Sentence examples for be a random field from inspiring English sources

Corollary 2 Let { X n, n ∈ Z + d } be a random field as in Theorem  1.

Corollary 1 Let { X n, n ∈ Z + d } be a random field being blockwise M-orthogonal as in Theorem  1 with centered and integrable random variables.

Let the n(x) be a random field of floodplain manning's roughness coefficient, and N x) = ln[n x, ω)], where x ϵ D and ω ϵ Ω (a probabilistic space).

Theorem 1 Let { X n, n ∈ Z + d } be a random field with centered and integrable random variables being blockwise M-orthogonal with respect to the blocks [ 2 n 1, 2 n 1 + 1 ) × [ 2 n 2, 2 n 2 + 1 ) × ⋯ × [ 2 n d, 2 n d + 1 ).

Let { X n, n ∈ Z + d } be a random field with M-orthogonal, centered random variables, if E X n 2 < ∞ for all n ∈ Z + d, then we have E ( max k ≺ n | S k | ) 2 ≤ ( m + 1 ) d ⋅ ( ∏ i = 1 d ( log 2 2 n i ) 2 ) ⋅ ∑ k ≺ n E X k 2. (1.2..

Corollary 3 Let { X n, n ∈ Z + d } be a random field with blockwise M-orthogonal, centered random variables satisfying E | X n | 2 ≤ M < ∞ for all n ∈ Z + d, then for any δ > 0, we have lim n → ∞ 1 n 1 1 2 ( log n 1 ) 1 2 + δ ⋯ n d 1 2 ( log n d ) 1 2 + δ ∑ j ≺ n X j = 0 a.s.s

So what we need is a random field, that is, a random variable with values in the set of smooth maps from (mathbb {R}^{2}) to (mathbb {C}).

The first example involves the linear buckling of an arch structure for which the thickness is a random field.

Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion.

The properties of the beam are described as a random field and are expressed using the Karhunen-Loève expansion.

The surface charge density is modeled as a random field, and is discretized both in the random dimension and space using polynomial chaos and classical boundary element method, respectively.