on non-Euclidean geometry of Nikolai Lobachevsky from 1823, and he rejected it, when it was submitted for publication in the Saint Petersburg Academy of Sciences. Ostrogradsky was a teacher of the children of Emperor Nicholas I. === Divergence theorem === In 1826, Ostrogradsky gave the first general proof of the divergence theorem, which was discovered by Lagrange in 1762. This theorem may be expressed using Ostrogradsky's equation: ∭ V ( ∂ P ∂ x + ∂ Q ∂ y + ∂ R ∂ z ) d x d y d z = ∬ Σ ( P cos λ + Q cos μ + R cos ν ) d Σ {\displaystyle \iiint _{V}\left({\partial P \over \partial x}+{\partial Q \over \partial y}+{\partial R \over \partial z}\right)dx\,dy\,dz=\iint _{\Sigma }\left(P\cos \lambda +Q\cos \mu +R\cos \nu \right)d\Sigma } ; where P, Q, and R are differentiable functions of x, y, and z defined on the compact region V bounded by a smooth closed surface Σ; λ, μ, and ν are the angles that the outward normal to Σ makes with the positive x, y, and z axes respectively; and dΣ is the surface area element on Σ. === Ostrogradsky's integration method === His method for integrating rational functions is well known. First, we separate the rational part of the integral of a fractional rational function, the sum of the rational part (algebraic fraction) and the transcendental part (with the logarithm and the arctangent). Second, we determine the rational part without integrating it, and we assign a given integral in Ostrogradsky's form: ∫ R ( x ) P ( x ) d x = T ( x ) S ( x ) + ∫ X ( x ) Y ( x ) d x , {\displaystyle \int {R(x) \over P(x)}\,dx={T(x) \over S(x)}+\int {X(x) \over Y(x)}\,dx,} where P ( x ) , S ( x ) , Y ( x ) {\displaystyle P(x),\,S(x),\,Y(x)} are known polynomials of degrees p, s, y respectively; R ( x ) {\displaystyle R(x)} is a known polynomial of degree not greater than p − 1 {\displaystyle p-1} ; and T ( x ) , X ( x ) {\displaystyle T(x),\,X(x)} are unknown polynomials of degrees not greater than s − 1 {\displaystyle s-1} and y − 1 {\displaystyle y-1} respectively. Third, S ( x ) {\displaystyle S(x)} is the greatest common divisor of P ( x ) {\displaystyle P(x)} and P ′ ( x ) {\displaystyle P'(x)} . Fourth, the denominator of the remaining integral Y ( x ) {\displaystyle Y(x)} can be calculated from the equation P ( x ) = S ( x ) Y ( x ) {\displaystyle P(x)=S(x)\,Y(x)} . When we differentiate both sides of the equation above, we get: R ( x ) = T ′ ( x ) Y ( x ) − T ( x ) H ( x ) + X ( x ) S ( x ) {\displaystyle R(x)=T'(x)Y(x)-T(x)H(x)+X(x)S(x)} , where H ( x ) = Y ( x ) S ′ ( x ) S ( x ) {\displaystyle H(x)={Y(x)S'(x) \over S(x)}} . It can be shown that H ( x ) {\displaystyle H(x)} is polynomial. == See also == Gauss-Ostrogradsky theorem Green's theorem Ostrogradsky instability == Notes == == References == Ostrogradsky, M. (1845a), "De l'intégration des fractions rationnelles", Bulletin de la classe physico-mathématique de l'Académie Impériale des Sciences de Saint-Pétersbourg, 4: 145–167. Ostrogradsky, M. (1845b), "De l'intégration des fractions rationnelles (fin)", Bulletin de la classe physico-mathématique de l'Académie Impériale des Sciences de Saint-Pétersbourg, 4: 286–300. Woodard, R.P. (9 August 2015). "The Theorem of Ostrogradsky". arXiv:1506.02210 [hep-th]. == External links == O'Connor, John J.; Robertson, Edmund F., "Mikhail Ostrogradsky", MacTutor History of Mathematics Archive, University of St Andrews Woodard, R.P. (9 Aug 2015). "The Theorem of Ostrogradsky". arXiv:1506.02210 [hep-th].
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Wikipedia:Mikhail Samuilovich Livsic#0
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Mikhail Samuilovich Livsic (4 July 1917 – 30 March 2007) was a Ukrainian-born Israeli mathematician who specialized in functional analysis. == Biography == Born in Pokotilova, Uman district on 4 July 1917, Livsic moved to Odessa with his family when he was four years old. His father was an associate professor of mathematics who frequented leading Soviet mathematicians in Odessa like Nikolai Chebotaryov, Veniamin Fedorovich Kagan, Mark Krein and Samuil O. Shatunovsky. In 1931 Livsic graduated from school was a close friend of the mathematician Israel Markowitsch Glasman (1916–1968) at the school. Both were very interested in philosophy and therefore decided to study natural science and mathematics. After graduation, Livsic attended first a school for radio engineers and from 1933 the newly created Departmenr of Physics and Mathematics of the Odessa State University, with Mark Krein, Mark Naimark and Boris Yakovlevich Lewin as teachers. Krein was one of the major figures of the Soviet school of functional analysis and headed a very active school of functional analysis. Lewin was a prominent Soviet mathematician who made significant contributions to function theory. Study colleagues of Livsic were the fellow mathematicians A.P. Artyomenko, David Milman, Vitold Shmulyan, M.A. Rutman and V.A. Potapov. Originally, he worked on the moment problem, at that time the main research area of Krein, and with quasi-analytical functions. Soon after, he worked on the theory of operators, inspired by the work of Marshall Stone, John von Neumann, Abraham Plessner and Naum Ilyich Akhiezer . Following the evacuation of the Odessa State University during the Second World War, Livsic received in 1942 in Maikop his Ph.D. on the application of Hermitian operators theory to the generalised moment problem under supervision of Mark Krein. In 1945, Livsic passed his habilitation thesis on generalisations of von Neumann's extension theory that was evaluated by prominent mathematicians, namely Stefan Banach, Israel Gelfand, Mark Naimark and Plessner at the Steklov Institute of Mathematics. Livsic could not return directly to the Odessa State University following the dismantling of Krein's school under the accusation of promoting too many Jewish mathematicians. Livsic himself was considered not being "suited for representing the Ukrainian culture". He taught until 1957 at the Hydrometeorological Institute in Odessa, then at the Mining Institute of Kharkiv. In 1962, he joined the department of mathematical physics at the University of Kharkiv at the invitation of Naum Akhiezer. His study focused on the applications of functional analysis to quantum theory. He worked on the physical interpretation of non-self adjoining operators and he developed a theory of open systems which are physical systems which interact with the environment. These research is compiled in two monographs. After moving to Tbilissi with his family, he started working on a generalisation of the Cayley-Hamilton theorem. He moved to Israel in 1978 and settled in Beersheba. He became a professor at Ben-Gurion University of the Negev and started to work with Naftali Kravitsky on a theory of several commuting operators. == References ==
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Wikipedia:Mikhail Shlyomovich Birman#0
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Mikhail Shlyomovich Birman (Russian: Михаил Шлёмович Бирман; born 17 January 1928 in Leningrad; died 2 July 2009) was a Russian mathematician and university professor. His research included functional analysis, partial differential equations and mathematical physics. In particular, he did research in the fields of scattering theory, operators in Hilbert spaces and the spectral theory of differential operators. Together with Mikhail Zakharovich Solomyak he developed the theory of double operator integrals. == Life == Birman was born in 1928 in Leningrad (now St. Petersburg) to a father that was a professor and specialist in theoretical mechanics and a mother that was a school teacher. During the Second World War the family fled to Sverdlovsk (now Yekaterinburg). After the war the family went back to Leningrad. In 1950 Birman graduated from the Mathematical-Mechanical Faculty of Leningrad University with a diploma. Although Birman was one of the best students, he was denied a doctorate at Leningrad University because of his Jewish origins and the state antisemitic policy of the time. In 1947 he married Tatyana Petrovna Ilyina. Eventually in 1954 he did his doctorate at the Leningrad Mining University. In 1956 he joined the Department of Mathematical Physics at the Physics Faculty of Leningrad University. In 1962 he received a Sc.D. (Doctor of Science) for his work The Spectrum of Singular Boundary Value Problems. In 1974 he was an invited speaker at the International Congress of Mathematicians in Vancouver, but was prohibited to leave the Soviet Union. Birman was the supervisor for more than 20 students, including Boris Pavlov and Timo Weidl. He was also editor of the Russian mathematics journals "Algebra i Analiz" and "Funktsional'Nyi Analiz i Ego Prilozheniya". He remained in the Department of Mathematical Physics at Saint Petersburg University until his death in 2009. == Scientific work == Birman wrote three monographs, six books and over 160 scholarly papers. === Books (selection) === Birman, M. Sh.; Vilenkin, N. Ya.; Gorin, E. A. (1972). Functional analysis (in Russian). Nauka. Birman, M. Sh.; Solomyak, M. Z. (1980). Spectral theory of selfadjoint operators in Hilbert space (in Russian). Leningrad. Univ. Birman, M. Sh.; Solomyak, M. Z. Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory. Amer. Math. Soc. Transl. Ser. 2. Vol. 114. American Mathematical Society. == Literature == Buslaev, V. S.; Vershik, A. M. (2000). "Mikhail Shlyomovich Birman (on his 70th birthday)". Russian Mathematical Surveys. 55 (1): 201. doi:10.1070/RM2000v055n01ABEH000266. S2CID 250826373. Solomyak, M. Z.; Suslina, T. A.; Yafaew, D. R. (2012). "On the mathematical work of M. Sh. Birman" (PDF). St. Petersburg Mathematical Journal. 23 (1): 1–38. doi:10.1090/S1061-0022-2011-01184-7. S2CID 119595913. == References == == External links == Biography from the St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
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Wikipedia:Milan Kolibiar#0
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Milan Kolibiar (born 14 February 1922 in Detvianska Huta, died 9 July 1994 in Bratislava) was a Slovak mathematician. He worked mostly in lattice theory and universal algebra. == External links == Milan Kolibiar's entry at biographies of Slovak mathematicians on the website of Mathematical Institute of Slovak academy of science == References == T. Katriňák: Professor Milan Kolibiar šesťdesiatročný, Math. Slovaca 32 (2), 1982, 189–194 [1]
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Wikipedia:Millennium Mathematics Project#0
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The Millennium Mathematics Project (MMP) was set up within the University of Cambridge in England as a joint project between the Faculties of Mathematics and Education in 1999. The MMP aims to support maths education for pupils of all abilities from ages 5 to 19 and promote the development of mathematical skills and understanding, particularly through enrichment and extension activities beyond the school curriculum, and to enhance the mathematical understanding of the general public. The project was directed by John Barrow from 1999 until September 2020. == Programmes == The MMP includes a range of complementary programmes: The NRICH website publishes free mathematics education enrichment material for ages 5 to 19. NRICH material focuses on problem-solving, building core mathematical reasoning and strategic thinking skills. In the academic year 2004/5 the website attracted over 1.7 million site visits (more than 49 million hits). Plus Magazine is a free online maths magazine for age 15+ and the general public. In 2004/5, Plus attracted over 1.3 million website visits (more than 31 million hits). The website won the Webby award in 2001 for the best Science site on the Internet. The Motivate video-conferencing project links university mathematicians and scientists to primary and secondary schools in areas of the UK from Jersey and Belfast to Glasgow and inner-city London, with international links to Pakistan, South Africa, India and Singapore. The project has also developed a Hands On Maths Roadshow presenting creative methods of exploring mathematics, and in 2004 took on the running of Simon Singh's Enigma schools workshops, exploring maths through cryptography and codebreaking. Both are taken to primary and secondary schools and public venues such as shopping centres across the UK and Ireland. James Grime is the Enigma Project Officer and gives talks in schools and to the general public about the history and mathematics of code breaking - including the demonstration of a genuine World War II Enigma Machine. In November 2005, the MMP won the Queen's Anniversary Prize for Higher and Further Education. == References == == External links == MMP Official Website Physics & Mathematics
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Wikipedia:Miloš Zahradník#0
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Miloš Zahradník is a Czech mathematician who works on statistical mechanics in Charles University in Prague. He is also known for the book We Grow Linear Algebra (in Czech) that he wrote with Luboš Motl. == References ==
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Wikipedia:Milutin Milanković#0
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Milutin Milanković (sometimes anglicised as Milutin Milankovitch; Serbian Cyrillic: Милутин Миланковић, pronounced [milǔtin milǎːnkoʋitɕ]; 28 May 1879 – 12 December 1958) was a Serbian mathematician, astronomer, climatologist, geophysicist, civil engineer, university professor, popularizer of science and academic. Milanković gave two fundamental contributions to global science. The first contribution is the "Canon of the Earth's Insolation", which characterizes the climates of all the planets of the Solar System. The second contribution is the explanation of Earth's long-term climate changes caused by changes in the position of the Earth in comparison to the Sun, now known as Milankovitch cycles. This partly explained the ice ages occurring in the geological past of the Earth, as well as the climate changes on the Earth which can be expected in the future. He founded planetary climatology by calculating temperatures of the upper layers of the Earth's atmosphere as well as the temperature conditions on planets of the inner Solar System, Mercury, Venus, Mars, and the Moon, as well as the depth of the atmosphere of the outer planets. He demonstrated the interrelatedness of celestial mechanics and the Earth sciences and enabled a consistent transition from celestial mechanics to the Earth sciences and transformation of descriptive sciences into exact ones. A distinguished professor of applied mathematics and celestial mechanics at the University of Belgrade, Milanković was a director of the Belgrade Observatory, member of the Commission 7 for celestial mechanics of the International Astronomical Union and vice-president of Serbian Academy of Sciences and Arts. Beginning his career as a construction engineer, he retained an interest in construction throughout his life, and worked as a structural engineer and supervisor on a series of reinforced concrete constructions throughout Yugoslavia. He registered multiple patents related to this area. == Life == === Early life === Milutin Milanković was born in the village of Dalj, a settlement on the banks of the Danube in what was then part of Austro-Hungarian Empire. Milutin and his twin sister were the oldest of seven children raised in a Serb family. Their father was a merchant, landlord and a local politician who died when Milutin was seven. As a result, Milutin and his siblings were raised by his mother, grandmother, and an uncle. His three brothers died of tuberculosis at a young age. As his health was fickle, Milutin received his elementary education at home, learning from his father Milan, private teachers, and from numerous relatives and friends of the family, some of whom were renowned philosophers, inventors, and poets. He attended secondary school in nearby Osijek, completing it in 1896. In 1896, he moved to Vienna to study Civil Engineering at the TU Wien and graduated in 1902. In his third year of studies, Milanković found more free time for wider education. He paid his full attention to the monumental buildings of Vienna, thereby gradually understanding all the beauty of architecture. He also visited Viennese museums and galleries, after which he became an admirer of Raphael's Madonna del Prato. He showed great interest in the Vienna Opera, which he visited regularly. In addition, he devoted his attention to learning the French language by taking private lessons and attending summer French language courses in Geneva. In the Viennese ″Café Elisabethbrücke″, which was not fashionable but served only for reading, he spent an hour or two daily reading numerous newspapers and magazines. The professor of the science of the building bridges, Johann Brik, the top expert of Viennese Mechanics of that time, taught the most important subject of the fifth school year. In Brikʼs teaching, young Milankovitch found strong inspiration for later scientific work, as he describe it: ″Brikʼs lectures were very interesting to me. His mastering of mathematical analysis was excellent and would constantly apply it in his lectures. To a good mathematician it gives certain independence and freedom in solving problems.″ After graduating and spending his obligatory year in military service, Milankovitch borrowed money from an uncle to pay for additional schooling at TU Wien in engineering. He researched concrete and wrote a theoretical evaluation of it as a building material. At age twenty-five, his PhD thesis was entitled Contribution to the Theory of Pressure Curves (Beitrag zur Theorie der Druckkurven) and its implementation allowed assessment of pressure curves' shape and properties when continuous pressure is applied, which is very useful in bridge, cupola and abutment construction. His thesis was successfully defended on December 8, 1904; examination committee members were Johann Emanuel Brik, Josef Finger, Emanuel Czuber and Ludwig von Tetmajer. He then worked for an engineering firm in Vienna, using his knowledge to design structures. === Middle years === ==== Structural engineering ==== At the beginning of 1905, Milanković took up practical work and joined the firm of Adolf Baron Pittel Betonbau-Unternehmung in Vienna. He built dams, bridges, viaducts, aqueducts, and other structures in reinforced concrete throughout Austria-Hungary. So Milanković verified his theoretical knowledge and design tools on numerous reinforced concrete structures that he built during his engineering service in Vienna. Milankovitch participated with structural calculations and practical work in the construction of a total of ten hydroelectric power plants. Among them, the most notable is the one built in Sebeș (present-day Romania) in the Transylvania region. Milankovitch's specific task was to design a reinforced concrete aqueduct 1200 m long, which would bring water to the dam above the hydroelectric power plant turbines. After that, he was engaged in the construction of the viaduct in Hirschwang (Semmering) in 1906 and in Pitten near Vienna in 1907. He also participated in the construction of bridges in Krainburg, Banhilda and Bad Ischl, then the Belgrade and Košice sewage system, and Krupp's metal factory in Berndorf. The bridge in Krainburg (130 meters long and seven meters wide) was particularly beautiful, set on three pillars with four arches each, 30 meters apart. It was built of reinforced concrete, but was later destroyed during World War II. Milankovitch's great reputation was certainly contributed to by inventions of a new technology of building reinforced concrete ceiling, under the name "System Milankovitch - Kreutz", with which he became famous throughout the Austria-Hungary. He developed and patented the mentioned system of building ceilings with Theodor Kreutz. Compared to the existing ones, this ceiling stood out due to its simpler design, lower consumption of materials and the fact that it had integrated thermal and sound insulation, which made it more aesthetically elegant. The ″Milankovitch - Kreutz" construction system was protected by four patents for three inventions. In 1908, Milankovitch invented and patented new and useful Improvement in the Production of hollow reinforced-concrete slabs AT 42720 B. This patent is the equivalent of Milanovitch's US patent US 940041 A. In 1905, he published the first paper on armored concrete named Contribution to the theory of reinforced armored pillars. He published the second paper on the same subject based on new results in 1906. In 1908, he published a paper titled "On membranes of same opposition" in which he proves that the ideal shape for a water reservoir of equally thick walls is that of a drop of water. His six patents were officially recognized and his reputation in the profession was enormous, bringing abundant financial wealth. Milanković continued to practice civil engineering in Vienna until 1 October 1909 when he was received an offer University of Belgrade to work as an associate professor at the Department of Applied Mathematics that comprised three basic branches: rational, celestial mechanics, and theoretical physics. Though he continued to pursue his investigations of various problems pertaining to the application of reinforced concrete, he decided to concentrate on fundamental research. Although this was the turning point in Milankovitch's career, he still does not abandon his "passion for the entire range of construction work, from theoretical ideas to craftsmanship", and continues to engage in design and construction, in parallel with his scientific work. Thus, after arriving in the Kingdom of Serbia, Milanković accepted the design and construction of the first reinforced concrete bridges on the Niš - Knjaževac railway, in the Timok Valley through the Nisevac Gorg, at the request of his friend and collegemate from TU Wien and civil engineer Petar Putnik. This undertaking was unique in that, at the suggestion of engineer Putnik, the type construction of a reinforced concrete bridge was applied for the first time in Serbia. The project of the 30-meter-span bridge, which rests on rocky shores, was done by Milanković with the aim of easier and faster construction of the railway on the route of which the construction of 19 bridges was planned. Thanks to this simple approach, the construction of all 19 bridges is solved with one project. That is precisely why Putnik's construction company won this job at the public procurement in 1912, when construction began. Milanković participated in the construction of the first of the nineteen bridges, which was located near Svrljig, where he fully immersed himself in the work and took care of how "the concrete is mixed, distributed over the formwork and compacted". ==== Planet's insolation ==== While studying the works of the contemporaneous climatologist Julius von Hann, Milanković noticed a significant issue, which became one of the major objects of his scientific research: a mystery ice age. The idea of possible astronomically-related climate changes was first considered by astronomers (John Herschel, 1792–1871) and geologists (Louis Agassiz, 1807–1873). Milanković studied the works of Joseph Adhemar whose pioneering theory on the astronomical origins of ice ages were formally rejected by his contemporaries and the amateur scientist James Croll (1821-1890), whose work was effectively forgotten after initial acceptance by contemporaries such as Charles Darwin. Despite having valuable data on the distribution of ice on the Alps across various glaciations, climatologists and geologists had not established the root causes of these cycles. Milanković decided to attempt correctly to calculate the magnitude of such influences. Milanković sought the solution of these complex problems in the field of spherical geometry, celestial mechanics, and theoretical physics. His first papers were in the field of celestial mechanics, Properties of motion in a specialized three-body problem (1910), On general integrals of the n-body problem (1911), On kinematic symmetry and its application to qualitative solutions of dynamics problem (1912), but from 1912 Milankovitch began to be interested in cosmic climatology or solar climate. He began working on it in 1912, after he had realized that "most of meteorology is nothing but a collection of innumerable empirical findings, mainly numerical data, with traces of physics used to explain some of them... Mathematics was even less applied, nothing more than elementary calculus... Advanced mathematics had no role in that science..." His first work described the present climate on Earth and how the Sun's rays determine the temperature on Earth's surface after passing through the atmosphere. He published the first paper on the subject entitled "Contribution to the mathematical theory of climate" in Belgrade in April 1912. His next paper was entitled "Distribution of the sun radiation on the earth's surface" and was published on June 1913. In December of that year, this paper was read by Wilhelm Wien, and was soon published in the German journal Annalen der Physik. He correctly calculated the intensity of insolation and developed a mathematical theory describing Earth's climate zones. His aim was an integral, mathematically accurate theory which connects thermal regimes of the planets to their movement around the Sun. He wrote: "...such a theory would enable us to go beyond the range of direct observations, not only in space, but also in time... It would allow reconstruction of the Earth's climate, and also its predictions, as well as give us the first reliable data about the climate conditions on other planets." He published a paper entitled "The problem of the astronomical theory of ice ages" in 1914. On 14 June 1914, Milanković married Kristina Topuzović and went on his honeymoon to his native village of Dalj in Austro-Hungary, where he heard about the Sarajevo assassination which was the cause of the July crisis. Meanwhile, the Austro-Hungarian Empire began massing troops in the Balkans near the border with the Kingdom of Serbia in preparation for an invasion. At that time, he was arrested as a citizen of Serbia, and at first he spent six weeks under house arrest, but was eventually imprisoned and later sent to a prisoner-of-war camp (K. u. K. Interienirungslager) in Nezsider, Hungary (today Neusiedl am See, Austria). He described his first day in prison, where he waited to be taken to the Esseg fortress as a prisoner of war, in the following words: ... Sat on the bed, I looked around and started synchronizing with my new social position .... In the suitcase I had my printed works and my notes on the cosmic problem, there was clean paper too and I started writing. It was far past midnight when I stopped. I looked around the room, wondering where I was. It felt like I was in a roadhouse on my trip through the Universe. His wife went to Vienna