Differentiable function

Abstract: A new algorithm is presented for obtaining points on a steepest descent path from the transition state of the reactants and products. In mass‐weighted coordinates, this path corresponds to the intrinsic reaction coordinate. Points on the reaction path are found by constrained optimizations involving all internal degrees of freedom of the molecule. The points are optimized so that the segment of the reaction path between any two adjacent points is given by an arc of a circle, and so that the gradient at each point is tangent to the path. Only the transition vector and the energy gradients are needed to construct the path. The resulting path is continuous, differentiable and piecewise quadratic. In the limit of small step size, the present algorithm is shown to take a step with the correct tangent vector and curvature vector; hence, it is a second order algorithm. The method has been tested on the following reactions: HCN→CNH, SiH2+H2→SiH4, CH4+H→CH3+H2, F−+CH3F→FCH3+F−, and C2H5F→C2H4+HF. Reaction paths calculated with a step size of 0.4 a.u. are almost identical to those computed with a step size of 0.1 a.u. or smaller.

Abstract: Our previous algorithm for following reaction paths downhill (J. Chem. Phys. 1989, 90, 2154), has been extended to use mass-weighted internal coordinates. Points on the reaction path are round by constrained optimizations involving the internal degrees or freedom or the molecule. The points are optimized so that the segment or the reaction path between any two adjacent points is described by an arc or a circle in mass-weighted internal coordinates, and so that the gradients (in mass-weighted internals) at the end points or the arc are tangent to the path. The algorithm has the correct tangent vector and curvature vectors in the limit or small step size but requires only the transition vector and the energy gradients; the resulting path is continuous, differentiable, and piecewise quadratic

Abstract: Preliminary Tools and Foundations. The Basic Sample Statistics. Transformations of Given Statistics. Asymptotic Theory in Parametric Inference. U--Statistics. Von Mises Differentiable Statistical Functions. M--Estimates. L--Estimates. R--Estimates. Asymptotic Relative Efficiency. Appendix. References. Author Index. Subject Index.

Abstract: A large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum 2. With f(2) - f(x) ~ Ix - 21" (for Ix - 21 sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio c~ (a = 2.5029078750957... for z = 2). This structure is determined by a universal function g*(x), where the 2"th iterate off, f("~, converges locally to ~-"g*(~nx) for large n. For ithe class of f's considered, there exists a A~ such that a 2"-point stable limit cycle including :7 exists; A~ - ~ ~ ~-" (~ = 4.669201609103... for z = 2). The numbers = and have been computationally determined for a range of z through their definitions, for a variety off's for each z. We present a recursive mechanism that explains these results by determining g* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.