Encyclopedia - Vector Bundle: Encyclopedia - Vector Bundle
In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety...   » Read the article

Encyclopedia - Coproduct: Encyclopedia - Coproduct
In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definitio...   » Read the article

Encyclopedia - Angular Momentum: Encyclopedia - Angular Momentum
In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia - Adjoint Functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiqu...   » Read the article

Encyclopedia - Flanders Family: Encyclopedia Ii - Flanders Family - Household
The Flanders' house is situated at 740 Evergreen Terrace, Springfield and is next door to 742 Evergreen Terrace, home of the Simpson fami...   » Read the article

Encyclopedia - Vector Bundle: Encyclopedia Ii - Vector Bundle - Definition And First Consequences
A real vector bundle is given by the following data: topological spaces X (the "base space") and E (the "total space") a continuous map ...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia Ii - Adjoint Functors - Motivation
Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the ...   » Read the article

Encyclopedia - Pullback: Encyclopedia Ii - Pullback - Pullback On Tensors
Let be a linear map between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback f * T on V ca...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Properties
Clifford algebra - Relation to the exterior algebra. Given a vector space V one can construct the exterior algebra Λ(V), whose definit...   » Read the article

Encyclopedia - Hom Functor: Encyclopedia Ii - Hom Functor - Formal Definition
Let C be a locally small category (i.e. a category for which Hom-classes are actually sets and not proper classes). For all objects A in ...   » Read the article

Encyclopedia - Pointed Space: Encyclopedia Ii - Pointed Space - Category Of Pointed Spaces
The class of all pointed spaces forms a category Top• with basepoint preserving continuous maps as morphisms. Another way to think abou...   » Read the article

Encyclopedia - Time-invariant System: Encyclopedia Ii - Time-invariant System - Abstract Example
We can denote the shift operator by where r is the amount by which a vector's index set should be shifted. For example, the "advance-by-...   » Read the article

Encyclopedia - Product Category Theory: Encyclopedia Ii - Product Category Theory - Discussion
The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of...   » Read the article

Encyclopedia - Pullback Category Theory: Encyclopedia Ii - Pullback Category Theory - Examples
In the category of sets the pullback of f and g is the set X ×Z Y = {(x, y) ∈ X × Y | f(x) = g(y)}, together with the restrictions ...   » Read the article

Encyclopedia - Product Topology: Encyclopedia Ii - Product Topology - Definition
Let I be a (possibly infinite) index set and suppose Xi is a topological space for every i in I. Set X = Π Xi, the Cartesian product of ...   » Read the article

Encyclopedia - Yamhill County Oregon: Encyclopedia Ii - Yamhill County Oregon - Economy
Yamhill County ranks seventh out of Oregon's thirty-six counties in annual market value of its agricultural production. This agricultural...   » Read the article

Encyclopedia - Galilean Transformation: Encyclopedia Ii - Galilean Transformation - Translation One Dimension
The Galilean transformation is nothing more than careful addition and subtraction of velocity vectors. Unlike the Galilean transformation...   » Read the article

Encyclopedia - Comma Category: Encyclopedia Ii - Comma Category - Definition
The most general comma category construction involves two converging functors. Typically, one of these will be a "selection" or "constant...   » Read the article

Encyclopedia - Angular Momentum: Encyclopedia Ii - Angular Momentum - Angular Momentum In Classical Mechanics
Angular momentum - Definition. The traditional mathematical definition of the angular momentum of a particle about some origin is: ...   » Read the article

Encyclopedia - Exponential Object: Encyclopedia Ii - Exponential Object - Definition
Let C be a category with binary products and let Y and Z be objects of C. The exponential object ZY can be defined as a universal morphis...   » Read the article

Encyclopedia - Exponential Map: Encyclopedia Ii - Exponential Map - Lie Theory
The exponential map is a fundamental construction in the theory of Lie groups. It is a map from the Lie algebra of a Lie group to the gro...   » Read the article

Encyclopedia - Hamiltonian Quantum Mechanics: Encyclopedia Ii - Hamiltonian Quantum Mechanics - The Quantum Hamiltonian
As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a vecto...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Properties
Universal property - Existence and uniqueness. Defining a quantity does not guarantee its existence. Given a functor U and an object X ...   » Read the article

