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Extra resources for Sutherland. Diffuse Matter in the Universe

Example text

Diﬀerentiating formally the initial equation with respect to α, we obtain duα ∂f = uα , dt ∂y © 2000 by CRC Press LLC uα = ∂y(t, α, β) . 25), we derive the initial conditions uα (t0 ) = −f (y0 , t0 , β) dt0 . 59) with initial conditions uβ (t0 ) = dy0 . 57), we obtain Uαβ (t0 ) = duα (t0 ) =− dβ ∂f (y0 , t0 , β) dy0 ∂f (y0 , t0 , β) + ∂y0 dβ ∂β dt0 . 61) As follows from the above reasoning, if the derivatives ∂2f , ∂y 2 ∂2f ∂y∂β are continuous and so are the functions ∂f , ∂β ∂y0 , ∂β ∂t0 , , ∂α the function uαβ exists and is independent of the order of diﬀerentiation by α and β.

Consider a general statement of the problem. 28), a classical optimization problem can be formulated as follows: it is required to ﬁnd a vector α0 ∈ Rα , where Rα is a given set of parameters, such that J1 (α0 ) ≥ J(α), α ∈ Rα , Y (t, α0 ) ∈ MY . 60) Solution of the problem ensures optimality of the system only for α = α0 . 61) where ∆α is parameters ﬂuctuation belonging to a region ∆Rα , which depends on α0 in the general case. 60) by Y (t, α + ∆α) ∈ MY where ∆α ∈ ∆Rα . 63) where ∆Y (t, α) = Y (t, α0 +∆α)−Y (t, α0 ) is the relevant additional motion.

Nevertheless, for ﬁnite-dimensional systems we presented suﬃcient conditions for existence of sensitivity functions and general equations for their determination. Moreover, the relation between the error of the n-th approximation and sensitivity functions of n + 1-th order has been established. 74) i=1 which is very useful for investigation of general properties of additional motion. 21) in a closed interval Jα : α1 ≤ α ≤ α2 . Let α = α0 be some ﬁxed base value of the parameter from the interval Jα , and Y (t, α0 ) be the corresponding solution.