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Movements, leaders, victories and the continuing fight for equality.
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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Olga Taussky-Todd | |
|---|---|
 1932 in Göttingen | |
| Born | August 30, 1906 Olomouc, Austria-Hungary |
| Died | October 7, 1995 (aged 89) Pasadena, California, U.S. |
| Education | University of Vienna |
| Known for | Torchbearer for matrix theory; supervised Caltech's first female Ph.D. in Math, Lorraine Foster; corrected David Hilbert's papers |
| Spouse | John Todd |
| Awards | Fellow of Girton College, Bryn Mawr College, and the AAAS, a Noether Lecturer and a recipient of the Austrian Cross of Honour for Science and Art, 1st class |
| Scientific career | |
| Fields | Mathematics |
| Workplaces | |
| Thesis | Über eine Verschärfung des Hauptidealsatzes (1930) |
| Philipp Furtwängler | |
Doctoral students | |
Olga Taussky-Todd (August 30, 1906 – October 7, 1995) was an Austrian and American mathematician.[1][2] She published more than 300 research papers on algebraic number theory, integral matrices, and matrices in algebra and analysis.
Olga Taussky was born[3] into a Jewish family in what is now Olomouc in the Czech Republic, on August 30, 1906.[4] Her father, Julius David Taussky, was an industrial chemist and her mother, Ida Pollach, was a housewife. She was the second of three children.[2] Her father preferred that, if his daughters had careers, they be in the arts, but they all went into the sciences. Ilona, three years older than Olga, became a consulting chemist in the glyceride industry, and Hertha, three years younger than Olga, became a pharmacist and later a clinical chemist at Cornell University Medical College in New York City.[5]
At the age of three, her family moved to Vienna and lived there until the middle of World War I. Later Taussky's father accepted a position as director of a vinegar factory at Linz in Upper Austria. At a young age, Taussky displayed a keen interest in mathematics. After her father died during her last year at school, she worked through the summer at her father's vinegar factory and was pressured by her family to study chemistry in order to take over her father's work. Her elder sister, however, qualified in chemistry and took over her father's work. In "Red Vienna" of the day, the Social Democratic Party of Austria encouraged woman to pursue higher education,[6] and Taussky enrolled at the University of Vienna in the fall of 1925 to study mathematics.[2]
Taussky worked first in algebraic number theory, with a doctorate at the University of Vienna supervised by Philipp Furtwängler, a number theorist from Germany.[7] During that time, she attended meetings of the so-called Vienna Circle, the group of philosophers and logicians developing the philosophy of logical positivism. Taussky, like Olga Hahn-Neurath and Rose Rand, was one of the first women to join the group, which included Otto Neurath, Rudolf Carnap, and Kurt Gödel and which was strongly influenced by Ludwig Wittgenstein.[6]
Taussky is best known for her work in matrix theory (in particular the computational stability of complex matrices), algebraic number theory, group theory, and numerical analysis.
According to Gian-Carlo Rota, as a young mathematician she was hired by a group of German mathematicians to find and correct the many mathematical errors in the works of David Hilbert, so that they could be collected into a volume to be presented to him on his birthday. There was only one paper, on the continuum hypothesis, that she was unable to repair.[8]
In 1935, she moved to England and became a Fellow at Girton College, Cambridge University, as well as at Bryn Mawr College. Soon after, in 1938, she married the Irish mathematician Jack Todd, a colleague at the University of London.
Later, she started to use matrices to analyze vibrations of airplanes during World War II, at the National Physical Laboratory in the United Kingdom. During this time she wrote several articles that were published by the Ministry of Aircraft Production in London. She later described herself as a torchbearer for matrix theory.
In 1945 the Todds emigrated to the United States and worked for the National Bureau of Standards. In 1957 she and her husband both joined the faculty of California Institute of Technology (Caltech) in Pasadena, California. She also supervised Caltech's first female Ph.D. in Math, Lorraine Foster, as well as Hanna Neumann, Philip J. Hanlon, and Charles Royal Johnson.
Taussky retired from teaching in 1977,[9] but continued her correspondence with other mathematicians regarding her work in matrix theory.
Taussky received the Ford Prize for an article on sum of squares published in 1970 in American Mathematical Monthly. She went on to receive an honorary doctorate from the University of Vienna and an honorary DSc by the University of Southern California in 1988.
She was a Fellow of the AAAS, a Noether Lecturer and a recipient of the Austrian Cross of Honour for Science and Art, 1st class (1978).
In 1993, the International Linear Algebra Society (ILAS) established a lecture series to honor the contributions to the field of linear algebra made by Taussky-Todd and her husband. In 2021, the Taussky–Todd lecture series was converted to the ILAS Taussky–Todd Prize.
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
Olga Taussky-Todd (August 30, 1906 – October 7, 1995) was an Austrian and American mathematician. She published more than 300 research papers on algebraic number theory, integral matrices, and matrices in algebra and analysis.
