Calculus from the past: multiple delay systems arising in cancer cell modelling

Non-local calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of non-local problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including the model of the cell cycle developed in Simms K, Bean N, & Koeber A. [10]. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents

A cell growth model revisited

In this paper a stochastic model for the simultaneous growth and division of a cell-population cohort structured by size is formulated. This probabilistic approach gives straightforward proof of the existence of the steady-size distribution and a simple derivation of the functional-differential equation for it. The latter one is the celebrated pantograph equation (of advanced type). This firmly establishes the existence of the steady-size distribution and gives a form for it in terms of a sequence of probability distribution functions. Also it shows that the pantograph equation is a key equation for other situations where there is a distinct stochastic framework

Hydrological Investigations at Biafo Glacier, Karakoram Range, Himalaya; an Important Source of Water For the Indus River

Over 80% of the flow of the Upper Indus River is derived from less than 20% of its area: essentially from zones of heavy snowfall and glacierized basins above 3500 m elevation. The trans-Himalaya n contribution comes largely from an area of some 20000 km2 of glacierized basins, mostly along the axis of the Greater Karakoram range and especially from 20-30 of the largest glacier basins. Very few glaciological investigations have so far been undertaken in this the major glacierized region of Central Asia. Biafo Glacier, one of the largest of the Karakoram glaciers, drains south-eastwards from the central Karakoram crest. Its basin covers a total area of 853 km2 , 628 km2 of which are permanent snow and ice, with 68% of the glacier area forming the accumulation zone. This paper describes investigations of snow accumulation, ablation , glacier movement, and glacier depth undertaken in the period 1985-87 , set against a background of investigations carried out over the last 130 yea rs. Biafo Glacier differs from most of the other Karakoram glaciers in being nourished mainly by direct snowfall rather than by avalanching; this has the advantage of allowing extensive investigation of accumulation over a broad range of altitude. Snow-accumulation studies in the Biafo Glacier basin have indicated that annual accumulation varies from 0.9 to 1.9 m of water equivalent between 4650 and 5450 m a .. s.l. This suggests an annual moisture input above the equilibrium line of approximately 0.6 km3. Monopulse radar measurements indicate the presence of ice thickness as great as 1400 m at the equilibrium line, although these results may not be completely reliable . Mean surface velocity during the summer of 0.8 m d -I has been measured near to the equilibrium line. Calculations of annual ice flux through the vertical cross-profile at the equilibrium line indicate a throughput of 0.7 km3 a-I Estimates from stake ablation measurements also suggest that ice loss on Biafo Glacier is about 0.7 km3 a-I. The close agreement between these three sets of measurements is reassuring, indicating that the ablation zone of Biafo Glacier, whose area covers 0.09% of the whole Upper Indus basin, produces approximately 0.9% of the total run-off. However. it should be mentioned that this estimate does not include water originating from seasonal snow melt, e either above or below the equilibrium line, or from rainfall. Net annual ice losses due to wastage of the glacier since 1910 are probably of the order of 0.4-{).5 m a-I; this would represent between 12 and 15% of annual water yield from melting ice

Estableciendo diálogos entre diferentes marcos teóricos: de los procesos narrativos a la teoría antropológica de lo didáctico

El punto de partida de este artículo es el fenómeno de la co-existencia de diferentes marcos teóricos en didáctica de las matemáticas. Proponemos reformular este fenómeno en términos de un problema científico: el problema del establecimiento de diálogos entre praxeologías de investigación. Iniciamos un diálogo entre dos praxeología de investigación partiendo de una secuencia de clase de una profesora inglesa. El problema de investigación inicial, comprender la acción didáctica de la profesora, fue formulado y abordado, desde la tradición anglosajona, usando el marco de los procesos narrativos. Exploramos aquí los mismos hechos empíricos desde la Teoría Antropológica de lo Didáctico, mostrando que cada praxeología de investigación construye sus propios problemas de investigación y ofreciendo resultados nuevos y complementarios

Numerical Determination of Critical Conditions for Thermal Ignition

The determination of ignition or thermal explosion in an oxidizing porous body of material, as described by a dimensionless reaction-diffusion equation of the form .tu = .2u + .e-1/u over the bounded region O, is critically reexamined from a modern perspective using numerical methodologies. First, the classic stationary model is revisited to establish the proper reference frame for the steady-state solution space, and it is demonstrated how the resulting nonlinear two-point boundary value problem can be reexpressed as an initial value problem for a system of first-order differential equations, which may be readily solved using standard algorithms. Then, the numerical procedure is implemented and thoroughly validated against previous computational results based on sophisticated path-following techniques. Next, the transient nonstationary model is attacked, and the full nonlinear form of the reaction-diffusion equation, including a generalized convective boundary condition, is discretized and expressed as a system of linear algebraic equations. The numerical methodology is implemented as a computer algorithm, and validation computations are carried out as a prelude to a broad-ranging evaluation of the assembly problem and identification of the watershed critical initial temperature conditions for thermal ignition. This numerical methodology is then used as the basis for studying the relationship between the shape of the critical initial temperature distribution and the corresponding spatial moments of its energy content integral and an attempt to forge a fundamental conjecture governing this relation. Finally, the effects of dynamic boundary conditions on the classic storage problem are investigated and the groundwork is laid for the development of an approximate solution methodology based on adaptation of the standard stationary model

Mass growth and mergers: direct observations of the luminosity function of LRG satellite galaxies out to z=0.7 from SDSS and BOSS images

