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Fundamental Theorems
In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Some of these are classification theorems of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describe classification of regular curves in space up to translation, rotation. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself. Fundamental theorems of mathematical topics * Fundamental theorem of algebra * Fundamental theorem of algeb ... [...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] |
Fundamental Theorem Of Boolean Algebra
Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: F = x \cdot F_x + x' \cdot F_, where F is any Boolean function, x is a variable, x' is the complement of x, and F_xand F_ are F with the argument x set equal to 1 and to 0 respectively. The terms F_x and F_ are sometimes called the positive and negative Shannon cofactors, respectively, of F with respect to x. These are functions, computed by restrict operator, \operatorname(F, x, 0) and \operatorname(F, x, 1) (see valuation (logic) and partial application). It has been called the "fundamental theorem of Boolean algebra". Besides its theoretical importance, it paved the way for binary decision diagrams (BDDs), satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits. In such engineering contexts (especially in BDDs), the expansion is interpreted as a if-then-else, with the variable x being the condition and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Ideal Theory In Number Fields
In number theory, the fundamental theorem of ideal theory in number fields states that every nonzero proper ideal in the ring of integers of a number field admits unique factorization into a product of nonzero prime ideals. In other words, every ring of integers of a number field is a Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily .... References * Keith ConradIdeal factorization* Algebraic numbers Theorems in algebraic number theory Factorization {{Numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem On Homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems. Group theoretic version Given two groups ''G'' and ''H'' and a group homomorphism , let ''N'' be a normal subgroup in ''G'' and φ the natural surjective homomorphism (where ''G''/''N'' is the quotient group of ''G'' by ''N''). If ''N'' is a subset of ker(''f'') then there exists a unique homomorphism such that . In other words, the natural projection φ is universal among homomorphisms on ''G'' that map ''N'' to the identity element. The situation is described by the following commutative diagram: : ''h'' is injective if and only if . Therefore, by setting we immediately get the first isomorphism theorem. We can write the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Geometric Calculus
In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms. Differentiation With a geometric algebra given, let a and b be vectors and let F be a multivector-valued function of a vector. The directional derivative of F along b at a is defined as :(\nabla_b F)(a) = \lim_, provided that the limit exists for all b, where the limit is taken for scalar \epsilon. This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued. Next, choose a set of basis vectors \ and consider the operators, denoted \partial_i, that perform directional derivatives in the directions of e_i: :\partial_i : F \mapsto (x\mapsto (\nabla_ F)(x)). Then, using the Einstein summation notation, consider the operator: :e^i\partial_i, which means :F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Galois Theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension ''E''/''F'' that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. (''Intermediate fields'' are fields ''K'' satisfying ''F'' ⊆ ''K'' ⊆ ''E''; they are also called ''subextensions'' of ''E''/''F''.) Explicit description of the correspondence For finite extensions, the correspondence can be described explicitly as follows. * For any subgroup ''H'' of Gal(''E''/''F''), the corresponding fixed field, denoted ''EH'', is the set of those elements of ''E'' which are fixed by every automorphism in ''H''. * For any intermediate field ''K'' of ''E''/''F'', the corresponding subgroup is Aut(''E''/''K''), that is, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Finite Distributive Lattices
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (other).'' In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.. The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called ''fields of sets'', closely related to the ''rings of sets'' used by Birkho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Finitely Generated Modules Over A Principal Ideal Domain
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields. Statement When a vector space over a field ''F'' has a finite generating set, then one may extract from it a basis consisting of a finite number ''n'' of vectors, and the space is therefore isomorphic to ''F''''n''. The corresponding statement with the ''F'' generalized to a principal ideal domain ''R'' is no longer true, since a basis for a finitely generated module over ''R'' might not exist. However such a module is still isomorphic to a quotient of some module ''Rn'' with ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Finitely Generated Abelian Groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_s. In this case, we say that the set \ is a ''generating set'' of G or that x_1,\dots, x_s ''generate'' G. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples * The integers, \left(\mathbb,+\right), are a finitely generated abelian group. * The integers modulo n, \left(\mathbb/n\mathbb,+\right), are a finite (hence finitely generated) abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group \left(\mathbb,+\right) of rational numbers is not finitely generated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Exterior Calculus
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, and Stokes' theorem is the case of a surface in \R^3. Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus. Stokes' theorem says that the integral of a differential form \omega over the boundary \partial\Omega of some orientable manifold \Omega is equal to the integral of its exterior derivative d\omega over the whole of \Omega, i.e., \int_ \omega = \int_\Omega d\omega\,. Stokes' theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Equivalence Relations
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Cyclic Groups
In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order ''n'', every subgroup's order is a divisor of ''n'', and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups. Finite cyclic groups For every finite group ''G'' of order ''n'', the following statements are equivalent: * ''G'' is cyclic. * For every divisor ''d'' of ''n'', ''G'' has at most one subgroup of order ''d''. If either (and thus both) are true, it follows that there exists exactly one subgroup of order ''d'', for any divisor of ''n''. This statement is known by various names such as characterization by subgroups. (See also cyclic group for some characterization.) There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example. However, the Klein group has more than one subgroup of order 2, so it does not meet the conditions of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
