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By I Prigogine

Advances in Chemical Physics, quantity 127 covers fresh advances on the innovative of study relative to chemical physics. The sequence, Advances in Chemical Physics, presents a discussion board for severe, authoritative reviews of advances in each region of the self-discipline.

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M is bijective. 3 Dual representations of exponential family members The above reflexive Legendre transform relationship implies that each density from an exponential family admits a dual representation involving either the natural parameters α or the mean value or moment parameters τ = Eα (T(x)); one parameter set uniquely determines the other. Amari6 refers to the Legendre transform pair ψm (α) and φ(τ ) as the convex potential functions associated with the natural and moment-parametric representations p(x, α) and q(x, τ ).

5) For any other (x, u), Fenchel’s inequality holds: ψ(x) + φ(u) > x, u . 6) Provided that the original function is strictly convex, we may interpret the Legendre transform as a mapping from this function to a family of its tangents expressed in functional form. Figs. 1(b) show two examples of applying the transform to convex functions. 2 (partial Legendre transform). Given the following convex function ψ(x, y) : Rk × Rl → R, the partial Legendre transform of ψ(x, y) with respect to x alone is defined as φ(u, y) sup x, u − ψ(x, y) = x0 (u, y), u − ψ(x0 , y), x∈Rk where, for each (u, y), x0 (u, y) is the solution of u = ∇x ψ(x, y), ∇x (·) being the gradient with respect to x alone.

Several sufficient conditions3 do exist for an infinite sequence of moments to uniquely define a density. One such condition guarantees uniqueness when the true density has finite support. Another sufficient condition requires a moment sequence of monomial statistics {xk } to be completely monotonic:4 k ∆k µn (−1)m m=0 k µn+m > 0 n, k ∈ Z+ . , the moment problem is typically ill posed. Consequently, candidate solutions must be selected according to their fitness under some additional optimality criterion.

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