q-Enumeration of words by their total variation

Combinatoric

The solution of the perturbed Tanaka-equation is pathwise unique

The Tanaka equation $dX_t={\operatorname{sign}}(X_t)\,dB_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion B, then the solution of the obtained equation is pathwise unique.Comment: Published in at http://dx.doi.org/10.1214/11-AOP716 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type $A$ are found. This leads to new models of card shuffling and has interesting combinatorial and number theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory.Comment: One change: we fix a typo in definition of f(m,k,i,d) on page 1

Factors influencing the accuracy of the plating method used to enumerate low numbers of viable micro-organisms in food

This study aims to assess several factors that influence the accuracy of the plate count technique to estimate low numbers of micro-organisms in liquid and solid food. Concentrations around 10 CFU/mL or 100 CFU/g in the original sample, which can still be enumerated with the plate count technique, are considered as low numbers. The impact of low plate counts, technical errors, heterogeneity of contamination and singular versus duplicate plating were studied. Batches of liquid and powdered milk were artificially contaminated with various amounts of Cronobacter sakazakii strain ATCC 29544 to create batches with accurately known levels of contamination. After thoroughly mixing, these batches were extensively sampled and plated in duplicate. The coefficient of variation (CV) was calculated for samples from both batches of liquid and powdered product as a measure of the dispersion within the samples. The impact of technical errors and low plate counts were determined theoretically, experimentally, as well as with Monte Carlo simulations. CV-values for samples of liquid milk batches were found to be similar to their theoretical CV-values established by assuming Poisson distribution of the plate counts. However, CV-values of samples of powdered milk batches were approximately five times higher than their theoretical CV-values. In particular, powdered milk samples with low numbers of Cronobacter spp. showed much more dispersion than expected which was likely due to heterogeneity. The impact of technical errors was found to be less prominent than that of low plate counts or of heterogeneity. Considering the impact of low plate counts on accuracy, it would be advisable to keep to a lower limit for plate counts of 25 colonies/plate rather than to the currently advocated 10 colonies/plate. For a powdered product with a heterogeneous contamination, it is more accurate to use 10 plates for 10 individual samples than to use the same 10 plates for 5 samples plated in duplicat

Walks on the slit plane: other approaches

Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i belongs to S for all i, and none of the points w_i, i>0, lie on the half-line H= {(k,0): k =< 0}. In a recent paper, G. Schaeffer and the author computed the length generating function S(t) of walks on the slit plane for several sets S. All the generating functions thus obtained turned out to be algebraic: for instance, on the ordinary square lattice, S(t) =\frac{(1+\sqrt{1+4t})^{1/2}(1+\sqrt{1-4t})^{1/2}}{2(1-4t)^{3/4}}. The combinatorial reasons for this algebraicity remain obscure. In this paper, we present two new approaches for solving slit plane models. One of them simplifies and extends the functional equation approach of the original paper. The other one is inspired by an argument of Lawler; it is more combinatorial, and explains the algebraicity of the product of three series related to the model. It can also be seen as an extension of the classical cycle lemma. Both methods work for any set of steps S. We exhibit a large family of sets S for which the generating function of walks on the slit plane is algebraic, and another family for which it is neither algebraic, nor even D-finite. These examples give a hint at where the border between algebraicity and transcendence lies, and calls for a complete classification of the sets S.Comment: 31 page