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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. For example, | x | + 3 = | x + 3 | only when x ≥ 0. The picture shows another example. The composition of one-to-one (injective) functions is always one ...
Observation: If g : S → Y is a function and G is the canonical set-valued function induced by g (i.e. G : S → 2 Y is defined by G(s) := { g(s) } for every s ∈ S) then since Gr g = Gr G, g has a closed (resp. sequentially closed, open, sequentially open) graph in X × Y if and only if the same is true of G.
Open E with dot below: George Herzog’s Jabo transcription Ɛ̣̀ ɛ̣̀: Open E with dot below and grave: Ɛ̣́ ɛ̣́: Open E with dot below and acute: Ɛ̣̂ ɛ̣̂: Open E with dot below and circumflex: Ɛ̣̃ ɛ̣̃: Open E with dot below and tilde: Ɛ̣̈ ɛ̣̈: Open E with dot below and diaeresis: Ɛ̣̈̀ ɛ̣̈̀: Open E with dot ...
1. Inner semidirect product: if N and H are subgroups of a group G, such that N is a normal subgroup of G, then = and = mean that G is the semidirect product of N and H, that is, that every element of G can be uniquely decomposed as the product of an element of N and an element of H.
Language scripts or transcription schemes that use the dot above a letter as a diacritical mark: In some forms of Arabic romanization, ġ stands for ghayn (غ).; The Latin orthography for Chechen includes ċ, ç̇, ġ, q̇ and ẋ, corresponding to Cyrillic цӏ, чӏ, гӏ, къ and хь and representing /tsʼ/, /t͡ʃʼ/, /ɣ/, /qʼ/ and /ħ/ respectively.
Latin Capital Letter F with dot above 0651 ISO 8859-14: U+1E1F ḟ Latin Small Letter F with dot above 0652 U+1E20 Ḡ Latin Capital Letter G with macron U+1E21 ḡ Latin Small Letter G with macron U+1E22 Ḣ Latin Capital Letter H with dot above U+1E23 ḣ Latin Small Letter H with dot above U+1E24 Ḥ Latin Capital Letter H with dot below: U+1E25
Taking F 0 = diag (–1,1,1) as base point, every F can be written in the form g F 0 g −1. Given a path F(t), the ordinary differential equation = /, with initial condition () =, has a unique C 1 solution g(t) with values in G, giving the lift by parallel transport of F.
Let G be a graph with vertex set V. Let F be a field, and f a function from V to F k such that xy is an edge of G if and only if f(x)·f(y) ≥ t. This is the dot product representation of G. The number t is called the dot product threshold, and the smallest possible value of k is called the dot product dimension. [1]