Sentence examples for exists a function of the from inspiring English sources

Suppose that the following conditions are satisfied: F is ( ≼ I ∗, ≼ I ∗ ) -increasing; there exists a function of the contractive factor φ : [ 0, ∞ ) → [ 0, 1 ) such that, for any x, y ∈ R m with x ≼ I ( m ) y or y ≼ I ( m ) x, the inequalities | g ( k ) ( s, x ) + λ x ( k ) − g ( k ) ( s, y ) − λ y ( k ) | ≤ ( min k = 1, …, m | x ( k ) − y ( k ) | ) ⋅ ϕ ¯ ∗ ( x ( k ), y ( k ) ) (81).

Suppose that the following conditions are satisfied: F is ( ≼ I ∗, ≼ I ∗ ) -increasing; there exists a function of the contractive factor φ : [ 0, ∞ ) → [ 0, 1 ) such that, for any x, y ∈ R m with x ≼ I ( m ) y or y ≼ I ( m ) x, the inequalities | g ( k ) ( s, x ) + λ x ( k ) − g ( k ) ( s, y ) − λ y ( k ) | ≤ ( min k = 1, …, m | x ( k ) − y ( k ) | ) ⋅ ϕ ¯ ( x, y ).

Let α be a positive real number, such that (m-1<alphale m), (minmathbb{N}) and (f^{(m)}(x)) <span class="lh">exists, a function of class C. Then the Caputo fractional derivative of f is defined as ^{mathrm{C}}D^{alpha}f(x)=frac{1}{Gamma(m - alpha)} int_{0}^{t} (t - s)^{m - alpha- 1} f^{(m)} (s),ds.

Assume that there exists a function of contractive factor such that for any (3.45).

Assume that there exists a function of contractive factor such that for any with, (3.6).

Weak irrepresentability: for any positive integer, there exists a function of which cannot be represented as any -time nested superposition constructed from several functions of.

Strong irrepresentability: there exists a function of which cannot be represented as any finite-time nested superposition constructed from several functions of.

Fixed (varepsilon>0), for each positive integer (m=1,2,ldots) , there exists a function (u_{m}) of the form (3.28) satisfying (3.29).

We have thus proved that the mapping satisfies the assumptions of Schauder's fixed point theorem and hence there exists a function with The proof of existence of a solution of (1.1) is complete.

If C is PAC learnable with membership queries under the uniform distribution or exact learnable in randomized polynomial-time, we prove that there exists a function f∈BPEXP (the exponential time analog of BPP) such that f∉C.

A sequence { μ n } n = 0 ∞ is called a moment sequence if there exists a function α ( t ) of bounded variation on the interval [ 0, 1 ] such that μ n = ∫ 0 1 t n d α ( t ), n ∈ N 0. (1).