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Anisotropic Quadratic Form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector space ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that is quadratic space and ''W'' is a subspace of ''V''. Then ''W'' is called an isotropic subspace of ''V'' if ''some'' vector in it is isotropic, a totally isotropic subspace if ''all'' vectors in it are isotropic, and an anisotropic subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma had ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Numbers
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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Direct Sum
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. Construction for vector spaces and abelian groups We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product Space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the picture); so, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt's Decomposition Theorem
:''"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.'' In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field ''k'' may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms over arbitrary fields. The theorem applies to classification of quadratic forms over ''k'' and in particular allows one to define the Witt group ''W''(''k'') which describes the "stable" theory of quadratic forms over the field ''k''. Statement Let be a finite-dimensional vector space over a field ''k'' of characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If is an isometry between two subspaces of ''V'' then ''f'' extends to an isometry of ''V''. Witt's ... [...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 internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergebnisse Der Mathematik Und Ihrer Grenzgebiete
''Ergebnisse der Mathematik und ihrer Grenzgebiete''/''A Series of Modern Surveys in Mathematics'' is a series of scholarly monographs published by Springer Science+Business Media. The title literally means "Results in mathematics and related areas". Most of the books were published in German or English, but there were a few in French and Italian. There have been several sequences, or ''Folge'': the original series, neue Folge, and 3.Folge. Some of the most significant mathematical monographs of 20th century appeared in this series. Original series The series started in 1932 with publication of ''Knotentheorie'' by Kurt Reidemeister as "Band 1" (English: volume 1). There seems to have been double numeration in this sequence. Neue Folge This sequence started in 1950 with the publication of ''Transfinite Zahlen'' by Heinz Bachmann. The volumes are consecutively numbered, designated as either "Band" or "Heft". A total of 100 volumes was published, often in multiple editions, but pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every vector in ''W''. Informally, it is called the perp, short for perpendicular complement. It is a subspace of ''V''. Example Let V = (\R^5, \langle \cdot, \cdot \rangle) be the vector space equipped with the usual dot product \langle \cdot, \cdot \rangle (thus making it an inner product space), and let W = \, with A = \begin 1 & 0\\ 0 & 1\\ 2 & 6\\ 3 & 9\\ 5 & 3\\ \end. then its orthogonal complement W^\perp = \ can also be defined as W^\perp = \, being \tilde = \begin -2 & -3 & -5 \\ -6 & -9 & -3 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end. The fact that every column vector in A is orthogonal to every column vector in \tilde can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic-orthogonal
In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity. Geometry Two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola. Two particular hyperbolas are frequently used in the plane: The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote ''A'', a pair of lines (''a'', ''b'') are hyperbolic orthogonal if there is a pair (''c'', ''d'') such that a \rVert c ,\ b \rVert d , and ''c'' is the reflection of ''d'' across ''A''. Similar to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
