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More formally, a quotient graph is a quotient object in the category of graphs. The category of graphs is concretizable – mapping a graph to its set of vertices makes it a concrete category – so its objects can be regarded as "sets with additional structure", and a quotient graph corresponds to the graph induced on the quotient set V / R of ...
Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R. If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties: such that [1] (i) The map π {\displaystyle \pi } is surjective, and its fibers are exactly the G-orbits in X.
In order for the definition to make sense the category in question must have zero morphisms. The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0 XY : X → Y. Explicitly, this means the following. The cokernel of f : X → Y is an object Q together with a morphism q : Y → Q such that the diagram
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and () The quotient rule states that the derivative of h(x) is
In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra. However, the use of the term for the more general cases ...
The direct approach can be made, by means of the function field of a variety (i.e. rational functions): take the G-invariant rational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put it in the ...
Analytic continuation of natural logarithm (imaginary part) Analytic continuation is a technique to extend the domain of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.