to talk to Emanuel Czuber, who was his mentor and a good friend. Through his social connections, Professor Czuber arranged Milanković's release from prison and permission to spend his captivity in Budapest with the right to work. After six months spent in the prison camp, Milanković was released on December 24, 1914. Immediately after arriving in Budapest, Milanković met the Director of the Library of the Hungarian Academy of Science, Kálmán Szily who, as a mathematician, eagerly accepted Milanković and enabled him to work undisturbed in the Academy's library and the Central Meteorological Institute. Milanković spent four years in Budapest, almost the entire war. His was only restricted not to leave town and to report to police office once a week. In 1915, Milanković's son Vasilije-Vasko was born in Budapest. He used mathematical methods to study the current climate of inner planets of the solar system. He shared the general opinion at the time that Mars and Venus contained water on their surface. This was logical thinking, since Earth has water, Mars has polar cap, and Venus has white clouds that associate on the water vapor. This significantly influenced his calculations for the basic thermal climate characteristics of these two planets. In 1916 he published a paper entitled "Investigation of the climate of the planet Mars". He knew the size of Mars and its distance from the Sun, but also that it has a similar rotation speed and axis orientation as Earth. Milanković calculated that the average temperature in the lower layers the atmosphere on Mars is −45 °C (−49 °F) and the average surface temperature is −17 °C (1 °F). Also, he concluded that: "This large temperature difference between the ground and lower layers of the atmosphere is not unexpected. Great transparency for solar radiation makes that is the climate of Mars very similar to altitudes climate of our Earth." In any case, Milanković's work suggested that Mars has a harsh climate, and calmed mounting enthusiasm concerning the prospect of discovering the presence of liquid water on the surface of Mars. He discussed the possibility of life on Mars and was skeptical that it could have complex life forms as well and vegetation. In addition to considering Mars, he dealt with the climatic conditions prevailing on Venus and Mercury. According to his own words, Milankovitch did not know the speed of rotation of Venus, the orientation of the axis, as well as the thickness and composition of the atmosphere. He was awere with Schiaparelli's suggestion that Venus has a slow rotation period equal to the duration of its orbits around the Sun, but he was skeptical because he thought that Venus would lose its atmosphere during a long-term day due to the effects of Solar Radiation. At the last, he accepted spectroscopy observations from that time that suggested a shorter rotation period similar to Earth's. So he considered a greenhouse effect (water vapor) on Venus calculated the temperature in the outer limit of the atmosphere +25 °C (77 °F), the upper layer +54 °C (129 °F), the middle layer +70 °C (158 °F) and the lower layer of the atmosphere +80 °C (176 °F) as well as a ground temperature of +97 °C (207 °F). In his literary work Through Distant Worlds and Times, he described of Venus in the following words: Here we are in the temple of Isis and Osiris, more magnificent than Schinkel himself imagined. From its huge dome, covered with a gently mother-of-pearl mosaic, a white mysterious light spills over the interior of this home. That dome, that's the sky of Venus. The Sun is never visible on it, only the Sun's silvery glow. Not a single star twinkles in this sky; no messenger of the universe reaches this sanctuary...What is this? A storm is raging in my head, blood vessels are beating like sledgehammers, I'm out of breath. You are pale, dear miss, your legs are wobbly - you have completely fainted... Half unconscious, I carry you, in my arms, to our Earth... He also discussed the possibility of life on Venus. He thought that the mystery of this planet lies in the answer to the question about its axis, the speed of rotation or how long a day lasts on Venus. His calculations of the surface temperature conditions on the neighboring Moon are particularly significant. Milankovitch knew that the moon rotates on its axis in 27.32 days, so lunar daytime on one side of the moon last about 13.5 Earth days. Milankovitch calculated that the temperature after a long moon night, in the early morning on the Moon, or before the rise of the Sun over horizon, was −53.8 °C (−64.8 °F). At noon, it rises on +97 °C (207 °F), only to reach its maximum value one Earth day later +100.5 °C (212.9 °F). At sunset, the temperature drops −8.8 °C (16.2 °F). According to Milankovitch, a sudden cooling occurs during the night. From 1912 to 1917, he wrote and published seven papers on mathematical theories of climate both on the Earth and on the other planets. He formulated a precise, numerical climatological model with the capacity for reconstruction of the past and prediction of the future, and established the astronomical theory of climate as a generalized mathematical theory of insolation. When these most important problems of the theory were solved, and a firm foundation for further work built, Milanković finished the manuscript under the title Mathematische Grundlagen der kosmischen Strahlungslehre that he sent to his Professor Czuber in Vienna at the summer of 1917. Czuber contacted a publishing house in Leipzig, but since there was a shortage of paper in early 1918, the printing of the book was cancelled. In the fall of 1917, Milankovitch got a job in a construction bureau in Budapest, where he worked on detailed projects of reinforced concrete constructions of a new six-story tuberculosis sanatorium built in the High Tatras, as well as on other important projects. After the Great War, the Austro-Hungarian Empire disintegrated and new states such as the Kingdom of Serbs, Croats and Slovenes, Republic of Austria, Kingdom of Hungary and Czechoslovak Republic were formed on its remains. Milanković returned from Budapest to Belgrade with his family after a three-day trip by steamboat ″Gizella″ on 19 March 1919. He continued his professorial career, becoming a full professor at the University of Belgrade. Milanković then, with the help of Professor Ivan Đaja, prepared the French text of this work and it was published under the title "Théorie mathématique des phénomènes thermiques produits par la radiation solaire" (Mathematical Theory of Heat Phenomena Produced by Solar Radiation) in 1920 in the edition of the Yugoslav Academy of Sciences and Arts (today HAZU) from Zagreb and the Gauthier-Villars in Paris. That same year, he was elected a corresponding member of the Serbian Royal Academy of Sciences in Belgrade and the Yugoslav Academy of Science and Arts in Zagreb. ==== Orbital variations and ice ages ==== As a consequence of the Russian Civil War, with the arrival of Russian scientists – emigrants, the personnel base of the Faculty of Philosophy at the University of Belgrade was expanded. Thus, from 1920 Anton Bilimovich (1879–1970), a distinguished scientist, who came from Odessa, took over the lectures on rational mechanics, and from 1925 the lectures on theoretical physics and vector theory were taken over by the newly elected assistant professor Wenceslas S. Jardetzky (1896–1962). Between the two wars, Milankovitch taught celestial mechanics and occasionally the theory of relativity, and after the Second World War until 1955, when he retired, he taught celestial mechanics and the history of astronomy. Milankovitch's works on astronomical explanations of ice ages, especially his curve of insolation for the past 130,000 years, received support from the climatologist Wladimir Köppen and from the geophysicist Alfred Wegener. Köppen noted the usefulness of Milanković's theory for paleoclimatological researchers. Milanković received a letter on 22 September 1922 from Köppen, who asked him to expand his studies from 130,000 years to 600,000 years. He accepted Köppen's suggestion that cool summers were a crucial factor for glaciation and agreed to calculate the secular progress of insolation of the Earth at the outer limit of the atmosphere for the past 650,000 years for parallels of 55°, 60° and 65° northern latitude, where the most important events of the Quaternary glaciations occurred. After developing the mathematical machinery enabling him to calculate the insolation in any given geographical latitude and for any annual season, Milanković was ready to start the realization of the mathematical description of climate of the Earth in the past. Milanković spent 100 days doing the calculations and prepared a graph of solar radiation changes at geographical latitudes of 55°, 60° and 65° north for the past 650,000 years. Milankovitch, in his early works, used the astronomical values of Stockwell-Pilgram. These curves showed the variations in insolation which correlated with four Alpine glaciations known at the time (Gunz, Mindel, Riss and Würm glaciation). Köppen felt that Milanković's theoretical approach to solar energy was a logical approach to the problem. His solar curve was introduced in a work entitled "Climates of the geological past", published by Wladimir Köppen and his son-in-law Alfred Wegener in 1924. In September of that year, he attended the lecture given by Alfred Wegener at Congress of German Naturalist in Innsbruck. That same year, he was elected a full member of the Serbian Royal Academy of Sciences. The Meteorological service of the Kingdom of Yugoslavia became a member of International Meteorological Organization – IMO (founded in Brussels in 1853 and in Vienna in 1873) as a predecessor of present World Meteorological Organization, WMO. Milanković served as a representative of the Kingdom of Yugoslavia there for many years. Milanković put the Sun at the center of his theory, as the only source of heat and light in the Solar System. He considered three cyclical movements of the Earth: eccentricity, axial tilt, and precession. Each cycle works on a different time-scale and each affects the amount of solar energy received by the planets. Such changes in the geometry of an orbit lead to the changes in the insolation – the quantity of heat received by any spot at the surface of a planet. These orbital variations, which are influenced by gravity of the Moon, Sun, Jupiter, and Saturn, form the basis of the Milankovitch cycle. Between 1925 and 1928 Milanković wrote the popular-science book Through Distant Worlds and Times in the form of letters to an anonymous woman. The work discusses the history of astronomy, climatology and science via a series of imaginary visits to various points in space and time by the author and his unnamed companion, encompassing the formation of the Earth, past civilizations, famous ancient and renaissance thinkers and their achievements, and the work of his contemporaries, Köppen and Wegener. In the "letters", Milanković expanded on some of his own theories on astronomy and climatology, and described the complicated problems of celestial mechanics in a simplified manner. Köppen proposed to Milanković on 14 December 1926 to extend his calculations to a million years and to send his results to Barthel Eberl, a geologist studying the Danube basin, as Eberl's research had unearthed some evidence of previous Ice Ages from before over 650,000 years ago. Eberl published all this in Augsburg in 1930 together with Milanković's curves. In 1927, Milanković received an offer from Köppen to collaborate on the Handbook of Climatology (Handbuch der Klimatologie), which was edited by Köppen himself. That same year, Milanković asked his colleague and friend, Vojislav Mišković, to collaborate in the work and calculate astronomical values based on the Le Verrier method. Mišković was a well-established astronomer from the Nice Observatory, who became the head of the Astronomical Observatory of the University of Belgrade and a professor of Theoretical and Practical Astronomy. After almost three years, Mišković and his staff completed the calculation of astronomical values based on the Le Verrier method and using the masses of the planets as known at that time. Milanković used these values in his later works. Subsequently, Milanković wrote the introductory portion of Mathematical science of climate and astronomical theory of the variations of the climate (Mathematische Klimalehre und Astronomische Theorie der Klimaschwankungen), published by Köppen (Handbook of Climatology; Handbuch der Klimalogie Band 1) in 1930 in German and translated into Russian in 1939. In 1935 Milanković published the book Celestial Mechanics. This textbook used vector calculus systematically to solve problems of celestial mechanics. His original contribution to celestial mechanics is called Milanković's system of vector elements of planetary orbits. He reduced six Lagrangean-Laplacian elliptical elements to two vectors determining the mechanics of planetary movements. The first specifies the planet's orbital plane, the sense of revolution of the planet, and the orbital ellipse parameter; the second specifies the axis of the orbit in its plane and the orbital eccentricity. By applying those vectors he significantly simplified the calculation and directly obtained all the formulas of the classical theory of secular perturbations. Milanković, in a simple but original manner, first deduced Newton's law of gravitation from Kepler's laws. Then Milanković treated the two-body and the many-body problems of celestial mechanics. He applied vector calculus from quantum mechanics to celestial mechanics. Meanwhile, in 1936 he attended the Third symposium of the International Union for Quaternary Research (INQUA) in Vienna. In the period from 1935 to 1938 Milanković calculated that ice cover depended on changes in insolation. He succeeded in defining the mathematical relationship between summer insolation and the altitude of the snow line. In this way he defined the increase of snow which would occur as a consequence of any given change in summer insolation. He published his results in the study "New Results of the Astronomic Theory of Climate Changes" in 1938. Geologists received a graph presenting bordering altitudes of ice cover for any period of time during the last 600,000 years. . ==== Polar wandering ==== Conversations with Wegener, the father of continental drift theory, got Milanković interested in the interior of the Earth and the movement of the poles, so he told his friend that he would investigate polar wandering. In November 1929, Milanković received an invitation from Professor Beno Gutenberg of Darmstadt to collaborate on a ten volume handbook on geophysics and to publish his views on the problem of the secular variations of the Earth's rotational poles. In the meantime, Wegener died in November 1930 during his fourth expedition to Greenland. Milanković became convinced that the continents 'float' on a somewhat fluid subsurface and that the positions of the continents with respect to the axis of rotation affect the centrifugal force of the rotation and can throw the axis off balance and force it to move. Wegener's tragedy additionally motivated Milankovich to persevere in solving the problem of polar wandering. Milanković began working on the problem of the shape of the Earth and the position of the Earth's poles in 1932 and 1933 at the suggestion of Alfred Wegener. The Earth as a whole he considered as a fluid body, which in the case of short-duration forces behaves as a solid body, but under an influence behaves as an elastic body. Using vector analysis he made a mathematical model of the Earth to create a theory of secular motion of the terrestrial poles. He derived the equation of secular trajectory of a terrestrial pole and also the equation of pole motion along this trajectory. His equation, also known as Milankovitch's theorem, is v = c grad Ω. He drew a map of the path of the poles over the past 300 million years and stated that changes happen in the interval of 5 million years (minimum) to 30 million years (maximum). He found that the secular pole trajectory depends only on the configuration of the terrestrial outer shell and the instantaneous pole position on it, more precisely on geometry of the Earth mass. On this basis he could calculate the secular pole trajectory. Also, based on Milanković's model, the continental blocks sink into their underlying "fluidal" base, and slide around, 'aiming to achieve' isostatic equilibrium. In his conclusion about this problem, he wrote: For an extraterrestrial observer, the displacement of the pole takes place in such a way that the ... Earth's axis maintains its orientation in space, but the Earth's crust is displaced on its substratum. Milankovitch published his paper on the subject entitled "Numerical trajectory of secular changes of pole’s rotation" in Belgrade in 1932. Milanković wrote four sections of Gutenberg's "Handbook of Geophysics" (Handbuch der Geophysik): Stellung und Bewegung der Erde im Weltall, No I,2 - 1931, (The Earth's Position and Movement in Space) Drehbewegungen der Erde, No. I,6 - 1933, (Rotational Movement of the Earth) Säkulare Polverlagerungen, No. I,7 - 1933, (Secular shift of the Poles) Astronomiche Mittel zur Erforschung der erdgeschichtlichen Klimate, No. IX, 7 - 1938, (Astronomic Means for Climate Study during the Earth's history) The lecture on the apparent shift of poles was held at a congress of Balkan mathematicians in Athens in 1934. That same year, held a lecture dedicated to the work of Alfred Wegener under the title Moving of the Earth's Poles – A Memory to Alfred Wegener in Belgrade, which was also published under the same name. Wegener's untimely death ended the collaboration between them on this subject. Milankovitch's work on this topic was criticized from the beginning. Milankovitch's trajectory of polar wandering was a topic of discussion after World War II. In the 1950s, paleomagnetic data showed different results than Milankovitch's theoretical numerical values for polar wandering trajectory. === Later life === To collect his scientific work on the theory of solar radiation that was scattered in many books and papers, Milanković began his life's work in 1939. This tome was entitled "Canon of Insolation of the Earth and Its Application to the Problem of the Ice Ages", which covered his nearly three decades of research, including a large number of formulas, calculations and schemes, but also summarized universal laws through which it was possible to explain cyclical climate change – his namesake Milankovitch cycles. Milanković spent two years arranging and writing the "Canon". The manuscript was submitted to print on 2 April 1941 – four days before the attack of Nazi Germany and its allies on the Kingdom of Yugoslavia. In the bombing of Belgrade on 6 April 1941, the printing house where his work was being printed was destroyed; however, almost all of the printed sheet paper remained undamaged in the printing warehouse. After the successful occupation of Serbia on 15 May 1941, two German officers and geology students came to Milanković in his house and brought greetings from Professor Wolfgang Soergel of Freiburg. Milanković gave them the only complete printed copy of the "Canon" to send to Soergel, to make certain that his work would be preserved. Milanković did not take part in the work of the university during the occupation, and after the war he was reinstated as professor. The "Canon" was issued in 1941 by the Royal Serbian Academy, 626 pages in quarto, and was printed in German as "Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem". The titles of the six parts of the book are: "The planets' motion around the Sun and their mutual perturbations" "The rotation of the Earth" "Secular wanderings of the rotational poles of the Earth" "The Earth's insolation and its secular changes" "The connection between insolation and the temperature of the Earth and its atmosphere. The mathematical climate of the Earth" "The ice age, its mechanism, structure and chronology". During the German occupation of Serbia from 1941 to 1944, Milanković withdrew from public life and decided to write a "history of his life and work" going beyond scientific matters, including his personal life and the love of his father who died in his youth. His autobiography would be published after the war, entitled "Recollection, Experiences and Vision" in Belgrade in 1952. ==== Tower of Babel ==== After the war, in 1947, Milanković's only son emigrated from the new communist Yugoslavia via Paris, London and Egypt to Australia. Milanković would never see his son again and the only way of correspondence between them would be through letters. Milanković was vice president of the Serbian Academy of Sciences (1948–1958). In 1948, the General Assembly of the International Astronomical Union was held in Zürich. Milankovich is listed as a member of Commission 7 for Celestial Mechanics, and “V. Mishkovitch” as member of Commission 19 for Latitude Variation and Commission 20 for Minor Planets. For a short period, he was the head of the Belgrade Observatory (1948 - 1951). At that time, the Cold War between nuclear powers began. In 1953, he was at the Congress of the International Union for Quaternary Research (INQUA) held in Rome where he was interrupted during his speech by numerous opponents since radiocarbon dating at that time showed different results than his theory. In the same year, he became a member of the Italian Institute of Paleontology. In November 1954, fifty years after receiving his original diploma, he received the Golden Doctor's diploma from the Technical University of Vienna. In 1955, he was also elected as a corresponding member to the Academy of Naturalists "Leopoldina" in Halle, Saxony-Anhalt, East Germany. At the same time, Milankovitch began publishing numerous books and textbooks on the history of science, including Isaac Newton and Newton's Principia (1946), The founders of the natural science Pythagoras – Democritus – Aristotle – Archimedes (1947), History of astronomy – from its beginnings up to 1727 (1948), Through empire of science – images from the lives of great scientists (1950), Twenty-two centuries of Chemistry (1953), and Technology in Ancient times (1955). In 1955, Milankovitch retired from the position of professor of celestial mechanics and the history of astronomy at the University of Belgrade. In the same year, he published his last work, which is not from the natural sciences, but from his original profession of structural engineering. The paper was titled The Tower of Babel of modern technology. Milankovitch in this work calculated the highest building possible on our Earth. He was inspired by work of Pieter Bruegel the Elder's Tower of Babel (older version in Vienna). The building would have a base radius of 112.84 km and a height of 21646 m. Since the building penetrates the Earth 1.4 km, it would have a height of 20.25 km above the Earth's surface. At the very top, there would be a wide platform for a meteorological and astronomical station. In September 1957, Milutin suffered a stroke and died in Belgrade in 1958. He is buried in his family cemetery in Dalj. == On the speed of light == Milanković authored two papers on relativity. He wrote his first paper "On the theory of Michelson's experiment" in 1924. He was doing research in this theory from 1912. His papers on this matter were on special relativity and both are on the Michelson experiment (now known as the Michelson–Morley experiment) which produced strong evidence against aether theory. In the light of the Michelson experiment he discussed on the validity of the second postulate of special theory of relativity, that the speed of light is the same in every reference frame. == Revised Julian calendar == Milanković proposed a revised Julian calendar in 1923. It made centennial years leap years if division by 900 left a remainder of 200 or 600, unlike the Gregorian rule which required that division by 400 left no remainder. (In both systems, the years 2000 and 2400 are leap years.) In May 1923 a congress of some Eastern Orthodox churches adopted the calendar; however, only the removal of 1–13 October 1923 and the revised leap year algorithm were adopted by a number of Eastern Orthodox churches. The dates of Easter and related holy days are still computed using the Julian calendar. At the time of Milanković's proposal, it was suspected the period of rotation of Earth might not be constant, but it was not until the development of quartz and atomic clocks beginning in the 1930s that this could be proven and quantified. The variation in the period of rotation of Earth is the chief cause of long-term inaccuracy in both the Gregorian and Revised Julian calendars. == Awards and honors == On June 25, 1923 he was conferred the Saint Sava Order, 3rd degree. On 1925, he was awarded Tunisian Nichan Iftikhar Order, 3rd degree. On 1929, he was awarded, at the proposal of the Ministry of Finances the White Eagle Order, 5th degree. On 1935, he was awarded Greek decoration - Phoenix Battalion Commander's Cross. On December 20, 1938 he was awarded the Royal Order of the Yugoslav Crown 3rd degree. In 1965, the Academy of Sciences of the Soviet Union named an impact crater on the far side of the Moon as Milankovic, which was later confirmed at the 14th IAU General Assembly in 1970. His name is also given to a crater on Mars at the 15th IAU General Assembly in 1973. Since 1993 the Milutin Milankovic Medal has been awarded by the European Geophysical Society (called the EGU since 2003) for contributions in the area of long-term climate and modeling. A main belt asteroid discovered in 1936 has also been dubbed 1605 Milankovitch. At NASA, in their edition of "On the Shoulders of Giants", Milanković has been ranked among the top fifteen minds of all time in the field of earth sciences. == Personal life == Milankovitch was a materialistic monist and determinist, according to whom nature is "unique", "boundless, eternal mother of life" and in "the boundless universe, which has no beginning or end in space and time, the same natural laws rule" as on Earth. As a materialist, Milankovitch stood on the position that the universe is eternal, uncreated and indestructible and that the question of its beginning has no meaning. Milanković was a great admirer of Nikola Tesla. On behalf of five academics, Milutin Milanković wrote a recommendation that Nikola Tesla be elected a full member of the Royal Serbian Academy, which was done at a ceremonial meeting on March 7, 1937. == Selected works == Théorie mathématique des phénomènes thermiques produits par la radiation solaire, XVI, 338 S. – Paris: Gauthier-Villars, 1920 Kroz vasionu i vekove. Novi Sad: Matica srpska, 1928 Reforma julijanskog kalendara. Srpska Kr. Akad. Pos. Izda’na 47: 52 S., Beograd: Sv. Sava, 1923 Mathematische Klimalehre und astronomische Theorie der Klimaschwankungen. In: Köppen, W.; Geiger R. (Hrsg.): Handbuch der Klimatologie, Bd. 1: Allgemeine Klimalehre, Berlin: Borntraeger, 1930 Mathematische Klimalehre. In: Gutenberg, B. (Hrsg.) Handbuch der Geophysik, Berlin: Borntraeger, 1933 Durch ferne Welten und Zeiten, Briefe eines Weltallbummlers. 389 S. – Leipzig: Koehler & Amelang, 1936 Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem. Académie royale serbe. Éditions speciales; 132 [vielm. 133]: XX, 633, Belgrad, 1941 Canon of insolation and the ice-age problem. English translation by the Israel Program for Scientific Translations, published for the U.S. Department of Commerce and National Science Foundation, Washington, D.C.: 633 S., 1969 Canon of Insolation and the Ice-Age Problem. Pantic, N. (Hrsg.), Beograd: Zavod Nastavna Sredstva, 634 S., 1998 == See also == History of climate change science == References == == External links == Orbit and Insolation (The Milankovitch-Theory) at the Freie Universität Berlin, Zentraleinrichtung für AV-Medien Divine glow of Milanković's canon on Youtube Molinek, Rudy (26 August 2024). "This World War I Prisoner of War Solved the Mystery of the Ice Ages". Smithsonian Magazine. Janc, Natalija; Gavrilov, Milivoj B.; Marković, Slobodan B.; Protić-Benišek, Vojislava; Popović, Luka Č.; Benišek, Vladimir (2021). "Correspondence Between Milutin Milanković and Else Wegener-Köppen" (PDF). Publ. Astron. Obs. Belgrade. 100: 375–386. Rusov, Lazar (2009). "Milanković's analysis of Newton's law of universal gravitation" (PDF). FME Transcations. 37 (4): 211–217. Berger, W.H. "Miklankovitch Theory - Hits and Misses". UC San Diego. "Milankovitch Cycles". climatedata.info. Archived from the original on 19 April 2012. Life and Scientific Work of Milutin Milanković at Tesla Society "Solar Radiation and Milanković". Open Universitiet. Archived from the original on 14 February 2004. Precession and the Milanković Theory at NASA "On the shoulders of giants" series article on Milankovitch at the NASA Earth Observatory