In linear algebra, a circulant matrix is a square matrix in which each row is a cyclic shift, by one position, of the row above it. Equivalently, its entries depend only on the difference of the row and column indices modulo the matrix size: if rows and columns are indexed from 0 {\displaystyle 0} to n − 1 {\displaystyle n-1} , the ( r , s ) {\displaystyle (r,s)} entry equals c ( r − s ) mod n {\displaystyle c_{(r-s){\bmod {n}}}} , where ( c 0 , c 1 , … , c n − 1 ) {\displaystyle (c_{0},c_{1},\ldots ,c_{n-1})} is a single generating vector. Circulant matrices are a special class of Toeplitz matrices. Every complex circulant matrix is diagonalized by the discrete Fourier transform (DFT). As a consequence, circulant matrices form a commutative algebra of normal matrices; the eigenvalues of a circulant matrix are the DFT of its generating vector; and matrix–vector products and nonsingular linear systems involving a circulant matrix can be computed with a fast Fourier transform (FFT) in O ( n log n ) {\displaystyle O(n\log n)} arithmetic operations. Algebraically, multiplication by a circulant matrix is circular convolution on the cyclic group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } . Circulant matrices arise wherever a linear operation is invariant under cyclic shifts. They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group C n {\displaystyle C_{n}} , and hence appear in signal and image processing with periodic boundary conditions, in discretizations of translation-invariant problems on periodic domains, and in the study of stationary stochastic processes. In orthogonal frequency-division multiplexing, the cyclic prefix prepended to each transmitted block makes the effective channel matrix circulant, so that channel equalization can be carried out separately on each component in the frequency domain. In cryptography, a circulant matrix is used in the MixColumns step of the Advanced Encryption Standard.
In numerical mathematics, hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension n {\displaystyle n} can be represented efficiently in O ( n ) {\displaystyle O(n)} units of storage by storing only its non-zero entries, a non-sparse matrix would require O ( n 2 ) {\displaystyle O(n^{2})} units of storage, and using this type of matrices for large problems would therefore be prohibitively expensive in terms of storage and computing time. Hierarchical matrices provide an approximation requiring only O ( n k log ( n ) ) {\displaystyle O(nk\,\log(n))} units of storage, where k {\displaystyle k} is a parameter controlling the accuracy of the approximation. In typical applications, e.g., when discretizing integral equations, preconditioning the resulting systems of linear equations, or solving elliptic partial differential equations, a rank proportional to log ( 1 / ϵ ) γ {\displaystyle \log(1/\epsilon )^{\gamma }} with a small constant γ {\displaystyle \gamma } is sufficient to ensure an accuracy of ϵ {\displaystyle \epsilon } . Compared to many other data-sparse representations of non-sparse matrices, hierarchical matrices offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in O ( n k α log ( n ) β ) {\displaystyle O(nk^{\alpha }\,\log(n)^{\beta })} operations, where α , β ∈ { 1 , 2 , 3 } . {\displaystyle \alpha ,\beta \in \{1,2,3\}.}
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter σ {\displaystyle \sigma } (sigma), and occasionally by τ {\displaystyle \tau } (tau) when used in connection with isospin symmetries. σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 1 ) . {\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\\\end{aligned}}} These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the identity matrix I {\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrix σ 0 {\displaystyle \sigma _{0}} ), the Pauli matrices form a basis of the vector space of 2 × 2 {\displaystyle 2\times 2} Hermitian matrices over the real numbers, under addition. This means that any 2 × 2 {\displaystyle 2\times 2} Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. The Pauli matrices satisfy the useful product relation: σ i σ j = δ i j I + i ε i j k σ k , {\displaystyle {\begin{aligned}\sigma _{i}\ \sigma _{j}=\delta _{ij}\ \mathbb {I} +i\ \varepsilon _{ijk}\ \sigma _{k}\ ,\end{aligned}}} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta, which equals + 1 {\displaystyle +1} if i = j {\displaystyle i=j} otherwise 0 {\displaystyle 0} , and the Levi-Civita symbol ε i j k {\displaystyle \varepsilon _{ijk}} is used. Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, σ k {\displaystyle \sigma _{k}} represents the observable corresponding to spin along the k {\displaystyle k} th coordinate axis in three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} . The Pauli matrices (after multiplication by i {\displaystyle i} to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: The matrices i σ 1 {\displaystyle i\sigma _{1}} , i σ 2 {\displaystyle i\sigma _{2}} , and i σ 3 {\displaystyle i\sigma _{3}} form a basis for the real Lie algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , which exponentiates to the special unitary group SU(2). The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of R 3 {\displaystyle \ \mathbb {R} ^{3}} and the (unital) associative algebra generated by i σ 1 {\displaystyle i\sigma _{1}} , i σ 2 {\displaystyle i\sigma _{2}} , and i σ 3 {\displaystyle i\sigma _{3}} functions identically (is isomorphic) to that of quaternions ( H {\displaystyle \mathbb {H} } ).
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Mae Jemison, aboard Space Shuttle Endeavour in 1992.