We present a statistical study of the luminosity functions of galaxies surrounding luminous red galaxies (LRGs) at average redshifts =0.34 and =0.65. The luminosity functions are derived by extracting source photometry around more than 40,000 LRGs and subtracting foreground and background contamination using randomly selected control fields. We show that at both studied redshifts the average luminosity functions of the LRGs and their satellite galaxies are poorly fitted by a Schechter function due to a luminosity gap between the centrals and their most luminous satellites. We utilize a two-component fit of a Schechter function plus a log-normal distribution to demonstrate that LRGs are typically brighter than their most luminous satellite by roughly 1.3 magnitudes. This luminosity gap implies that interactions within LRG environments are typically restricted to minor mergers with mass ratios of 1:4 or lower. The luminosity functions further imply that roughly 35% of the mass in the environment is locked in the LRG itself, supporting the idea that mass growth through major mergers within the environment is unlikely. Lastly, we show that the luminosity gap may be at least partially explained by the selection of LRGs as the gap can be reproduced by sparsely sampling a Schechter function. In that case LRGs may represent only a small fraction of central galaxies in similar mass halos.Comment: ApJ accepted versio

Modeling the color evolution of luminous red galaxies - improvements with empirical stellar spectra

Predicting the colors of Luminous Red Galaxies (LRGs) in the Sloan Digital Sky Survey (SDSS) has been a long-standing problem. The g,r,i colors of LRGs are inconsistent with stellar population models over the redshift range 0.1<z<0.7. The g-r colors in the models are on average redder than the data while the r-i colors in the models are bluer towards low redshift. Beyond redshift 0.4, the predicted r-i color becomes instead too red, while the predicted g-r agrees with the data. We provide a solution to this problem, through a combination of new astrophysics and a fundamental change to the stellar population modeling. We find that the use of the empirical library of Pickles (1998) instead of theoretical spectra modifies the predicted colors exactly in the way suggested by the data. The reason is a lower flux in the empirical libraries, with respect to the theoretical ones, in the wavelength range 5500-6500 AA. The discrepancy increases with decreasing effective temperature independently of gravity. This result has general implications for a variety of studies from globular clusters to high-redshift galaxies. The astrophysical part of our solution regards the composition of the stellar populations of these massive Luminous Red Galaxies. We find that on top of the previous effect one needs to consider a model in which ~3% of the stellar mass is in old metal-poor stars. Other solutions such as substantial blue Horizontal Branch at high metallicity or young stellar populations can be ruled out by the data. Our new model provides a better fit to the g-r and r-i colors of LRGs and gives new insight into the formation histories of these most massive galaxies. Our model will also improve the k- and evolutionary corrections for LRGs which are critical for fully exploiting present and future galaxy surveys.Comment: Submitted to ApJ Letters. High resolution version available at http://www.maraston.eu/Maraston_etal_2008.pd

The Clustering of AGN in the Sloan Digital Sky Survey

We present the two--point correlation function (2PCF) of narrow-line active galactic nuclei (AGN) selected within the First Data Release of the Sloan Digital Sky Survey. Using a sample of 13605 AGN in the redshift range 0.055 < z < 0.2, we find that the AGN auto--correlation function is consistent with the observed galaxy auto--correlation function on scales 0.2h^{-1}Mpc to >100h^{-1}Mpc. The AGN hosts trace an intermediate population of galaxies and are not detected in either the bluest (youngest) disk--dominated galaxies or many of the reddest (oldest) galaxies. We show that the AGN 2PCF is dependent on the luminosity of the narrow [OIII] emission line (L_{[OIII]}), with low L_{[OIII]} AGN having a higher clustering amplitude than high L_{[OIII]} AGN. This is consistent with lower activity AGN residing in more massive galaxies than higher activity AGN, and L_{[OIII]} providing a good indicator of the fueling rate. Using a model relating halo mass to black hole mass in cosmological simulations, we show that AGN hosted by ~ 10^{12} M_{odot} dark matter halos have a 2PCF that matches that of the observed sample. This mass scale implies a mean black hole mass for the sample of M_{BH} ~ 10^8 M_{odot}.Comment: 5 pages, 4 figures. Accepted for publication in ApJ

Spatially Extended Low Ionization Emission Regions (LIERs) at z∼0.9z\sim0.9

We present spatially resolved emission diagnostics for eight $z\sim0.9$ galaxies that demonstrate extended low ionization emission-line regions (LIERs) over kpc scales. Eight candidates are selected based on their spatial extent and emission line fluxes from slitless spectroscopic observations with the HST/WFC3 G141 and G800L grisms in the well-studied GOODS survey fields. Five of the candidates (62.5%) are matched to X-ray counterparts in the \textit{Chandra X-Ray Observatory} Deep Fields. We modify the traditional Baldwin-Philips-Terlevich (BPT) emission line diagnostic diagram to use [SII]/(H$\alpha$+[NII]) instead of [NII]/H$\alpha$ to overcome the blending of [NII] and H$\alpha$+[NII] in the low resolution slitless grism spectra. We construct emission line ratio maps and place the individual pixels in the modified BPT. The extended LINER-like emission present in all of our candidates, coupled with X-Ray properties consistent with star-forming galaxies and weak [OIII]$\lambda$5007\AA\ detections, is inconsistent with purely nuclear sources (LINERs) driven by active galactic nuclei. While recent ground-based integral field unit spectroscopic surveys have revealed significant evidence for diffuse LINER-like emission in galaxies within the local universe $(z\sim0.04)$, this work provides the first evidence for the non-AGN origin of LINER-like emission out to high redshifts.Comment: 11 pages, 1 table, 6 figures, accepted for publication in the Astrophysics Journal (ApJ