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Wikipedia:Milü#0
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Milü (Chinese: 密率; pinyin: mìlǜ; lit. 'close ratio'), also known as Zulü (Zu's ratio), is the name given to an approximation of π (pi) found by the Chinese mathematician and astronomer Zu Chongzhi during the 5th century. Using Liu Hui's algorithm, which is based on the areas of regular polygons approximating a circle, Zu computed π as being between 3.1415926 and 3.1415927 and gave two rational approximations of π, 22/7 and 355/113, which were named yuelü (约率; yuēlǜ; 'approximate ratio') and milü respectively. 355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places. It is within 0.000009% of the value of π, or in terms of common fractions overestimates π by less than 1/3748629. The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as 86953/27678. For eight, 102928/32763 is needed. The accuracy of milü to the true value of π can be explained using the continued fraction expansion of π, the first few terms of which are [3; 7, 15, 1, 292, 1, 1, ...]. A property of continued fractions is that truncating the expansion of a given number at any point will give the best rational approximation of the number. To obtain milü, truncate the continued fraction expansion of π immediately before the term 292; that is, π is approximated by the finite continued fraction [3; 7, 15, 1], which is equivalent to milü. Since 292 is an unusually large term in a continued fraction expansion (corresponding to the next truncation introducing only a very small term, 1/292, to the overall fraction), this convergent will be especially close to the true value of π: π = 3 + 1 7 + 1 15 + 1 1 + 1 292 + ⋯ ≈ 3 + 1 7 + 1 15 + 1 1 + 0 = 355 113 {\displaystyle \pi =3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\color {magenta}{\cfrac {1}{292+\cdots }}}}}}}}}\quad \approx \quad 3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\color {magenta}0}}}}}}}={\frac {355}{113}}} Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called 'harmonization of the divisor of the day' (调日法; diaorifa) to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions. Zu's approximation of π ≈ 355/113 can be obtained with He Chengtian's method. == See also == Continued fraction expansion of π and its convergents Approximations of π Pi Approximation Day == Notes == == References == == External links == Fractional Approximations of Pi
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Wikipedia:Mina Aganagić#0
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Mina Aganagić is a mathematical physicist who works as a professor in the Center for Theoretical Physics, the Department of Mathematics, the Department of Physics at the University of California, Berkeley. == Career == Aganagić was raised in Sarajevo, Bosnia and Herzegovina, Yugoslavia. She has a bachelor's degree and a doctorate from the California Institute of Technology, in 1995 and 1999 respectively; her PhD advisor was John Henry Schwarz. She was a postdoctoral fellow at the Harvard University physics department from 1999 to 2003. She then joined the physics faculty at the University of Washington, where she became a Sloan Research Fellow and a DOE Outstanding Junior Investigator. She moved to UC Berkeley in 2004. In 2016 the Simons Foundation gave her a Simons Investigator Award and the same year American Physical Society had awarded her with its fellowship. == Research == She is known for applying string theory to various problems in mathematics, including knot theory (refined Chern–Simons theory),[3] enumerative geometry,[2] mirror symmetry,[1][4] and the geometric Langlands correspondence.[5] == Selected publications == Aganagić, Mina; Vafa, Cumrun (2000), Mirror symmetry, D-branes and counting holomorphic discs, arXiv:hep-th/0012041, Bibcode:2000hep.th...12041A Aganagić, Mina; Klemm, Albrecht; Mariño, Marcos; Vafa, Cumrun (2005), "The topological vertex", Communications in Mathematical Physics, 254 (2): 425–478, arXiv:hep-th/0305132, Bibcode:2005CMaPh.254..425A, doi:10.1007/s00220-004-1162-z, MR 2117633, S2CID 12221793 Aganagić, Mina; Shakirov, Shamil (2011), Knot homology from refined Chern–Simons theory, arXiv:1105.5117, Bibcode:2011arXiv1105.5117A Aganagić, Mina; Vafa, Cumrun (2012), Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots, arXiv:1204.4709, Bibcode:2012arXiv1204.4709A Aganagić, Mina; Frenkel, Edward; Okounkov, Andrei (2017), Quantum q-Langlands Correspondence, arXiv:1701.03146, Bibcode:2017arXiv170103146A == References == == External links == Physics Department home page Math Department home page
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Wikipedia:Mina Teicher#0
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Mina Teicher (Hebrew: מינה טייכר) is an Israeli mathematician at Bar-Ilan University, specializing in algebraic geometry. Teicher earned bachelor's, masters, and doctoral degrees from Tel Aviv University in 1974, 1976, and 1981 respectively. Her dissertation, Birational Transformation Between 4-folds, was supervised by Ilya Piatetski-Shapiro. Since 1999, she has directed the Emmy Noether Research Institute for Mathematics at Bar-Ilan University. In 2001–2002 she was the inaugural Emmy Noether Visiting professor at the University of Göttingen, where she lectured about braid groups. She has held leadership roles in academia and science, including serving from 2005 to 2007 as chief scientist at Israel's Ministry of Science and Technology, and chairing the board of governors of the United States – Israel Binational Science Foundation from 2012 to 2013. == References == == External links == Home page
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Wikipedia:Ming Antu's infinite series expansion of trigonometric functions#0
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Ming Antu's infinite series expansion of trigonometric functions. Ming Antu, a court mathematician of the Qing dynasty did extensive work on the infinite series expansion of trigonometric functions in his masterpiece Geyuan Milü Jiefa (Quick Method of Dissecting the Circle and Determination of The Precise Ratio of the Circle). Ming Antu built geometrical models based on a major arc of a circle and the nth dissection of the major arc. In Fig 1, AE is the major chord of arc ABCDE, and AB, BC, CD, DE are its nth equal segments. If chord AE = y, chord AB = BC = CD = DE = x, the task was to find chord y as the infinite series expansion of chord x. He studied the cases of n = 2, 3, 4, 5, 10, 100, 1000 and 10000 in great detail in volumes 3 and 4 of Geyuan Milü Jiefa. == Historical background == In 1701, French Jesuit missionary Pierre Jartoux (1669-1720) came to China, and he brought along three infinite series expansions of trigonometric functions by Isaac Newton and J. Gregory: π = 3 ( 1 + 1 4 ⋅ 3 ! + 3 2 4 2 ⋅ 5 ! + 3 2 ⋅ 5 2 4 3 ⋅ 7 ! + ⋯ ) {\displaystyle \pi =3\left(1+{\frac {1}{4\cdot 3!}}+{\frac {3^{2}}{4^{2}\cdot 5!}}+{\frac {3^{2}\cdot 5^{2}}{4^{3}\cdot 7!}}+\cdots \right)} sin x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ {\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots } vers x = x 2 2 ! − x 4 4 ! + x 6 6 ! + ⋯ . {\displaystyle \operatorname {vers} x={\frac {x^{2}}{2!}}-{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots .} These infinite series stirred up great interest among Chinese mathematicians, as the calculation of π with these "quick methods" involved only multiplication, addition or subtraction, being much faster than classic Liu Hui's π algorithm which involves taking square roots. However, Jartoux did not bring along the method for deriving these infinite series. Ming Antu suspected that the Europeans did not want to share their secrets, and hence he was set to work on it. He worked on and off for thirty years and completed a manuscript called Geyuan Milü Jiefa. He created geometrical models for obtaining trigonometric infinite series, and not only found the method for deriving the above three infinite series, but also discovered six more infinite series. In the process, he discovered and applied Catalan numbers. == Two-segment chord == Figure 2 is Ming Antu's model of a 2-segment chord. Arc BCD is a part of a circle with unit (r = 1) radius. AD is the main chord, arc BCD is bisected at C, draw lines BC, CD, let BC = CD = x and let radius AC = 1. Apparently, B D = 2 x − G H {\displaystyle BD=2x-GH} Let EJ = EF, FK = FJ; extend BE straight to L, and let EL = BE; make BF = BE, so F is inline with AE. Extended BF to M, let BF = MF; connect LM, LM apparently passes point C. The inverted triangle BLM along BM axis into triangle BMN, such that C coincident with G, and point L coincident with point N. The Invert triangle NGB along BN axis into triangle; apparently BI = BC. A B : B C : C I = 1 : x : x 2 {\displaystyle AB:BC:CI=1:x:x^{2}} BM bisects CG and let BM = BC; join GM, CM; draw CO = CM to intercept BM at O; make MP = MO; make NQ = NR, R is the intersection of BN and AC. ∠EBC = 1/2 ∠CAE = 1/2 ∠EAB; ∴ {\displaystyle \therefore } ∠EBM = ∠EAB; thus we obtain a series of similar triangles: ABE, BEF, FJK, BLM, CMO, MOP, CGH and triangle CMO = triangle EFJ; A B : B E : E F : F J : J K = 1 : p : p 2 : p 3 : p 4 {\displaystyle AB:BE:EF:FJ:JK=1:p:p^{2}:p^{3}:p^{4}} 1 : B E = B E : E F ; {\displaystyle 1:BE=BE:EF;} namely E F = B E 2 {\displaystyle EF=BE^{2}} 1 : B E 2 = x : G H {\displaystyle 1:BE^{2}=x:GH} So G H = x ⋅ B E 2 = x p 2 {\displaystyle GH=x\cdot BE^{2}=xp^{2}} , and B D = 2 x − x p 2 {\displaystyle BD=2x-xp^{2}} Because kite-shaped ABEC and BLIN are similar,. E F = L C = C M = M G = N G = I N {\displaystyle EF=LC=CM=MG=NG=IN} L M + M N = C M + M N + I N = C I + O P = J K + C I {\displaystyle LM+MN=CM+MN+IN=CI+OP=JK+CI} ∴ A B : ( B E + E C ) = B L : ( L M + M N ) {\displaystyle \therefore AB:(BE+EC)=BL:(LM+MN)} and A B : B L = B L : ( C I + J K ) {\displaystyle AB:BL=BL:(CI+JK)} Let B L = q {\displaystyle BL=q} A B : B L : ( C I + J K ) = 1 : q : q 2 {\displaystyle AB:BL:(CI+JK)=1:q:q^{2}} J K = p 4 {\displaystyle JK=p^{4}} C I = y 2 {\displaystyle CI=y^{2}} C I + J K = q 2 = B L 2 = ( 2 B E ) 2 = ( 2 p ) 2 = 4 p 2 {\displaystyle CI+JK=q^{2}=BL^{2}=(2BE)^{2}=(2p)^{2}=4p^{2}} Thus q 2 = 4 p 2 {\displaystyle q^{2}=4p^{2}} or p = q 2 {\displaystyle p={\frac {q}{2}}} Further: C I + J K = x 2 + p 4 = q 2 {\displaystyle CI+JK=x^{2}+p^{4}=q^{2}} . x 2 + q 4 16 = q 2 , {\displaystyle x^{2}+{\frac {q^{4}}{16}}=q^{2},} then x 2 = q 2 − q 4 16 {\displaystyle x^{2}=q^{2}-{\frac {q^{4}}{16}}} Square up the above equation on both sides and divide by 16: ( x 2 ) 2 16 = ( q 2 − q 4 16 ) 2 16 = ∑ j = 0 2 ( − 1 ) j ( 2 j ) q 2 ( 2 + j ) 16 j {\displaystyle {\frac {(x^{2})^{2}}{16}}={\frac {(q^{2}-{\frac {q^{4}}{16}})^{2}}{16}}=\sum _{j=0}^{2}(-1)^{j}{2 \choose j}{\frac {q^{2(2+j)}}{16^{j}}}} x 4 16 = q 4 16 − q 6 128 + q 8 4096 16 {\displaystyle {\frac {x^{4}}{16}}={\frac {q^{4}}{16}}-{\frac {q^{6}}{128}}+{\frac {q^{8}}{4096}}{16}} And so on x 2 n 16 n − 1 = ∑ j = 0 n ( − 1 ) j ( n j ) q 2 ( n + j ) 16 n + j − 1 {\displaystyle {\frac {x^{2n}}{16^{n-1}}}=\sum _{j=0}^{n}(-1)^{j}{n \choose j}{\frac {q^{2(n+j)}}{16^{n+j-1}}}} . Add up the following two equations to eliminate q 4 {\displaystyle q^{4}} items: x 2 = q 2 − q 4 16 {\displaystyle x^{2}=q^{2}-{\frac {q^{4}}{16}}} x 4 16 = q 4 16 − 2 q 6 16 2 + q 8 4096 16 {\displaystyle {\frac {x^{4}}{16}}={{\frac {q^{4}}{16}}-{\frac {2q^{6}}{16^{2}}}+{\frac {q^{8}}{4096}}}{16}} x 2 + x 4 16 = q 2 − q 6 128 + q 8 4096 {\displaystyle x^{2}+{\frac {x^{4}}{16}}=q^{2}-{\frac {q^{6}}{128}}+{\frac {q^{8}}{4096}}} x 2 + x 4 16 + 2 x 6 16 2 = q 2 − 5 q 8 4096 + 3 q 10 32768 − q 12 524288 , {\displaystyle x^{2}+{\frac {x^{4}}{16}}+{\frac {2x^{6}}{16^{2}}}=q^{2}-{\frac {5q^{8}}{4096}}+{\frac {3q^{10}}{32768}}-{\frac {q^{12}}{524288}},} (after eliminated q 6 {\displaystyle q^{6}} item). ...................................... x 2 + x 4 16 + 2 x 6 16 2 + 5 x 8 16 3 + 14 x 10 16 4 + 42 x 12 16 5 + 132 x 14 16 6 + 429 x 16 16 7 + 1430 x 18 16 8 + 4862 x 20 16 9 + 16796 x 22 16 10 + 58786 x 24 16 11 + 208012 x 26 16 12 + 742900 x 28 16 13 + 2674440 x 30 16 14 + 9694845 x 32 16 15 + 35357670 x 34 16 16 + 129644790 x 36 16 17 + 477638700 x 38 16 18 + 1767263190 x 40 16 19 + 6564120420 x 42 16 20 = q 2 + 62985 8796093022208 q 24 {\displaystyle {\begin{aligned}&x^{2}+{\frac {x^{4}}{16}}+{\frac {2x^{6}}{16^{2}}}+{\frac {5x^{8}}{16^{3}}}+{\frac {14x^{10}}{16^{4}}}+{\frac {42x^{12}}{16^{5}}}\\[10pt]{}&+{\frac {132x^{14}}{16^{6}}}+{\frac {429x^{16}}{16^{7}}}+{\frac {1430x^{18}}{16^{8}}}+{\frac {4862x^{20}}{16^{9}}}\\[10pt]&{}+{\frac {16796x^{22}}{16^{10}}}+{\frac {58786x^{24}}{16^{11}}}+{\frac {208012x^{26}}{16^{12}}}\\[10pt]&{}+{\frac {742900x^{28}}{16^{13}}}+{\frac {2674440x^{30}}{16^{14}}}+{\frac {9694845x^{32}}{16^{15}}}\\[10pt]&{}+{\frac {35357670x^{34}}{16^{16}}}+{\frac {129644790x^{36}}{16^{17}}}\\[10pt]&{}+{\frac {477638700x^{38}}{16^{18}}}+{\frac {1767263190x^{40}}{16^{19}}}+{\frac {6564120420x^{42}}{16^{20}}}\\[10pt]&=q^{2}+{\frac {62985}{8796093022208}}q^{24}\end{aligned}}} Expansion coefficients of the numerators: 1, 1, 2, 5, 14, 42, 132 ...... (see Figure II Ming Antu original figure bottom line, read from right to left) are the Catalan numbers; Ming Antu discovered the Catalan number. Thus: q 2 = ∑ n = 1 ∞ C n x 2 n 4 2 n − 2 {\displaystyle q^{2}=\sum _{n=1}^{\infty }C_{n}{\frac {x^{2n}}{4^{2n-2}}}} in which C n = 1 n + 1 ( 2 n n ) {\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}} is Catalan number. Ming Antu pioneered the use of recursion relations in Chinese mathematics C n = ∑ k ( − 1 ) k ( n k k + 1 ) C n k {\displaystyle C_{n}=\sum _{k}(-1)^{k}{nk \choose k+1}C_{nk}} ∵ B C : C G : G H = A B : B E : E F = 1 : p : p 2 = x : p x : p 2 x {\displaystyle \because BC:CG:GH=AB:BE:EF=1:p:p^{2}=x:px:p^{2}x} ∴ G H := p 2 x = ( q 2 ) 2 x = q 2 x 4 {\displaystyle \therefore GH:=p^{2}x=({\frac {q}{2}})^{2}x={\frac {q^{2}x}{4}}} substituted into B D = 2 x − G H {\displaystyle BD=2x-GH} Finally he obtained B D = 2 x − x 4 q 2 {\displaystyle BD=2x-{\frac {x}{4}}q^{2}} = 2 x − ∑ n = 1 ∞ C n x 2 n + 1 4 2 n − 1 {\displaystyle =2x-\sum _{n=1}^{\infty }C_{n}{\frac {x^{2n+1}}{4^{2n-1}}}} In Figure 1 BAE angle = α, BAC angle = 2α × x = BC = sinα × q = BL = 2BE = 4sin (α /2) × BD = 2sin (2α) Ming Antu obtained B D = 2 x − x ⋅ B E 2 {\displaystyle BD=2x-x\cdot BE^{2}} That is sin ( 2 α ) = 2 sin α − ∑ n = 1 ∞ C n ( sin α ) 2 n + 1 4 n − 1 {\displaystyle \sin(2\alpha )=2\sin \alpha -\sum _{n=1}^{\infty }C_{n}{\frac {(\sin \alpha )^{2n+1}}{4^{n-1}}}} = 2 sin ( α ) − 2 sin ( α ) 3 1 + cos ( α ) {\displaystyle =2\sin(\alpha )-{\frac {2\sin(\alpha )^{3}}{1+\cos(\alpha )}}} q 2 = B L 2 = ∑ n = 1 ∞ C n x 2 n 4 2 n − 2 {\displaystyle q^{2}=BL^{2}=\sum _{n=1}^{\infty }C_{n}{\frac {x^{2n}}{4^{2n-2}}}} Ie sin ( α 2 ) 2 = ∑ n = 1 ∞ C n ( s i n α ) 2 n 4 2 n {\displaystyle \sin({\frac {\alpha }{2}})^{2}=\sum _{n=1}^{\infty }C_{n}{\frac {(sin\alpha )^{2n}}{4^{2n}}}} == Three-segment chord == As shown in Fig 3, BE is a whole arc chord, BC = CE = DE = an are three arcs of equal portions. Radii AB = AC = AD = AE = 1. Draw lines BC, CD, DE, BD, EC; let BG=EH = BC, Bδ = Eα = BD, then triangle Cαβ = Dδγ; while triangle Cαβ is similar to triangle BδD. As such: A B : B C = B C : C G = C G : G F {\displaystyle AB:BC=BC:CG=CG:GF} , B C : F G = B D : δ γ {\displaystyle BC:FG=BD:\delta \gamma } 2 B D = B E + δ α {\displaystyle 2BD=BE+\delta \alpha } 2 B D − δ γ = B E + B C {\displaystyle 2BD-\delta \gamma =BE+BC} ∴ 2 × B D − δ γ − B C = B E {\displaystyle \therefore 2\times BD-\delta \gamma -BC=BE} Eventually, he obtained B E = 3 × a − a 3 {\displaystyle BE=3\times a-a^{3}} == Four-segment chord == Let y 4 {\displaystyle y_{4}} denotes the length of the main chord, and let the length of four equal segment chord =x, y 4 = 4 × a − 10 × a 3 4 + 14 × a 5 4 3 − 12 × a 7 4 5 {\displaystyle y_{4}=4\times a-{\frac {10\times a^{3}}{4}}+{\frac {14\times a^{5}}{4^{3}}}-{\frac {12\times a^{7}}{4^{5}}}} +...... 4 a − 10 × a 3 / 4 + ∑ n = 1 ∞ ( 16 C n − 2 C n + 1 ) × a 2 n + 1 4 2 n − 1 {\displaystyle 4a-10\times a^{3}/4+\sum _{n=1}^{\infty }(16C_{n}-2C_{n+1})\times {\frac {a^{2n+1}}{4^{2n-1}}}} . Trigonometry meaning: sin ( 4 × α ) = 4 × sin ( α ) − 10 × sin 3 α {\displaystyle \sin(4\times \alpha )=4\times \sin(\alpha )-10\times \sin ^{3}\alpha } + ∑ n = 1 ∞ ( 16 × C n − 2 C n + 1 ) × sin 2 n + 3 ( α ) 4 n {\displaystyle +\sum _{n=1}^{\infty }(16\times C_{n}-2C_{n+1})\times {\frac {\sin ^{2n+3}(\alpha )}{4^{n}}}} . == Five-segment chord == y 5 = 5 a − 5 a 3 + a 5 {\displaystyle y_{5}=5a-5a^{3}+a^{5}} that is sin ( 5 α ) = 5 sin ( α ) − 20 sin 3 ( α ) + 16 sin 5 ( α ) {\displaystyle \sin(5\alpha )=5\sin(\alpha )-20\sin ^{3}(\alpha )+16\sin ^{5}(\alpha )} 。 == Ten-segment chord == From here on, Ming Antu stop building geometrical model, he carried out his computation by pure algebraic manipulation of infinite series. Apparently ten segments can be considered as a composite 5 segment, with each segment in turn consist of two subsegments. ∴ y 10 = y 5 ( y 2 ) {\displaystyle \therefore y_{10}=y_{5}(y_{2})} y 10 ( a ) = 5 × y 2 − 5 × ( y 2 ) 3 + ( y 2 ) 5 {\displaystyle y_{10}(a)=5\times y_{2}-5\times (y_{2})^{3}+(y_{2})^{5}} , He computed the third and fifth power of infinite series y 2 {\displaystyle y_{2}} in the above equation, and obtained: y 10 ( a ) = 10 × a − 165 × a 3 4 + 3003 × a 5 4 3 − 21450 × a 7 4 5 {\displaystyle y_{10}(a)=10\times a-{\frac {165\times a^{3}}{4}}+{\frac {3003\times a^{5}}{4^{3}}}-{\frac {21450\times a^{7}}{4^{5}}}} +...... == Hundred-segment chord == A hundred segment arc's chord can be considered as composite 10 segment-10 subsegments, thus substituting a = y 10 {\displaystyle a=y_{10}} into y 10 {\displaystyle y_{10}} , after manipulation with infinite series he obtained: y 100 = y 10 ( a = y 10 ) {\displaystyle y_{100}=y_{10}(a=y_{10})} y 100 ( a ) = 100 × a − 166650 × a 3 4 + 333000030 × a 5 4 × 16 − 316350028500 × a 7 4 × 16 2 + 17488840755750 × a 9 4 × 16 3 + … {\displaystyle {\begin{aligned}y_{100}(a)=&100\times a-166650\times {\frac {a^{3}}{4}}+333000030\times {\frac {a^{5}}{4\times 16}}\\&-316350028500\times {\frac {a^{7}}{4\times 16^{2}}}+17488840755750\times {\frac {a^{9}}{4\times 16^{3}}}+\ldots \end{aligned}}} == Thousand-segment chord == y 1000 = y 100 ( y 10 ) {\displaystyle y_{1000}=y_{100}(y_{10})} y 1000 ( a ) = 1000 × a − 1666666500 × a 3 4 + 33333000000300 × a 5 4 × 16 − 3174492064314285000 a 7 4 × 16 2 + {\displaystyle y_{1000}(a)=1000\times a-1666666500\times {\frac {a^{3}}{4}}+33333000000300\times {\frac {a^{5}}{4\times 16}}-3174492064314285000{\frac {a^{7}}{4\times 16^{2}}}+} ...... == Ten-thousand-segment chord == y 10000 = 10000 × a − 166666665000 × a 3 4 + 33333330000000300 × a 5 4 3 + {\displaystyle y_{10000}=10000\times a-{\frac {166666665000\times a^{3}}{4}}+{\frac {33333330000000300\times a^{5}}{4^{3}}}+} ............ == When number of segments approaches infinity == After obtained the infinite series for n=2, 3, 5, 10, 100, 1000, and 10000 segments, Ming Antu went on to handle the case when n approaches infinity. y100, y1000 and y10000 can be rewritten as: y 100 = 100 a − ( 100 a ) 3 24.002400240024002400 + ( 100 a ) 5 24.024021859697730358 × 80 + {\displaystyle y100=100a-{\frac {(100a)^{3}}{24.002400240024002400}}+{\frac {(100a)^{5}}{24.024021859697730358\times 80}}+} .......... y 1000 := 1000 a − ( 1000 a ) 3 24.000024000024000024 + ( 1000 a ) 5 24.000240002184019680 × 80 + {\displaystyle y1000:=1000a-{\frac {(1000a)^{3}}{24.000024000024000024}}+{\frac {(1000a)^{5}}{24.000240002184019680\times 80}}+} .............. y 10000 := 10000 a − ( 10000 a ) 3 24.000000240000002400 + ( 10000 a ) 5 24.000002400000218400 × 80 + {\displaystyle y10000:=10000a-{\frac {(10000a)^{3}}{24.000000240000002400}}+{\frac {(10000a)^{5}}{24.000002400000218400\times 80}}+} .................. He noted that when n approaches infinity, the denominators 24.000000240000002400, 24.000002400000218400×80 approach 24 and 24×80 respectively, and when n -> infinity, na (100a, 1000a, 1000a) becomes the length of the arc; hence c h o r d = a r c − a r c 3 4 × 3 ! + a r c 5 4 2 × 5 ! − a r c 7 4 3 × 7 ! + {\displaystyle chord=arc-{\frac {arc^{3}}{4\times 3!}}+{\frac {arc^{5}}{4^{2}\times 5!}}-{\frac {arc^{7}}{4^{3}\times 7!}}+} ..... = ∑ n = 1 ∞ ( − 1 ) n − 1 × a r c 2 × n − 1 ( 4 n − 1 × ( 2 × n − 1 ) ! ) {\displaystyle =\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}\times arc^{2\times n-1}}{(4^{n-1}\times (2\times n-1)!)}}} Ming Antu then performed an infinite series reversion and expressed the arc in terms of its chord a r c := c h o r d + c h o r d 3 24 + 3 × c h o r d 5 640 + 5 × c h o r d 7 7168 + {\displaystyle arc:=chord+{\frac {chord^{3}}{24}}+{\frac {3\times chord^{5}}{640}}+{\frac {5\times chord^{7}}{7168}}+} ............ == References == Luo A Modern Chinese Translation of Ming Antu's Geyuan Milv Jifa, translated and annotated by Luo Jianjin, Inner Mongolia Education Press 1998(明安图原著 罗见今译注 《割圆密率捷法》 内蒙古教育出版社 This is the only modern Chinese translation of Ming Antu's book, with detailed annotation with modern mathematical symbols). ISBN 7-5311-3584-1 Yoshio Mikami The Development of Mathematics in China and Japan, Leipzig, 1912
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Wikipedia:Mingarelli identity#0
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In the field of ordinary differential equations, the Mingarelli identity is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order. == The identity == Consider the n solutions of the following (uncoupled) system of second order linear differential equations over the t–interval [a, b]: ( p i ( t ) x i ′ ) ′ + q i ( t ) x i = 0 , x i ( a ) = 1 , x i ′ ( a ) = R i {\displaystyle (p_{i}(t)x_{i}^{\prime })^{\prime }+q_{i}(t)x_{i}=0,\,\,\,\,\,\,\,\,\,\,x_{i}(a)=1,\,\,x_{i}^{\prime }(a)=R_{i}} where i = 1 , 2 , … , n {\displaystyle i=1,2,\ldots ,n} . Let Δ {\displaystyle \Delta } denote the forward difference operator, i.e. Δ x i = x i + 1 − x i . {\displaystyle \Delta x_{i}=x_{i+1}-x_{i}.} The second order difference operator is found by iterating the first order operator as in Δ 2 ( x i ) = Δ ( Δ x i ) = x i + 2 − 2 x i + 1 + x i , {\displaystyle \Delta ^{2}(x_{i})=\Delta (\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i},} , with a similar definition for the higher iterates. Leaving out the independent variable t for convenience, and assuming the xi(t) ≠ 0 on (a, b], there holds the identity, x n − 1 2 Δ n − 1 ( p 1 r 1 ) ] a b = ∫ a b ( x n − 1 ′ ) 2 Δ n − 1 ( p 1 ) − ∫ a b x n − 1 2 Δ n − 1 ( q 1 ) − ∑ k = 0 n − 1 C ( n − 1 , k ) ( − 1 ) n − k − 1 ∫ a b p k + 1 W 2 ( x k + 1 , x n − 1 ) / x k + 1 2 , {\displaystyle {\begin{aligned}x_{n-1}^{2}\Delta ^{n-1}(p_{1}r_{1})]_{a}^{b}=&\int _{a}^{b}(x_{n-1}^{\prime })^{2}\Delta ^{n-1}(p_{1})-\int _{a}^{b}x_{n-1}^{2}\Delta ^{n-1}(q_{1})\\&-\sum _{k=0}^{n-1}C(n-1,k)(-1)^{n-k-1}\int _{a}^{b}p_{k+1}W^{2}(x_{k+1},x_{n-1})/x_{k+1}^{2},\end{aligned}}} where r i = x i ′ / x i {\displaystyle r_{i}=x_{i}^{\prime }/x_{i}} is the logarithmic derivative, W ( x i , x j ) = x i ′ x j − x i x j ′ {\displaystyle W(x_{i},x_{j})=x_{i}^{\prime }x_{j}-x_{i}x_{j}^{\prime }} , is the Wronskian determinant, C ( n − 1 , k ) {\displaystyle C(n-1,k)} are binomial coefficients. When n = 2 this equality reduces to the Picone identity. == An application == The above identity leads quickly to the following comparison theorem for three linear differential equations, which extends the classical Sturm–Picone comparison theorem. Let pi, qi i = 1, 2, 3, be real-valued continuous functions on the interval [a, b] and let ( p 1 ( t ) x 1 ′ ) ′ + q 1 ( t ) x 1 = 0 , x 1 ( a ) = 1 , x 1 ′ ( a ) = R 1 {\displaystyle (p_{1}(t)x_{1}^{\prime })^{\prime }+q_{1}(t)x_{1}=0,\,\,\,\,\,\,\,\,\,\,x_{1}(a)=1,\,\,x_{1}^{\prime }(a)=R_{1}} ( p 2 ( t ) x 2 ′ ) ′ + q 2 ( t ) x 2 = 0 , x 2 ( a ) = 1 , x 2 ′ ( a ) = R 2 {\displaystyle (p_{2}(t)x_{2}^{\prime })^{\prime }+q_{2}(t)x_{2}=0,\,\,\,\,\,\,\,\,\,\,x_{2}(a)=1,\,\,x_{2}^{\prime }(a)=R_{2}} ( p 3 ( t ) x 3 ′ ) ′ + q 3 ( t ) x 3 = 0 , x 3 ( a ) = 1 , x 3 ′ ( a ) = R 3 {\displaystyle (p_{3}(t)x_{3}^{\prime })^{\prime }+q_{3}(t)x_{3}=0,\,\,\,\,\,\,\,\,\,\,x_{3}(a)=1,\,\,x_{3}^{\prime }(a)=R_{3}} be three homogeneous linear second order differential equations in self-adjoint form, where pi(t) > 0 for each i and for all t in [a, b] , and the Ri are arbitrary real numbers. Assume that for all t in [a, b] we have, Δ 2 ( q 1 ) ≥ 0 {\displaystyle \Delta ^{2}(q_{1})\geq 0} , Δ 2 ( p 1 ) ≤ 0 {\displaystyle \Delta ^{2}(p_{1})\leq 0} , Δ 2 ( p 1 ( a ) R 1 ) ≤ 0 {\displaystyle \Delta ^{2}(p_{1}(a)R_{1})\leq 0} . Then, if x1(t) > 0 on [a, b] and x2(b) = 0, then any solution x3(t) has at least one zero in [a, b]. == Notes == == References == Clark D.N.; G. Pecelli & R. Sacksteder (1981). Contributions to Analysis and Geometry. Baltimore, USA: Johns Hopkins University Press. pp. ix+357. ISBN 0-80182-779-5. Mingarelli, Angelo B. (1979). "Some extensions of the Sturm–Picone theorem". Comptes Rendus Mathématique. 1 (4). Toronto, Ontario, Canada: The Royal Society of Canada: 223–226.
