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Joel Lee Brenner
Joel Lee Brenner ( – ) was an American mathematician who specialized in matrix theory, linear algebra, and group theory. He is known as the translator of several popular Russian texts. He was a teaching professor at some dozen colleges and universities and was a Senior Mathematician at Stanford Research Institute from 1956 to 1968. He published over one hundred scholarly papers, 35 with coauthors, and wrote book reviews.LeRoy B. Beasley (1987) "The Mathematical Work of Joel Lee Brenner", Linear Algebra and its Applications 90:1–13 Academic career In 1930 Brenner earned a B.A. degree with major in chemistry from Harvard University. In graduate study there he was influenced by Hans Brinkmann, Garrett Birkhoff, and Marshall Stone. He was granted the Ph.D. in February 1936. Brenner later described some of his reminiscences of his student days at Harvard and of the state of American mathematics in the 1930s in an article for American Mathematical Monthly. In 1951 Brenner publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boston
Boston (), officially the City of Boston, is the state capital and most populous city of the Commonwealth of Massachusetts, as well as the cultural and financial center of the New England region of the United States. It is the 24th- most populous city in the country. The city boundaries encompass an area of about and a population of 675,647 as of 2020. It is the seat of Suffolk County (although the county government was disbanded on July 1, 1999). The city is the economic and cultural anchor of a substantially larger metropolitan area known as Greater Boston, a metropolitan statistical area (MSA) home to a census-estimated 4.8 million people in 2016 and ranking as the tenth-largest MSA in the country. A broader combined statistical area (CSA), generally corresponding to the commuting area and including Providence, Rhode Island, is home to approximately 8.2 million people, making it the sixth most populous in the United States. Boston is one of the oldest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reminiscence
Reminiscence is the act of recollecting past experiences or events. An example of the typical use of reminiscence is when people share their personal stories with others or allows other people to live vicariously through stories of family, friends, and acquaintances while gaining an authentic meaningful relationship with the people. An example of reminiscence may be grandparents remembering past events with friends or their grandchildren, sharing their individual experience of what the past was like. Psychological usage Reminiscence therapy Reminiscence can be defined as the act or process of recalling past experiences, events, or memories. Anyone can reminiscence about the past or a certain event, but reminiscence is often used in the older population, particularly the elderly population with forms of dementia as a therapeutic tool. This type of reminiscence is called reminiscence therapy. Reminiscence therapy is a non-pharmacological intervention that improves self-esteem and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Ostrowski
Alexander Markowich Ostrowski ( uk, Олександр Маркович Островський; russian: Алекса́ндр Ма́ркович Остро́вский; 25 September 1893, in Kiev, Russian Empire – 20 November 1986, in Montagnola, Lugano, Switzerland) was a mathematician. His father Mark having been a merchant, Alexander Ostrowski attended the Kiev College of Commerce, not a high school, and thus had an insufficient qualification to be admitted to university. However, his talent did not remain undetected: Ostrowski's mentor, Dmitry Grave, wrote to Landau and Hensel for help. Subsequently, Ostrowski began to study mathematics at Marburg University under Hensel's supervision in 1912. During World War I he was interned, but thanks to the intervention of Hensel, the restrictions on his movements were eased somewhat, and he was allowed to use the university library. After the war ended Ostrowski moved to Göttingen where he wrote his doctoral dissertation a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. 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 this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing 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 meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roots Of Unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation :z^n = 1. Unless otherwise specified, the roots of unity may be taken to be complex numbers (incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exercise (mathematics)
A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Mathematics teachers assign mathematical exercises to develop the skills of their students. Early exercises deal with addition, subtraction, multiplication, and division of integers. Extensive courses of exercises in school extend such arithmetic to rational numbers. Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions. Usually instructors prepare students with worked examples: the exercise is stated, then a model answer is provided. Often several worked examples are demonstrated before students are prepared to attempt exercises on their own. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksei Fedorovich Filippov
Aleksei Fedorovich Filippov (russian: Алексей Фёдорович Филиппов; 29 September 1923 – 10 October 2006) was a Russian mathematician who worked on differential equations, differential inclusions, diffraction theory and numerical methods. Born in Moscow in 1923, Filippov served in the Red Army during the Second World War, then attended Moscow State University ( Faculty of Mechanics and Mathematics). After graduating in 1950, he remained to work at the school. He got his Ph.D. under the supervision of I. G. Petrovsky, and became a professor in 1978. He taught until his death in 2006. Filippov showed interest in continuous loops in 1950 when he constructed a proof that they divide a plane into interior and exterior parts. Known as the Jordan curve theorem, it exemplifies a mathematical proposition easily stated but difficult to prove. In 1955 Filippov and V. S. Ryaben'kii became interested in difference equations and wrote ''On the Stability of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolay Krasovsky
Nikolay Nikolayevich Krasovsky (russian: Никола́й Никола́евич Красо́вский; 7 September 1924 – 4 April 2012) was a Russian mathematician who worked in the mathematical theory of control, the theory of dynamical systems, and the theory of differential games. He was the author of Krasovskii-LaSalle principle and the chief of the Ural scientific school in mathematical theory of control and the theory of differential games. Biography Nikolay Krasovsky was born in Yekaterinburg, Soviet Union (renamed later to Sverdlovsk) in the family of a doctor. In 1949, he graduated summa cum laude from the department of metallurgical science at the Ural State Technical University. In 1954, he presented his first thesis and received his ''kandidat nauk'' degree in mathematics. In 1957, he defended his second thesis for the degree of ''doktor nauk'' and became a professor of mathematics. From 1949 to 1959, he worked at the Ural State Technical University. Since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science (journal)
''Science'', also widely referred to as ''Science Magazine'', is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals. It was first published in 1880, is currently circulated weekly and has a subscriber base of around 130,000. Because institutional subscriptions and online access serve a larger audience, its estimated readership is over 400,000 people. ''Science'' is based in Washington, D.C., United States, with a second office in Cambridge, UK. Contents The major focus of the journal is publishing important original scientific research and research reviews, but ''Science'' also publishes science-related news, opinions on science policy and other matters of interest to scientists and others who are concerned with the wide implications of science and technology. Unlike most scientific journals, which focus on a specific field, ''Science'' and its rival ''Nature (journal), Nature'' c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Gantmacher
Felix Ruvimovich Gantmacher (russian: Феликс Рувимович Гантмахер) (23 February 1908 – 16 May 1964) was a Soviet mathematician, professor at Moscow Institute of Physics and Technology, well known for his contributions in mechanics, linear algebra and Lie group theory. In 1925–1926 he participated in seminar guided by Nikolai Chebotaryov in Odessa and wrote his first research paper in 1926. His book ''Theory of Matrices'' (1953) is a standard reference of linear algebra. It has been translated into various languages including a two-volume version in English prepared by Joel Lee Brenner, Donald W. Bushaw, and S. Evanusa. George Herbert Weiss noted that "this book cannot be recommended too highly as it contains material otherwise unavailable in book form". Gantmacher collaborated with Mark Krein on ''Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems''. In 1939 he contributed to the classification problem of the real Lie algebras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
