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Kolchin, V. F., Sevast'Yanov, B. A. and Chistyakov, V. P., Random allocations , V. H. Winston & Sons, Washington D.C. (1978). MR 57 #10758b Random allocations

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Moment inqualities and strong laws of large numbers are proved for random allocations of balls into boxes. Random broken lines and random step lines are constructed using partial sums of i.i.d. random variables that are modified by random allocations. Functional limit theorems for such random processes are obtained.

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The problem of random allocation is that of placing n balls independently with equal probability to N boxes. For several domains of increasing numbers of balls and boxes, the final number of empty boxes is known to be asymptotically either normally or Poissonian distributed. In this paper we first derive a certain two-index transfer theorem for mixtures of the domains by considering random numbers of balls and boxes. As a consequence of a well known invariance principle this enables us to prove a corresponding general almost sure limit theorem. Both theorems inherit a mixture of normal and Poisson distributions in the limit. Applications of the general almost sure limit theorem for logarithmic weights complement and extend results of Fazekas and Chuprunov [10] and show that asymptotic normality dominates.

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Abstract  

Activation energy is calculated from a single curve of a derivative of mass loss perturbed by a sinusoidal modulation of a temperature-time relationship. The method is based on a prediction of a hypothetical derivative of mass loss that corresponds to the absence of this modulation (perturbation). Simple considerations show that the unperturbed derivative coincides with the modulated derivative at inflection points of the modulated temperature-time relationship. The ratio of the perturbed and unperturbed derivatives at the points of time corresponding to maxima and minima of the sinusoidal component of the modulated temperature immediately leads to activation energy. Accuracy of the method grows with decreasing in the amplitude of the modulation. All illustrations are prepared numerically. It makes possible to objectively test the method and to investigate its errors. Two-stage decomposition kinetics with two independent (parallel) reactions is considered as an example. The kinetic parameters are chosen so that the derivative of mass loss would represent two overlapping peaks. The errors are introduced into the modulated derivative by the random-number generator with the normal distribution. Standard deviation for the random allocation of errors is selected with respect to maximum of the derivative. If the maximum of the derivative is observed within the region from 200 to 600C and the amplitude of the temperature modulation is equal to 5C, the error in the derivative 0.5% leads to the error in activation energy being equal to 2-6 kJ mol-1. As the derivative vanishes, the error grows and tends to infinity in the regions of the start and end of decomposition. With the absolute error 0.5% evaluations of activation energy are impossible beyond the region from 5 to 95% of mass loss.

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Journal of Behavioral Addictions
Authors: Philip Nielsen, Maxwell Christensen, Craig Henderson, Howard A Liddle, Marina Croquette-Krokar, Nicolas Favez, and Henk Rigter

information and informed consent materials to take home to read and sign if willing to participate. One week later, the consenting families attended a meeting with the research assistant for the baseline assessment and for random allocation to MDFT or FTAU, in

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