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Wikipedia:Minggatu#0
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Minggatu (Mongolian script: ᠮᠢᠩᠭᠠᠲᠦ; Chinese: 明安图; pinyin: Míng'āntú, c.1692-c. 1763), full name Sharavyn Myangat (Mongolian: Шаравын Мянгат), also known as Ming Antu, was a Mongolian astronomer, mathematician, and topographic scientist at the Qing court. His courtesy name was Jing An (静安). Minggatu was born in Plain White Banner (now Plain and Bordered White Banner, Xilin Gol League, Inner Mongolia) of the Qing Empire. He was of the Sharaid clan. His name first appeared in official Chinese records in 1713, among the Kangxi Emperor's retinue, as a shengyuan (state-subsidized student) of the Imperial Astronomical Bureau. He worked there at a time when Jesuit missionaries were in charge of calendar reforms. He also participated in the work of compiling and editing three very important books in astronomy and joined the team of China's area measurement. From 1724 up to 1759, he worked at the Imperial Observatory. He participated in drafting and editing the calendar and the study of the armillary sphere. His seminal work The Quick Method for Obtaining the Precise Ratio of Division of a Circle (Chinese: 割圜密率捷法; pinyin: Gēyuán Mìlǜ Jiéfǎ), which was completed after his death by his son Mingshin, and students (among them his most gifted pupil Chen Jihin and an intendant in the minister of finance, Zhang), was a significant contribution to the development of mathematics in China. He was the first person in Inner Mongolia who calculated infinite series and obtained more than 10 formulae. In the 1730s, he first established and used what was later to be known as Catalan numbers. The Jesuit missionaries' influence can be seen by many traces of European mathematics in his works, including the use of Euclidean notions of continuous proportions, series addition, subtraction, multiplication and division, series reversion, and the binomial theorem. Minggatu's work is remarkable in that expansions in series, trigonometric and logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus. In 1742 he participated in the revision of the Compendium of Observational and Computational Astronomy. In 1756, he participated in the surveying of the Dzungar Khanate (renamed Xinjiang), which was incorporated into the Qing Empire by the Qianlong Emperor. It was due to his geographical surveys in Xinjiang that the Complete Atlas of the Empire (the first atlas of China drawn with scientific methods) was finished. From 1760 to 1763, shortly before his death, he was administrator of the Imperial Astronomical Bureau. == Later recognition == In 1910, Japanese mathematician Yoshio Mikami mentioned that Minggatu was the first Mongolian who had ever entered into the field of analytical research methods. Mathematician Dr. P. J. Larcombe of Derby University published 7 papers on Minggatu and his work in 1999, including the stimulus of Jesuit missionary, engineer, mathematician and geographer Pierre Jartoux, who brought three infinite series to China early in the 1700s. On May 26, 2002, the minor planet 28242 was named after Minggatu as 28242 Mingantu. The nomination ceremony and traditional meeting were held in Minggatu's hometown in August 2002. More than 500 delegates and 20,000 local residents gathered together to celebrate and a conference on "The Science Contribution of Ming Antu" was held. The Chinese government named Minggatu’s hometown as "Ming Antu Town". == See also == Ming Antu's infinite series expansion of trigonometric functions == References ==
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Wikipedia:Minimal algebra#0
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Algebraic notation is the standard method of chess notation, used for recording and describing moves. It is based on a system of coordinates to identify each square on the board uniquely. It is now almost universally used by books, magazines, newspapers and software, and is the only form of notation recognized by FIDE, the international chess governing body. An early form of algebraic notation was invented by the Syrian player Philip Stamma in the 18th century. In the 19th century, it came into general use in German chess literature and was subsequently adopted in Russian chess literature. Descriptive notation, based on abbreviated natural language, was generally used in English language chess publications until the 1980s. Similar descriptive systems were in use in Spain and France. A few players still use descriptive notation, but it is no longer recognized by FIDE, and may not be used as evidence in the event of a dispute. The term "algebraic notation" may be considered a misnomer, as the system is unrelated to algebra. == Naming the squares == Each square of the board is identified by a unique coordinate pair—a letter and a number—from White's point of view. The vertical columns of squares, called files, are labeled a through h from White's left (the queenside) to right (the kingside). The horizontal rows of squares, called ranks, are numbered 1 to 8 starting from White's side of the board. Thus each square has a unique identification of file letter followed by rank number. For example, the initial square of White's king is designated as "e1". == Naming the pieces == Each piece type (other than pawns) is identified by an uppercase letter. English-speaking players use the letters K for king, Q for queen, R for rook, B for bishop and N for knight. Different initial letters are used by other languages. In modern chess literature, especially that intended for an international audience, the language-specific letters are usually replaced by universally recognized piece symbols; for example, ♞c6 in place of Nc6. This style is known as figurine algebraic notation. The Unicode Miscellaneous Symbols set includes all the symbols necessary for figurine algebraic notation. == Notation for moves == In standard (or short-form) algebraic notation, each move of a piece is indicated by the piece's uppercase letter, plus the coordinates of the destination square. For example, Be5 (bishop moves to e5), Nf3 (knight moves to f3). For pawn moves, a letter indicating pawn is not used, only the destination square is given. For example, c5 (pawn moves to c5). === Captures === When a piece makes a capture, an "x" is inserted immediately before the destination square. For example, Bxe5 (bishop captures the piece on e5). When a pawn makes a capture, the file from which the pawn departed is used to identify the pawn. For example, exd5 (pawn on the e-file captures the piece on d5). En passant captures are indicated by specifying the capturing pawn's file of departure, the "x", the destination square (not the square of the captured pawn) and (optionally) the suffix "e.p." indicating the capture was en passant. For example, exd6 e.p. Sometimes a multiplication sign (×) or a colon (:) is used instead of "x", either in the middle (B:e5) or at the end (Be5:). Some publications, such as the Encyclopaedia of Chess Openings (ECO), omit any indication that a capture has been made; for example, Be5 instead of Bxe5; ed6 instead of exd6 or exd6 e.p. When it is unambiguous to do so, a pawn capture is sometimes described by specifying only the files involved (exd or even ed). These shortened forms are sometimes called abbreviated algebraic notation or minimal algebraic notation. === Disambiguating moves === When two (or more) identical pieces can move to the same square, the moving piece is uniquely identified by specifying the piece's letter, followed by (in descending order of preference): the file of departure (if they differ); the rank of departure (if the files are the same but the ranks differ). If neither file nor rank alone is sufficient to identify the piece (such as when three or more pieces of the same type can move to the same square), then both are specified (double disambiguation). In the diagram, both black rooks could legally move to f8, so the move of the d8-rook to f8 is disambiguated as Rdf8. For the white rooks on the a-file which could both move to a3, it is necessary to provide the rank of the moving piece, i.e., R1a3. In the case of the white queen on h4 moving to e1, neither the rank nor file alone are sufficient to disambiguate from the other white queens. As such, this move is written Qh4e1. As above, an "x" can be inserted to indicate a capture; for example, if the final case were a capture, it would be written as Qh4xe1. === Pawn promotion === When a pawn promotes, the piece promoted to is indicated at the end. For example, a pawn on e7 promoting to a queen on e8 may be variously rendered as e8Q, e8=Q, e8(Q), e8/Q etc. === Castling === Castling is indicated by the special notations 0-0 (for kingside castling) and 0-0-0 (queenside castling). O-O and O-O-O (letter O rather than digit 0) are also commonly used. === Check === A move that places the opponent's king in check usually has the symbol "+" appended. Alternatively, sometimes a dagger (†) or the abbreviation "ch" is used. Some publications indicate a discovered check with an abbreviation such as "dis ch", or with a specific symbol. Double check is usually indicated the same as check, but is sometimes represented specifically as "dbl ch" or "++", particularly in older chess literature. Some publications such as ECO omit any indication of check. === Checkmate === Checkmate at the completion of moves is represented by the symbol "#" in standard FIDE notation and PGN. The word mate is commonly used instead; occasionally a double dagger (‡) or a double plus sign (++) is used, although the double plus sign is also used to represent "double check" when a king is under attack by two enemy pieces simultaneously. A checkmate is represented by "≠" (the not equal sign) in the macOS chess application. In Russian and ex-USSR publications, where captures are indicated by ":", checkmate can also be represented by "X" or "x". === Draw offer === FIDE specifies draw offers to be recorded by an equals sign with parentheses "(=)" after the move on the score sheet. This is not usually included in published game scores. === End of game === The notation 1–0 at the completion of moves indicates that White won, 0–1 indicates that Black won and ½–½ indicates a draw. In case of forfeit, the scores 0–0, ½–0 and 0–½ are also possible. In case of loss by default, results are +/−, −/+ or −/−. Except in the case of checkmate, there is no information in the notation regarding the circumstance of the final result. Merely 1–0 or 0–1 is written whether a player resigned, lost due to time control or forfeited; in the case of a draw ½–½ is written whether the draw was decided by mutual agreement, repetition, stalemate, 50-move rule or dead position. Sometimes direct information is given by words such as "resigns", "draw agreed" etc., but this is not considered part of the notation, rather a part of the narrative text. == Similar notations == Besides standard (or short form) algebraic notation already described, several similar systems have been used. === Long algebraic notation === In long algebraic notation, also known as fully expanded algebraic notation, both the starting and ending squares are specified, for example: e2e4. Sometimes these are separated by a hyphen, e.g. Nb1-c3, while captures are indicated by an "x", e.g. Rd3xd7. Long algebraic notation takes more space and is no longer commonly used in print; however, it has the advantage of clarity. Both short and long algebraic notation are acceptable for keeping a record of the moves on a scoresheet, as is required in FIDE rated games. A form of long algebraic notation (without piece names) is also used by the Universal Chess Interface (UCI) standard, which is a common way for graphical chess programs to communicate with chess engines, e.g. e2e4, e1g1 (castling), e7e8q (promotion). === ICCF numeric notation === In international correspondence chess the use of algebraic notation may cause confusion, since different languages employ different names (and therefore different initial letters) for the pieces, and some players may be unfamiliar with the Latin alphabet. Hence, the standard for transmitting moves by post or email is ICCF numeric notation, which identifies squares using numerical coordinates, and identifies both the departure and destination squares. For example, the move 1.e4 is rendered as 1.5254. In recent years, the majority of correspondence games have been played on on-line servers rather than by email or post, leading to a decline in the use of ICCF numeric notation. === PGN === Portable Game Notation (PGN) is a text-based file format for storing chess games, which uses standard English algebraic notation and a small amount of markup. PGN can be processed by almost all chess software, as well as being easily readable by humans. For example, the Game of the Century could be represented as follows in PGN: == Formatting == A game or series of moves is generally written in one of two ways; in two columns, as White/Black pairs, preceded by the move number and a period: 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 or horizontally: 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 Moves may be interspersed with commentary, called annotations. When the game score resumes with a Black move, an ellipsis (...) fills the position of the White move, for example: 1. e4 e5 2. Nf3 White attacks the black e-pawn. 2... Nc6 Black defends and develops simultaneously. 3. Bb5 White plays the Ruy Lopez. 3... a6 Black elects Morphy's Defense. == Annotation symbols == Though not technically a part of algebraic notation, the following are some symbols commonly used by annotators, for example in publications Chess Informant and Encyclopaedia of Chess Openings, to give editorial comment on a move or position. The symbol chosen is appended to the end of the move notation, for example, in the Soller Gambit: 1.d4 e5?! 2.dxe5 f6 3.e4! Nc6 4.Bc4+/−. === Moves === === Positions === == History == Descriptive notation was usual in the Middle Ages in Europe. A form of algebraic chess notation that seems to have been borrowed from Muslim chess, however, appeared in Europe in a 12th-century manuscript referred to as "MS. Paris Fr. 1173 (PP.)". The files run from a to h, just as they do in the current standard algebraic notation. The ranks, however, are also designated by letters, with the exception of the 8th rank which is distinct because it has no letter. The ranks are lettered in reverse – from the 7th to the 1st: k, l, m, n, o, p, q. Another system of notation using only letters appears in a book of Mediaeval chess, Rechenmeister Jacob Köbel's Schachzabel Spiel of 1520. Algebraic notation exists in various forms and languages and is based on a system developed by Philipp Stamma in the 1730s. Stamma used the modern names of the squares (and may have been the first to number the ranks), but he used p for pawn moves and the capital original file of a piece (A through H) instead of the initial letter of the piece name as used now. Piece letters were introduced in the 1780s by Moses Hirschel, and Johann Allgaier with Aaron Alexandre developed the modern castling notation in the 1810s. Algebraic notation was described in 1847 by Howard Staunton in his book The Chess-Player's Handbook. Staunton credits the idea to German authors, and in particular to "Alexandre, Jaenisch and the Handbuch [des Schachspiels]". While algebraic notation has been used in German and Russian chess literature since the 19th century, the Anglosphere was slow to adopt it, using descriptive notation for much of the 20th century. Beginning in the 1970s, algebraic notation gradually became more common in English language publications, and by 1980 it had become the prevalent notation. In 1981, FIDE stopped recognizing descriptive notation, and algebraic notation became the accepted international standard. == Piece names in various languages == The table contains names for all the pieces as well as the words for chess, check and checkmate in several languages. Several languages use the Arabic loanword alfil for the piece called bishop in English; in this context it is a chess-specific term which no longer has its original meaning of "elephant". == See also == Chess notation Chess annotation symbols == Notes == == References == == External links == FIDE Laws of Chess (see Appendix C. Algebraic Notation)
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Wikipedia:Minimax#0
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The Team Mini-MAX is a large family of single-seat, mid-wing, strut-braced, single engine aircraft, available in kit form for amateur construction. The first Mini-MAX had its first flight in 1984. Its name indicates its original design goals: a minimum-cost aircraft that requires a minimum of building space, time and skill, but which provides a maximum of enjoyment and performance. The Mini-MAX family was originally produced by TEAM Incorporated of Bradyville, Tennessee. After that company was bankrupted by a lawsuit, production passed to Ison Aircraft also of Bradyville, Tennessee and next to JDT Mini-MAX of Nappanee, Indiana. The company was renamed Team Mini-Max LLC in 2012, with production in Niles, Michigan. == Development == The Mini-MAX models are all predominantly constructed from wood truss with plywood gussets and covered with doped aircraft fabric. The construction time to complete a Mini-MAX varies depending on the model chosen. Many models feature open cockpits equipped with windshields. All versions feature a short-span wing of only 25 ft (7.6 m), except the V-MAX and 1600R EROS, which have a 26.5 ft (8.1 m) wingspan. The wing and horizontal stabilizer are both strut-braced: the wing is braced to the landing gear and the tail is braced from the horizontal tail surface to the fin. All models have conventional landing gear, with wheel pants as an option. Since the wing is braced to the mainwheels and the mainwheels are connected by a rigid axle, the pneumatic tires provide the only suspension. The aircraft was originally intended to meet the requirements of the US FAR 103 Ultralight Vehicles category, including that category's maximum 254 lb (115 kg) empty weight. The original ultralight models of the Mini-MAX were equipped with the 28 hp (21 kW) Rotax 277 engine to achieve acceptable empty weights. Today the 1030F MAX 103 and 1100F Mini-MAX achieve an acceptable FAR 103 empty weight if they are equipped with the 28 hp (21 kW) Hirth F-33 powerplant. Other models use heavier engines which place them in the US Experimental - Amateur-built category. The Mini-MAX was also developed into a high winged version, called the Hi-MAX. The two designs share much in the way of parts and design concept commonality. == Variants == 1030F MAX-103 Single seat, open cockpit, mid-wing aircraft with the 28 hp (21 kW) Hirth F-33 engine. Still in production. Manufacturer claimed construction time 300-350 hours. 1030R MAX-103 Single seat, open cockpit, mid-wing aircraft with the 28 hp (21 kW) Rotax 277. First flight 1993, out of production, replaced by the 1030F. Manufacturer claimed construction time 350 hours. 250 completed and flown by 2011. 1100F Mini-MAX Single seat, open cockpit, mid-wing aircraft with the 28 hp (21 kW) Hirth F-33 engine. Still in production. Manufacturer claimed construction time 250-300 hours. 1100R Mini-MAX Single seat, open cockpit, mid-wing aircraft with the 40 hp (30 kW) Rotax 447 engine. First flight 1984, still in production. Manufacturer claimed construction time 250-300 hours. 600 completed and flown by 2011. 1200Z Z-MAX Single seat, open cockpit, mid-wing aircraft with the 45 hp (34 kW) Zenoah G-50 engine. First flight 1991, out of production. Manufacturer claimed construction time 350 hours. 124 completed and flown by 2001. As this is a US aircraft the name is pronounced "Zee-Max". 1300Z Z-MAX Single seat, enclosed cockpit, mid-wing aircraft with the 45 hp (34 kW) Zenoah G-50 engine. First flight 1990, out of production. Manufacturer claimed construction time 400 hours. 231 completed and flown by 2001. As this is a US aircraft the name is pronounced "Zee-Max". 1500R Sport Single seat, open cockpit, mid-wing aircraft with the 40 hp (30 kW) Rotax 447 engine. First flight 1987, still in production. Manufacturer claimed construction time 300-350 hours. 200 completed and flown by 2011. 1550V V-MAX Single seat, open cockpit, mid-wing aircraft with the 50 hp (37 kW) Volkswagen air-cooled engine and 26.5 ft (8.1 m) wingspan. First flight 1993, still in production. Manufacturer claimed construction time 325-400 hours. 250 completed and flown by 2011. 1600R Sport Single seat, enclosed cockpit, mid-wing aircraft with the 40 hp (30 kW) Rotax 447. First flight 1989, still in production. Manufacturer claimed construction time 325-400 hours. 315 completed and flown by 2011. 1650R EROS Single seat, enclosed cockpit, mid-wing aircraft with the 50 hp (37 kW) Rotax 503 and 26.5 ft (8.1 m) wingspan. Still in production. Manufacturer claimed construction time 325-400 hours. 300 completed and flown by 2011. == Specifications (1650R EROS) == Data from Aerocrafter, Kitplanes & JDT websiteGeneral characteristics Crew: one Length: 16 ft 0 in (4.88 m) Wingspan: 26 ft 6 in (8.08 m) Height: 5 ft 0 in (1.52 m) Wing area: 118 sq ft (11.0 m2) Empty weight: 400 lb (181 kg) Gross weight: 700 lb (318 kg) Fuel capacity: 10 US gallons (38 litres) Powerplant: 1 × Rotax 503 twin cylinder, two-stroke aircraft engine, 50 hp (37 kW) Performance Maximum speed: 80 mph (130 km/h, 70 kn) Cruise speed: 75 mph (121 km/h, 65 kn) Stall speed: 36 mph (58 km/h, 31 kn) Never exceed speed: 110 mph (180 km/h, 96 kn) Range: 144 mi (232 km, 125 nmi) Service ceiling: 12,000 ft (3,700 m) g limits: +4.0/-2.0 Rate of climb: 1,200 ft/min (6.1 m/s) == See also == Aircraft of comparable role, configuration, and era Airdrome Eindecker E-III Ameri-Cana Eureka Fisher Avenger Fisher FP-303 Hummel Bird Hummel CA-2 == References == == External links == Official website
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Wikipedia:Minimum rank of a graph#0
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In mathematics, the minimum rank is a graph parameter mr ( G ) {\displaystyle \operatorname {mr} (G)} for a graph G. It was motivated by the Colin de Verdière graph invariant. == Definition == The adjacency matrix of an undirected graph is a symmetric matrix whose rows and columns both correspond to the vertices of the graph. Its elements are all 0 or 1, and the element in row i and column j is nonzero whenever vertex i is adjacent to vertex j in the graph. More generally, a generalized adjacency matrix is any symmetric matrix of real numbers with the same pattern of nonzeros off the diagonal (the diagonal elements may be any real numbers). The minimum rank of G {\displaystyle G} is defined as the smallest rank of any generalized adjacency matrix of the graph; it is denoted by mr ( G ) {\displaystyle \operatorname {mr} (G)} . == Properties == Here are some elementary properties. The minimum rank of a graph is always at most equal to n − 1, where n is the number of vertices in the graph. For every induced subgraph H of a given graph G, the minimum rank of H is at most equal to the minimum rank of G. If a graph is disconnected, then its minimum rank is the sum of the minimum ranks of its connected components. The minimum rank is a graph invariant: isomorphic graphs necessarily have the same minimum rank. == Characterization of known graph families == Several families of graphs may be characterized in terms of their minimum ranks. For n ≥ 2 {\displaystyle n\geq 2} , the complete graph Kn on n vertices has minimum rank one. The only graphs that are connected and have minimum rank one are the complete graphs. A path graph Pn on n vertices has minimum rank n − 1. The only n-vertex graphs with minimum rank n − 1 are the path graphs. A cycle graph Cn on n vertices has minimum rank n − 2. Let G {\displaystyle G} be a 2-connected graph. Then mr ( G ) = | G | − 2 {\displaystyle \operatorname {mr} (G)=|G|-2} if and only if G {\displaystyle G} is a linear 2-tree. A graph G {\displaystyle G} has mr ( G ) ≤ 2 {\displaystyle \operatorname {mr} (G)\leq 2} if and only if the complement of G {\displaystyle G} is of the form ( K s 1 ∪ K s 2 ∪ K p 1 , q 1 ∪ ⋯ ∪ K p k , q k ) ∨ K r {\displaystyle (K_{s_{1}}\cup K_{s_{2}}\cup K_{p_{1},q_{1}}\cup \cdots \cup K_{p_{k},q_{k}})\vee K_{r}} for appropriate nonnegative integers k , s 1 , s 2 , p 1 , q 1 , … , p k , q k , r {\displaystyle k,s_{1},s_{2},p_{1},q_{1},\ldots ,p_{k},q_{k},r} with p i + q i > 0 {\displaystyle p_{i}+q_{i}>0} for all i = 1 , … , k {\displaystyle i=1,\ldots ,k} . == Notes == == References == Fallat, Shaun; Hogben, Leslie, "The minimum rank of symmetric matrices described by a graph: A survey", Linear Algebra and its Applications 426 (2007) (PDF), pp. 558–582.
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Wikipedia:Minkowski content#0
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The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets. It is typically applied to fractal boundaries of domains in the Euclidean space, but it can also be used in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure. == Definition == For A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} , and each integer m with 0 ≤ m ≤ n {\displaystyle 0\leq m\leq n} , the m-dimensional upper Minkowski content is M ∗ m ( A ) = lim sup r → 0 + μ ( { x : d ( x , A ) < r } ) α ( n − m ) r n − m {\displaystyle M^{*m}(A)=\limsup _{r\to 0^{+}}{\frac {\mu (\{x:d(x,A)<r\})}{\alpha (n-m)r^{n-m}}}} and the m-dimensional lower Minkowski content is defined as M ∗ m ( A ) = lim inf r → 0 + μ ( { x : d ( x , A ) < r } ) α ( n − m ) r n − m {\displaystyle M_{*}^{m}(A)=\liminf _{r\to 0^{+}}{\frac {\mu (\{x:d(x,A)<r\})}{\alpha (n-m)r^{n-m}}}} where α ( n − m ) r n − m {\displaystyle \alpha (n-m)r^{n-m}} is the volume of the (n−m)-ball of radius r and μ {\displaystyle \mu } is an n {\displaystyle n} -dimensional Lebesgue measure. If the upper and lower m-dimensional Minkowski content of A are equal, then their common value is called the Minkowski content Mm(A). == Properties == The Minkowski content is (generally) not a measure. In particular, the m-dimensional Minkowski content in Rn is not a measure unless m = 0, in which case it is the counting measure. Indeed, clearly the Minkowski content assigns the same value to the set A as well as its closure. If A is a closed m-rectifiable set in Rn, given as the image of a bounded set from Rm under a Lipschitz function, then the m-dimensional Minkowski content of A exists, and is equal to the m-dimensional Hausdorff measure of A. == See also == Gaussian isoperimetric inequality Geometric measure theory Isoperimetric inequality in higher dimensions Minkowski–Bouligand dimension == Footnotes == == References == Federer, Herbert (1969), Geometric Measure Theory, Springer-Verlag, ISBN 3-540-60656-4. Krantz, Steven G.; Parks, Harold R. (1999), The geometry of domains in space, Birkhäuser Advanced Texts: Basler Lehrbücher, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-4097-5, MR 1730695.
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Wikipedia:Minkowski–Bouligand dimension#0
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In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a bounded set S {\textstyle S} in a Euclidean space R n {\textstyle \mathbb {R} ^{n}} , or more generally in a metric space ( X , d ) {\textstyle (X,d)} . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand. To calculate this dimension for a fractal S {\textstyle S} , imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm. Suppose that N ( ε ) {\textstyle N(\varepsilon )} is the number of boxes of side length ε {\textstyle \varepsilon } required to cover the set. Then the box-counting dimension is defined as dim box ( S ) := lim ε → 0 log N ( ε ) log ( 1 / ε ) . {\displaystyle \dim _{\text{box}}(S):=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}.} Roughly speaking, this means that the dimension is the exponent d {\textstyle d} such that N ( ε ) ≈ C ε − d {\textstyle N(\varepsilon )\approx C\varepsilon ^{-d}} , which is what one would expect in the trivial case where S {\textstyle S} is a smooth space (a manifold) of integer dimension d {\textstyle d} . If the above limit does not exist, one may still take the limit superior and limit inferior, which respectively define the upper box dimension and lower box dimension. The upper box dimension is sometimes called the entropy dimension, Kolmogorov dimension, Kolmogorov capacity, limit capacity or upper Minkowski dimension, while the lower box dimension is also called the lower Minkowski dimension. The upper and lower box dimensions are strongly related to the more popular Hausdorff dimension. Only in very special applications is it important to distinguish between the three (see below). Yet another measure of fractal dimension is the correlation dimension. == Alternative definitions == It is possible to define the box dimensions using balls, with either the covering number or the packing number. The covering number N covering ( ε ) {\textstyle N_{\text{covering}}(\varepsilon )} is the minimal number of open balls of radius ε {\textstyle \varepsilon } required to cover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number N covering ′ ( ε ) {\textstyle N'_{\text{covering}}(\varepsilon )} , which is defined the same way but with the additional requirement that the centers of the open balls lie in the set S. The packing number N packing ( ε ) {\textstyle N_{\text{packing}}(\varepsilon )} is the maximal number of disjoint open balls of radius ε {\textstyle \varepsilon } one can situate such that their centers would be in the fractal. While N {\textstyle N} , N covering {\textstyle N_{\text{covering}}} , N covering ′ {\textstyle N'_{\text{covering}}} and N packing {\textstyle N_{\text{packing}}} are not exactly identical, they are closely related to each other and give rise to identical definitions of the upper and lower box dimensions. This is easy to show once the following inequalities are proven: N packing ( ε ) ≤ N covering ′ ( ε ) ≤ N covering ( ε / 2 ) ≤ N covering ′ ( ε / 2 ) ≤ N packing ( ε / 4 ) . {\displaystyle N_{\text{packing}}(\varepsilon )\leq N'_{\text{covering}}(\varepsilon )\leq N_{\text{covering}}(\varepsilon /2)\leq N'_{\text{covering}}(\varepsilon /2)\leq N_{\text{packing}}(\varepsilon /4).} These, in turn, follow either by definition or with little effort from the triangle inequality. The advantage of using balls rather than squares is that this definition generalizes to any metric space. In other words, the box definition is extrinsic – one assumes the fractal space S is contained in a Euclidean space, and defines boxes according to the external geometry of the containing space. However, the dimension of S should be intrinsic, independent of the environment into which S is placed, and the ball definition can be formulated intrinsically. One defines an internal ball as all points of S within a certain distance of a chosen center, and one counts such balls to get the dimension. (More precisely, the Ncovering definition is extrinsic, but the other two are intrinsic.) The advantage of using boxes is that in many cases N(ε) may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal. The logarithm of the packing and covering numbers are sometimes referred to as entropy numbers and are somewhat analogous to the concepts of thermodynamic entropy and information-theoretic entropy, in that they measure the amount of "disorder" in the metric space or fractal at scale ε and also measure how many bits or digits one would need to specify a point of the space to accuracy ε. Another equivalent (extrinsic) definition for the box-counting dimension is given by the formula dim box ( S ) = n − lim r → 0 log vol ( S r ) log r , {\displaystyle \dim _{\text{box}}(S)=n-\lim _{r\to 0}{\frac {\log {\text{vol}}(S_{r})}{\log r}},} where for each r > 0, the set S r {\textstyle S_{r}} is defined to be the r-neighborhood of S, i.e. the set of all points in R n {\textstyle R^{n}} that are at distance less than r from S (or equivalently, S r {\textstyle S_{r}} is the union of all the open balls of radius r which have a center that is a member of S). == Properties == The upper box dimension is finitely stable, i.e. if {A1, ..., An} is a finite collection of sets, then dim upper box ( A 1 ∪ ⋯ ∪ A n ) = max { dim upper box ( A 1 ) , … , dim upper box ( A n ) } . {\displaystyle \dim _{\text{upper box}}(A_{1}\cup \dotsb \cup A_{n})=\max\{\dim _{\text{upper box}}(A_{1}),\dots ,\dim _{\text{upper box}}(A_{n})\}.} However, it is not countably stable, i.e. this equality does not hold for an infinite sequence of sets. For example, the box dimension of a single point is 0, but the box dimension of the collection of rational numbers in the interval [0, 1] has dimension 1. The Hausdorff dimension by comparison, is countably stable. The lower box dimension, on the other hand, is not even finitely stable. An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If A and B are two sets in a Euclidean space, then A + B is formed by taking all the pairs of points a, b where a is from A and b is from B and adding a + b. One has dim upper box ( A + B ) ≤ dim upper box ( A ) + dim upper box ( B ) . {\displaystyle \dim _{\text{upper box}}(A+B)\leq \dim _{\text{upper box}}(A)+\dim _{\text{upper box}}(B).} == Relations to the Hausdorff dimension == The box-counting dimension is one of a number of definitions for dimension that can be applied to fractals. For many well-behaved fractals all these dimensions are equal; in particular, these dimensions coincide whenever the fractal satisfies the open set condition (OSC). For example, the Hausdorff dimension, lower box dimension, and upper box dimension of the Cantor set are all equal to log(2)/log(3). However, the definitions are not equivalent. The box dimensions and the Hausdorff dimension are related by the inequality dim Haus ≤ dim lower box ≤ dim upper box . {\displaystyle \dim _{\text{Haus}}\leq \dim _{\text{lower box}}\leq \dim _{\text{upper box}}.} In general, both inequalities may be strict. The upper box dimension may be bigger than the lower box dimension if the fractal has different behaviour in different scales. For example, examine the set of numbers in the interval [0, 1] satisfying the condition The digits in the "odd place-intervals", i.e. between digits 22n+1 and 22n+2 − 1, are not restricted and may take any value. This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating N(ε) for ε = 10 − 2 n {\displaystyle \varepsilon =10^{-2^{n}}} and noting that their values behave differently for n even and odd. Another example: the set of rational numbers Q {\textstyle \mathbb {Q} } , a countable set with dim Haus = 0 {\textstyle \dim _{\text{Haus}}=0} , has dim box = 1 {\textstyle \dim _{\text{box}}=1} because its closure, R {\textstyle \mathbb {R} } , has dimension 1. In fact, dim box { 0 , 1 , 1 2 , 1 3 , 1 4 , … } = 1 2 . {\displaystyle \dim _{\text{box}}\left\{0,1,{\frac {1}{2}},{\frac {1}{3}},{\frac {1}{4}},\ldots \right\}={\frac {1}{2}}.} These examples show that adding a countable set can change box dimension, demonstrating a kind of instability of this dimension. == See also == Correlation dimension Packing dimension Uncertainty exponent Weyl–Berry conjecture Lacunarity == References == Falconer, Kenneth (1990). Fractal geometry: mathematical foundations and applications. Chichester: John Wiley. pp. 38–47. ISBN 0-471-92287-0. Zbl 0689.28003. Weisstein, Eric W. "Minkowski-Bouligand Dimension". MathWorld. == External links == FrakOut!: an OSS application for calculating the fractal dimension of a shape using the box counting method (Does not automatically place the boxes for you). FracLac: online user guide and software ImageJ and FracLac box counting plugin; free user-friendly open source software for digital image analysis in biology
