Sentence examples for where f is a smooth function from inspiring English sources

Mirollo and Strogatz further assume that the dynamics of each (uncoupled) oscillator is governed by (v(t)=f phi)) where f is a smooth function satisfying (f(0)=0), (f(1)=1), (f^{prime} phi >0) and (f^{primeprime} phi <0) for all (phiin[0,1]).

An almost contact structure is said to be normal if the almost complex structure J on the product manifold M ¯ × R given by J X X, f d d t ) = ( ϕ X − f ξ, η ( X ) d d t ), where f is a smooth function on M ¯ × R, has no torsion, i.e., J is integrable, the condition for normality in terms of ϕ, ξ and η is + 2 d η ⊗ ξ = 0 on M ¯, where is the Nijenhuis tensor of ϕ.

The simplest quasi-linear equation is ut + a(f u))x = 0, where a is a given discontinuous coefficient function and f is a smooth function.

The purpose of this article, is to study the Dirichlet problems of the sub-Laplace equation Lu + f ξ, u) = 0, where L is the sub-Laplacian on the Carnot group G and f is a smooth function.

Since f is a smooth function in view of (3.4), it follows that f(x) = a · x, where a ∈ ℂ m.

Consider the initial boundary value problem begin{aligned} Lu&= 0 quad hbox {in} D, end{aligned} (1.9) begin{aligned} u&= 0 quad hbox {for} x_0 ll 0 hbox {in} D, end{aligned} (1.10) begin{aligned} ubig |_S&=f, end{aligned} (1.11 where f is a smooth function on (S=0) with compact support, (Lu=0) is the same as in (1.1).

If f is a smooth function on the sphere, we may extend it to a smooth function (tilde{f}) on (mathbb {R}^{3} -O) that is constant on every ray issued from the origin: (tilde{f}(x) = f(x/|x|)).

Let f be a smooth function with compact support.

Let ((M, g)) be an n-dimensional complete Riemannian manifold and f be a smooth function defined on M. Then the triple ((M, g,e^{-f}, dv)) is called a smooth metric measure space, where dv denotes the volume element of the metric g and (e^{-f},dv) is called the weighted measure.

The method is developed for the solution of min{f(x): x∈Rn}, where f is a nonlinear function that is sufficiently smooth to possess a Hessian matrix that is continuous.

where F is a hypergeometric function.