Bounds on the diameter of Cayley graphs of the symmetric group

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.Comment: 17 pages, 6 table

Baby-Step Giant-Step Algorithms for the Symmetric Group

We study discrete logarithms in the setting of group actions. Suppose that $G$ is a group that acts on a set $S$. When $r,s \in S$, a solution $g \in G$ to $r^g = s$ can be thought of as a kind of logarithm. In this paper, we study the case where $G = S_n$, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets $A, B \subseteq S_n$ such that every permutation of $S_n$ can be written as a product $ab$ of elements $a \in A$ and $b \in B$. Our deterministic procedure is optimal up to constant factors, in the sense that $A$ and $B$ can be computed in optimal asymptotic complexity, and $|A|$ and $|B|$ are a small constant from $\sqrt{n!}$ in size. We also analyze randomized "collision" algorithms for the same problem

Scientific investigations in the Gulf of Mexico and Caribbean Sea during the 1974-1975 Calypso cruise, parts 1 and 2

The distribution and concentrations of the standing crop of phytoplankton and nutrient salts in the Gulf of Mexico and the Caribbean Sea were investigated to provide ground truth for correlating temperature and chlorophyll-a measurements with observations from NASA U-2 aircraft equipped with specially designed sensors for measuring ocean color phenomena. The physical, chemical, and biological data obtained is summarized. Sampling procedures and methods used for determining plant pigments, species composition of phytoplankton, nutrient salt analysis, and the euphotic zones are described

The incidence of prostate cancer in Iran: Results of a population-based cancer registry

Background: Little is known about the epidemiology of prostate cancer in Iranian men. We carried out an active prostate cancer surveillance program in five provinces of Iran. Methods: Data used in this study were obtained from population-based cancer registries between 1996 and 2000. Results: The age-standardized incidence rate of prostate carcinoma in the five provinces was 5.1 per 100,000 person-years. No significant difference was seen in the age-standardized incidence rate of prostate cancer within the provinces studied. The mean±SD age of patients with prostate cancer was 67±13.5 years. Conclusion: The incidence of prostate cancer in Iran is very low as compared to the Western countries. This can partly be explained by lack of nationwide screening program, younger age structure and quality of cancer registration system in Iran

Revisiting the value of information sharing in two-stage supply chains

There is a substantive amount of literature showing that demand information sharing can lead to considerable reduction of the bullwhip effect/inventory costs. The core argument/analysis underlying these results is that the downstream supply-chain member (the retailer) quickly adapts its inventory position to an updated end-customer demand forecast. However, in many real-life situations, retailers adapt slowly rather than quickly to changes in customer demand as they cannot be sure that any change is structural. In this paper, we show that the adaption speed and underlying (unknown) demand process crucially effect the value of information sharing. For the situation with a single upstream supply-chain member (manufacturer) and a single retailer, we consider two demand processes: stationary or random walk. These represent two extremes where a change in customer demand is never or always structural, respectively. The retailer and manufacturer both forecast demand using a moving average, where the manufacturer bases its forecast on retailer demand without information sharing, but on end-customer demand with information sharing. In line with existing results, the value of information turns out to be positive under stationary demand. One contribution, though, is showing that some of the existing papers have overestimated this value by making an unfair comparison. Our most striking and insightful finding is that the value of information is negative when demand follows a random walk and the retailer is slow to react. Slow adaptation is the norm in real-life situations and deserves more attention in future research - exploring when information sharing indeed pays off

On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201

Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

The hydrodynamics of swimming microorganisms

Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection, and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming (tens of microns and below). The focus is on the fundamental flow physics phenomena occurring in this inertia-less realm, and the emphasis is on the simple physical picture. We review the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming, such as resistance matrices for solid bodies, flow singularities, and kinematic requirements for net translation. Then we review classical theoretical work on cell motility: early calculations of the speed of a swimmer with prescribed stroke, and the application of resistive-force theory and slender-body theory to flagellar locomotion. After reviewing the physical means by which flagella are actuated, we outline areas of active research, including hydrodynamic interactions, biological locomotion in complex fluids, the design of small-scale artificial swimmers, and the optimization of locomotion strategies.Comment: Review articl