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Definition and notation: When we write f : D(f) ⊆ X → Y then we mean that f is a Y-valued function with domain D(f) where D(f) ⊆ X. If we say that f : D(f) ⊆ X → Y is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of f is closed (resp. sequentially closed ...
That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h. [3] Since the parentheses do not change the result, they are generally omitted. In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g ; in a wider sense, it is sufficient that the former be an improper subset of ...
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]
Isaac Newton's notation for differentiation (also called the dot notation, fluxions, or sometimes, crudely, the flyspeck notation [12] for differentiation) places a dot over the dependent variable. That is, if y is a function of t , then the derivative of y with respect to t is
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
The Theory of Functional Connections (TFC) is a mathematical framework designed for functional interpolation.It introduces a method to derive a functional— a function that operates on another function—capable of transforming constrained optimization problems into equivalent unconstrained problems.
In a nozzle or other constriction, the discharge coefficient (also known as coefficient of discharge or efflux coefficient) is the ratio of the actual discharge to the ideal discharge, [1] i.e., the ratio of the mass flow rate at the discharge end of the nozzle to that of an ideal nozzle which expands an identical working fluid from the same initial conditions to the same exit pressures.
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.