Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

Stereotype deduction about bisexual women

Bisexuals are an invisible sexual minority. However, at the same time, bisexuals are stereotypically associated with confusion and promiscuity. Stereotype learning theories suggest that individuals who are unfamiliar with a social group are less likely to have stereotypical beliefs about its members. In contrast, it has been recently hypothesized that stereotypes about bisexuality are not necessarily learned but rather deduced based on common conceptualizations of sexuality. Because stereotypes are suppressed only if they are recognized as offensive, lack of knowledge regarding bisexual stereotypes should actually enhance their adoption. To assess the strength of the two competing accounts, we examined the relationship between explicit knowledge of bisexual stereotypes and stereotypical evaluation of bisexual individuals. Heterosexual participants (N = 261) read a description of two characters on a date and evaluated one of them. Bisexual women were evaluated as more confused and promiscuous relative to nonbisexual women. Moreover, the stereotypical evaluations of bisexual women were inversely related to knowledge about these stereotypes. The findings support the notion that bisexual stereotypes are not learned but rather deduced from shared assumptions about sexuality. Consequently, public invisibility not only exists alongside bisexual stereotypes but might also exacerbate their uninhibited adoption

Theory of continuum percolation II. Mean field theory

I use a previously introduced mapping between the continuum percolation model and the Potts fluid to derive a mean field theory of continuum percolation systems. This is done by introducing a new variational principle, the basis of which has to be taken, for now, as heuristic. The critical exponents obtained are $\beta= 1$, $\gamma= 1$ and $\nu = 0.5$, which are identical with the mean field exponents of lattice percolation. The critical density in this approximation is \rho_c = 1/\ve where \ve = \int d \x \, p(\x) \{ \exp [- v(\x)/kT] - 1 \}. p(\x) is the binding probability of two particles separated by \x and v(\x) is their interaction potential.Comment: 25 pages, Late

Non-equilibrium dynamics of gene expression and the Jarzynski equality

In order to express specific genes at the right time, the transcription of genes is regulated by the presence and absence of transcription factor molecules. With transcription factor concentrations undergoing constant changes, gene transcription takes place out of equilibrium. In this paper we discuss a simple mapping between dynamic models of gene expression and stochastic systems driven out of equilibrium. Using this mapping, results of nonequilibrium statistical mechanics such as the Jarzynski equality and the fluctuation theorem are demonstrated for gene expression dynamics. Applications of this approach include the determination of regulatory interactions between genes from experimental gene expression data

A genomic map of the effects of linked selection in Drosophila

Natural selection at one site shapes patterns of genetic variation at linked sites. Quantifying the effects of 'linked selection' on levels of genetic diversity is key to making reliable inference about demography, building a null model in scans for targets of adaptation, and learning about the dynamics of natural selection. Here, we introduce the first method that jointly infers parameters of distinct modes of linked selection, notably background selection and selective sweeps, from genome-wide diversity data, functional annotations and genetic maps. The central idea is to calculate the probability that a neutral site is polymorphic given local annotations, substitution patterns, and recombination rates. Information is then combined across sites and samples using composite likelihood in order to estimate genome-wide parameters of distinct modes of selection. In addition to parameter estimation, this approach yields a map of the expected neutral diversity levels along the genome. To illustrate the utility of our approach, we apply it to genome-wide resequencing data from 125 lines in Drosophila melanogaster and reliably predict diversity levels at the 1Mb scale. Our results corroborate estimates of a high fraction of beneficial substitutions in proteins and untranslated regions (UTR). They allow us to distinguish between the contribution of sweeps and other modes of selection around amino acid substitutions and to uncover evidence for pervasive sweeps in untranslated regions (UTRs). Our inference further suggests a substantial effect of linked selection from non-classic sweeps. More generally, we demonstrate that linked selection has had a larger effect in reducing diversity levels and increasing their variance in D. melanogaster than previously appreciated

Boolean versus continuous dynamics on simple two-gene modules

We investigate the dynamical behavior of simple modules composed of two genes with two or three regulating connections. Continuous dynamics for mRNA and protein concentrations is compared to a Boolean model for gene activity. Using a generalized method, we study within a single framework different continuous models and different types of regulatory functions, and establish conditions under which the system can display stable oscillations. These conditions concern the time scales, the degree of cooperativity of the regulating interactions, and the signs of the interactions. Not all models that show oscillations under Boolean dynamics can have oscillations under continuous dynamics, and vice versa.Comment: 8 pages, 10 figure

A new method for constructing small-bias spaces from Hermitian codes

We propose a new method for constructing small-bias spaces through a combination of Hermitian codes. For a class of parameters our multisets are much faster to construct than what can be achieved by use of the traditional algebraic geometric code construction. So, if speed is important, our construction is competitive with all other known constructions in that region. And if speed is not a matter of interest the small-bias spaces of the present paper still perform better than the ones related to norm-trace codes reported in [12]

Analysis and Resynthesis of the Handpan Sound

(Abstract to follow