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Infinitedimensional Vector Space In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.[1] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis,[a] and all bases of a vector space have equal cardinality;[b] as a result, the dimension of a vector space is uniquely defined. We say V is finitedimensional if the dimension of V is finite, and infinitedimensional if its dimension is infinite. The dimension of the vector space V over the field F can be written as dimF(V) or as [V : F], read "dimension of V over F" [...More...]  "Infinitedimensional Vector Space" on: Wikipedia Yahoo Parouse 

Mathematics Mathematics Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist [...More...]  "Mathematics" on: Wikipedia Yahoo Parouse 

Jinvariant In mathematics, Felix Klein's jinvariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper halfplane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that j ( e 2 π i / 3 ) = 0 , j ( i ) = 1728 = 12 3 . displaystyle jleft(e^ 2pi i/3 right)=0,quad j(i)=1728=12^ 3 . Rational functions of j are modular, and in fact give all modular functions [...More...]  "Jinvariant" on: Wikipedia Yahoo Parouse 

Bialgebra In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.) Similar bialgebras are related by bialgebra homomorphisms [...More...]  "Bialgebra" on: Wikipedia Yahoo Parouse 

Trace Class In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis [...More...]  "Trace Class" on: Wikipedia Yahoo Parouse 

Hilbert Space The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinitedimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz [...More...]  "Hilbert Space" on: Wikipedia Yahoo Parouse 

Nuclear Operator In mathematics, a nuclear operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace) [...More...]  "Nuclear Operator" on: Wikipedia Yahoo Parouse 

Banach Space In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.[1] Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis [...More...]  "Banach Space" on: Wikipedia Yahoo Parouse 

Representation Theory Representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.[1] In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras [...More...]  "Representation Theory" on: Wikipedia Yahoo Parouse 

Character (mathematics) In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings.[1] Other uses of the word "character" are almost always qualified.Contents1 Multiplicative character 2 Character of a representation 3 See also 4 References 5 External linksMultiplicative character[edit] Main article: multiplicative character A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field (Artin 1966), usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of G [...More...]  "Character (mathematics)" on: Wikipedia Yahoo Parouse 

Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It 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. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.[1][2] Groups share a fundamental kinship with the notion of symmetry [...More...]  "Group (mathematics)" on: Wikipedia Yahoo Parouse 

Monstrous Moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a twodimensional conformal field theory, allowing physics to form a bridge between two mathematical areas [...More...]  "Monstrous Moonshine" on: Wikipedia Yahoo Parouse 

Graded Dimension In mathematics, a graded vector space is a vector space that has the extra structure of a grading or a gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.Contents1 ℕgraded vector spaces 2 General Igraded vector spaces 3 Homomorphisms 4 Operations on graded vector spaces 5 See also 6 Referencesℕgraded vector spaces[edit] Let ℕ be the set of nonnegative integers. An ℕgraded vector space, often called simply a graded vector space without the prefix ℕ, is a vector space V which decomposes into a direct sum of the form V = ⨁ n ∈ N V n displaystyle V=bigoplus _ nin mathbb N V_ n where each V n displaystyle V_ n is a vector space [...More...]  "Graded Dimension" on: Wikipedia Yahoo Parouse 

Algebra Over A Field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure, which consists of a set, together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, and satisfies the axioms implied by "vector space" and "bilinear".[1] The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and nonassociative algebras. Given an integer n, the ring of real square matrices of order n is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative [...More...]  "Algebra Over A Field" on: Wikipedia Yahoo Parouse 

Monster Group In the area of modern algebra known as group theory, the Monster group M (also known as the Fischer– Griess Griess Monster, or the Friendly Giant) is the largest sporadic simple group, having order 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8×1053.The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern [...More...]  "Monster Group" on: Wikipedia Yahoo Parouse 

McKay–Thompson Series In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a twodimensional conformal field theory, allowing physics to form a bridge between two mathematical areas [...More...]  "McKay–Thompson Series" on: Wikipedia Yahoo Parouse 