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Wikipedia:Miodrag Petković#0
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Miodrag S. Petković (born 10 February 1948 in Niš, Serbia in the former Yugoslavia) is a mathematician and computer scientist. In 1991 he became a full professor of mathematics at the Faculty of Electronic Engineering, University of Niš in Serbia. == Biography == Petković specializes in the theory of iterative processes for solving nonlinear equations and Interval mathematics. He wrote 270 academic papers (153 in Clarivate Analytics' SCI journals) and 28 books, including four monographs Iterative Methods for Simultaneous Inclusion of Polynomial Zeros (Springer-Verlag 1989), Complex Interval Arithmetic and Its Applications (Wiley-VCH 1998), Point Estimation of Root Finding Methods (Springer-Verlag 2008), and Multipoint Methods for Solving Nonlinear Equations (Elsevier 2013). Petković papers were cited 1193 times with Hirsch index h=20, while Elsevier's Reference Manager Mendeley displays 1565 citations and h=21. He was visiting professor at the University of Oldenburg from 1989 to 2001, the Louis Pasteur University, Strasbourg (France, 1992), the University of Tsukuba (Japan, 2001), and a scientific researcher/invited lecturer at Columbia University, Harvard University, and at the universities of Freiburg, Zurich (ETH), Oldenburg, Berlin (Humboldt University), London, Sofia, Kiel, Tokyo, Tsukuba, Nagoya and Vienna. He took part at 60 conferences and congresses, and he was the invited lecturer on two world's congresses in 1992 and 1996, and several international conferences. He was a co-organizer of the international conference at the University of Kiel (Germany) 1998. Petković is an Associate Editor in Journal of Computational and Applied Mathematics and Applied Mathematics and Computation and a member of editorial board of Reliable Computing, Journal of Applied Mathematics, Journal of Mathematics and Computing Systems, Journal of Complex Analysis, Mathematical Aeterna, and Novi Sad J. Math. Petković is a member of the Serbian Scientific Society, New York Academy of Science, American Mathematical Society, GAMM, and was a member of the Serbian National Council of Science from 2010 to 2015. == Publications == Publications include: Iterative Methods for Simultaneous Inclusion of Polynomial Zeros. Lecture Notes in Mathematics. Vol. 1387. Springer-Verlag. 1989. ISBN 978-3-540-51485-5. Mathematics and Chess. Dover Publications. 1997. ISBN 0-486-29432-3. (a collection of 110 problems in algebra, geometry, and combinatorics based on the rules of the chess game – part of this book can be read on Google books) Point Estimation of Root Finding Methods. Lecture Notes in Mathematics. Vol. 1933. Springer-Verlag. 2008. ISBN 978-3-540-77850-9. Famous puzzles of great mathematicians. Providence, R.I. : American Mathematical Society. 2009. ISBN 978-0-8218-4814-2. Complex interval arithmetic and its applications. Wiley-VCH. 1998. ISBN 3-527-40134-2. (preview readable on Google books) Multipoint Methods for Solving Nonlinear Equations. Elsevier/Academic Press. 2013. ISBN 978-0-12-397013-8. == References == == External links == Personal website, list of Web of Science citation ResearchGate website, list of publications ResearchGate website, list of Web of Science citation Academic profile on Mendeley.com
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Wikipedia:Mir Maswood Ali#0
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Mir Maswood Ali (Bengali: মীর মসূদ আলী; 12 March 1929 – 18 August 2009) was a Canadian statistician and mathematician of Bengali origin. He is known for co-discovering the Ali-Mikhail-Haq copula, which is a topic of active research, both in theory and application. Ali played a key role in establishing the Journal of Statistical Research, of which the first issue appeared in 1970. The December 2008 issue of the Journal of Statistical Research was dedicated in honor of Ali. In 2008, Ali received the Qazi Motahar Husain Gold Medal Award in recognition of his contributions to statistics. Ali's research interests in statistics and mathematics included order statistics, distribution theory, characterizations, spherically symmetric and elliptically contoured distributions, multivariate statistics, and n-dimensional geometry. He published articles in well-known statistical journals, such as the Annals of Mathematical Statistics, the Journal of the Royal Statistical Society, the Journal of Multivariate Analysis, and Biometrika. Two of his most highly rated papers are in geometry, and appeared in the Pacific Journal of Mathematics. == References ==
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Wikipedia:Miranda Teboh-Ewungkem#0
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Miranda Ijang Teboh-Ewungkem (born 1974) is a Cameroonian-American applied mathematician, mathematical biologist and university professor. Her research focuses on ordinary and partial differential equations and statistical methods for modeling the dynamics and transmission of infectious diseases. == Life and work == Teboh-Ewungkem was born into a family of seven children and grew up in Cameroon, West Africa. Because her parents encouraged her to study medicine she enrolled in a biology class as an undergraduate, but after one month, she tired of her studies and switched to her best subject, math. It was during her first year at the university that Teboh-Ewungkem came down with the "worst case of malaria" she ever contracted. Later, she said, "The goal at the time was to survive, ... but ultimately the experience would inspire a master’s thesis." Since then her life's work has been to use mathematics to "examine a mosquito’s transmission of the disease-carrying parasite from one person to another." At the University of Buea in Cameroon, Teboh-Ewungkem earned a Bachelor of Science in 1996 in mathematics with a minor in computer science and a Master of Science in mathematics in 1998. From 1998 to 2002 she conducted research at Lehigh University in Bethlehem, Pennsylvania, and there she received a Master of Science in statistics in January 2003. Shortly thereafter, in May 2003, she received her doctorate under Eric Paul Salathe with a dissertation entitled: Mathematical Analysis of Oxygen and Substrate Transport Within a Multicapillary System in Skeletal Muscle. She then worked as Hsiung Visiting Assistant Professor at Lehigh University until 2004. She joined the faculty at Lafayette College in 2004 and taught as a Visiting Assistant Professor until 2006 when she was appointed Assistant Professor. She has served as an associate editor of the International Journal of Applied Mathematics & Statistics (IJAMAS) since 2006. She has given keynote lectures as an invited speaker at many conferences, including as the keynote lecture at the 2010 Southern Africa Mathematical Sciences Association (SAMSA) conference in Gaborone, Botswana, which was attended by government officials and the head of the University of Botswana. In 2009, with a grant from the U.S. National Science Foundation, she organized an international workshop and conference at the University of Buea for colleges and universities in Africa, the United States, and Europe to exchange ideas on the use of applied mathematics for health problems including AIDS and malaria. She has said that "in African countries, more than a million people die of malaria each year and 20 percent of them are children." In addition to malaria, Teboh-Ewungkem's research contributes to investigations into other mosquito-borne diseases such as Dengue fever, Zika fever, Chikungunya and lymphatic filariasis. == Memberships == Association for Women in Mathematics American Mathematical Society Society for Mathematical Biology Society of Industrial and Applied Mathematicians Mathematical Association of America Honor Society for International students Scholars-Beta Pi Chapter of Phi Beta Delta Black Women in Mathematics == Selected honors == 2003: Best Professor, Summer Excel Program, Lehigh University 2020: Fellow of the African Scientific Institute == References ==
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Wikipedia:Mireille Capitaine#0
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Mireille Capitaine is a French mathematician whose research focuses on random matrices and free probability theory. In 2012 she was a recipient of the G. de B. Robinson Award for a paper she coauthored that introduced free Bessel laws, a two-parameter family of generalizations of the free Poisson distribution. She received her PhD in 1996 from Paul Sabatier University, where she was advised by Michel Ledoux. She is currently a researcher for the French National Centre for Scientific Research (CNRS), associated with the Toulouse Institute of Mathematics. == References ==
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Wikipedia:Mireille Martin-Deschamps#0
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Mireille Martin-Deschamps is a French mathematician who studies the algebraic geometry of space curves. She was president of the Société mathématique de France. == Education and career == Martin-Deschamps studied at the École normale supérieure de jeunes filles from 1965 to 1969, and completed a doctorate in 1976 at Paris-Sud University, supervised by Pierre Samuel. She was a researcher for the French National Centre for Scientific Research from 1969 until 2003, when she became a professor at Versailles Saint-Quentin-en-Yvelines University. She retired in 2010, and the university held a colloquium in honor of her retirement. She was president of the Société mathématique de France from 1998 to 2001. As well, she served on the executive committee of the European Mathematical Society beginning in 2006. == Research == Martin-Deschamps's doctoral work was in algebraic geometry in the style of Alexander Grothendieck. Her later work involved Hilbert schemes of space curves in projective space. == References ==
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Wikipedia:Miriam Cohen#0
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Miriam Cohen (Hebrew: מרים כהן; born October 1941 died October 2023) was an Israeli mathematician and a professor in the Department of Mathematics at Ben-Gurion University of the Negev whose main areas of research are Hopf algebras, quantum groups and Noncommutative rings. == Biography == Miriam Cohen (née Hirsch) was born in Ramat Gan, British Mandate of Palestine. Her parents Dr. Hanna and Jusin Hirsch fled Nazi Germany to Palestine in 1939. She lived in Petah Tikva and joined the IDF communication corps in 1959. In 1961 she married Yair Cohen and in 1962 they left to study in the US. Miriam received her B.Sc. in Mathematics with High Honors from California State University and continued for her Ph.D. in Mathematics at UCLA where she received her M.Sc. After returning to Israel she completed her Ph.D. at Tel Aviv University under the supervision of Prof. Israel Nathan Herstein from the University of Chicago and Prof. A.A. Klein from Tel Aviv University. During these years the couple had four children (Omer, Ira, Alma and Adaya). In 1978 the family moved for idealistic reasons from Herzliya Pituach to the development town of Yeruham and lived there for 10 years. Miriam joined the faculty of the department of Mathematics at BGU and has been a member ever since. During her time as a researcher and lecturer at BGU she volunteered in educational projects in Yeruham (see below). In 1983–5 Cohen was a visiting Associate Professor at the University of Southern California and at UCLA in Los Angeles. In 1997–8 she was a visiting professor at the Mathematics Institute of Fudan University in Shanghai. She was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to Hopf algebras and their representations, and for service to the mathematical community". == Functions in the Israeli mathematical community == In 1992–94 she served as President of the Israel Mathematical Union. In 1996 she initiated and established the special track in Bioinformatics in the Department of Mathematics and Computer Sciences Department at BGU. In 1998–2002 she served as elected dean of the Faculty of Natural Sciences at BGU (the first woman in Israel to serve in this capacity) and was a member of the central steering committee and a member of the executive committee of BGU In 2001 she founded the Center of Advanced Studies in Mathematics at BGU and has serves as its director since June 2003 She serves as associate editor in Communications in Algebra . In 1993–95 she served as Chairman of the Mathematics and Computer Science Department == Additional activities == Initiation of the joint Israel Mathematical Union (IMU) and the American Mathematical Society (AMS) International meeting in 1995 and serves as member of the scientific committee of the second IMU and AMS meeting to take place in Israel in 2014. Since 2002 chair and member of the organizing and scientific committee of the Moshe Flato Colloquia series at BGU. Speaker at the Women Scientists Forum at the Israel Academy of Sciences and Humanities. Israeli delegate to the council of the European Mathematical Society (attended in this capacity the 1994 Council Meeting in Zurich). Chair of the session on Einstein and Quantum mechanics, Albert Einstein Legacy Symposium organized by the Israel Academy of Sciences and Humanities. Member of various professional committees such as the grant committee of the Clore Foundation for outstanding Ph.D. students, the professional evaluation committee of the Emet prize, the Harvey Award committee of the Technion and the advisory committee of the rector of Haifa University. == Research grants == U.S.–Israel Binational Fund (BSF) (with S. Montgomery). Quality of the completed work was rated excellent by the U.S. National Science Foundation. 1986–1989 A 3-year grant from The Basic Science Foundation, The Israel National Academy of Sciences and Humanities. 1991–1994 A 3-year grant from the Israel Ministry of Science together with Prof. Davidson of the Department of Physics at Ben-Gurion University. Prof. Nissimov and Prof. Pacheva. 1991–1994 A 3-year Grant from the Basic Science Foundation, The Israel National Academy of Sciences and Humanities. 1996–1999 A 3-year grant (with S. Westreich) from the Basic Science Foundation, the Israel National Academy of Sciences and Humanities. Research project on "Computer Algebra" jointly with the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. 2001–2005 A 3-year grant from the US-Israel Binational Fund (BSF) (jointly with A. Yekutieli, M. Artin, J.Zhang) Since 2012 A 3-year grant (with S. Westreich) from the Basic Science Foundation, the Israel National Academy of Sciences and Humanities. == Contribution to the community == Founder of the computer summer camps projects in 1980. The camps catered to children from all over Israel and were a pioneering activity at that time. Chairman of the Education Subcommittee, Project Renewal, Yeroham. Voluntary high school teacher in mathematics in Yeruham . She served as a member of the Ministry of Education's Committee on Mathematics. Chair of its subcommittee on the Junior High level. Member of the National Committee on the Information Technology Society, appointed by the Prime Minister of Israel. == References == == External links == Miriam Cohen's profile Center for Advanced Studies in Mathematics Department of Mathematics Faculty
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Wikipedia:Mirka Miller#0
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Mirka Miller (née Koutova, 9 May 1949 – 2 January 2016) was a Czech-Australian mathematician and computer scientist interested in graph theory and data security. She was a professor of electrical engineering and computer science at the University of Newcastle. == Life == Miller was born on 9 May 1949 in Rumburk, then part of Czechoslovakia, as the oldest in a family of five children. After attempting to escape Czechoslovakia in 1968, stopped because of her companion's illness, she became a student at Charles University before successfully escaping in 1969 and becoming a refugee in Australia. Miller earned a bachelor's degree from the University of Sydney in 1976, both in mathematics and computer science, and as a student also played volleyball for the New South Wales team and then the Australia women's national volleyball team. She married ornithologist Ben Miller, became a computer programmer working with the Sydney Morning Herald and for NSW Parks and Wildlife on Lord Howe Island, and began raising a son with Miller. She separated from her husband and returned to graduate study, earning two master's degrees from the University of New England in 1983 and 1986; her mentors in these degrees were Ernie Bowen and Ivan Friš. She completed a PhD from the University of New South Wales in 1990. Her dissertation, Security of Statistical Databases, was supervised by Jennifer Seberry. She held academic positions at the University of New England from 1982 to 1991, but after marrying graph theorist Joe Ryan they both moved to the University of Newcastle. She was a faculty member at the University of Newcastle from 1992 to 2004, when she temporarily moved to the University of Ballarat, and returned to Newcastle as a research professor from 2008 until her retirement. At Newcastle, she spent many years as the only woman in the Faculty of Engineering. She retired as a professor emeritus in 2014. She also held a position at the University of West Bohemia as Conjoint Professor since 2001. She died of gastroesophageal cancer on 2 January 2016. A special issue of the Australasian Journal of Combinatorics was published in her honour in 2017, and special issues of the European Journal of Combinatorics and Journal of Discrete Algorithms followed in 2018. == Contributions == Miller was the author of two books on magic graphs, Super Edge-Antimagic Graphs: A Wealth of Problems and Some Solutions (with Martin Bača, BrownWalker Press, 2008), and (posthumously) Magic and Antimagic Graphs: Attributes, Observations and Challenges in Graph Labelings (with Bača, Joe Ryan, and Andrea Semaničová-Feňovčíková, Springer, 2019). She wrote over 200 research publications, including a widely cited survey of the degree diameter problem, supervised 20 doctoral students before her death, was the supervisor of six more at the time of her death, and helped found four workshop series on algorithms, graph theory, and networks. She was also influential in the history of graph theory in Indonesia, where she visited twice and supervised six doctoral students. An infinite family of vertex-transitive graphs with diameter two and a large number of vertices relative to their degree and diameter, the McKay–Miller–Širáň graphs, are named after Miller and her co-authors Brendan McKay and Jozef Širáň, who first constructed them in 1998. They include the Hoffman–Singleton graph as a special case. == References == == External links == Home page Mirka Miller publications indexed by Google Scholar
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Wikipedia:Miron Nicolescu#0
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Miron Nicolescu (Romanian: [miˈron nikoˈlesku]; August 27, 1903 – June 30, 1975) was a Romanian mathematician, best known for his work in real analysis and differential equations. He was president of the Romanian Academy and vice-president of the International Mathematical Union. Born in Giurgiu, the son of a teacher, he attended the Matei Basarab High School in Bucharest. After completing his undergraduate studies at the Faculty of Mathematics of the University of Bucharest in 1924, he went to Paris, where he enrolled at the École Normale Supérieure and the Sorbonne. In 1928, he completed his doctoral dissertation, Fonctions complexes dans le plan et dans l'espace, under the direction of Paul Montel. Upon returning to Romania, he taught at the University of Cernăuți until 1940, when he was named professor at the University of Bucharest. In 1936, he was elected an associate member of the Romanian Academy, and, in 1953, full member. After King Michael's Coup of August 23, 1944, Nicolescu joined the Social Democratic Party, and later became a member of the Romanian Communist Party. In 1963, he became director of the Institute of Mathematics of the Romanian Academy, a position he held until 1973. From 1966 until his death, he served as president of the Romanian Academy. Peter Freund (who met Nicolescu when he gave a lecture in Timișoara), described him as an "affable, debonair man, and a very handsome ladies' man." Nicolescu was awarded the Legion of Honour, Commander rank, and was elected in 1972 member of the German National Academy of Sciences Leopoldina. At the International Congress of Mathematicians held in Vancouver, British Columbia, Canada in 1974, he was elected vice-president of the International Mathematical Union, a position he held from 1975 until his death (his term was completed by Gheorghe Vrănceanu). A technical high school in Sector 4 of Bucharest bears his name, and so does a boulevard in Giurgiu. == Publications == Nicolesco, Miron (1935). "Recherches sur les fonctions polyharmoniques". Annales Scientifiques de l'École Normale Supérieure. 3e série (in French). 52: 183–220. doi:10.24033/asens.848. Nicolescu, Miron (1992). Opera matematică. Ecuații eliptice și parabolice (in Romanian). With a preface by Solomon Marcus. București: Editura Academiei Republicii Socialiste România. ISBN 973-27-0312-1. MR 1254660. OCLC 7248296. Nicolescu, Miron (1995). Opera matematică. Analiză reală (in Romanian). With a preface by Solomon Marcus. București: Editura Academiei Republicii Socialiste România. ISBN 973-27-0523-X. MR 1473972. OCLC 758797290. == See also == Heat equation == References ==
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Wikipedia:Miroslav Fiedler#0
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Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory. His article, "Algebraic Connectivity of Graphs", published in the Czechoslovak Math Journal in 1973, established the use of the eigenvalues of the Laplacian matrix of a graph to create tools for measuring algebraic connectivity in algebraic graph theory. Fiedler is honored by the Fiedler eigenvalue (the second smallest eigenvalue of the graph Laplacian), with its associated Fiedler eigenvector, as the names for the quantities that characterize algebraic connectivity. Since Fiedler's original contribution, this structure has become essential to large areas of research in network theory, flocking, distributed control, clustering, multi-robot applications and image segmentation. == References == == External links == Home page at the Academy of Sciences of the Czech Republic. O'Connor, John J.; Robertson, Edmund F., "Miroslav Fiedler", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Mischa Cotlar#0
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Mischa Cotlar (1913, Sarny, Russian Empire – January 16, 2007, Buenos Aires, Argentina) was a mathematician who started his scientific career in Uruguay and worked most of his life on it in Argentina and Venezuela. His contributions to mathematics are in the fields of harmonic analysis, ergodic theory and spectral theory. He introduced the Cotlar–Stein lemma. He was the author or co-author of over 80 articles in refereed journals. According to Alberto Calderón, Cotlar showed in 1955 "that theorems on singular integrals can be generalized and put in the framework of ergodic theory." According to Krause, Lacey, and Wierdl, Karl E. Petersen in 1983 published an "especially direct proof" of Cotlar's 1955 theorem. In January 1994 in Caracas, an international conference was held in his honor. == Selected publications == Aritmética abstracta, Boletín de la Facultad de Ingeniería, Montevideo, Uruguay, 1937 Teoría de anágenos, An. Soc. Ci. Argentina, 127, 1939 Familias normales de funciones no analíticas, An. Soc. Ci. Argentina 129, 1940 Un método para obtener congruencias de números de Bernoulli, Math. Notae 7, 1947 On The Foundation Of The Ergodic Theory, Actas Symposia, UNESCO, 1951 Cotlar, Mischa (July 1954). "On a theorem of Beurling and Kaplansky" (PDF). Pacific Journal of Mathematics. 4 (3): 459–466. doi:10.2140/pjm.1954.4.459. with R. Ricabarra: Cotlar, M.; Ricabarra, R. (1954). "On the Existence of Characters in Topological Groups". American Journal of Mathematics. 76 (2): 375–388. doi:10.2307/2372579. JSTOR 2372579. A combinatorial inequality and its application to L2 spaces, Math. Cuyana, 1, 1955 "Sobre lat Teoria algebraica de la Medida y el Teorema de Hahn-Banach" (PDF). Revista de la Unión Matemática Argentina. 17: 9–24. 1956. with R. Panzone: "Generalized potential operators" (PDF). Revista de la Unión Matemática Argentina. 19: 3–41. 1960. Convolution Operators and Factorization, McGill Analysis Seminar, McGill University, Montreal, 1972 "Moment Theory and Continuity of Hilbert and Poisson Transforms in L2 Spaces by M. Cotlar". Functional Analysis, Holomorphy, and Approximation Theory (1st ed.). CRC Press. 1983. doi:10.1201/9781003072577-4. ISBN 9781003072577. S2CID 236816646. "The Helson-Szegö Theorem in Lp by M. Cotlar and C. Sadosky". In: Harmonic Analysis and Partial Differential Equations: Proceedings of a Conference Held April 4-5, 1988. Contemporary Mathematics, vol. 107. American Mathematical Soc. 1990. pp. 19–37. ISBN 9780821851135. with C. Sadosky: Cotlar, M.; Sadosky, C. (1994). "Nehari and Nevanlinna-Pick Problems and Holomorphic Extensions in the Polydisk in Terms of Restricted BMO". Journal of Functional Analysis. 124: 205–210. doi:10.1006/jfan.1994.1105. with Pedro Alegría: Albgría, Pedro; Cotlar, Mischa (1998). "Generalized Toeplitz Forms and Interpolation Colligations". Mathematische Nachrichten. 190: 5–29. doi:10.1002/mana.19981900102. S2CID 121531260. == References == == External links == Mischa Cotlar at the Mathematics Genealogy Project Horváth, John (2009), "Encounters with Mischa Cotlar", Notices of the American Mathematical Society, 56 (5): 616–620, ISSN 0002-9920, MR 2509066 Zygmund, A. (2002) [1935], Trigonometric Series. Vol. I, II, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3, MR 1963498 Analysis Center, Faculty of Sciences, Central University of Venezuela Mischa Cotlar: A Biography "Barry Simon | Tales of Our Forefathers". YouTube. Rocky Mountain Mathematical Physics Seminar. November 19, 2020. (section on Mischa Cotlar from 51:04 to 57:57)
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Wikipedia:Misha Verbitsky#0
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Misha Verbitsky (Russian: Ми́ша Верби́цкий, born June 20, 1969, in Moscow) is a Russian mathematician. He works at the Instituto Nacional de Matemática Pura e Aplicada in Rio de Janeiro. He is primarily known to the general public as a controversial critic, political activist and independent music publisher. == Scientific activities == Verbitsky graduated from a Math class at the Moscow State School 57 in 1986, and has been active in mathematics since then. His principal area of interest in mathematics is differential geometry, especially geometry of hyperkähler manifolds and locally conformally Kähler manifolds. He proved an analogue of the global Torelli theorem for hyperkähler manifolds and the mirror conjecture in hyperkähler case. He also contributed to the theory of Hodge structures. His PhD thesis, titled Cohomology of compact Hyperkaehler Manifolds, was defended in 1995 at Harvard University under the supervision of David Kazhdan. He has held different positions, most prominently at the Independent University of Moscow (since 1996), the University of Glasgow (2002–2007), and the HSE Faculty of Mathematics (since 2010). He currently works at IMPA in Rio de Janeiro. == Lenin == Verbitsky's webzine :LENIN:, started around 1997, is one of the oldest Russian online projects and has been hugely influential in the shaping of Russian counter-culture. It was the first website in Russian to openly discuss topics considered taboo at the time, such as pornography and Right-wing extremism, and to create a milieu for the emerging counter-culture aesthetic. The site also contains the largest single collection of rare underground music from the ex-USSR and contemporary Russia. == Western counter-culture == While studying mathematics at Harvard University in the early 90s, Verbitsky was heavily influenced by Western counter-culture, especially Thelema and industrial music, and was the first to introduce these concepts to post-Soviet Russia via his webzine. At the same time, Verbitsky developed his political views which can be described as a mixture of Social Darwinism, National Bolshevism and Anarchism. He is also a prominent supporter of the anti-copyright movement, and has given lectures on the subject at various locations, including Oxford University. His work Anticopyright: The Book is the only Russian publication placing concepts such as open source and copyleft into historical and cultural context. == National Bolshevik Party == After graduating from Harvard with a PhD, Verbitsky moved to Russia and became a close associate (though not a member) of Eduard Limonov's National Bolshevik Party. His articles were published in a variety of newspapers and magazines, including Russkij Zhurnal, Zhurnal.Ru, Zavtra and Limonka. When the National Bolsheviks split in 1998, he joined the Eurasia Party of Alexandr Dugin. Verbitsky has given numerous talks at the Novyi Universitet, Dugin's educational vehicle, and contributed to a variety of his publications such as Elementy and Vtorzhenie. == Writing style == Verbitsky's provocative writing style can be described as both aggressive and ironic, a mixture of gonzo journalism, profanity and surreal exaggeration which instantly captures the reader's attention. The critical response to his writings ranges from anger and disgust to fascination and widespread imitation (for example, his catchphrases "So it goes, Misha" and "Kill, Kill, Kill" have been plagiarised all over the Russian web). == Ur-Realist Records == In 1998, Verbitsky founded the independent label UR-Realist Records to publish experimental and controversial underground music. Since then, over 40 albums have been released, including those of punk legend Grazhdanskaya Oborona and the neofolk band Rada i Ternovnik. == Personal life == Verbitsky currently lives in Rio de Janeiro and teaches at IMPA. In December 2009, Yuri Kuklachev, the founder of the Moscow Cat Theatre, filled a defamation lawsuit against Verbitsky. Kuklachev was seeking to recover alleged damages caused by Verbitsky quoting previously published allegations of Kuklachev's animal cruelty in one of his blog posts; moreover, Kuklachev holds Verbitsky responsible for the expletive-ridden content of anonymous comments to the post in question. In 2012, Verbitsky was convicted without his knowledge of copyright infringement. He had posted a photograph of a Russian politician who claims that his beard is trademarked. The case has been appealed. == Bibliography == Anticopyright: The Book (2002) :LENIN: an offering to Gods Unknown webzine Against Culture + Pt. 2 Chaos and Underground Culture == References == == External links == Official website List of publications in Russ.ru Biography at Marat Guelman's art page (in Russian)
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Wikipedia:Misir Mardanov#0
