Sentence examples for be defined by formula from inspiring English sources

Proof of Corollary 3.6 Let the operator ℓ be defined by formula (102).

The operator H 00 can be defined by formula (5.1) on C ∞ functions vanishing in some neighbourhoods of the points 0, t 0 / 2 and t 0. Therefore the operators H and H ~ have the same asymptotics of spectra.

Let − ℓ ∈ P a b, and let the functional h be defined by formula (4), where λ > 0 and h 0, h 1 ∈ P F a b are such that inequalities (11) are fulfilled.

Let ℓ ∈ P a b be a b-Volterra operator, and let the functional h be defined by formula (4), where λ > 0 and h 0, h 1 ∈ P F a b are such that inequalities (11) are fulfilled.

Proof of Corollary 3.4 Let the operator ℓ 0 be defined by formula (87), and let ℓ 1 ≡ 0. It is easy to verify that conditions (81) and (82) yield 1 3 ℓ 0 ∈ V ˜ a b − ( h ), − 1 3 ℓ 0 ∈ V ˜ a b − ( h ).

Let p ∈ L ( [ a, b ] ; R + ), τ : [ a, b ] → [ a, b ] be a measurable function, and let the functional h be defined by formula (4), where λ > 0 and h 0, h 1 ∈ P F a b are such that the inequalities h ( 1 ) > 1, 0 < h 0 ( 1 ) < 1. are fulfilled.

Let a kernel h ( t ) be defined by formulas (1.2) and (3.1).

Let kernel h ( t ) be defined by formulas (1.2) and (3.1) where p k = p ¯ k for k = 0, 1, …, K.

Let kernel h ( t ) be defined by formulas (1.2) and (3.1) where p k = p ¯ k for k = 0, 1, …, K. Suppose that K is even and p K > 0. Let Q ( x ) be polynomial (3.4) with coefficients (3.3).

if, where is defined by formula (6.17).

where and are defined by formula (2.19) respectively.