[permalink] [id link]
In algebra, for such entities as toposes and Hopf algebras, the structure supports an internal language that is a constructive theory ; working within the constraints of that language is often more intuitive and flexible than working externally by such means as reasoning about the set of possible concrete algebras and their homomorphisms.
from
Wikipedia
Some Related Sentences
algebra and for
solution for the additive coefficients by simple algebra rather than
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.
100 cc and P = 2. 50E6 pascals, so we can solve for temperature by simple algebra:
# Topological algebra, infinity-stacks, ' dérivateurs ', cohomological formalism of toposes as an inspiration for a new homotopic algebra
for all x and y in the algebra.
The left and right alternative identities for an algebra are equivalent to
Conversely, any algebra for which this is true is clearly alternative.
The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008: He shows ( page 162 ) the relation for an algebra A with unit element e and an involutive anti-automorphism such that a + a * and aa * are on the line spanned by e for all a in A.
An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I.
The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets ( for spaces only requiring finite additivity, such as the ba space ).
AKCL formed the foundation for Axiom, another computer algebra system.
A Banach algebra is called " unital " if it has an identity element for the multiplication whose norm is 1, and " commutative " if its multiplication is commutative.
Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering and then applying the outcome in the original algebra.
For example, the spectrum of an element of a complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).
The binomial theorem also holds for two commuting elements of a Banach algebra.
If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.
The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics ( for example Venn diagrams and symbolic reasoning about their Boolean algebra ), and the everyday usage of set theory concepts in most contemporary mathematics.
Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
algebra and such
The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic.
The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.
Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.
An important example of such an algebra is a commutative C *- algebra.
Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.
The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra.
Commonly used types of data include life history, fecundity, and survivorship, and these are analysed using mathematical techniques such as matrix algebra.
Especially, the fact that the integers and any polynomial ring in one variable over a field are Euclidean domains such that the Euclidean division is easily computable is of basic importance in computer algebra.
* Computer algebra systems such as Mathematica and Maxima can often handle irrational numbers like or in a completely " formal " way, without dealing with a specific encoding of the significand.
Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.
The spacetime algebra and the conformal geometric algebra are specific examples of such geometric algebras.
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures ( such as groups, rings, or vector spaces ).
Al-Khwarizmi also used the word algebra (' al-jabr ') to describe the mathematical operations he introduced, such as balancing equations, which helped in several problems.
In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
Informally we can think of elements of the Lie algebra as elements of the group that are " infinitesimally close " to the identity, and the Lie bracket is something to do with the commutator of two such infinitesimal elements.
The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker – Campbell – Hausdorff formula: there exists a neighborhood U of the zero element of, such that for u, v in U we have
Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces.
Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.
algebra and entities
A key feature of GA is its emphasis on geometric interpretations of certain elements of the algebra as geometric entities.
However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.
0.675 seconds.