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Misir Jumayil oglu Mardanov (born October 3, 1946). Member of ANAS (2017), director of the AMEA Institute of Mathematics and Mechanics, former Minister of Education of Azerbaijan, doctor of physical and mathematical sciences and university professor. == Pedagogical activity == Between 1973 and 1998, he gave lectures on analytical geometry, differential geometry, mathematical methods of optimal control at the BSU Faculty of Mechanics and Mathematics. He also lectured and conducted seminars in specialist courses on optimal management and supervised students' diploma studies. == Socio-political activity == He was a member of the Board of Directors of the New Azerbaijan Party until 2013 and a member of the Political Council of the New Azerbaijan Party until 2020. == Rewards == In 2011, by the order of the President of the Republic of Dagestan, he was awarded the honorary title "Honored Scientist of the Republic of Dagestan" for his great contributions to the development of scientific cooperation between the Republics of Dagestan and Azerbaijan. In 2021, he was awarded the 1st degree "For Service to the Motherland" order. == References ==
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Wikipedia:Misiurewicz point#0
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In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. This term makes less sense for maps in greater generality that have more than one free critical point because some critical points might be periodic and others not. These points are named after the Polish-American mathematician Michał Misiurewicz, who was the first to study them. == Mathematical notation == A parameter c {\displaystyle c} is a Misiurewicz point M k , n {\displaystyle M_{k,n}} if it satisfies the equations: f c ( k ) ( z c r ) = f c ( k + n ) ( z c r ) {\displaystyle f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})} and: f c ( k − 1 ) ( z c r ) ≠ f c ( k + n − 1 ) ( z c r ) {\displaystyle f_{c}^{(k-1)}(z_{cr})\neq f_{c}^{(k+n-1)}(z_{cr})} so: M k , n = c : f c ( k ) ( z c r ) = f c ( k + n ) ( z c r ) {\displaystyle M_{k,n}=c:f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})} where: z c r {\displaystyle z_{cr}} is a critical point of f c {\displaystyle f_{c}} , k {\displaystyle k} and n {\displaystyle n} are positive integers, f c ( k ) {\displaystyle f_{c}^{(k)}} denotes the k {\displaystyle k} -th iterate of f c {\displaystyle f_{c}} . == Name == The term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent; that is, in which there exists a neighbourhood for every critical point that is not visited by the orbit of this critical point. This meaning is firmly established in the context of the dynamics of iterated interval maps. Only in very special cases does a quadratic polynomial have a strictly periodic and unique critical point. In this restricted sense, the term is used in complex dynamics; a more appropriate one would be Misiurewicz–Thurston points (after William Thurston, who investigated post-critically finite rational maps). == Quadratic maps == A complex quadratic polynomial has only one critical point. By a suitable conjugation any quadratic polynomial can be transformed into a map of the form P c ( z ) = z 2 + c {\displaystyle P_{c}(z)=z^{2}+c} which has a single critical point at z = 0 {\displaystyle z=0} . The Misiurewicz points of this family of maps are roots of the equations: P c ( k ) ( 0 ) = P c ( k + n ) ( 0 ) , {\displaystyle P_{c}^{(k)}(0)=P_{c}^{(k+n)}(0),} Subject to the condition that the critical point is not periodic, where: k is the pre-period n is the period P c ( n ) = P c ( P c ( n − 1 ) ) {\displaystyle P_{c}^{(n)}=P_{c}(P_{c}^{(n-1)})} denotes the n-fold composition of P c ( z ) = z 2 + c {\displaystyle P_{c}(z)=z^{2}+c} with itself i.e. the nth iteration of P c {\displaystyle P_{c}} . For example, the Misiurewicz points with k= 2 and n= 1, denoted by M2,1, are roots of: P c ( 2 ) ( 0 ) = P c ( 3 ) ( 0 ) ⇒ c 2 + c = ( c 2 + c ) 2 + c ⇒ c 4 + 2 c 3 = 0. {\displaystyle {\begin{aligned}&P_{c}^{(2)}(0)=P_{c}^{(3)}(0)\\\Rightarrow {}&c^{2}+c=(c^{2}+c)^{2}+c\\\Rightarrow {}&c^{4}+2c^{3}=0.\end{aligned}}} The root c= 0 is not a Misiurewicz point because the critical point is a fixed point when c= 0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point M2,1 at c = −2. === Properties of Misiurewicz points of complex quadratic mapping === Misiurewicz points belong to, and are dense in, the boundary of the Mandelbrot set. If c {\displaystyle c} is a Misiurewicz point, then the associated filled Julia set is equal to the Julia set and means the filled Julia set has no interior. If c {\displaystyle c} is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling (in particular the cycle that the critical orbit falls onto). The Mandelbrot set and Julia set J c {\displaystyle J_{c}} are locally asymptotically self-similar around Misiurewicz points. ==== Types ==== Misiurewicz points in the context of the Mandelbrot set can be classified based on several criteria. One such criterion is the number of external rays that converge on such a point. Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more external arguments (or angles). Non-branch points have exactly two external rays (these correspond to points lying on arcs within the Mandelbrot set). These non-branch points are generally more subtle and challenging to identify in visual representations. End points, or branch tips, have only one external ray converging on them. Another criterion for classifying Misiurewicz points is their appearance within a plot of a subset of the Mandelbrot set. Misiurewicz points can be found at the centers of spirals as well as at points where two or more branches meet. According to the Branch Theorem of the Mandelbrot set, all branch points of the Mandelbrot set are Misiurewicz points. Most Misiurewicz parameters within the Mandelbrot set exhibit a "center of a spiral". This occurs due to the behavior at a Misiurewicz parameter where the critical value jumps onto a repelling periodic cycle after a finite number of iterations. At each point during the cycle, the Julia set exhibits asymptotic self-similarity through complex multiplication by the derivative of this cycle. If the derivative is non-real, it implies that the Julia set near the periodic cycle has a spiral structure. Consequently, a similar spiral structure occurs in the Julia set near the critical value, and by Tan Lei's theorem, also in the Mandelbrot set near any Misiurewicz parameter for which the repelling orbit has a non-real multiplier. The visibility of the spiral shape depends on the value of this multiplier. The number of arms in the spiral corresponds to the number of branches at the Misiurewicz parameter, which in turn equals the number of branches at the critical value in the Julia set. Even the principal Misiurewicz point M 3 , 1 {\displaystyle M_{3,1}} in the 1/3-limb, located at the end of the parameter rays at angles 9/56, 11/56, and 15/56, is asymptotically a spiral with infinitely many turns, although this is difficult to discern without magnification. ==== External arguments ==== External arguments of Misiurewicz points, measured in turns are: Rational numbers Proper fractions with an even denominator Dyadic fractions with denominator = 2 b {\displaystyle =2^{b}} and finite (terminating) expansion: 1 2 10 = 0.5 10 = 0.1 2 {\displaystyle {\frac {1}{2}}_{10}=0.5_{10}=0.1_{2}} Fractions with a denominator = a ⋅ 2 b ; a , b ∈ N ; 2 ∤ b {\displaystyle =a\cdot 2^{b};a,b\in \mathbb {N} ;2\nmid b} and repeating expansion: 1 6 10 = 1 2 × 3 10 = 0.16666... 10 = 0.0 ( 01 ) . . . 2 . {\displaystyle {\frac {1}{6}}_{10}={\frac {1}{2\times 3}}_{10}=0.16666..._{10}=0.0(01)..._{2}.} The subscript number in each of these expressions is the base of the numeral system being used. === Examples of Misiurewicz points of complex quadratic mapping === ==== End points ==== Point c = M 2 , 2 = i {\displaystyle c=M_{2,2}=i} is considered an end point as it is a tip of a filament, and the landing point of the external ray for the angle 1/6. Its critical orbit is { 0 , i , i − 1 , − i , i − 1 , − i . . . } {\displaystyle \{0,i,i-1,-i,i-1,-i...\}} . Point c = M 2 , 1 = − 2 {\displaystyle c=M_{2,1}=-2} is considered an end point as it is the endpoint of the main antenna of the Mandelbrot set. and the landing point of only one external ray (parameter ray) of angle 1/2. It is also considered an end point because its critical orbit is { 0 , − 2 , 2 , 2 , 2 , . . . } {\displaystyle \{0,-2,2,2,2,...\}} , following the Symbolic sequence = C L R R R ... with a pre-period of 2 and period of 1. ==== Branch points ==== Point c = − 0.10109636384562... + i 0.95628651080914... = M 3 , 1 {\displaystyle c=-0.10109636384562...+i\,0.95628651080914...=M_{3,1}} is considered a branch point because it is a principal Misiurewicz point of the 1/3 limb and has 3 external rays: 9/56, 11/56 and 15/56. ==== Other points ==== These are points which are not-branch and not-end points. Point c = − 0.77568377 + i 0.13646737 {\displaystyle c=-0.77568377+i\,0.13646737} is near a Misiurewicz point M 23 , 2 {\displaystyle M_{23,2}} . This can be seen because it is a center of a two-arms spiral, the landing point of 2 external rays with angles: 8388611 25165824 {\displaystyle {\frac {8388611}{25165824}}} and 8388613 25165824 {\displaystyle {\frac {8388613}{25165824}}} where the denominator is 3 ∗ 2 23 {\displaystyle 3*2^{23}} , and has a preperiodic point with pre-period k = 23 {\displaystyle k=23} and period n = 2 {\displaystyle n=2} . Point c = − 1.54368901269109 {\displaystyle c=-1.54368901269109} is near a Misiurewicz point M 3 , 1 {\displaystyle M_{3,1}} , as it is the landing point for pair of rays: 5 12 {\displaystyle {\frac {5}{12}}} , 7 12 {\displaystyle {\frac {7}{12}}} and has pre-period k = 3 {\displaystyle k=3} and period n = 1 {\displaystyle n=1} . == See also == Arithmetic dynamics Feigenbaum point Dendrite (mathematics) == References == == Further reading == == External links == Preperiodic (Misiurewicz) points in the Mandelbrot set by Evgeny Demidov M & J-sets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel Misiurewicz Point of the logistic map by J. C. Sprott
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Wikipedia:Mitchell's embedding theorem#0
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Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd. == Details == The precise statement is as follows: if A is a small abelian category, then there exists a ring R (with 1, not necessarily commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all left R-modules). The functor F yields an equivalence between A and a full subcategory of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod. Such an equivalence is necessarily additive. The theorem thus essentially says that the objects of A can be thought of as R-modules, and the morphisms as R-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective R-modules. == Sketch of the proof == Let L ⊂ Fun ( A , A b ) {\displaystyle {\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)} be the category of left exact functors from the abelian category A {\displaystyle {\mathcal {A}}} to the category of abelian groups A b {\displaystyle Ab} . First we construct a contravariant embedding H : A → L {\displaystyle H:{\mathcal {A}}\to {\mathcal {L}}} by H ( A ) = h A {\displaystyle H(A)=h^{A}} for all A ∈ A {\displaystyle A\in {\mathcal {A}}} , where h A {\displaystyle h^{A}} is the covariant hom-functor, h A ( X ) = Hom A ( A , X ) {\displaystyle h^{A}(X)=\operatorname {Hom} _{\mathcal {A}}(A,X)} . The Yoneda Lemma states that H {\displaystyle H} is fully faithful and we also get the left exactness of H {\displaystyle H} very easily because h A {\displaystyle h^{A}} is already left exact. The proof of the right exactness of H {\displaystyle H} is harder and can be read in Swan, Lecture Notes in Mathematics 76. After that we prove that L {\displaystyle {\mathcal {L}}} is an abelian category by using localization theory (also Swan). This is the hard part of the proof. It is easy to check that the abelian category L {\displaystyle {\mathcal {L}}} is an AB5 category with a generator ⨁ A ∈ A h A {\displaystyle \bigoplus _{A\in {\mathcal {A}}}h^{A}} . In other words it is a Grothendieck category and therefore has an injective cogenerator I {\displaystyle I} . The endomorphism ring R := Hom L ( I , I ) {\displaystyle R:=\operatorname {Hom} _{\mathcal {L}}(I,I)} is the ring we need for the category of R-modules. By G ( B ) = Hom L ( B , I ) {\displaystyle G(B)=\operatorname {Hom} _{\mathcal {L}}(B,I)} we get another contravariant, exact and fully faithful embedding G : L → R - M o d . {\displaystyle G:{\mathcal {L}}\to R\operatorname {-Mod} .} The composition G H : A → R - M o d {\displaystyle GH:{\mathcal {A}}\to R\operatorname {-Mod} } is the desired covariant exact and fully faithful embedding. Note that the proof of the Gabriel–Quillen embedding theorem for exact categories is almost identical. == References ==
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Wikipedia:Mitrofan Cioban#0
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Mitrofan Cioban (5 January 1942 – 2 February 2021) was a Moldovan mathematician specializing in topology, a member of the Academy of Sciences of Moldova. == Early life and education == Cioban was born in Copceac (then in Tighina County, Romania, now in Ștefan Vodă District, Moldova), the forth child out of seven of farmers Mihail and Tecla Cioban. At age 17 he enrolled in the Faculty of Mathematics and Physics of Tiraspol State University. After one year Cioban transferred to Moscow State University, where he started attending the Topology seminar of Pavel Alexandrov. He obtained his PhD in 1969 with thesis Properties of Quotient Mappings and Classification of Spaces written under the direction of Alexander Arhangelskii. Despite initially not knowing Russian, German, or English, he became well-versed in mathematical literature in these languages. == Academic and professional career == Upon graduation, Cioban returned in 1970 to Tiraspol State University as a faculty member, where he directed 17 PhD theses and served as prorector and then rector. He published over 200 papers in academic journals from 1966 to 2020, mostly under the names of Choban or Čoban, and occasionally Cioban, Ciobanu, or Coban. Starting from 1999, he served as president of the Mathematical Society of the Republic of Moldova, with headquarters in Chișinău. Cioban supervised 22 PhD students and 4 doctors for their habilitation in mathematics. == Scientific contributions == Cioban helped found the Moldovan school of general topology and made substantial contributions to the areas of topology and topological algebra, descriptive set theory, functional analysis, optimization theory, and measure theory. He published his first major result in 1966 in Proceedings of the USSR Academy of Sciences, in which he generalized a theorem of Arthur Harold Stone on the metrizability of quotient spaces. == Death == Cioban died from COVID-19 on 2 February 2021 in Chișinău during the COVID-19 pandemic in Moldova. == Publications == Chaber, Józef; Čoban, Mitrofan; Nagami, Keiô (1974). "On monotonic generalizations of Moore spaces, Čech complete spaces and p-spaces". Fundamenta Mathematicae. 84 (2): 107–119. doi:10.4064/fm-84-2-107-119. MR 0343244. Arhangel'skii, Alexander V.; Choban, Mitrofan M. (2010). "Remainders of rectifiable spaces". Topology and Its Applications. 157 (4): 789–799. doi:10.1016/j.topol.2009.08.028. MR 2585412. == References == == External links == "Choban, Mitrofan Mihailovich". math-net.ru. All-Russian Mathematical Portal. Retrieved February 3, 2021.
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Wikipedia:Mixed linear complementarity problem#0
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In mathematical optimization theory, the mixed linear complementarity problem, often abbreviated as MLCP or LMCP, is a generalization of the linear complementarity problem to include free variables. == References == Complementarity problems Algorithms for complementarity problems and generalized equations An Algorithm for the Approximate and Fast Solution of Linear Complementarity Problems
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Wikipedia:Mizan Rahman#0
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Mizan Rahman (September 16, 1932 – January 5, 2015) was a Bangladeshi Canadian mathematician and writer. He specialized in fields of mathematics such as hypergeometric series and orthogonal polynomials. He also had interests encompassing literature, philosophy, scientific skepticism, freethinking and rationalism. He co-authored Basic Hypergeometric Series with George Gasper. This book is widely considered as the standard work of choice for that subject of study. He also published ten Bengali books. == Education and career == Rahman was born and grew up in East Bengal, British India (nowadays Bangladesh). He studied at the University of Dhaka, where he obtained his B.Sc degree in Mathematics and Physics in 1953, and his M.Sc. in Applied Mathematics in 1954. He received a B.A. in Mathematics from the University of Cambridge in 1958, and an M.A. in mathematics from the same university in 1963. He was a senior lecturer at the University of Dhaka from 1958 until 1962. Rahman went to the University of New Brunswick of Canada in 1962 and received his Ph.D. in 1965 with a thesis on the kinetic theory of plasma using singular integral equation techniques. After his Ph.D, he became an assistant professor, later a full professor, at Carleton University, where he spent the rest of his career, after his retirement as a Distinguished Professor Emeritus. He unexpectedly died in Ottawa on January 5, 2015, at the age of 82. == Writing and other activities == Apart from his teaching and academic activities, Rahman wrote on various issues, particularly on those related to Bangladesh. He contributed to Internet blogs and various internet e-magazines, mainly in the Bengali language, covering his interests. He was a prolific writer and a regular contributor to Porshi, a Bengali monthly publication based in Silicon Valley, California. He was also the member of the advisory board of the Mukto-Mona, an Internet congregation of freethinkers, rationalists, skeptics, atheists and humanists of mainly Bengali and South Asian descent. == Honors and awards == Best Teaching Award (1986) Life-time membership in the Bharat Ganita Parishad (Indian Mathematical Society) Fellow of the Bangladesh Academy of Sciences (2002) Award of Excellence from Bangladesh Publications (Ottawa) (1996) == Books == English Basic Hypergeometric Series (co-author) Special Functions, q-Series and Related Topics (co-editor) The Little Garden in the Corner (prose) Bengali তীর্থ আমার গ্রাম (Tirtha is my village) (1994) লাল নদী (The Red River) (2001) অ্যালবাম (Album) (2002) প্রসঙ্গ নারী (Context - Women) (2002) অনন্যা আমার দেশ (Ananya is my country) (2004) আনন্দ নিকেতন (Ananda Niketan) (2006) দুর্যোগের পূর্বাভাস (Premonition) (2007) শুধু মাটি নয় (Not just soil) (2009) ভাবনার আত্মকথন (Autobiography of thought) (2010) শূন্য (Zero) (2012) শূন্য থেকে মহাবিশ্ব With Avijit Roy(The universe from zero)(2015) == References == == External links == OP-SF NET 22.1, January 2015. Topic #1: Mizan Rahman 1932–2015 (a short obituary by Martin Muldoon) A photo album in memory of Mizan Rahman (maintained by Tom Koornwinder)
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Wikipedia:Mladen Bestvina#0
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Mladen Bestvina (born 1959) is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah. == Life and career == Mladen Bestvina is a three-time medalist at the International Mathematical Olympiad (two silver medals in 1976 and 1978 and a bronze medal in 1977). He received a B. Sc. in 1982 from the University of Zagreb. He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh. He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990–91. Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993. He was appointed a Distinguished Professor at the University of Utah in 2008. Bestvina received the Alfred P. Sloan Fellowship in 1988–89 and a Presidential Young Investigator Award in 1988–91. Bestvina gave an invited address at the International Congress of Mathematicians in Beijing in 2002, and gave a plenary lecture at virtual ICM 2022. He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago. Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society and as an associate editor of the Annals of Mathematics. Currently he is an editorial board member for Duke Mathematical Journal, Geometric and Functional Analysis, Geometry and Topology, the Journal of Topology and Analysis, Groups, Geometry and Dynamics, Michigan Mathematical Journal, Rocky Mountain Journal of Mathematics, and Glasnik Matematicki. In 2012 he became a fellow of the American Mathematical Society. Since 2012, he has been a correspondent member of the HAZU (Croatian Academy of Science and Art). == Mathematical contributions == A 1988 monograph of Bestvina gave an abstract topological characterization of universal Menger compacta in all dimensions; previously only the cases of dimension 0 and 1 were well understood. John Walsh wrote in a review of Bestvina's monograph: 'This work, which formed the author's Ph.D. thesis at the University of Tennessee, represents a monumental step forward, having moved the status of the topological structure of higher-dimensional Menger compacta from one of "close to total ignorance" to one of "complete understanding".' In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups. The theorem provides a set of sufficient conditions for amalgamated free products and HNN extensions of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in geometric group theory and has had many applications and generalizations (e.g.). Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine) In particular their paper gives a proof of the Morgan–Shalen conjecture that a finitely generated group G admits a free isometric action on an R-tree if and only if G is a free product of surface groups, free groups and free abelian groups. A 1992 paper of Bestvina and Handel introduced the notion of a train track map for representing elements of Out(Fn). In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture, which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Examples of applications of train tracks include: a theorem of Brinkmann proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups; and others. Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative, settling a long-standing open problem. In a 1997 paper Bestvina and Brady developed a version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In particular, they constructed an example of a group which provides a counter-example to either the Whitehead asphericity conjecture or to the Eilenberg−Ganea conjecture, thus showing that at least one of these conjectures must be false. Brady subsequently used their Morse theory technique to construct the first example of a finitely presented subgroup of a word-hyperbolic group that is not itself word-hyperbolic. == Selected publications == Bestvina, Mladen, Characterizing k-dimensional universal Menger compacta. Memoirs of the American Mathematical Society, vol. 71 (1988), no. 380 Bestvina, Mladen; Feighn, Mark, Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no. 3, pp. 449–469 Bestvina, Mladen; Mess, Geoffrey, The boundary of negatively curved groups. Journal of the American Mathematical Society, vol. 4 (1991), no. 3, pp. 469–481 Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51 M. Bestvina and M. Feighn, A combination theorem for negatively curved groups. Journal of Differential Geometry, Volume 35 (1992), pp. 85–101 M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287 321 Bestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445–470 Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). I. Dynamics of exponentially-growing automorphisms. Annals of Mathematics (2), vol. 151 (2000), no. 2, pp. 517–623 Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn). II. A Kolchin type theorem. Annals of Mathematics (2), vol. 161 (2005), no. 1, pp. 1–59 Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan, The dimension of the Torelli group. Journal of the American Mathematical Society, vol. 23 (2010), no. 1, pp. 61–105 == See also == Real tree Artin group Out(Fn) Train track map Pseudo-Anosov map Word-hyperbolic group Mapping class group Whitehead conjecture == References == == External links == Mladen Bestvina, personal webpage, Department of Mathematics, University of Utah
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Wikipedia:Modal algebra#0
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In algebra and logic, a modal algebra is a structure ⟨ A , ∧ , ∨ , − , 0 , 1 , ◻ ⟩ {\displaystyle \langle A,\land ,\lor ,-,0,1,\Box \rangle } such that ⟨ A , ∧ , ∨ , − , 0 , 1 ⟩ {\displaystyle \langle A,\land ,\lor ,-,0,1\rangle } is a Boolean algebra, ◻ {\displaystyle \Box } is a unary operation on A satisfying ◻ 1 = 1 {\displaystyle \Box 1=1} and ◻ ( x ∧ y ) = ◻ x ∧ ◻ y {\displaystyle \Box (x\land y)=\Box x\land \Box y} for all x, y in A. Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics. Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame. A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying ◻ ( − ◻ x ∨ x ) = ◻ x {\displaystyle \Box (-\Box x\lor x)=\Box x} . Magari algebras correspond to provability logic. == See also == Interior algebra Heyting algebra == References == A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Logic Guides vol. 35, Oxford University Press, 1997. ISBN 0-19-853779-4
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Wikipedia:Moderne Algebra#0
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In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory gives a unified framework to study properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the variety of groups. == History == Before the nineteenth century, algebra was defined as the study of polynomials. Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of algebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal axiomatic definitions of various algebraic structures such as groups, rings, and fields. This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's Moderne Algebra, which start each chapter with a formal definition of a structure and then follow it with concrete examples. === Elementary algebra === The study of polynomial equations or algebraic equations has a long history. c. 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as rhetorical algebra and was the dominant approach up to the 16th century. Al-Khwarizmi originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until François Viète's 1591 New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637 La Géométrie. The formal study of solving symbolic equations led Leonhard Euler to accept what were then considered "nonsense" roots such as negative numbers and imaginary numbers, in the late 18th century. However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century. George Peacock's 1830 Treatise of Algebra was the first attempt to place algebra on a strictly symbolic basis. He distinguished a new symbolical algebra, distinct from the old arithmetical algebra. Whereas in arithmetical algebra a − b {\displaystyle a-b} is restricted to a ≥ b {\displaystyle a\geq b} , in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as ( − a ) ( − b ) = a b {\displaystyle (-a)(-b)=ab} , by letting a = 0 , c = 0 {\displaystyle a=0,c=0} in ( a − b ) ( c − d ) = a c + b d − a d − b c {\displaystyle (a-b)(c-d)=ac+bd-ad-bc} . Peacock used what he termed the principle of the permanence of equivalent forms to justify his argument, but his reasoning suffered from the problem of induction. For example, a b = a b {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}} holds for the nonnegative real numbers, but not for general complex numbers. === Early group theory === Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to the Galois group of a polynomial. Gauss's 1801 study of Fermat's little theorem led to the ring of integers modulo n, the multiplicative group of integers modulo n, and the more general concepts of cyclic groups and abelian groups. Klein's 1872 Erlangen program studied geometry and led to symmetry groups such as the Euclidean group and the group of projective transformations. In 1874 Lie introduced the theory of Lie groups, aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group of Möbius transformations, and its subgroups such as the modular group and Fuchsian group, based on work on automorphic functions in analysis. The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term "group", signifying a collection of permutations closed under composition. Arthur Cayley's 1854 paper On the theory of groups defined a group as a set with an associative composition operation and the identity 1, today called a monoid. In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left cancellation property b ≠ c → a ⋅ b ≠ a ⋅ c {\displaystyle b\neq c\to a\cdot b\neq a\cdot c} , similar to the modern laws for a finite abelian group. Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation. Walther von Dyck in 1882 was the first to require inverse elements as part of the definition of a group. Once this abstract group concept emerged, results were reformulated in this abstract setting. For example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group. Otto Hölder was particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed the Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator of two elements. Burnside, Frobenius, and Molien created the representation theory of finite groups at the end of the nineteenth century. J. A. de Séguier's 1905 monograph Elements of the Theory of Abstract Groups presented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916 Abstract Theory of Groups. === Early ring theory === Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented biquaternions, Cayley introduced octonions, and Grassman introduced exterior algebras. James Cockle presented tessarines in 1848 and coquaternions in 1849. William Kingdon Clifford introduced split-biquaternions in 1873. In addition Cayley introduced group algebras over the real and complex numbers in 1854 and square matrices in two papers of 1855 and 1858. Once there were sufficient examples, it remained to classify them. In an 1870 monograph, Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of an associative algebra. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the Peirce decomposition. Frobenius in 1878 and Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over R {\displaystyle \mathbb {R} } were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimple Lie algebras could be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } uniquely decomposes into the direct sums of a nilpotent algebra and a semisimple algebra that is the product of some number of simple algebras, square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called the Wedderburn principal theorem and Artin–Wedderburn theorem. For commutative rings, several areas together led to commutative ring theory. In two papers in 1828 and 1832, Gauss formulated the Gaussian integers and showed that they form a unique factorization domain (UFD) and proved the biquadratic reciprocity law. Jacobi and Eisenstein at around the same time proved a cubic reciprocity law for the Eisenstein integers. The study of Fermat's last theorem led to the algebraic integers. In 1847, Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the cyclotomic fields were UFDs, yet as Kummer pointed out, Q ( ζ 23 ) ) {\displaystyle \mathbb {Q} (\zeta _{23}))} was not a UFD. In 1846 and 1847 Kummer introduced ideal numbers and proved unique factorization into ideal primes for cyclotomic fields. Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of prime ideals, a precursor of the theory of Dedekind domains. Overall, Dedekind's work created the subject of algebraic number theory. In the 1850s, Riemann introduced the fundamental concept of a Riemann surface. Riemann's methods relied on an assumption he called Dirichlet's principle, which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the direct method in the calculus of variations. In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially M. Noether studied algebraic functions and curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring R [ x , y ] {\displaystyle \mathbb {R} [x,y]} , although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory of algebraic function fields which allowed the first rigorous definition of a Riemann surface and a rigorous proof of the Riemann–Roch theorem. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit in E. Noether's work. Lasker proved a special case of the Lasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection of primary ideals. Macauley proved the uniqueness of this decomposition. Overall, this work led to the development of algebraic geometry. In 1801 Gauss introduced binary quadratic forms over the integers and defined their equivalence. He further defined the discriminant of these forms, which is an invariant of a binary form. Between the 1860s and 1890s invariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the Jacobian and the Hessian for binary quartic forms and cubic forms. In 1868 Gordan proved that the graded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis. Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to Hilbert's basis theorem. Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given by Abraham Fraenkel in 1914. His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element. In addition, he had two axioms on "regular elements" inspired by work on the p-adic numbers, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative. Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one. In 1920, Emmy Noether, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called Idealtheorie in Ringbereichen (Ideal theory in rings'), analyzing ascending chain conditions with regard to (mathematical) ideals. The publication gave rise to the term "Noetherian ring", and several other mathematical objects being called Noetherian. Noted algebraist Irving Kaplansky called this work "revolutionary"; results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom. Artin, inspired by Noether's work, came up with the descending chain condition. These definitions marked the birth of abstract ring theory. === Early field theory === In 1801 Gauss introduced the integers mod p, where p is a prime number. Galois extended this in 1830 to finite fields with p n {\displaystyle p^{n}} elements. In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893. In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms. The first clear definition of an abstract field was due to Heinrich Martin Weber in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry. In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by their characteristic, and proved many theorems commonly seen today. === Other major areas === Solving of systems of linear equations, which led to linear algebra === Modern algebra === The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group. Questions of structure and classification of various mathematical objects came to the forefront. These processes were occurring throughout all of mathematics but became especially pronounced in algebra. Formal definitions through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. Hence such things as group theory and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of the 20th century were systematically exposed in Bartel van der Waerden's Moderne Algebra, the two-volume monograph published in 1930–1931 that reoriented the idea of algebra from the theory of equations to the theory of algebraic structures. == Basic concepts == By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer nontrivial theorems and fewer applications. Examples of algebraic structures with a single binary operation are: Magma Quasigroup Monoid Semigroup Group Examples involving several operations include: == Branches of abstract algebra == === Group theory === A group is a set G {\displaystyle G} together with a "group product", a binary operation ⋅ : G × G → G {\displaystyle \cdot :G\times G\rightarrow G} . The group satisfies the following defining axioms (cf. Group (mathematics) § Definition): Identity: there exists an element e {\displaystyle e} such that, for each element a {\displaystyle a} in G {\displaystyle G} , it holds that e ⋅ a = a ⋅ e = a {\displaystyle e\cdot a=a\cdot e=a} . Inverse: for each element a {\displaystyle a} of G {\displaystyle G} , there exists an element b {\displaystyle b} so that a ⋅ b = b ⋅ a = e {\displaystyle a\cdot b=b\cdot a=e} . Associativity: for each triplet of elements a , b , c {\displaystyle a,b,c} in G {\displaystyle G} , it holds that ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)} . === Ring theory === A ring is a set R {\displaystyle R} with two binary operations, addition: ( x , y ) ↦ x + y , {\displaystyle (x,y)\mapsto x+y,} and multiplication: ( x , y ) ↦ x y {\displaystyle (x,y)\mapsto xy} satisfying the following axioms. R {\displaystyle R} is a commutative group under addition. R {\displaystyle R} is a monoid under multiplication. Multiplication is distributive with respect to addition. == Applications == Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem. In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian. == See also == Coding theory Group theory List of publications in abstract algebra == References == === Bibliography === == Further reading == Allenby, R. B. J. T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2 Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra Gilbert, Jimmie; Gilbert, Linda (2005), Elements of Modern Algebra, Thomson Brooks/Cole, ISBN 978-0-534-40264-8 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 Sethuraman, B. A. (1996), Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94848-5 Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave, ISBN 978-0-333-79447-0 W. Keith Nicholson (2012) Introduction to Abstract Algebra, 4th edition, John Wiley & Sons ISBN 978-1-118-13535-8 . John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons == External links == Charles C. Pinter (1990) [1982] A Book of Abstract Algebra, second edition, from University of Maryland