Encyclopedia - Pushout Category Theory: Encyclopedia Ii - Pushout Category Theory - Examples Of Pushouts
Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in t...   » Read the article

Encyclopedia - Injective Module: Encyclopedia Ii - Injective Module - Facts
Any product of (even infinitely many) injective modules is injective. Every direct sum of finitely many injective modules is injective. I...   » Read the article

Encyclopedia - Coequalizer: Encyclopedia Ii - Coequalizer - Definition
The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects ...   » Read the article

Encyclopedia - Hopf Algebra: Encyclopedia Ii - Hopf Algebra - Examples
Group algebra. Suppose G is a group. The group algebra KG is a unital associative algebra over K. It turns into a Hopf algebra if we defi...   » Read the article

Encyclopedia - Exponential Object: Encyclopedia Ii - Exponential Object - Examples
In the category of sets, the exponential object ZY is the set of all functions from Y to Z. The map is just the evaluation map which sen...   » Read the article

Encyclopedia - Flanders Family: Encyclopedia Ii - Flanders Family - The Adventures Of Ned Flanders
The Adventures of Ned Flanders was a short that appeared at the end of the episode "The Front". Here, the Flanders dedication to religion...   » Read the article

Encyclopedia - Pullback Category Theory: Encyclopedia Ii - Pullback Category Theory - Universal Property
Explicitly, the pullback of the morphisms f and g consists of an object P and two morphisms p1 : P → X and p2 : P → Y for w...   » Read the article

Encyclopedia - Hom Functor: Encyclopedia Ii - Hom Functor - Yoneda's Lemma
Referring to the above commutative diagram, one observes that every morphism h : A′ → A gives rise to a natural transformation ...   » Read the article

Encyclopedia - Pullback: Encyclopedia Ii - Pullback - Pullback On Tensor Bundles
More generally, one can construct the pullback map between tensor bundles of rank (0,n); the construction proceeds entirely analogously t...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Examples
We give a few worked examples to highlight the general idea. The reader can construct numerous other examples by consulting the articles ...   » Read the article

Encyclopedia - Universal Property: Encyclopedia Ii - Universal Property - Formal Definition
Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consis...   » Read the article

Encyclopedia - Yamhill County Oregon: Encyclopedia Ii - Yamhill County Oregon - Geography
According to the U.S. Census Bureau, the county has a total area of 1,861 km² (718 mi²). 1,853 km² (716 mi²) of it is land and 7 km²...   » Read the article

Encyclopedia - Yamhill County Oregon: Encyclopedia Ii - Yamhill County Oregon - Demographics
As of the census2 of 2000, there are 84,992 people, 28,732 households, and 21,376 families residing in the county. The population density...   » Read the article

Encyclopedia - Yamhill County Oregon: Encyclopedia Ii - Yamhill County Oregon - History
The earliest known inhabitants of the area were the Yamhill Indians, who have inhabited the area for over 8000 years. They are one of the...   » Read the article

Encyclopedia - Pushout Category Theory: Encyclopedia Ii - Pushout Category Theory - Construction Via Coproducts And Coequalizers
All of the above examples may be regarded as special cases of the following very general construction, which works in any category C sati...   » Read the article

Encyclopedia - Pushout Category Theory: Encyclopedia Ii - Pushout Category Theory - Universal Property
Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X → P and i2 : Y → P for wh...   » Read the article

Encyclopedia - Product Category Theory: Encyclopedia Ii - Product Category Theory - Examples
Given the Set (the category of sets), the product in the category theoretic sense is the cartesian product. Given a family of sets Xi the...   » Read the article

Encyclopedia - Pullback: Encyclopedia Ii - Pullback - Pullback Of Cotangent Bundles
The pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism f* : T*N → T*M, fo...   » Read the article

Encyclopedia - Exponential Map: Encyclopedia Ii - Exponential Map - Relationships
The two notions of the exponential map coincide in the case of Lie groups equipped with bi-invariant metrics (i.e. Riemannian metrics inv...   » Read the article

Encyclopedia - Pullback: Encyclopedia Ii - Pullback - Pullback Of Diffeomorphisms
When the map f between manifolds is a diffeomorphism, that is, it is both smooth and invertible, then the pullback can be defined for the...   » Read the article

