A Classical Sequential Growth Dynamics for Causal Sets

Starting from certain causality conditions and a discrete form of general covariance, we derive a very general family of classically stochastic, sequential growth dynamics for causal sets. The resulting theories provide a relatively accessible ``half way house'' to full quantum gravity that possibly contains the latter's classical limit (general relativity). Because they can be expressed in terms of state models for an assembly of Ising spins living on the relations of the causal set, these theories also illustrate how non-gravitational matter can arise dynamically from the causal set without having to be built in at the fundamental level. Additionally, our results bring into focus some interpretive issues of importance for causal set dynamics, and for quantum gravity more generally.Comment: 28 pages, 9 figures, LaTeX, added references and a footnote, minor correction

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

High degree graphs contain large-star factors

We show that any finite simple graph with minimum degree $d$ contains a spanning star forest in which every connected component is of size at least $\Omega((d/\log d)^{1/3})$. This settles a problem of J. Kratochvil

Good Random Matrices over Finite Fields

The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and vice versa. Further examples of k-good random matrices are derived from homogeneous weights on matrix modules. Several applications of k-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a k-dense set of m-by-n matrices is studied, identifying such sets as blocking sets with respect to (m-k)-dimensional flats in a certain m-by-n matrix geometry and determining their minimum size in special cases.Comment: 25 pages, publishe

Maximum union-free subfamilies

An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. We show that every family of m sets contains a union-free subfamily of size at least \lfloor \sqrt{4m+1}\rfloor - 1 and that this bound is tight. This solves Moser's problem and proves a conjecture of Erd\H{o}s and Shelah from 1972. More generally, a family of sets is a-union-free if there are no a+1 distinct sets in the family such that one of them is equal to the union of a others. We determine up to an absolute multiplicative constant factor the size of the largest guaranteed a-union-free subfamily of a family of m sets. Our result verifies in a strong form a conjecture of Barat, F\"{u}redi, Kantor, Kim and Patkos.Comment: 10 page

Daily rhythms of the sleep-wake cycle

The amount and timing of sleep and sleep architecture (sleep stages) are determined by several factors, important among which are the environment, circadian rhythms and time awake. Separating the roles played by these factors requires specific protocols, including the constant routine and altered sleep-wake schedules. Results from such protocols have led to the discovery of the factors that determine the amounts and distribution of slow wave and rapid eye movement sleep as well as to the development of models to determine the amount and timing of sleep. One successful model postulates two processes. The first is process S, which is due to sleep pressure (and increases with time awake) and is attributed to a 'sleep homeostat'. Process S reverses during slow wave sleep (when it is called process S'). The second is process C, which shows a daily rhythm that is parallel to the rhythm of core temperature. Processes S and C combine approximately additively to determine the times of sleep onset and waking. The model has proved useful in describing normal sleep in adults. Current work aims to identify the detailed nature of processes S and C. The model can also be applied to circumstances when the sleep-wake cycle is different from the norm in some way. These circumstances include: those who are poor sleepers or short sleepers; the role an individual's chronotype (a measure of how the timing of the individual's preferred sleep-wake cycle compares with the average for a population); and changes in the sleep-wake cycle with age, particularly in adolescence and aging, since individuals tend to prefer to go to sleep later during adolescence and earlier in old age. In all circumstances, the evidence that sleep times and architecture are altered and the possible causes of these changes (including altered S, S' and C processes) are examined

The early bee catches the flower - circadian rhythmicity influences learning performance in honey bees, Apis mellifera

Circadian rhythmicity plays an important role for many aspects of honey bees’ lives. However, the question whether it also affects learning and memory remained unanswered. To address this question, we studied the effect of circadian timing on olfactory learning and memory in honey bees Apis mellifera using the olfactory conditioning of the proboscis extension reflex paradigm. Bees were differentially conditioned to odours and tested for their odour learning at four different “Zeitgeber” time points. We show that learning behaviour is influenced by circadian timing. Honey bees perform best in the morning compared to the other times of day. Additionally, we found influences of the light condition bees were trained at on the olfactory learning. This circadian-mediated learning is independent from feeding times bees were entrained to, indicating an inherited and not acquired mechanism. We hypothesise that a co-evolutionary mechanism between the honey bee as a pollinator and plants might be the driving force for the evolution of the time-dependent learning abilities of bees