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Wikipedia:Modeshape#0
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Mode shapes in physics are specific patterns of vibration that a structure or system can exhibit when it oscillates at its natural frequencies. These patterns describe the relative displacement of different parts of the system during vibration. In applied mathematics, mode shapes are a manifestation of eigenvectors which describe the relative displacement of two or more elements in a mechanical system or wave front. A mode shape is a deflection pattern related to a particular natural frequency and represents the relative displacement of all parts of a structure for that particular mode. == Mathematical derivation == Mode shapes have a mathematical meaning as 'eigenvectors' or 'eigenfunctions' of the eigenvalue problem which arises, studying particular solutions of the partial differential equation of a system. == See also == Normal mode Harmonic oscillator == References ==
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Wikipedia:Modular equation#0
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In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli. The most frequent use of the term modular equation is in relation to the moduli problem for elliptic curves. In that case the moduli space itself is of dimension one. That implies that any two rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two variables over the complex numbers. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular curve. This can be qualified by saying that P, in the worst case, will be of high degree and the plane curve it defines will have singular points; and the coefficients of P may be very large numbers. Further, the 'cusps' of the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read off from knowledge of P. In that sense a modular equation becomes the equation of a modular curve. Such equations first arose in the theory of multiplication of elliptic functions (geometrically, the n2-fold covering map from a 2-torus to itself given by the mapping x → n·x on the underlying group) expressed in terms of complex analysis. == See also == Modular lambda function Ramanujan's lost notebook == References ==
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Wikipedia:Modularity (networks)#0
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Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks. Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities. == Motivation == Many scientifically important problems can be represented and empirically studied using networks. For example, biological and social patterns, the World Wide Web, metabolic networks, food webs, neural networks and pathological networks are real world problems that can be mathematically represented and topologically studied to reveal some unexpected structural features. Most of these networks possess a certain community structure that has substantial importance in building an understanding regarding the dynamics of the network. For instance, a closely connected social community will imply a faster rate of transmission of information or rumor among them than a loosely connected community. Thus, if a network is represented by a number of individual nodes connected by links which signify a certain degree of interaction between the nodes, communities are defined as groups of densely interconnected nodes that are only sparsely connected with the rest of the network. Hence, it may be imperative to identify the communities in networks since the communities may have quite different properties such as node degree, clustering coefficient, betweenness, centrality, etc., from that of the average network. Modularity is one such measure, which when maximized, leads to the appearance of communities in a given network. == Definition == Modularity is the fraction of the edges that fall within the given groups minus the expected fraction if edges were distributed at random. The value of the modularity for unweighted and undirected graphs lies in the range [ − 1 / 2 , 1 ] {\displaystyle [-1/2,1]} . It is positive if the number of edges within groups exceeds the number expected on the basis of chance. For a given division of the network's vertices into some modules, modularity reflects the concentration of edges within modules compared with random distribution of links between all nodes regardless of modules. There are different methods for calculating modularity. In the most common version of the concept, the randomization of the edges is done so as to preserve the degree of each vertex. Consider a graph with n {\displaystyle n} nodes and m {\displaystyle m} links (edges) such that the graph can be partitioned into two communities using a membership variable s {\displaystyle s} . If a node v {\displaystyle v} belongs to community 1, s v = 1 {\displaystyle s_{v}=1} , or if v {\displaystyle v} belongs to community 2, s v = − 1 {\displaystyle s_{v}=-1} . Let the adjacency matrix for the network be represented by A {\displaystyle A} , where A v w = 0 {\displaystyle A_{vw}=0} means there's no edge (no interaction) between nodes v {\displaystyle v} and w {\displaystyle w} and A v w = 1 {\displaystyle A_{vw}=1} means there is an edge between the two. Also for simplicity we consider an undirected network. Thus A v w = A w v {\displaystyle A_{vw}=A_{wv}} . (It is important to note that multiple edges may exist between two nodes, but here we assess the simplest case). Modularity Q {\displaystyle Q} is then defined as the fraction of edges that fall within group 1 or 2, minus the expected number of edges within groups 1 and 2 for a random graph with the same node degree distribution as the given network. The expected number of edges shall be computed using the concept of a configuration model. The configuration model is a randomized realization of a particular network. Given a network with n {\displaystyle n} nodes, where each node v {\displaystyle v} has a node degree k v {\displaystyle k_{v}} , the configuration model cuts each edge into two halves, and then each half edge, called a stub, is rewired randomly with any other stub in the network, even allowing self-loops (which occur when a stub is rewired to another stub from the same node) and multiple-edges between the same two nodes. Thus, even though the node degree distribution of the graph remains intact, the configuration model results in a completely random network. == Expected Number of Edges Between Nodes == Now consider two nodes v {\displaystyle v} and w {\displaystyle w} , with node degrees k v {\displaystyle k_{v}} and k w {\displaystyle k_{w}} respectively, from a randomly rewired network as described above. We calculate the expected number of full edges between these nodes. Let us consider each of the k v {\displaystyle k_{v}} stubs of node v {\displaystyle v} and create associated indicator variables I i ( v , w ) {\displaystyle I_{i}^{(v,w)}} for them, i = 1 , … , k v {\displaystyle i=1,\ldots ,k_{v}} , with I i ( v , w ) = 1 {\displaystyle I_{i}^{(v,w)}=1} if the i {\displaystyle i} -th stub happens to connect to one of the k w {\displaystyle k_{w}} stubs of node w {\displaystyle w} in this particular random graph. If it does not, then I i ( v , w ) = 0 {\displaystyle I_{i}^{(v,w)}=0} . Since the i {\displaystyle i} -th stub of node v {\displaystyle v} can connect to any of the 2 m − 1 {\displaystyle 2m-1} remaining stubs with equal probability (while m {\displaystyle m} is the number of edges in the original graph), and since there are k w {\displaystyle k_{w}} stubs it can connect to associated with node w {\displaystyle w} , evidently p ( I i ( v , w ) = 1 ) = E [ I i ( v , w ) ] = k w 2 m − 1 {\displaystyle p(I_{i}^{(v,w)}=1)=E[I_{i}^{(v,w)}]={\frac {k_{w}}{2m-1}}} The total number of full edges J v w {\displaystyle J_{vw}} between v {\displaystyle v} and w {\displaystyle w} is just J v w = ∑ i = 1 k v I i ( v , w ) {\displaystyle J_{vw}=\sum _{i=1}^{k_{v}}I_{i}^{(v,w)}} , so the expected value of this quantity is E [ J v w ] = E [ ∑ i = 1 k v I i ( v , w ) ] = ∑ i = 1 k v E [ I i ( v , w ) ] = ∑ i = 1 k v k w 2 m − 1 = k v k w 2 m − 1 {\displaystyle E[J_{vw}]=E\left[\sum _{i=1}^{k_{v}}I_{i}^{(v,w)}\right]=\sum _{i=1}^{k_{v}}E[I_{i}^{(v,w)}]=\sum _{i=1}^{k_{v}}{\frac {k_{w}}{2m-1}}={\frac {k_{v}k_{w}}{2m-1}}} Many texts then make the following approximations, for random networks with a large number of edges. When m {\displaystyle m} is large, they drop the subtraction of 1 {\displaystyle 1} in the denominator above and simply use the approximate expression k v k w 2 m {\displaystyle {\frac {k_{v}k_{w}}{2m}}} for the expected number of edges between two nodes. Additionally, in a large random network, the number of self-loops and multi-edges is vanishingly small. Ignoring self-loops and multi-edges allows one to assume that there is at most one edge between any two nodes. In that case, J v w {\displaystyle J_{vw}} becomes a binary indicator variable, so its expected value is also the probability that it equals 1 {\displaystyle 1} , which means one can approximate the probability of an edge existing between nodes v {\displaystyle v} and w {\displaystyle w} as k v k w 2 m {\displaystyle {\frac {k_{v}k_{w}}{2m}}} . == Modularity == Hence, the difference between the actual number of edges between node v {\displaystyle v} and w {\displaystyle w} and the expected number of edges between them is A v w − k v k w 2 m {\displaystyle A_{vw}-{\frac {k_{v}k_{w}}{2m}}} Summing over all node pairs gives the equation for modularity, Q {\displaystyle Q} . It is important to note that Eq. 3 holds good for partitioning into two communities only. Hierarchical partitioning (i.e. partitioning into two communities, then the two sub-communities further partitioned into two smaller sub communities only to maximize Q) is a possible approach to identify multiple communities in a network. Additionally, (3) can be generalized for partitioning a network into c communities. where eij is the fraction of edges with one end vertices in community i and the other in community j: e i j = ∑ v w A v w 2 m 1 v ∈ c i 1 w ∈ c j {\displaystyle e_{ij}=\sum _{vw}{\frac {A_{vw}}{2m}}1_{v\in c_{i}}1_{w\in c_{j}}} and ai is the fraction of ends of edges that are attached to vertices in community i: a i = k i 2 m = ∑ j e i j {\displaystyle a_{i}={\frac {k_{i}}{2m}}=\sum _{j}e_{ij}} == Example of multiple community detection == We consider an undirected network with 10 nodes and 12 edges and the following adjacency matrix. The communities in the graph are represented by the red, green and blue node clusters in Fig 1. The optimal community partitions are depicted in Fig 2. == Matrix formulation == An alternative formulation of the modularity, useful particularly in spectral optimization algorithms, is as follows. Define S v r {\displaystyle S_{vr}} to be 1 {\displaystyle 1} if vertex v {\displaystyle v} belongs to group r {\displaystyle r} and 0 {\displaystyle 0} otherwise. Then δ ( c v , c w ) = ∑ r S v r S w r {\displaystyle \delta (c_{v},c_{w})=\sum _{r}S_{vr}S_{wr}} and hence Q = 1 4 m ∑ v w ∑ r [ A v w − k v k w 2 m ] S v r S w r = 1 4 m T r ( S T B S ) , {\displaystyle Q={\frac {1}{4m}}\sum _{vw}\sum _{r}\left[A_{vw}-{\frac {k_{v}k_{w}}{2m}}\right]S_{vr}S_{wr}={\frac {1}{4m}}\mathrm {Tr} (\mathbf {S} ^{\mathrm {T} }\mathbf {BS} ),} where S {\displaystyle S} is the (non-square) matrix having elements S v {\displaystyle S_{v}} and B {\displaystyle B} is the so-called modularity matrix, which has elements B v w = A v w − k v k w 2 m . {\displaystyle B_{vw}=A_{vw}-{\frac {k_{v}k_{w}}{2m}}.} All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always 0 {\displaystyle 0} . For networks divided into just two communities, one can alternatively define s v = ± 1 {\displaystyle s_{v}=\pm 1} to indicate the community to which node v {\displaystyle v} belongs, which then leads to Q = 1 4 m ∑ v w B v w s v s w = 1 4 m s T B s , {\displaystyle Q={1 \over 4m}\sum _{vw}B_{vw}s_{v}s_{w}={1 \over 4m}\mathbf {s} ^{\mathrm {T} }\mathbf {Bs} ,} where s {\displaystyle s} is the column vector with elements s v {\displaystyle s_{v}} . This function has the same form as the Hamiltonian of an Ising spin glass, a connection that has been exploited to create simple computer algorithms, for instance using simulated annealing, to maximize the modularity. The general form of the modularity for arbitrary numbers of communities is equivalent to a Potts spin glass and similar algorithms can be developed for this case also. == Overfitting == Although the method of modularity maximization is motivated by computing a deviation from a null model, this deviation is not computed in a statistically consistent manner. Because of this, the method notoriously finds high-scoring communities in its own null model (the configuration model), which by definition cannot be statistically significant. Because of this, the method cannot be used to reliably obtain statistically significant community structure in empirical networks. == Resolution limit == Modularity compares the number of edges inside a cluster with the expected number of edges that one would find in the cluster if the network were a random network with the same number of nodes and where each node keeps its degree, but edges are otherwise randomly attached. This random null model implicitly assumes that each node can get attached to any other node of the network. This assumption is however unreasonable if the network is very large, as the horizon of a node includes a small part of the network, ignoring most of it. Moreover, this implies that the expected number of edges between two groups of nodes decreases if the size of the network increases. So, if a network is large enough, the expected number of edges between two groups of nodes in modularity's null model may be smaller than one. If this happens, a single edge between the two clusters would be interpreted by modularity as a sign of a strong correlation between the two clusters, and optimizing modularity would lead to the merging of the two clusters, independently of the clusters' features. So, even weakly interconnected complete graphs, which have the highest possible density of internal edges, and represent the best identifiable communities, would be merged by modularity optimization if the network were sufficiently large. For this reason, optimizing modularity in large networks would fail to resolve small communities, even when they are well defined. This bias is inevitable for methods like modularity optimization, which rely on a global null model. == Multiresolution methods == There are two main approaches which try to solve the resolution limit within the modularity context: the addition of a resistance r to every node, in the form of a self-loop, which increases (r>0) or decreases (r<0) the aversion of nodes to form communities; or the addition of a parameter γ>0 in front of the null-case term in the definition of modularity, which controls the relative importance between internal links of the communities and the null model. Optimizing modularity for values of these parameters in their respective appropriate ranges, it is possible to recover the whole mesoscale of the network, from the macroscale in which all nodes belong to the same community, to the microscale in which every node forms its own community, hence the name multiresolution methods. However, it has been shown that these methods have limitations when communities are very heterogeneous in size. == Software Tools == There are a couple of software tools available that are able to compute clusterings in graphs with good modularity. Original implementation of the multi-level Louvain method. The Leiden algorithm which additionally avoids unconnected communities. The Vienna Graph Clustering (VieClus) algorithm, a parallel memetic algorithm. == See also == Complex network Community structure Null model Percolation theory == References ==
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Wikipedia:Module homomorphism#0
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In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function f : M → N {\displaystyle f:M\to N} is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R, f ( x + y ) = f ( x ) + f ( y ) , {\displaystyle f(x+y)=f(x)+f(y),} f ( r x ) = r f ( x ) . {\displaystyle f(rx)=rf(x).} In other words, f is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If M, N are right R-modules, then the second condition is replaced with f ( x r ) = f ( x ) r . {\displaystyle f(xr)=f(x)r.} The preimage of the zero element under f is called the kernel of f. The set of all module homomorphisms from M to N is denoted by Hom R ( M , N ) {\displaystyle \operatorname {Hom} _{R}(M,N)} . It is an abelian group (under pointwise addition) but is not necessarily a module unless R is commutative. The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorphism. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules. == Terminology == A module homomorphism is called a module isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups. The isomorphism theorems hold for module homomorphisms. A module homomorphism from a module M to itself is called an endomorphism and an isomorphism from M to itself an automorphism. One writes End R ( M ) = Hom R ( M , M ) {\displaystyle \operatorname {End} _{R}(M)=\operatorname {Hom} _{R}(M,M)} for the set of all endomorphisms of a module M. It is not only an abelian group but is also a ring with multiplication given by function composition, called the endomorphism ring of M. The group of units of this ring is the automorphism group of M. Schur's lemma says that a homomorphism between simple modules (modules with no non-trivial submodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a division ring. In the language of the category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism. == Examples == The zero map M → N that maps every element to zero. A linear transformation between vector spaces. Hom Z ( Z / n , Z / m ) = Z / gcd ( n , m ) {\displaystyle \operatorname {Hom} _{\mathbb {Z} }(\mathbb {Z} /n,\mathbb {Z} /m)=\mathbb {Z} /\operatorname {gcd} (n,m)} . For a commutative ring R and ideals I, J, there is the canonical identification Hom R ( R / I , R / J ) = { r ∈ R | r I ⊂ J } / J {\displaystyle \operatorname {Hom} _{R}(R/I,R/J)=\{r\in R|rI\subset J\}/J} given by f ↦ f ( 1 ) {\displaystyle f\mapsto f(1)} . In particular, Hom R ( R / I , R ) {\displaystyle \operatorname {Hom} _{R}(R/I,R)} is the annihilator of I. Given a ring R and an element r, let l r : R → R {\displaystyle l_{r}:R\to R} denote the left multiplication by r. Then for any s, t in R, l r ( s t ) = r s t = l r ( s ) t {\displaystyle l_{r}(st)=rst=l_{r}(s)t} . That is, l r {\displaystyle l_{r}} is right R-linear. For any ring R, End R ( R ) = R {\displaystyle \operatorname {End} _{R}(R)=R} as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation R → ∼ End R ( R ) , r ↦ l r {\displaystyle R{\overset {\sim }{\to }}\operatorname {End} _{R}(R),\,r\mapsto l_{r}} . Similarly, End R ( R ) = R o p {\displaystyle \operatorname {End} _{R}(R)=R^{op}} as rings when R is viewed as a left module over itself. Textbooks or other references usually specify which convention is used. Hom R ( R , M ) = M {\displaystyle \operatorname {Hom} _{R}(R,M)=M} through f ↦ f ( 1 ) {\displaystyle f\mapsto f(1)} for any left module M. (The module structure on Hom here comes from the right R-action on R; see #Module structures on Hom below.) Hom R ( M , R ) {\displaystyle \operatorname {Hom} _{R}(M,R)} is called the dual module of M; it is a left (resp. right) module if M is a right (resp. left) module over R with the module structure coming from the R-action on R. It is denoted by M ∗ {\displaystyle M^{*}} . Given a ring homomorphism R → S of commutative rings and an S-module M, an R-linear map θ: S → M is called a derivation if for any f, g in S, θ(f g) = f θ(g) + θ(f) g. If S, T are unital associative algebras over a ring R, then an algebra homomorphism from S to T is a ring homomorphism that is also an R-module homomorphism. == Module structures on Hom == In short, Hom inherits a ring action that was not used up to form Hom. More precise, let M, N be left R-modules. Suppose M has a right action of a ring S that commutes with the R-action; i.e., M is an (R, S)-module. Then Hom R ( M , N ) {\displaystyle \operatorname {Hom} _{R}(M,N)} has the structure of a left S-module defined by: for s in S and x in M, ( s ⋅ f ) ( x ) = f ( x s ) . {\displaystyle (s\cdot f)(x)=f(xs).} It is well-defined (i.e., s ⋅ f {\displaystyle s\cdot f} is R-linear) since ( s ⋅ f ) ( r x ) = f ( r x s ) = r f ( x s ) = r ( s ⋅ f ) ( x ) , {\displaystyle (s\cdot f)(rx)=f(rxs)=rf(xs)=r(s\cdot f)(x),} and s ⋅ f {\displaystyle s\cdot f} is a ring action since ( s t ⋅ f ) ( x ) = f ( x s t ) = ( t ⋅ f ) ( x s ) = s ⋅ ( t ⋅ f ) ( x ) {\displaystyle (st\cdot f)(x)=f(xst)=(t\cdot f)(xs)=s\cdot (t\cdot f)(x)} . Note: the above verification would "fail" if one used the left R-action in place of the right S-action. In this sense, Hom is often said to "use up" the R-action. Similarly, if M is a left R-module and N is an (R, S)-module, then Hom R ( M , N ) {\displaystyle \operatorname {Hom} _{R}(M,N)} is a right S-module by ( f ⋅ s ) ( x ) = f ( x ) s {\displaystyle (f\cdot s)(x)=f(x)s} . == A matrix representation == The relationship between matrices and linear transformations in linear algebra generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right R-module U, there is the canonical isomorphism of the abelian groups Hom R ( U ⊕ n , U ⊕ m ) → ∼ f ↦ [ f i j ] M m , n ( End R ( U ) ) {\displaystyle \operatorname {Hom} _{R}(U^{\oplus n},U^{\oplus m}){\overset {f\mapsto [f_{ij}]}{\underset {\sim }{\to }}}M_{m,n}(\operatorname {End} _{R}(U))} obtained by viewing U ⊕ n {\displaystyle U^{\oplus n}} consisting of column vectors and then writing f as an m × n matrix. In particular, viewing R as a right R-module and using End R ( R ) ≃ R {\displaystyle \operatorname {End} _{R}(R)\simeq R} , one has End R ( R n ) ≃ M n ( R ) {\displaystyle \operatorname {End} _{R}(R^{n})\simeq M_{n}(R)} , which turns out to be a ring isomorphism (as a composition corresponds to a matrix multiplication). Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rank free modules, then a choice of an ordered basis corresponds to a choice of an isomorphism F ≃ R n {\displaystyle F\simeq R^{n}} . The above procedure then gives the matrix representation with respect to such choices of the bases. For more general modules, matrix representations may either lack uniqueness or not exist. == Defining == In practice, one often defines a module homomorphism by specifying its values on a generating set. More precisely, let M and N be left R-modules. Suppose a subset S generates M; i.e., there is a surjection F → M {\displaystyle F\to M} with a free module F with a basis indexed by S and kernel K (i.e., one has a free presentation). Then to give a module homomorphism M → N {\displaystyle M\to N} is to give a module homomorphism F → N {\displaystyle F\to N} that kills K (i.e., maps K to zero). == Operations == If f : M → N {\displaystyle f:M\to N} and g : M ′ → N ′ {\displaystyle g:M'\to N'} are module homomorphisms, then their direct sum is f ⊕ g : M ⊕ M ′ → N ⊕ N ′ , ( x , y ) ↦ ( f ( x ) , g ( y ) ) {\displaystyle f\oplus g:M\oplus M'\to N\oplus N',\,(x,y)\mapsto (f(x),g(y))} and their tensor product is f ⊗ g : M ⊗ M ′ → N ⊗ N ′ , x ⊗ y ↦ f ( x ) ⊗ g ( y ) . {\displaystyle f\otimes g:M\otimes M'\to N\otimes N',\,x\otimes y\mapsto f(x)\otimes g(y).} Let f : M → N {\displaystyle f:M\to N} be a module homomorphism between left modules. The graph Γf of f is the submodule of M ⊕ N given by Γ f = { ( x , f ( x ) ) | x ∈ M } {\displaystyle \Gamma _{f}=\{(x,f(x))|x\in M\}} , which is the image of the module homomorphism M → M ⊕ N, x → (x, f(x)), called the graph morphism. The transpose of f is f ∗ : N ∗ → M ∗ , f ∗ ( α ) = α ∘ f . {\displaystyle f^{*}:N^{*}\to M^{*},\,f^{*}(\alpha )=\alpha \circ f.} If f is an isomorphism, then the transpose of the inverse of f is called the contragredient of f. == Exact sequences == Consider a sequence of module homomorphisms ⋯ ⟶ f 3 M 2 ⟶ f 2 M 1 ⟶ f 1 M 0 ⟶ f 0 M − 1 ⟶ f − 1 ⋯ . {\displaystyle \cdots {\overset {f_{3}}{\longrightarrow }}M_{2}{\overset {f_{2}}{\longrightarrow }}M_{1}{\overset {f_{1}}{\longrightarrow }}M_{0}{\overset {f_{0}}{\longrightarrow }}M_{-1}{\overset {f_{-1}}{\longrightarrow }}\cdots .} Such a sequence is called a chain complex (or often just complex) if each composition is zero; i.e., f i ∘ f i + 1 = 0 {\displaystyle f_{i}\circ f_{i+1}=0} or equivalently the image of f i + 1 {\displaystyle f_{i+1}} is contained in the kernel of f i {\displaystyle f_{i}} . (If the numbers increase instead of decrease, then it is called a cochain complex; e.g., de Rham complex.) A chain complex is called an exact sequence if im ( f i + 1 ) = ker ( f i ) {\displaystyle \operatorname {im} (f_{i+1})=\operatorname {ker} (f_{i})} . A special case of an exact sequence is a short exact sequence: 0 → A → f B → g C → 0 {\displaystyle 0\to A{\overset {f}{\to }}B{\overset {g}{\to }}C\to 0} where f {\displaystyle f} is injective, the kernel of g {\displaystyle g} is the image of f {\displaystyle f} and g {\displaystyle g} is surjective. Any module homomorphism f : M → N {\displaystyle f:M\to N} defines an exact sequence 0 → K → M → f N → C → 0 , {\displaystyle 0\to K\to M{\overset {f}{\to }}N\to C\to 0,} where K {\displaystyle K} is the kernel of f {\displaystyle f} , and C {\displaystyle C} is the cokernel, that is the quotient of N {\displaystyle N} by the image of f {\displaystyle f} . In the case of modules over a commutative ring, a sequence is exact if and only if it is exact at all the maximal ideals; that is all sequences 0 → A m → f B m → g C m → 0 {\displaystyle 0\to A_{\mathfrak {m}}{\overset {f}{\to }}B_{\mathfrak {m}}{\overset {g}{\to }}C_{\mathfrak {m}}\to 0} are exact, where the subscript m {\displaystyle {\mathfrak {m}}} means the localization at a maximal ideal m {\displaystyle {\mathfrak {m}}} . If f : M → B , g : N → B {\displaystyle f:M\to B,g:N\to B} are module homomorphisms, then they are said to form a fiber square (or pullback square), denoted by M ×B N, if it fits into 0 → M × B N → M × N → ϕ B → 0 {\displaystyle 0\to M\times _{B}N\to M\times N{\overset {\phi }{\to }}B\to 0} where ϕ ( x , y ) = f ( x ) − g ( x ) {\displaystyle \phi (x,y)=f(x)-g(x)} . Example: Let B ⊂ A {\displaystyle B\subset A} be commutative rings, and let I be the annihilator of the quotient B-module A/B (which is an ideal of A). Then canonical maps A → A / I , B / I → A / I {\displaystyle A\to A/I,B/I\to A/I} form a fiber square with B = A × A / I B / I . {\displaystyle B=A\times _{A/I}B/I.} == Endomorphisms of finitely generated modules == Let ϕ : M → M {\displaystyle \phi :M\to M} be an endomorphism between finitely generated R-modules for a commutative ring R. Then ϕ {\displaystyle \phi } is killed by its characteristic polynomial relative to the generators of M; see Nakayama's lemma#Proof. If ϕ {\displaystyle \phi } is surjective, then it is injective. See also: Herbrand quotient (which can be defined for any endomorphism with some finiteness conditions.) == Variant: additive relations == An additive relation M → N {\displaystyle M\to N} from a module M to a module N is a submodule of M ⊕ N . {\displaystyle M\oplus N.} In other words, it is a "many-valued" homomorphism defined on some submodule of M. The inverse f − 1 {\displaystyle f^{-1}} of f is the submodule { ( y , x ) | ( x , y ) ∈ f } {\displaystyle \{(y,x)|(x,y)\in f\}} . Any additive relation f determines a homomorphism from a submodule of M to a quotient of N D ( f ) → N / { y | ( 0 , y ) ∈ f } {\displaystyle D(f)\to N/\{y|(0,y)\in f\}} where D ( f ) {\displaystyle D(f)} consists of all elements x in M such that (x, y) belongs to f for some y in N. A transgression that arises from a spectral sequence is an example of an additive relation. == See also == Mapping cone (homological algebra) Smith normal form Chain complex Pairing == Notes ==
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Wikipedia:Module spectrum#0
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In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra. The ∞-category of (say right) module spectra is stable; hence, it can be considered as either analog or generalization of the derived category of modules over a ring. == K-theory == Lurie defines the K-theory of a ring spectrum R to be the K-theory of the ∞-category of perfect modules over R (a perfect module being defined as a compact object in the ∞-category of module spectra.) == See also == G-spectrum == References == J. Lurie, Lecture 19: Algebraic K-theory of Ring Spectra
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Wikipedia:Moedomo Soedigdomarto#0
