By Vladimir S. Lerner

ISBN-10: 1614700923

ISBN-13: 9781614700920

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**Extra resources for Information Path Functional and Informational Macrodynamics**

**Example text**

3. 29), let us find such ε i ∈ (0,1), i = 1, 2, 3 at which the conditions for the maximums of these probabilities: max{ε i exp[ − (2Si /(1 − ε i ))1/2 ]} are satisfied. 1. 71) is reached on the solutions of equation (1 − x)3 = Sx 2 at x ∈(0,1) , S >0, S ∈ Si . 72) Proof. 72a) we will analyze its maximum considering its first derivative: f ' (x) = exp[−( 2S 1/ 2 ) ] + x exp[−( 2S ) ] ( −1 ) 1/ 2 1− x 1− x 1 1 x 1/ 2 1 ( −1) = 0,1 = (2S) ( ) . − (2S) (− ) × 3/ 2 2 (1 − x) 2 (1 − x)3/ 2 1/ 2 3 We come to equation (1 − x) = Sx 2 , x ∈(0,1) , S>0, which has a unique real root.

2. 33) 2b = σσ T , a u (t , xt* ) = a* (t , xt* , ut ) , ut = u(t,x t ,x t1 ) , where functions a u (t, x) = a(t,x,u) , a u (t , xt* ) = a* (t , xt* , ut ) , u n and σ ij (t, x) , ai (t, x) are measured and limited on Δ × R and Δ × R × U accordingly. 33), belongs to the same class of functions on n Δ × R ×U . The Initial Mathematical Models Since xt* is a function, measured on 21 ω ∈Ω with Px and is a continuous for ∀t ∈ Δ , we conclude that the function L = Lu (t , xt* ) is also measured, limited, and therefore is a summable function by measure mesΔ × Px , with the Lebeg measure mesΔ on Δ × Ω.

50) is reduced to Inf of the upper limit of P{ A} . yt (At this case, G y = Gy =0 , K y = K y = 0 ). 49) independently on the values y˜s , ys . (The numerical values of these limitations are different in the cases of ys ≠ y˜s , and ys = y˜s accordingly). 1. 27) applied to ρ L2 ( y˜t , yt ). 59) s which is fulfilled at the solution of equation y = q(t , y , u ) . 49) become the nearest to the most probable solutions (in the L -metric) by upper evaluation ( m ) as well as by lower ( m ) evaluation of the above probabilities.

### Information Path Functional and Informational Macrodynamics by Vladimir S. Lerner

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