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Superdeterminant
In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold. Definition The Berezinian is uniquely determined by two defining properties: *\operatorname(XY) = \operatorname(X)\operatorname(Y) *\operatorname(e^X) = e^\, where str(''X'') denotes the supertrace of ''X''. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices. The simplest case to consider is the Berezinian of a supermatrix with entries in a field ''K''. Such supermatrices represent linear transformations of a super vector space over ''K''. A particular even supermatrix is a block matrix of the form :X = \beginA & 0 \\ 0 & D\end Such a matrix is invertible if and only if both ''A'' and ''D'' are invertible matrices over ''K''. The Berezin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berezin Integration
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions. Definition Let \Lambda^n be the exterior algebra of polynomials in anticommuting elements \theta_,\dots,\theta_ over the field of complex numbers. (The ordering of the generators \theta_1,\dots,\theta_n is fixed and defines the orientation of the exterior algebra.) One variable The ''Berezin integral'' over the sole Grassmann variable \theta = \theta_1 is defined to be a linear functional : \int f(\theta)+bg(\theta)\, d\theta = a\int f(\theta) \, d\theta + b\int g(\theta) \, d\theta, \qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berezin Integral In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions. Definition Let \Lambda^n be the exterior algebra of polynomials in anticommuting elements \theta_,\dots,\theta_ over the field of complex numbers. (The ordering of the generators \theta_1,\dots,\theta_n is fixed and defines the orientation of the exterior algebra.) One variable The ''Berezin integral'' over the sole Grassmann variable \theta = \theta_1 is defined to be a linear functional : \int f(\theta)+bg(\theta)\, d\theta = a\int f(\theta) \, d\theta + b\int g(\theta) \, d\theta, ... [...More Info...] [...Related Items...] |
