Sentence examples similar to system p x from inspiring English sources

The application of the relative Greenberg functor to the smooth affine group scheme P x defines a projective system of affine k -groups P x, n ( n ≥ 1 ) such that P x, n ( k ) = P x ( O / π n e 0 O ).

As for the initial data, we consider u 0 ( x ) ≡ 0, ρ 0 ( x ) = ρ h ( x ) and, in the case of the full system, p 0 ( x ) = p h ( x ).

The main purpose of this paper is to establish existence results for the second-order Dirichlet system ( P ) { x i Δ Δ = f i ( t, x σ ( t ) ), t ∈ ( J κ ) o, x i ( a ) = A i, x i ( σ 2 ( b ) ) = B i, i = 1, 2, …, n. with f = ( f 1, …, f n ), i = 1, 2, …, n, where f i : ( J κ ) o × A → R, A ⊂ R n and J is a time scale interval.

(iv) Equation (4), where 1 / p, q ∈ C prd, can be written as the symplectic dynamic system ( x p x Δ ) Δ = ( 0 1 / p q μ q / p ) ( x p x Δ ).  .

This paper is devoted to investigating the following p-Laplacian Liénard neutral differential system: ( φ p ( ( x ( t ) − B x ( t − τ ) ) ′ ) ) ′ + f ( x ( t ) ) x ′ ( t ) + g ( x ( t − γ ( t ) ) ) = e ( t ), (1.1).

In this article, we mainly consider the existence of positive weak solutions for the system ( P ) - Δ p ( x ) u = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] in  Ω, - Δ p ( x ) v = λ p ( x ) [ g ( x ) b ( v ) + h ( u ) ] in  Ω, u = v = 0 on  ∂ Ω, where Ω ⊂ ℝ N is a bounded domain with C2 boundary ∂ Ω, 1 < p(x) ∈ C1 is a function.

El Manouni and Kbiri Alaoui [25] obtained the existence of at least three solutions of system (1) when p(x) ≡ p in Ω by the three critical points theorem obtained by Ricceri [26].

Let us now consider the case when the function g is affine, that is, g ( y ) = C y + c, where C is a k × m matrix, c is a k vector, and the graph of T is a convex polyhedral set, that is, y ∈ T ( x ) if and only if x and y solve a linear system P y ≤ Q x + q, where P is a k × m matrix, Q is a k × n matrix and q is a k vector.

In [8], Bin Ge and Ji-Hong Shen obtained multiple solutions for a class of differential inclusion systems involving the ( p ( x ), q ( x ) ) -Laplacian.

The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) given its current state depends only on the current state of the system, and not additionally on the state of the system at previous steps: P ( X n + 1 X 1, X 2, …, X n ) = P ( X n + 1 X n ).

In this section, we consider a linear (VRP) as mentioned in the preceding section, namely we wish to find x ¯ ∈ R n such that A x ¯ ≤ C y + c for every y solution to the system P y ≤ Q x ¯ + q.