axioms

A variety of topologies can be placed on a set to form a topological space. When every set in a topology T1 is also in a topology T2, we say that T2 is finer than T1, and T1 is coarser than T2. A proof which relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger a ...

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Topological space, Topological space - Definition, Topological space - Comparison of topologies, Topological space - Continuous functions, Topological space - Alternative definitions, Topological space - Examples of topological spaces, Topological space - Topological constructions, Topological space - Classification of topological spaces, Topological space - Topological spaces with algebraic structure, Topological space - Topological spaces with order structure

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Early in the 20th century, parallel to the development of mathematical logic as a stand-alone branch of mathematics, many parts of mathematics began to treated by delineating useful sets of axioms and then studying their consequences. Thus for example the studies of "hypercomplex numbers", popular at the turn of the century, were put onto an axiomatic footing as branches of ring theory (in this case, sp ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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A well-known example was the development of analytic geometry, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curves and surfaces of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques (that is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.) At the end of the 19th century, Felix Klein noted that the many branches of geometry which had been developed during th ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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Once you have defined the operations and axioms for your algebra, you can now define the notion of homomorphism between two algebras A and B. A homomorphism h: A → B is simply a function from the set A to the set B such that, for every operation f (of arity, say, n), h(fA(x1,...,xn)) = fB(h(x1),...,h(xnSee also:

Universal algebra, Universal algebra - Basic idea, Universal algebra - Examples, Universal algebra - Groups, Universal algebra - Modules, Universal algebra - Further issues

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The T0 axiom is special in that it cannot only be added to a property (so that regular plus T0 is T3) but also subtracted from a property (so that Hausdorff minus T0 is preregular), in a fairly precise sense; see Kolmogorov quotient for more information. When applied to the separation axioms, this leads to the relationships in the table below: In this table, you go from the right side to the left side by adding the requirement of T0, and you go from the left side to the right side b ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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In the above definition of an associative algebra, the definition of associativity was made with regard to all of the elements of A. It is sometimes more convenient to have a definition of associativity that does not need to refer to the elements of A. This can be done as follows. An algebra is defined as a map M (multiplication) on a vector space A: An associative algebra is an algebra w ...

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Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

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For example the group of projective geometry in n dimensions is the symmetry group of n-dimensional projective space (the matrix group of size n+1, quotiented by scalar matrices). The affine group will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity. This subgroup has a known structure (semidirect product of the matrix group of size n with the subgroup of translations). This description then tells us which properties are 'affine'. In Euclidean plane geo ...

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Erlangen program, Erlangen program - The problems of nineteenth century geometry, Erlangen program - Homogeneous spaces, Erlangen program - Examples: affine geometry, Erlangen program - Influence on later work, Erlangen program - Abstract returns from the Erlangen program

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In other words, the "traditional spaces" are homogeneous spaces; but not for a uniquely determined group. Changing the group changes the appropriate geometric language. In today's language, the groups concerned in classical geometry are all very well-known as Lie groups: the classical groups. The specific relationships are quite simply described, using technical language. ...

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Erlangen program, Erlangen program - The problems of nineteenth century geometry, Erlangen program - Homogeneous spaces, Erlangen program - Examples: affine geometry, Erlangen program - Influence on later work, Erlangen program - Abstract returns from the Erlangen program

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Theorem 1.5: The inverse of each element in (G,*) is unique; equivalently, for all a in G, a*x = e if and only if x=a -1. If x=a -1, then a*x = e by A4. Apply theorem 1.3, with b = e. Alternative proof: Suppose that an element g of G has two inverses, h and k say. Then h = h*e = h*(g*k) = (h*g)*k = e*k = k (equalities justifie ...

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Elementary group theory, Elementary group theory - R+ is a group, Elementary group theory - R* is not a group, Elementary group theory - R^#* is a group, Elementary group theory - Inverses work on either side, Elementary group theory - An identity works on either side, Elementary group theory - Latin square property, Elementary group theory - The identity is unique, Elementary group theory - Inverses are unique, Elementary group theory - Inverting twice gets you back where you started, Elementary group theory - The inverse of ab, Elementary group theory - Cancellation, Elementary group theory - Repeated use of *

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The long-term effects of the Erlangen programme can be seen all over pure mathematics (see tacit use at congruence (geometry), for example); and the idea of transformations and of synthesis using groups of symmetry is of course now standard too in physics. When topology is routinely described in terms of properties invariant under homeomorphism, one can see the underlying idea in operation. The groups involved will be infinite-dimensional in almost all cases - and not Lie groups - but the philosophy is the same. Of course this mostly ...

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Erlangen program, Erlangen program - The problems of nineteenth century geometry, Erlangen program - Homogeneous spaces, Erlangen program - Examples: affine geometry, Erlangen program - Influence on later work, Erlangen program - Abstract returns from the Erlangen program

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Nowadays, most algorithms are implemented and run on a computer. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library. A different approach is taken by the Numerical Recipes library, which emphasizes understanding ...

