A note on the probability of generating alternating or symmetric groups

We improve on recent estimates for the probability of generating the alternating and symmetric groups $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$. In particular we find the sharp lower bound, if the probability is given by a quadratic in $n^{-1}$. This leads to improved bounds on the largest number $h(\mathrm{Alt}(n))$ such that a direct product of $h(\mathrm{Alt}(n))$ copies of $\mathrm{Alt}(n)$ can be generated by two elements

Friction at the tennis shoe-court interface: how biomechanically informed lab-based testing can enhance understanding

This paper presents some of the methodology, observations and findings from a 30-month study, aiming to improve the understanding of tennis shoe-court interactions and the biomechanical implications of changes in friction between the shoe and surface. A detailed programme of biomechanical player testing on different court surfaces provided the boundary conditions with which to develop a lab-based rig capable of simulating the key aspects of shoe-surface interaction that are required for acceptable performance (e.g. push-off to accelerate) within expected levels of consistency (e.g. for a controlled slide). Large- scale parametric testing could then be carried out for a variety of surface types and components under a range of loading conditions, without the risk of injury to human participants. Two case studies are described to demonstrate the value of a combined approach of biomechanical field testing and lab-based rigs that simulate shoe-court interactions. These include a study that compared different artificial clay court designs; and a study that examined the effect of different acrylic hard court parameters on friction and the tribological mechanisms that explain the observed interactio

General relativistic spinning fluids with a modified projection tensor

An energy-momentum tensor for general relativistic spinning fluids compatible with Tulczyjew-type supplementary condition is derived from the variation of a general Lagrangian with unspecified explicit form. This tensor is the sum of a term containing the Belinfante-Rosenfeld tensor and a modified perfect-fluid energy-momentum tensor in which the four-velocity is replaced by a unit four-vector in the direction of fluid momentum. The equations of motion are obtained and it is shown that they admit a Friedmann-Robertson-Walker space-time as a solution.Comment: Submitted to General Relativity and Gravitatio

The orbifold transform and its applications

We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the theory of permutation orbifolds is addressed, and the general results illustrated on the example of torus partition functions

Collineation group as a subgroup of the symmetric group

Let $\Psi$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_{\Psi}$ of the set $\Psi$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $\Psi$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $\Psi$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_{\Psi}$ of $\mathfrak{S}_{\Psi}$. We show in Theorem \ref{density} below that $H=\mathfrak{S}_{\Psi}$, if $\Psi$ is infinite.Comment: 9 page

Distribution of Mesoscale Convective Complex Rainfall in the United States

Several annual mesoscale convective complex (MCC) summaries have been compiled since Maddox strictly defined their criteria in 1980. These previous studies have largely been independent of each other and therefore have not established the extended spatial and temporal patterns associated with these large, quasi-circular, and, typically, severe convective systems. This deficiency is primarily due to the difficulty of archiving enough satellite imagery to accurately record each MCC based on Maddox’s criteria. Consequently, this study utilizes results from each of the MCC summaries compiled between 1978 and 1999 for the United States in order to develop a more complete climatology, or description of long-term means and interannual variation, of these storms. Within the 22-yr period, MCC summaries were compiled for a total of 15 yr. These 15 yr of MCC data are employed to establish estimated tracks for all MCCs documented and, thereafter, are utilized to determine MCC populations on a monthly, seasonal, annual, and multiyear basis. Subsequent to developing an extended climatology of MCCs, the study ascertains the spatial and temporal patterns of MCC rainfall and determines the precipitation contributions made by MCCs over the central and eastern United States. Results indicate that during the warm season, significant portions of the Great Plains receive, on average, between 8% and 18% of their total precipitation from MCC rainfall. However, there is large yearly and even monthly variability in the location and frequency of MCC events that leads to highly variable precipitation contributions

Empirical Agent-Based Modelling for exploring Intergroup Contact in a Segregated Society

Agent-based modelling has a long history of application in the study of segregation, but is rarely deployed beyond the examination of residential segregation. This study leverages multiple datasets: including census, survey, PGIS and GPS traces; in order to create an empirical agent-based model for the exploration of mobility practices between segregated communities in Belfast (Northern Ireland). In doing so, we are able to conduct novel examinations into the impact of day-to-day mobility choices upon intergroup attitudes and activity-space segregation; with policy implications for understanding and combatting segregation in cities around the world

A characterization of those automata that structurally generate finite groups

Antonenko and Russyev independently have shown that any Mealy automaton with no cycles with exit--that is, where every cycle in the underlying directed graph is a sink component--generates a fi- nite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is proved in this paper