Speed and concentration of the covering time for structured coupon collectors

Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $ , where the $X_i$ are independent random variables with distribution given by X. The covering time T is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^t X_i=V$ . The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.

The Probabilistic Turn (1876–1884)

This chapter deals with writings in which Boltzmann expressed the statistical nature of the entropy law and temporarily made the relation between entropy and combinatorial probability a basic constructive tool of his theory. In 1881, he discovered that this relation derived from what we now call the microcanonical distribution, and he approved Maxwell’s recent foundation of the equilibrium problem on the microcanonical ensemble. Boltzmann also kept working on problems he had tackled in earlier years. He proposed a new solution to the problem of specific heats, and he performed enormous calculations for the viscosity and diffusion coefficients in the hard-ball model. In a lighter genre, he conceived a new way of determining molecular sizes, and he speculated on a gas model in which the molecular forces would be entirely attractive.

Mass isotopomer distribution analysis at eight years: theoretical, analytic, and experimental considerations

Mass isotopomer distribution analysis (MIDA) is a technique for measuring the synthesis of biological polymers. First developed approximately eight years ago, MIDA has been used for measuring the synthesis of lipids, carbohydrates, and proteins. The technique involves quantifying by mass spectrometry the relative abundances of molecular species of a polymer differing only in mass (mass isotopomers), after introduction of a stable isotope-labeled precursor. The mass isotopomer pattern, or distribution, is analyzed according to a combinatorial probability model by comparing measured abundances to theoretical distributions predicted from the binomial or multinomial expansion. For combinatorial probabilities to be applicable, a labeled precursor must therefore combine with itself in the form of two or more repeating subunits. MIDA allows dilution in the monomeric (precursor) and polymeric (product) pools to be determined. Kinetic parameters can then be calculated (e.g., replacement rate of the polymer, fractional contribution from the endogenous biosynthetic pathway, absolute rate of biosynthesis). Several issues remain unresolved, however. We consider here the impact of various deviations from the simple combinatorial probability model of biosynthesis and describe the analytic requirements for successful use of MIDA. A formal mathematical algorithm is presented for generating tables and equations ( ), on the basis of which effects of various confounding factors are simulated. These include variations in natural isotope abundances, isotopic disequilibrium in the precursor pool, more than one biosynthetic precursor pool, incorrect values for number of subunits present, and concurrent measurement of turnover from exogenously labeled polymers. We describe a strategy for testing whether isotopic inhomogeneity (e.g., an isotopic gradient or separate biosynthetic sites) is present in the precursor pool by comparing higher-mass (multiply labeled) to lower-mass (single- and double-labeled) isotopomer patterns. Also, an algebraic correction is presented for calculating fractional synthesis when an incomplete ion spectrum is monitored, and an approach for assessing the sensitivity of biosynthetic parameters to measurement error is described. The different calculation algorithms published for MIDA are compared; all share a common model, use overlapping solutions to computational problems, and generate identical results. Finally, we discuss the major practical issue for using MIDA at present: quantitative inaccuracy of instruments. The nature and causes of analytic inaccuracy, strategies for evaluating instrument performance, and guidelines for optimizing accuracy and reducing impact on biosynthetic parameters are suggested. Adherence to certain analytic guidelines, particularly attention to concentration effects on mass isotopomer ratios and maximizing enrichments in the isotopomers of interest, reduces error. Improving instrument accuracy for quantification of isotopomer ratios is perhaps the highest priority for this field. In conclusion, MIDA remains the “equation for biosynthesis,” but attention to potentially confounding factors and analytic performance is required for optimal application.