For the KGD, we have, after substituting expressions for the PMF and CDF (Equations (10) and (9)), h ( x ) = 1 - ( 1 - q x ) α 1 - 1 - q x + 1 α β - 1. (24).
Substituting expression (1.7) for into this equality, we get (3.1).
Using the same method of power series and polynomial expansion which has been used to derive the substituted expression of f Y (y|L c e n t e r ) in subsection 3.2, we could get that E [ Y L edge ] = α M 1 − α M 0 1 − γ 2 R β (39).
However, a definition of substitution for FOL (and in general, for an abstract syntax, that is, for a language with a variable binding operator) has to guarantee that no free occurrence of a variable in the substituted expression becomes bound in the resulting expression.
Substituting expression (5) into equation (4), after some operations, we obtain the following system of differential equations for determining (S (t )), (N (t )) and (omega (t )): begin{aligned} &dot{S} (t )=-F' (t )S (t )-S (t )F (t )+S (t )M (t )S (t )-Q (t ), &dot{N} (t )= bigl[S (t )M (t )-F' (t ) bigr]N (t ), &dot{omega} (t )= bigl[S (t )M (t )-F' (t ) bigr]omega (t )-S (t )v (t ).
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Note that substituting expression (4) into (1) resolves into the equation z threshold (μΦ + fσΦ, μ c - fσ c ) = -2 f.
Thus, substituting Expression 4 and Λ in the score 2 for each node ν we have (5) S c o r e = { Λ × Γ (s for f (w [ 1.. k ] ) ≥ 1 0 otherwise where s is the proper longest suffix of w = ⟨ ν⟩, that is, the word from the root to the node referenced by the suffix link that goes out from ν.
Expand (x t)) as a Fourier series and substitute the expressions into the integrals.
Now, substitute the expressions of u ′ ( 0 ) and u ″ ( 0 ) into (8) to get (7).
We can substitute these expressions in Eq. (6) in order to obtain a Schrödinger equation for Ψ.