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Numerical analysis, Numerical analysis - General introduction, Numerical analysis - Direct and iterative methods, Numerical analysis - Discretization, Numerical analysis - The generation and propagation of errors, Numerical analysis - Applications, Numerical analysis - Areas of study, Numerical analysis - Computing values of functions, Numerical analysis - Interpolation extrapolation and regression, Numerical analysis - Solving equations and systems of equations, Numerical analysis - Optimization, Numerical analysis - Evaluating integrals, Numerical analysis - Differential equations, Numerical analysis - History, Numerical analysis - Software

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Old: Abacus Napier's bones, slide rule Ruler and compass Mental calculation New: Calculators and computers Programming languages Computer algebra systems (listing) Internet shorthand notation statistical analysis software SPSS SAS programming language R programming language ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudosci ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups. There arises the question of reading the Erlangen program from the abstract group, to the geometry. One example: oriented (i.e., reflections not included) elliptic geometry (i.e., the surface of an n-sphere with opposite points identified) and oriented spherical geometry (the same nonEuclidean geometry, but with opposite points not identified) have isomorphic automorphism group, SO(n+1) for even n. These may appear to be distinct. It turns out, however, t ...

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Erlangen program, Erlangen program - The problems of nineteenth century geometry, Erlangen program - Homogeneous spaces, Erlangen program - Examples: affine geometry, Erlangen program - Influence on later work, Erlangen program - Abstract returns from the Erlangen program

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Theorem 1.4: The identity element of a group (G,*) is unique. a*e = a (by A3) Apply theorem 1.3, with b = a. Alternative proof: Suppose that G has two identity elements, e and f say. Then e*f = e, by A3', but also e*f = f, by Theorem 1.2. Hence e = f. As a result, we can speak of the identity element of (G,*) rather than an identity element. Where different groups are being discussed and compared, often e ...

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Elementary group theory, Elementary group theory - R+ is a group, Elementary group theory - R* is not a group, Elementary group theory - R^#* is a group, Elementary group theory - Inverses work on either side, Elementary group theory - An identity works on either side, Elementary group theory - Latin square property, Elementary group theory - The identity is unique, Elementary group theory - Inverses are unique, Elementary group theory - Inverting twice gets you back where you started, Elementary group theory - The inverse of ab, Elementary group theory - Cancellation, Elementary group theory - Repeated use of *

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Informally, the Peano axioms may be stated as follows: There is a natural number 0. Every natural number a has a successor, denoted by S(a) or a'. There is no natural number whose successor is 0. Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b). If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it i ...

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Peano axioms, Peano axioms - The axioms, Peano axioms - Peano arithmetic, Peano axioms - Existence and uniqueness, Peano axioms - Binary operations and ordering, Peano axioms - Categorical interpretation, Peano axioms - Metamathematical discussion

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Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be. Certainly the initial set of functions are intuitively computable (in their very simplicity), and the two operations by which one can create new primitive recursive functions are also very straightforward. However the set of primitive recursive functions does not include every possible computable function — this can be seen with a variant of Cantor's diagonal argument. This argument provides a computable function which is n ...

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Primitive recursive function, Primitive recursive function - Definition, Primitive recursive function - Examples, Primitive recursive function - Addition, Primitive recursive function - Subtraction, Primitive recursive function - Limitations, Primitive recursive function - Bibliography

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Take an arbitrary set S and form the K-vector space with basis S. The elements of this vector space are those functions from S to K that map all but finitely many elements of S to zero; we identify the element s of S with the function that maps s to 1 and all other elements of S to 0. We will denote this space by C. We define By linearity, both Δ and ε can then uniquely be extended to all of C. The vector space C becomes a coalgebra with comultiplication Δ and counit ε (you may want to ...

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Coalgebra, Coalgebra - Formal definition, Coalgebra - Examples, Coalgebra - Sweedler notation, Coalgebra - Further concepts and facts

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Formally, a coalgebra over a field K is a K-vector space C together with K-linear maps and such that . Equivalently, the following two diagrams commute: In the first diagram we silently identify with ; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces C, and are identified. The first diagram is the dual of the one expressing associativity of algebra multiplication (ca ...

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Coalgebra, Coalgebra - Formal definition, Coalgebra - Examples, Coalgebra - Sweedler notation, Coalgebra - Further concepts and facts

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Primitive recursive function - Addition. Intuitively we would like to define addition recursively as: add(0,x)=x add(n+1,x)=add(n,x)+1 In order to fit this into a strict primitive recursive definition, we define: add(0,x)=P11(x) add(S(n),x)=S(P13(add(n,x),n,x)) (Note: here P13 is a func ...

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Primitive recursive function, Primitive recursive function - Definition, Primitive recursive function - Examples, Primitive recursive function - Addition, Primitive recursive function - Subtraction, Primitive recursive function - Limitations, Primitive recursive function - Bibliography

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When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element c of the coalgebra (C,Δ,ε), we know that there exist elements c(1)(i) and c(2)(i) in C such that In Sweedler's notation, this is abbreviated to The fact that ε is a counit can then be expressed with the following formula The coassociativity ...

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Coalgebra, Coalgebra - Formal definition, Coalgebra - Examples, Coalgebra - Sweedler notation, Coalgebra - Further concepts and facts

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These axioms are given here in a second-order predicate calculus form. See first-order predicate calculus for a way to rephrase these axioms to be first-order. Dedekind proved, in his 1888 book Was sind und was sollen die Zahlen, that any model of the second order Peano axioms is isomorphic to the natural numbers. On the other hand, the last axiom listed above, the mathematical induction axiom, is not itself expressible in the first order language of arithmetic. If one replaces the last axiom with the schema: If P(0) is true; and for all x, P(x) implies P(x ...

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Peano axioms, Peano axioms - The axioms, Peano axioms - Peano arithmetic, Peano axioms - Existence and uniqueness, Peano axioms - Binary operations and ordering, Peano axioms - Categorical interpretation, Peano axioms - Metamathematical discussion

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