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Moedomo Soedigdomarto, also spelled Mudomo Sudigdomarto, (29 November 1927, Magetan – 5 November 2005, Bandung) was an Indonesian mathematician, educator and professor at the Bandung Institute of Technology, of which he was rector. Soedigdomarto was one of the first Indonesians to obtain a Ph.D. in mathematics, which he earned from the University of Illinois Urbana-Champaign, with a dissertation titled "A Representation Theory for the Laplace Transform of Vector-Valued Functions", in 1959 at the age of 32, under the orientation of Robert Gardner Bartle. Soedigdomarto was the first Indonesian to have a paper recorded in Mathematical Reviews (Moedomo and J. J. Uhl, Jr. "Radon-Nikodym theorems for the Bochner and Pettis integrals" published in the Pacific Journal of Mathematics in 1971). == References ==
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Wikipedia:Mohamed Amine Khamsi#0
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Mohamed Amine Khamsi (born 1959 in Morocco) is an American/Moroccan mathematician known for his work in nonlinear functional analysis, the fixed point theory, and metric spaces. He has made notable contributions to the fixed point theory of metric spaces, particularly in developing the theory of modular function spaces and their applications in data science. He graduated from the prestigious École Polytechnique in 1983 after attending the equally prestigious Lycée Louis-le-Grand in Paris, France. He completed his PhD, entitled "La propriété du point fixe dans les espaces de Banach et les espaces Métriques," at the Pierre-and-Marie-Curie University in May 1987 under the supervision of Gilles Godefroy. His early research laid the foundation for significant advances in fixed point theory. Since 1989, Khamsi has been a faculty member at the University of Texas at El Paso (UTEP), where he became a full professor of mathematics in 1999. During his tenure at UTEP, he was instrumental in reforming the Differential Equations course, significantly improving student success and retention. Currently, Khamsi is a professor of mathematics at Khalifa University in Abu Dhabi, where his research continues to impact fields such as nonlinear analysis, fixed point theory, and their applications in optimization and data science. == Academic Career == Dr. Khamsi has held visiting positions at the University of Southern California and the University of Rhode Island from 1987 to 1989. His research collaborations with mathematicians like W. M. Kozlowski have led to new developments in Modular Function Spaces and their applications to nonlinear problems. He is the co-author, with W. A. Kirk, of the widely cited book An Introduction to Metric Spaces and Fixed Point Theory, published by Wiley in 2001. == Editorial Roles == Khamsi has served as an associate editor for several prestigious journals, including Fixed Point Theory, the Arabian Journal of Mathematics, and the Moroccan Journal of Pure and Applied Analysis. == Awards and Recognitions == Dr. Khamsi has received numerous accolades throughout his career, recognizing his contributions to mathematics and his commitment to teaching and mentorship. == Selected Publications == Khamsi, M. A., and W. A. Kirk. An Introduction to Metric Spaces and Fixed Point Theory. Wiley, 2001. Khamsi, M. A., Kozlowski, W. M. "Fixed Point Theory in Modular Function Spaces." Birkhäuser, 2015. Khamsi, M. A., "Modular Uniform Convexity Structures and Applications to Boundary Value Problems with Non-Standard Growth," Journal of Mathematical Analysis and Applications, 2024. == Bibliography == Aksoy, Asuman; Khamsi, Mohamed A. (1990). Nonstandard Methods in fixed point theory. Springer Verlag. ISBN 0-387-97364-8. Kirk, William A.; Khamsi, Mohamed A. (2001). An Introduction to Metric Spaces and Fixed Point Theory. John Wiley, New York. ISBN 978-0-471-41825-2. == References == == External links == Mohamed A Khamsi's personal webpage at the University of Texas at El Paso S.O.S. Mathematics Personal website
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Wikipedia:Mohamed Amine Sbihi#0
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Mohamed Amine Sbihi (Arabic: محمد أمين الصبيحي - born 1954, Salé) is a Moroccan politician of the Party of Progress and Socialism. Between 3 January 2012 and 6 April 2017, he held the position of Minister of Culture in Abdelilah Benkirane's government. He was succeeded by Mohamed Laaraj. He was professor of Statistics and Mathematics at the Mohammed V University of Rabat and al-Akhawayn University of Ifrane. In December 2021, Sbihi was named ambassador to Greece and Cyprus by King Mohammed VI. On 19 January 2022, he presented credentials to the President of the Hellenic Republic Katerina Sakellaropoulou. == See also == Cabinet of Morocco == References == == External links == Ministry of Culture
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Wikipedia:Mohamed El-Amin Ahmed El-Tom#0
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Mohamed El-Amin Ahmed El-Tom (Arabic: محمد الأمين أحمد التوم; born October 1941), also known as Muhammad Al-Amin Al-Tom, is a Sudanese mathematician and the first Minister of Education after the Sudanese Revolution, serving between 2019 and 2022. During his tenure, he worked on various initiatives to improve education in Sudan, including the development of a comprehensive plan for the sector. However, El-Tom and his assistant, Omer al-Qarray, faced controversy over the inclusion of Michelangelo's famous painting, The Creation of Adam, in Sudanese school textbooks. The decision was met with strong opposition from some conservative Muslim groups, who argued that the image of God reaching out to Adam in the painting was inconsistent with Islamic beliefs and should not be included in textbooks. == Early life and education == Mohamed El-Amin Ahmed El-Tom was born in October 1941. He completed a Bachelor of Science degree in Mathematics from the University of Leeds with first class honours in 1965. This was followed by a Diploma in Advanced Mathematics from the University of Oxford in 1966 taking courses on Numerical Analysis, Functional Analysis, Group Theory, and Commutative Algebra. He then completed a Doctor of Philosophy (DPhil) in 1969, supervised by David Christopher Handscomb. == Academic career == After his bachelor's, El-Tom became a junior scholar at the University of Khartoum (1962-65). Following his DPhil, he returned as a senior scholar between 1965 and 1968, where he helped establish the Mathematical Sciences School. Subsequently, he took on a position as a research fellow at the Center de Calcul, University of Louvain, Belgium (1968-69). El-Tom advanced in his academic career, serving as a lecturer at the University of Ulster and later as a professor at Columbia University. His expertise also led him to roles at the European Center for Nuclear Research (CERN), Qatar University, and the Sudan Centre for Educational Research. Subsequently, he was appointed the dean of the University of Garden City. === Research === El-Tom has published over 50 papers on integration convergence, approximations, and interpolation. His work also focuses on the state and future of mathematics education and mathematical research in Sudan, Islamic countries, Africa, and North America. In March 1978, El-Tom chaired and organised the International Conference on Developing Mathematics in Third World Countries, in Khartoum, and The Status and Future of Higher Education in Sudan, in Cairo, in 1998. Additionally, he co-founded the African Institute for Mathematical Sciences in Dar es Salaam in 2003. == Minister of Education == After the Sudanese Revolution, the Sudanese Teachers Committee and the Northern Entity Alliance, one of the components of the Revolutionary Front, nominated El-Tom to be the minister of the Ministry of Education, due to his "deep patriotism, professionalism and high efficiency", and because "there is no person more qualified than him". On 5 September 2019, he was appointed Minister of Education. === Reforms === El-Tom, within the Council of Ministers, moved education from the ninth to second position in the government spending priorities, which were presented by the former Minister of Finance in charge of the Council. He championed implementing free education. Through donors, he provided $2 for each student per year. He also tried to secure school meals. El-Tom enacted new public and private education laws in 2020. El-Tom introduced an e-learning management system, linking all schools throughout Sudan. He was also able to provide virtual training for hundreds of teachers by Sudanese companies, organisations and volunteers from abroad, who initially provided training in the English language. He developed a plan for training in mathematics. The Ministry of Education also authorised a regulation banning corporal punishment in educational institutions. El-Tom proposed offering financial support to students pursuing education degrees as a means to boost the college's enrollment figures and guarantee the production of well-prepared teachers upon graduation. He is credited with improving the status of the teachers, by raising the salary of a new graduate from 3,000 SDG to more than 16,000 SDG per month. He planned for the construction Exemplary Schools, in terms of buildings and content, called the Twenty-First Century Skills Schools, which was funded by international donors. In March 2020, following the declaration of the COVID-19 pandemic, the Sudanese Ministry of Education delayed the secondary school exams initially set for 12 April. The new date was to be announced later. At a press conference, El-Tom stated that the decision prioritised the well-being of students and their families. He acknowledged the sudden nature of the decision and its impact on the preparations for the exams, which involved approximately 500,000 students. In August 2020, El-Tom argued against closing schools due to the pandemic, stating that many students would quit school and start working in markets, and a large part of the student body would forget previous lessons. However, he conceded to the Health Emergency Committee. In September 2020, the Ministry of Education declared a delay in the school opening date, initially set for the 27th of that month, to 22 November. This decision was made in response to the unpreparedness of many schools across different regions of Sudan, which were adversely affected by floods and heavy rainfall. In November 2020, the World Bank Board of Directors approved an education project supported by a $61.5 million grant to support preparatory education in Sudan to maintain and improve basic education for children, with significant support for teachers, schools and communities. The project was planned to enhance the government's ability to formulate policies and monitor progress across the education system. This grant for education constituted the largest funding to support basic education in Sudan. The project covered all public schools, prioritising investment in disadvantaged and marginalised areas. In early 2020, the Global Partnership for Education provided an additional $11 million grant to support Sudan in strengthening programs to respond to the country's education needs in light of the COVID-19 pandemic. In November 2020, as part of an initiative to give 50,000 out-of-school children access to high-quality formal and informal education, the Education Above All (EAA) Foundation and its partner UNICEF signed a Memorandum of Understanding (MoU) with the Federal Ministry of Education of Sudan. According to a statement from EAA, the project was planned to put a special emphasis on enhancing learning environments and building capacity to ensure that children who are deprived of education receive a quality education, as well as increasing community involvement and raising awareness of the value of access to and enrolment in primary education. === The new curriculum controversy === El-Tom's ministry had taken steps to reform the educational system in Sudan while advocating for "free education for all" by 2030. Regarding the curricula and their changes, El-Tom believed that the general trend, regardless of the subject, is to take into account the student's age and the readiness of their mind to absorb the material. El-Tom addressed the debate about adjusting the number of Qur’an surahs for a specific stage. He clarified that when choosing a surah for a six-year-old, it should align with specific objectives and the child's ability to memorise and understand it effortlessly. This task was carried out by the National Center for Educational Curricula, Training, Guidance and Research, led by Omer al-Qarray, which revised and developed elementary school curricula from grade one to grade six in 2020. A public debate ensued involving politicians, clerics, and journalists over proposals to change the school curricula, led by the Director of Curricula at the Ministry of Education, Omar Al-Qarai. At the same time, education experts and activists opposed the change. The controversy began after a proposal for a new history book for the sixth grade was leaked. Specifically, the painting The Creation of Adam by Renaissance artist Michelangelo caused controversy due to claims of it being heretical. In January 2021, discussions about the curricula emerged on social media among Sudanese individuals. Additionally, accounts and pages were created that either criticized or supported Omar Al-Qarai. The situation worsened as a video emerged featuring imam Muhammad Al-Amin Ismail, shedding tears during the Friday Khutbah. In the video, he expressed his disappointment regarding the content in the new curricula before launching a criticism against Al-Qarai. The head of the Sudan Liberation Movement/Army, Minni Minnawi, tweeted that the campaign led by some imams do not stem from motives to preserve religion, but rather was a political campaign aimed at obstructing change that begins with the educational curricula. The political secretary of the Justice and Equality Movement, Suleiman Sandal, stated in press statements that "a school curriculum will not be taught to our children that contains images that embody God while we are in the government". The leader of the National Umma Party, Abd al-Rahman al-Ghali, incited the government to dismiss al-Qarai pointing to Al-Qari's intellectual and political background as a member of the Republican Brotherhood, whose founder, Mahmoud Mohammed Taha, was executed in 1985 on charges of apostasy. The educational expert, Mubarak Yahya, head of the Sudanese Coalition for Education for All, criticised the regression of the curricula issue to the political arena and the quarrels that it entailed. He said that the curricula required a national conference to ensure proper construction of them at a high professional level and societal values and advice, far from politics. However a member of the Central Council of the Sudanese Gathering of the Teachers Committee, Ammar Yusef, believed that there is a campaign against Al-Qari behind which supporters of the former regime stand. But he pointed out that the Teachers' Committee did not participate in developing the new curricula. Omer al-Qarray blamed the Minister of Religious Affairs and Endowments, Nasr al-Din Mufreh, for his silence even when some clerics called for his death. Al-Qari had insisted that he would not step down unless a decision was taken to cancel the curricula in response to pressure. However, Al-Qarai resigned soon afterwards on 7 January 2021. The Sudanese Teachers Committee affirmed that there would be no turning back from the project of radical change in the education system that was initiated by the Ministry of Education, and great strides were made in it. Prime Minister Abdallah Hamdok formed a national committee to review the curricula, and to submit its report after two weeks. The committee ensured that specialists prepared each subject and that the curriculum met high-quality educational goals, professional and national standards, and was teachable. === Formation of a new government === On 8 February 2021, Sudanese Prime Minister Abdallah Hamdok issued a decision to relieve the ministers of the transitional government from their positions in preparation of a new government formation. The statement added that the ministers would continue in their positions as caretakers until the formation of the new government and the completion of handover procedures once the Transitional Sovereignty Council had announced the formation of the new government. The statement indicated that the new formation would not include the name of a Minister of Education, as consultations were still taking place regarding this ministry. El-Tom was excluded from the new cabinet's nomination list by the Prime Minister Abdalla Hamdok, who submitted the list to the Transitional Sovereignty Council under the pretext that El-Tom had failed a "security check". The Sudanese Teachers Committee protested the exclusion. El-Tom considered his exclusion from the ministry due to a failed security check a “disgrace to his reputation”, which caused him "psychological harm". He told Al-Ekhbari: "I am now an accused citizen, and those accusing me must prove whether I am an agent, or have I committed a crime, or been involved in corruption, or what?". He mentioned that he received calls questioning if he had committed any wrongdoing that might have led to his exclusion from the ministry. The Teachers Committee - the leading component of the Sudanese Professionals Association - announced its rejection of the partisan quota of the ministry, and supported El-Tom. In May 2021 and after two months of the post being vacant, a memorandum calling on the Prime Minister to name El-Tom as Minister of Education was signed by a group of civil society organisations, university professors and intellectuals. The memorandum also demanded the approval of the laws drawn up by the competent committees to reform education including the Public Education Law of 2020, the Private Education Law of 2020, and the Law of the National Center for Educational Curricula, Training, Guidance and Research 2020. In April 2021, the Ministry of Education denied that El-Tom has resigned, contrary to what was being circulated on social media. However, on 26 August 2021, El-Tom announced that he had submitted his resignation letter to Hamdok because Hamdok sanctioned the new curriculum, after it had been approved by the Minister of Education. Afterwards, Hamdok demanded that each book of the textbooks be carefully reviewed by authors, language specialists and designers, including illustrators and makers of pictures. Hamdok also requested that the draft of the book be sent to reviewers specialised in the subject of each book. Hamdok called for the rewriting of the history book for the sixth grade and the preservation of the contested surahs without removing any of the verses, and keeping all the units and lessons that were removed from the mathematics textbooks. Hamdok met with delegations of Islamic and Christian clerics, alongside the Minister of Religious Affairs, Nasr al-Din Mufreh, to discuss reviewing the curriculum developed under Omer al-Qarray's leadership at the National Curriculum Center. == Personal life == El-Tom is married with three children. == Awards and honours == El-Tom was elected a Fellow of the Institute of Mathematics and its Applications (FIMA) in 1978, and a Fellow of the African Academy of Sciences (FAAS) in 1986. He was a member of the International Centre for Theoretical Physics, Trieste, Italy, between 1984 and 1989. He is also a member of the Arab Thought Forum, Jordan since 1985, the Mathematical Association of America, USA since 1992, the American Mathematical Society, USA since 1995, and the Sudanese National Academy of Sciences since 2007. In response to an invitation from the Global Science Program at Uppsala University, El-Tom participated in attending the 2013 Nobel Prize event in Stockholm. In 2021, he was elected Sudan’s personality of the year by Al Khatim Adlan Center for Enlightenment and Human Development (KACE). == References == == External links == Interview with Professor Mohamed El Amin El-Tom, in Arabic by Sudan Bukra
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Wikipedia:Mohamed Hag Ali Hassan#0
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Mohamed Hag Ali Hag el Hassan OMRI GCONMC FAAS FIAS FTWAS (Arabic: محمد حاج علي حاج الحسن; born 21 November 1947) is a Sudanese-Italian mathematician and physicist who co-founded numerous scientific councils. He is the President of The World Academy of Sciences and Sudanese National Academy of Sciences. == Early life == Hassan was born in Elgetina, Sudan, on the 21st of November 1947. He obtained a bachelor’s degree (B.Sc.) with special honours from the University of Newcastle Upon Tyne in 1968, followed by an M.Sc in Advanced Mathematics from the University of Oxford in 1969. He obtained his DPhil in Plasma Physics from the University of Oxford in 1974. == Career == Hassan returned to Sudan later became Professor and Dean of the School of Mathematical Sciences, University of Khartoum from 1985 to 1986. Frustrated by scientific stagnation in Sudan, and at the request of his father, Hassan visited Italy, and was then motivated to re-peruse science again by Nobel Prize laureate, Abdus Salam who was (at time) working at the International Centre for Theoretical Physics (ICTP), Trieste. Abdus Salam offered Hassan an associate membership at ICTP to provide a conducive environment for research. Hassan has a long list of publications in theoretical plasma physics and fusion energy, environmental modelling of soil erosion in drylands, and geophysics, astrophysics and space physics. He has also published several articles on science and technology in the developing world. Hassan was the founding Executive Director of the Academy of Science for the Developing World (TWAS) in 1983, President of the African Academy of Sciences in 2000, President of the Network of Academies of Science in Africa (NASAC) in 2001 and Chairman, President of the Sudanese National Academy of Sciences, Director of Secretariat of InterAcademy Partnership (IAP) in 2001, Honorary Presidential Advisory Council for Science and Technology, Nigeria in 2001, and Chair of the Governing Council of the United Nations Technology Bank for the Least Developed Countries. He also is a founding member of the Lebanese Academy of Sciences. He is a member of several merit-based academies of science, including TWAS, the African Academy of Sciences, Islamic World Academy of Sciences, Academia Colombiana de Ciencias Exactas, Fisicas y Naturales, Académie Royale des Sciences d’Outre-Mer, Pakistan Academy of Sciences; Lebanese Academy of Sciences, Cuban Academy of Sciences, Pontifical Academy of Sciences, Grand challenges Canada, and Academy of Sciences of South Africa. Hassan is the co-chair of resident of the InterAcademy Partnership (IAP), and chairman of the Council of the United Nations University (UNU). He also serves on a number of Boards of international organizations worldwide, including the Board of Trustees of Bibliotheca Alexandrina, Egypt, the Council of Science and Technology in Society (STS ) Forum, Japan, the Board of the International Science Programme, Sweden, the Board of the Science Initiative Group (SIG), USA, the International Advisory Board of the Centre for International Development (ZEF), Germany, the advisory group of the Global Young Academy. == Awards and honours == Hassan is Comendador (1996) and Grand Cross (2005) of the Brazilian National Order of Scientific Merit, Officer of the Order of Merit of the Italian Republic (2003), and he is the recipient of the G77 Leadership Award and of the Abdus Salam Medal for Science and Technology. Hassan is a Founding Fellow of the African Academy of Sciences (1985), a Fellow of The World Academy of Sciences (1985), a Fellow of the Islamic World Academy of Sciences (1992), an Honorary Member of the Academia Colombiana de Ciencias Exactas, Fisicas y Naturales (1996), and a Foreign Fellow of the Pakistan Academy of Sciences (2002). == Personal life == Hassan is married with three children. == Selected publications == Hassan, Mohamed H. A. (2005). "Small Things and Big Changes in the Developing World". Science. 309 (5731). American Association for the Advancement of Science (AAAS): 65–66. doi:10.1126/science.1111138. ISSN 0036-8075. PMID 15994516. S2CID 166679567. Hassan, Mohamed H. A. (2001). "Can Science Save Africa?". Science. 292 (5522). American Association for the Advancement of Science (AAAS): 1609. doi:10.1126/science.292.5522.1609. ISSN 0036-8075. PMID 11387443. S2CID 36662994. El-Baz, Farouk; Hassan, M. H. A. (1986). Physics of desertification. Dordrecht: Springer Netherlands. ISBN 978-94-009-4388-9. OCLC 840305558. Hassan, Mohamed H.A. (2007). "Building Capacity in the Life Sciences in the Developing World". Cell. 131 (3). Elsevier BV: 433–436. doi:10.1016/j.cell.2007.10.020. ISSN 0092-8674. PMID 17981107. S2CID 33362062. Hassan, M.A. (2009). "Novel Fire Retardant Backcoatings for Textiles". In Hull, T. Richard; Kandola, Baljinder K. (eds.). Fire retardancy of polymers : new strategies and mechanisms. Cambridge: Royal Society of Chemistry. pp. 341–358. doi:10.1039/9781847559210. ISBN 978-1-84755-921-0. OCLC 319518106. == See also == Ahmed Hassan Fahal Sultan Hassan Nashwa Eassa == References == == External links == Mohamed Hag Ali Hassan publications indexed by Google Scholar
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Wikipedia:Moment problem#0
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In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ {\displaystyle \mu } to the sequence of moments m n = ∫ − ∞ ∞ x n d μ ( x ) . {\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\mu (x)\,.} More generally, one may consider m n = ∫ − ∞ ∞ M n ( x ) d μ ( x ) . {\displaystyle m_{n}=\int _{-\infty }^{\infty }M_{n}(x)\,d\mu (x)\,.} for an arbitrary sequence of functions M n {\displaystyle M_{n}} . == Introduction == In the classical setting, μ {\displaystyle \mu } is a measure on the real line, and M {\displaystyle M} is the sequence { x n : n = 1 , 2 , … } {\displaystyle \{x^{n}:n=1,2,\dotsc \}} . In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of μ {\displaystyle \mu } is allowed to be the whole real line; the Stieltjes moment problem, for [ 0 , ∞ ) {\displaystyle [0,\infty )} ; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as [ 0 , 1 ] {\displaystyle [0,1]} . The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matrices are replaced by Toeplitz matrices and the support of μ is the complex unit circle instead of the real line. == Existence == A sequence of numbers m n {\displaystyle m_{n}} is the sequence of moments of a measure μ {\displaystyle \mu } if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices H n {\displaystyle H_{n}} , ( H n ) i j = m i + j , {\displaystyle (H_{n})_{ij}=m_{i+j}\,,} should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional Λ {\displaystyle \Lambda } such that Λ ( x n ) = m n {\displaystyle \Lambda (x^{n})=m_{n}} and Λ ( f 2 ) ≥ 0 {\displaystyle \Lambda (f^{2})\geq 0} (non-negative for sum of squares of polynomials). Assume Λ {\displaystyle \Lambda } can be extended to R [ x ] ∗ {\displaystyle \mathbb {R} [x]^{*}} . In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional Λ {\displaystyle \Lambda } is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is Λ ( x n ) = ∫ − ∞ ∞ x n d μ {\displaystyle \Lambda (x^{n})=\int _{-\infty }^{\infty }x^{n}d\mu } . A condition of similar form is necessary and sufficient for the existence of a measure μ {\displaystyle \mu } supported on a given interval [ a , b ] {\displaystyle [a,b]} . One way to prove these results is to consider the linear functional φ {\displaystyle \varphi } that sends a polynomial P ( x ) = ∑ k a k x k {\displaystyle P(x)=\sum _{k}a_{k}x^{k}} to ∑ k a k m k . {\displaystyle \sum _{k}a_{k}m_{k}.} If m k {\displaystyle m_{k}} are the moments of some measure μ {\displaystyle \mu } supported on [ a , b ] {\displaystyle [a,b]} , then evidently Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend φ {\displaystyle \varphi } to a functional on the space of continuous functions with compact support C c ( [ a , b ] ) {\displaystyle C_{c}([a,b])} ), so that By the Riesz representation theorem, (2) holds iff there exists a measure μ {\displaystyle \mu } supported on [ a , b ] {\displaystyle [a,b]} , such that φ ( f ) = ∫ f d μ {\displaystyle \varphi (f)=\int f\,d\mu } for every f ∈ C c ( [ a , b ] ) {\displaystyle f\in C_{c}([a,b])} . Thus the existence of the measure μ {\displaystyle \mu } is equivalent to (1). Using a representation theorem for positive polynomials on [ a , b ] {\displaystyle [a,b]} , one can reformulate (1) as a condition on Hankel matrices. == Uniqueness (or determinacy) == The uniqueness of μ {\displaystyle \mu } in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on [ 0 , 1 ] {\displaystyle [0,1]} . For the problem on an infinite interval, uniqueness is a more delicate question. There are distributions, such as log-normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments. == Formal solution == When the solution exists, it can be formally written using derivatives of the Dirac delta function as d μ ( x ) = ρ ( x ) d x , ρ ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! δ ( n ) ( x ) m n {\displaystyle d\mu (x)=\rho (x)dx,\quad \rho (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)m_{n}} . The expression can be derived by taking the inverse Fourier transform of its characteristic function. == Variations == An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory. == Probability == The moment problem has applications to probability theory. The following is commonly used: By checking Carleman's condition, we know that the standard normal distribution is a determinate measure, thus we have the following form of the central limit theorem: == See also == Carleman's condition Hamburger moment problem Hankel matrix Hausdorff moment problem Moment (mathematics) Stieltjes moment problem Trigonometric moment problem == Notes == == References == Shohat, James Alexander; Tamarkin, Jacob D. (1943). The Problem of Moments. New York: American mathematical society. ISBN 978-1-4704-1228-9. {{cite book}}: ISBN / Date incompatibility (help) Akhiezer, Naum I. (1965). The classical moment problem and some related questions in analysis. New York: Hafner Publishing Co. (translated from the Russian by N. Kemmer) Kreĭn, M. G.; Nudel′man, A. A. (1977). The Markov Moment Problem and Extremal Problems. Translations of Mathematical Monographs. Providence, Rhode Island: American Mathematical Society. doi:10.1090/mmono/050. ISBN 978-0-8218-4500-4. ISSN 0065-9282. Schmüdgen, Konrad (2017). The Moment Problem. Graduate Texts in Mathematics. Vol. 277. Cham: Springer International Publishing. doi:10.1007/978-3-319-64546-9. ISBN 978-3-319-64545-2. ISSN 0072-5285.
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Wikipedia:Mona Merling#0
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Mona Merling is a Romanian-American mathematician specializing in algebraic topology, including algebraic K-theory and equivariant stable homotopy theory. She is an associate professor of mathematics at the University of Pennsylvania. == Education and career == Merling has dual Romanian and American citizenship. She credits a Romanian mathematics teacher, Mihaela Flamaropol, for sparking her interest in mathematics and in mathematics competitions. She majored in mathematics as an undergraduate at Bard College, where she received her bachelor's degree in 2009. At Bard, mathematician Lauren Lynn Rose became a faculty mentor. She continued her studies in mathematics at the University of Chicago, received a master's degree there in 2010, and completed her Ph.D. in 2014. Her dissertation, Equivariant algebraic K-theory, was supervised by J. Peter May. She was a postdoctoral researcher and J. J. Sylvester Assistant Professor at Johns Hopkins University from 2014 to 2018, working there under the mentorship of Jack Morava. She joined the University of Pennsylvania as an assistant professor of mathematics in 2018, and was tenured as an associate professor in 2024. == Recognition == Merling was the 2025 recipient of the Joan & Joseph Birman Research Prize in Topology and Geometry, given to her "for innovative and impactful research in algebraic K-theory, equivariant homotopy theory, and their applications to manifold theory". == References == == External links == Home page Mona Merling publications indexed by Google Scholar
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Wikipedia:Monad (homological algebra)#0
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In homological algebra, a monad is a 3-term complex A → B → C of objects in some abelian category whose middle term B is projective, whose first map A → B is injective, and whose second map B → C is surjective. Equivalently, a monad is a projective object together with a 3-step filtration B ⊃ ker(B → C) ⊃ im(A → B). In practice A, B, and C are often vector bundles over some space, and there are several minor extra conditions that some authors add to the definition. Monads were introduced by Horrocks (1964, p.698). == See also == ADHM construction == References == Barth, Wolf; Hulek, Klaus (1978), "Monads and moduli of vector bundles", Manuscripta Mathematica, 25 (4): 323–347, doi:10.1007/BF01168047, ISSN 0025-2611, MR 0509589, Zbl 0395.14007 Horrocks, G. (1964), "Vector bundles on the punctured spectrum of a local ring", Proceedings of the London Mathematical Society, Third Series, 14 (4): 689–713, doi:10.1112/plms/s3-14.4.689, ISSN 0024-6115, MR 0169877, Zbl 0126.16801
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