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Each pair (GX, ε X) is a terminal morphism from F to X in C; Each pair (FY, η Y) is an initial morphism from Y to G in D; In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. The equivalence ...
Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition. The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances.
In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox . It is defined by a Mellin–Barnes integral H p , q m , n [ z | ( a 1 , A 1 ) ( a 2 , A 2 ) …
Given the function symbols F and G, one can introduce a new function symbol F ∘ G, the composition of F and G, satisfying (F ∘ G)(X) = F(G(X)), for all X. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of F matches the codomain type of G , so this is required for the composition to be ...
The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings , and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph.
A unitary representation : of a locally compact group on a Hilbert space = (, , ) defines for each pair of vectors , a continuous function on , the matrix coefficient, by g ↦ h , ρ ( g ) k {\displaystyle g\mapsto \langle h,\rho (g)k\rangle } .
The sheaf of rational functions K X of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties , such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, K X ( U ) is the ...
If the tournament is 4‑strongly connected, then each pair of vertices can be connected with a Hamiltonian path. [9] For every set B {\displaystyle B} of at most k − 1 {\displaystyle k-1} arcs of a k {\displaystyle k} -strongly connected tournament T {\displaystyle T} , we have that T − B {\displaystyle T-B} has a Hamiltonian cycle. [ 10 ]