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Star-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert's space and Hermitian adjoints. However, it may happen that an algebra admits no involution. Definitions *-ring In mathematics, a *-ring is a ring with a map that is an antiautomorphism and an involution. More precisely, is required to satisfy the following properties: * * * * for all in . This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant. Elements such that are called ''self-adjoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if x^*=y then since y^*=x^=x in a star-algebra, the set is a self-adjoint set even though ''x'' and ''y'' need not be self-adjoint elements. In functional analysis, a linear operator A : H \to H on a Hilbert space is called self-adjoint if it is equal to its own adjoint ''A''. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator ''A'' is self-adjoint if and only if the matrix describing ''A'' with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger categor ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sesquilinear Form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix ''sesqui-'' meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector. A motivating special case is a sesquilinear form on a complex vector space, . This is a map that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asterisk
The asterisk ( ), from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star. Computer scientists and mathematicians often vocalize it as star (as, for example, in ''the A* search algorithm'' or ''C*-algebra''). In English, an asterisk is usually five- or six-pointed in sans-serif typefaces, six-pointed in serif typefaces, and six- or eight-pointed when handwritten. Its most common use is to call out a footnote. It is also often used to censor offensive words. In computer science, the asterisk is commonly used as a wildcard character, or to denote pointers, repetition, or multiplication. History The asterisk has already been used as a symbol in ice age cave paintings. There is also a two thousand-year-old character used by Aristarchus of Samothrace called the , , which he used when proofreading Homeric poetry to mark lines that were duplicated. Origen is know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Line
In typography, the mean line is the imaginary line at the top of the x-height. upright 2.0, alt=A diagram showing the line terms used in typography Round glyphs will tend to break ( overshoot) the mean line slightly in many typefaces, since this is aesthetically more pleasing, otherwise curved letters such as ''a'', ''c'', ''e'', ''m'', ''n'', ''o'', ''r'', ''s'', and ''u'' will appear visually smaller than flat-topped (or bottomed) characters of equal height, due to an optical illusion Within visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual perception, percept that arguably appears to differ from reality. Illusions come in a wide v .... References External links * Page 31The Complete Manual of Typography: A Guide to Setting Perfect Type, Second Editionby James Felici {{DEFAULTSORT:Mean line Typography ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unary Operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on . Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial ), functional notation (e.g. or ), and superscripts (e.g. transpose ). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument. Examples Unary negative and positive As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation: :3 − −2 Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expression i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''Undergraduate algebra'', Springer, 2005; V.§3. (alternative notations: Mat''n''(''R'') and ). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an associative algebra over ''R'', and may be called a matrix algebra. In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''. Examples * The set of all matrices over ''R'', denoted M''n''(''R''). This is sometimes called the "full ring of ''n''-by-''n'' matrices". * The set of all upper triangular matrices over ''R''. * The set of all l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF(x) * F(x + y) = F(x) + F(y) * F(xy) = F(x) F(y) The first two conditions say that is a ''K''-linear map (or ''K''-module homomorphism if ''K'' is a commutative ring), and the last condition says that is a (non-unital) ring homomorphism. If admits an inverse homomorphism, or equivalently if it is bijective, is said to be an isomorphism between and . Unital algebra homomorphisms If ''A'' and ''B'' are two unital algebras, then an algebra homomorphism F:A\rightarrow B is said to be ''unital'' if it maps the unity of ''A'' to the unity of ''B''. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded. A unital algebra homomorphism is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate-linear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y \in V and every complex number s, where \overline denotes the complex conjugate of s. Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces. Definitions and characterizations A function is called or if it is additive and conjugate homogeneous. An on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rng (algebra)
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term ''rng'' (IPA: ) is meant to suggest that it is a ring without ''i'', that is, without the requirement for an identity element. There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see ). The term ''rng'' was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity. A number of algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space. Definition Formally, a rng is a set ''R'' with two binary operations called ''addition'' and ''multiplication'' such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