Encyclopedia - Product Category Theory: Encyclopedia Ii - Product Category Theory - Definition
Let C be a category and let {Xi | i ∈ I} be an indexed family of objects in C. The product of the set {Xi} is an object X together with...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Structure Of Clifford Algebras
In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple...   » Read the article

Encyclopedia - Injective Module: Encyclopedia Ii - Injective Module - Definition
More formally, a left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent condition...   » Read the article

Encyclopedia - Hopf Algebra: Encyclopedia Ii - Hopf Algebra - Related Concepts
Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure of the totality of all homology or co...   » Read the article

Encyclopedia - Injective Module: Encyclopedia Ii - Injective Module - Examples
Trivially, the zero module {0} is injective. Every vector space Q is injective. Reason: if Q is a subspace of V, we can find a basis of Q...   » Read the article

Encyclopedia - Product Topology: Encyclopedia Ii - Product Topology - Examples
If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one o...   » Read the article

Encyclopedia - Product Topology: Encyclopedia Ii - Product Topology - Properties
The product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topolog...   » Read the article

Encyclopedia - Hamiltonian Quantum Mechanics: Encyclopedia Ii - Hamiltonian Quantum Mechanics - Energy Eigenket Degeneracy Symmetry And Conservation Laws
In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstat...   » Read the article

Encyclopedia - Galilean Transformation: Encyclopedia Ii - Galilean Transformation - Galilean Transformations
Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated ...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia Ii - Adjoint Functors - Formal Definitions
A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural is...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia Ii - Adjoint Functors - Examples
Free objects and forgetful functors. If F : Set → Grp is the functor assigning to each set X the free group over X, and if G ...   » Read the article

Encyclopedia - Angular Momentum: Encyclopedia Ii - Angular Momentum - Angular Momentum In Relativistic Mechanics
In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angul...   » Read the article

Encyclopedia - Angular Momentum: Encyclopedia Ii - Angular Momentum - Angular Momentum In Quantum Mechanics
In quantum mechanics, angular momentum is defined like momentum - not as a quantity but as an operator on the wave function: where r a...   » Read the article

Encyclopedia - Vector Bundle: Encyclopedia Ii - Vector Bundle - Vector Bundle Morphisms
A morphism from the vector bundle π1 : E1 → X1 to the vector bundle π2 : E2 → X2 is given by a pair of continuous maps f&...   » Read the article

Encyclopedia - Vector Bundle: Encyclopedia Ii - Vector Bundle - Sections And Locally Free Sheaves
Given a vector bundle π : E → X and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s ...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Spin And Pin Groups
In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies tha...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - The Clifford Group Γ
In this section we assume that V is finite dimensional and the bilinear form of Q is non-singular. The Clifford group Γ is defined to be...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Spinors
Suppose that p+q=2n is even. Then the Clifford algebra Cℓp,q(C) is a matrix algebra, and so has a complex representation of dimension 2...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Applications
Clifford algebra - Differential geometry. One of the principal applications of the exterior algebra is in differential geometry where i...   » Read the article

Encyclopedia - Coequalizer: Encyclopedia Ii - Coequalizer - Special Cases
In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism. In p...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Examples: Real And Complex Clifford Algebras
The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. Every non...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Basis And Dimension
If the dimension of V is n and {e1,…,en} is a basis of V, then the set is a basis for Cℓ(V,Q). The empty product (k = 0) is define...   » Read the article

Encyclopedia - Time-invariant System: Encyclopedia Ii - Time-invariant System - Simple Example
To demonstrate how to determine if a system is time-invariant then consider the two systems: System A: System B: Since system A expli...   » Read the article

Encyclopedia - Time-invariant System: Encyclopedia Ii - Time-invariant System - Formal Example
A more formal proof of why system A & B from above is now presented. To perform this proof, the second definition will be used. Syste...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Introduction And Basic Properties
Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quad...   » Read the article

Encyclopedia - Clifford Algebra: Encyclopedia Ii - Clifford Algebra - Universal Property And Construction
Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. In most cases of interest the field K is eithe...   » Read the article

Encyclopedia - Exponential Map: Encyclopedia Ii - Exponential Map - Riemannian Geometry
In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself. Exponen...   » Read the article