Counting fixed points and rooted closed walks of the singular map x↦xxnx \mapsto x^{x^n} modulo powers of a prime

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.Comment: 18 pages. Version 2 shortens proofs, reduces redundancy, and introduces new technique for counting rooted closed walks. Version 3 updates title to agree with journal publicatio

Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator-prey pursuit and evasion example

We have studied properties of nonlinear waves in a mathematical model of a predator-prey system with pursuit and evasion. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the ``taxis'', represented by nonlinear ``cross-diffusion'' terms in the mathematical formulation. We have shown that the dependence of the velocity of wave propagation on the taxis has two distinct forms, ``parabolic'' and ``linear''. Transition from one form to the other correlates with changes in the shape of the wave profile. Dependence of the propagation velocity on diffusion in this system differs from the square-root dependence typical of reaction-diffusion waves. We demonstrate also that, for systems with negative and positive taxis, for example, pursuit and evasion, there typically exists a large region in the parameter space, where the waves demonstrate quasisoliton interaction: colliding waves can penetrate through each other, and waves can also reflect from impermeable boundaries.Comment: 15 pages, 18 figures, submitted to Physica

Pursuit-evasion predator-prey waves in two spatial dimensions

We consider a spatially distributed population dynamics model with excitable predator-prey dynamics, where species propagate in space due to their taxis with respect to each other's gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wavetips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatio-temporal activity, unknown in reaction-diffusion population models.Comment: 15 pages, 15 figures, submitted to Chao

X-ray Observations of Distant Optically Selected Cluster

We have measured fluxes or flux limits for 31 of the 79 cluster candidates in the Palomar Distant Cluster Survey (PDCS) using archival ROSAT/PSPC pointed observations. Our X-ray survey reaches a flux limit of $\simeq 3 \times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ (0.4 - 2.0 keV), which corresponds to luminosities of $L_x\simeq 5 \times 10^{43}$ erg s$^{-1}$ ($H_o$ = 50 km s$^{-1}$ Mpc$^{-1}$, $q_o$ = ${1/2}$), if we assume the PDCS estimated redshifts. Of the 31 cluster candidates, we detect six at a signal-to-noise greater than three. We estimate that $2.9^{+3.3}_{-1.4}$ (90% confidence limits) of these six detections are a result of X-ray emission from objects unrelated to the PDCS cluster candidates. The net surface density of X-ray emitting cluster candidates in our survey, $1.71^{+0.91}_{-2.19}$ clusters deg$^{-2}$, agrees with that of other, X-ray selected, surveys. It is possible, given the large error on our contamination rate, that we have not detected X-ray emission from any of our observed PDCS cluster candidates. We find no statistically significant difference between the X-ray luminosities of PDCS cluster candidates and those of Abell clusters of similar optical richness. This suggests that the PDCS contains objects at high redshift similar to the low redshift clusters in the Abell catalogs. We show that the PDCS cluster candidates are not bright X-ray sources, the average luminosity of the six detected candidates is only $\bar{L_x}=0.9\times10^{44}$ erg s$^{-1}$ (0.4-2.0 keV). This finding is in agreement with previous X-ray studies of high redshift, optically selected, rich clusters of galaxies.Comment: 19 pages, LaTeX with AAS Preprint Macros (v. 4), 3 embedded postscript figures, 3 Seperate Tables using aj_pt4.sty, Accepted by the Astronomical Journal for November 199

Genetic diversity at the FMR1 locus in the Indonesian population

We report an analysis of allelic diversity at short tandem repeat polymorphisms within the fragile XA locus in 1069 male volunteers from twelve Indonesian sub-populations. An odd numbered allele of DXS548 was found at high frequency in all Indonesian populations. Greater allelic diversity was identified at the loci under study than has been previously reported for an Asian population. These differences distinguish the Indonesian population from all previously reported Asian, European and African populations. A high frequency of small premutation alleles, 4/120 (3.3%, 95% CI 0.9–8.3%), was identified in the Moluccan population of Hiri Island

Irish Dairy Farming: Effects of Introducing a Maize Component on Grassland Management Over the Next 50 Years

Typical management of Irish dairy units is based on a low-cost spring-calving strategy with 90% of annual feed derived from grass grown on the farm. Almost 70% of feed is from grazed grass managed by rotational grazing, the remainder is conserved forage and concentrates. The objectives of the work were to examine how the management system has to be modified when part of the dairy unit land is allocated to maize silage instead of grass silage production, and to examine how climate change over the next 50 years will impact on grass and maize management within the production system

Controls on the spatial distribution of oceanic <i>δ</i>¹³C_DIC

We describe the design and evaluation of a large ensemble of coupled climate–carbon cycle simulations with the Earth system model of intermediate complexity GENIE. This ensemble has been designed for application to a range of carbon cycle questions, including the causes of late- Quaternary fluctuations in atmospheric CO2. Here we evaluate the ensemble by applying it to a transient experiment over the recent industrial era (1858 to 2008 AD). We employ singular vector decomposition and principal component emulation to investigate the spatial modes of ensemble variability of oceanic dissolved inorganic carbon (DIC) δ13C, considering both the spun-up pre-industrial state and the transient change. These analyses allow us to separate the natural (preindustrial) and anthropogenic controls on the δ13CDIC distribution. We apply the same dimensionally reduced emulation techniques to consider the drivers of the spatial uncertainty in anthropogenic DIC. We show that the sources of uncertainty related to the uptake of anthropogenic δ13CDIC and DIC are quite distinct. Uncertainty in anthropogenic δ13C uptake is controlled by air–sea gas exchange, which explains 63% of modelled variance. This mode of variability is largely absent from the ensemble variability in CO2 uptake, which is rather driven by uncertainties in thermocline ventilation rates. Although the need to account for air–sea gas exchange is well known, these results suggest that, to leading order, uncertainties in the ocean uptake of anthropogenic 13C and CO2 are governed by very different processes. This illustrates the difficulties in reconstructing one from the other, and furthermore highlights the need for careful targeting of both δ13CDIC and DIC observations to better constrain the ocean sink of anthropogenic CO2