Dimension Spectra of Lines

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim(a, b) is equal to the effective packing dimension Dim(a, b), then sp(L) contains a unit interval. We also show that, if the dimension dim(a, b) is at least one, then sp(L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite

Dual random fragmentation and coagulation and an application to the genealogy of Yule processes

The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes.Comment: 14 page

The free energy in the Derrida--Retaux recursive model

We are interested in a simple max-type recursive model studied by Derrida and Retaux (2014) in the context of a physics problem, and find a wide range for the exponent in the free energy in the nearly supercritical regime

A Brownian particle in a microscopic periodic potential

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle's momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, and so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein-Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle's momentum.Comment: 60 pages. We reorganized the article and made a few simplifications of the conten

The role of mentorship in protege performance

The role of mentorship on protege performance is a matter of importance to academic, business, and governmental organizations. While the benefits of mentorship for proteges, mentors and their organizations are apparent, the extent to which proteges mimic their mentors' career choices and acquire their mentorship skills is unclear. Here, we investigate one aspect of mentor emulation by studying mentorship fecundity---the number of proteges a mentor trains---with data from the Mathematics Genealogy Project, which tracks the mentorship record of thousands of mathematicians over several centuries. We demonstrate that fecundity among academic mathematicians is correlated with other measures of academic success. We also find that the average fecundity of mentors remains stable over 60 years of recorded mentorship. We further uncover three significant correlations in mentorship fecundity. First, mentors with small mentorship fecundity train proteges that go on to have a 37% larger than expected mentorship fecundity. Second, in the first third of their career, mentors with large fecundity train proteges that go on to have a 29% larger than expected fecundity. Finally, in the last third of their career, mentors with large fecundity train proteges that go on to have a 31% smaller than expected fecundity.Comment: 23 pages double-spaced, 4 figure

Toolbox model of evolution of metabolic pathways on networks of arbitrary topology

In prokaryotic genomes the number of transcriptional regulators is known to quadratically scale with the total number of protein-coding genes. Toolbox model was recently proposed to explain this scaling for metabolic enzymes and their regulators. According to its rules the metabolic network of an organism evolves by horizontal transfer of pathways from other species. These pathways are part of a larger "universal" network formed by the union of all species-specific networks. It remained to be understood, however, how the topological properties of this universal network influence the scaling law of functional content of genomes. In this study we answer this question by first analyzing the scaling properties of the toolbox model on arbitrary tree-like universal networks. We mathematically prove that the critical branching topology, in which the average number of upstream neighbors of a node is equal to one, is both necessary and sufficient for the quadratic scaling. Conversely, the toolbox model on trees with exponentially expanding, supercritical topology is characterized by the linear scaling with logarithmic corrections. We further generalize our model to include reactions with multiple substrates/products as well as branched or cyclic metabolic pathways. Unlike the original model the new version employs evolutionary optimized pathways with the smallest number of reactions necessary to achieve their metabolic tasks. Numerical simulations of this most realistic model on the universal network from the KEGG database again produced approximately quadratic scaling. Our results demonstrate why, in spite of their "small-world" topology, real-life metabolic networks are characterized by a broad distribution of pathway lengths and sizes of metabolic regulons in regulatory networks.Comment: 34 pages, 9 figures, 2 table

A hierarchical kinetic theory of birth, death, and fission in age-structured interacting populations

We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we develop a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies

Computed Tomography Imaging of Primary Lung Cancer in Mice Using a Liposomal-Iodinated Contrast Agent

To investigate the utility of a liposomal-iodinated nanoparticle contrast agent and computed tomography (CT) imaging for characterization of primary nodules in genetically engineered mouse models of non-small cell lung cancer.Primary lung cancers with mutations in K-ras alone (Kras(LA1)) or in combination with p53 (LSL-Kras(G12D);p53(FL/FL)) were generated. A liposomal-iodine contrast agent containing 120 mg Iodine/mL was administered systemically at a dose of 16 µl/gm body weight. Longitudinal micro-CT imaging with cardio-respiratory gating was performed pre-contrast and at 0 hr, day 3, and day 7 post-contrast administration. CT-derived nodule sizes were used to assess tumor growth. Signal attenuation was measured in individual nodules to study dynamic enhancement of lung nodules.A good correlation was seen between volume and diameter-based assessment of nodules (R(2)>0.8) for both lung cancer models. The LSL-Kras(G12D);p53(FL/FL) model showed rapid growth as demonstrated by systemically higher volume changes compared to the lung nodules in Kras(LA1) mice (p<0.05). Early phase imaging using the nanoparticle contrast agent enabled visualization of nodule blood supply. Delayed-phase imaging demonstrated significant differential signal enhancement in the lung nodules of LSL-Kras(G12D);p53(FL/FL) mice compared to nodules in Kras(LA1) mice (p<0.05) indicating higher uptake and accumulation of the nanoparticle contrast agent in rapidly growing nodules.The nanoparticle iodinated contrast agent enabled visualization of blood supply to the nodules during the early-phase imaging. Delayed-phase imaging enabled characterization of slow growing and rapidly growing nodules based on signal enhancement. The use of this agent could facilitate early detection and diagnosis of pulmonary lesions as well as have implications on treatment response and monitoring

Modeling the Spread of Methicillin-Resistant Staphylococcus aureus in Nursing Homes for Elderly

Methicillin-resistant Staphylococcus aureus (MRSA) is endemic in many hospital settings, including nursing homes. It is an important nosocomial pathogen that causes mortality and an economic burden to patients, hospitals, and the community. The epidemiology of the bacteria in nursing homes is both hospital- and community-like. Transmission occurs via hands of health care workers (HCWs) and direct contacts among residents during social activities. In this work, mathematical modeling in both deterministic and stochastic frameworks is used to study dissemination of MRSA among residents and HCWs, persistence and prevalence of MRSA in a population, and possible means of controlling the spread of this pathogen in nursing homes. The model predicts that: without strict screening and decolonization of colonized individuals at admission, MRSA may persist; decolonization of colonized residents, improving hand hygiene in both residents and HCWs, reducing the duration of contamination of HCWs, and decreasing the resident∶staff ratio are possible control strategies; the mean time that a resident remains susceptible since admission may be prolonged by screening and decolonization treatment in colonized individuals; in the stochastic framework, the total number of colonized residents varies and may increase when the admission of colonized residents, the duration of colonization, the average number of contacts among residents, or the average number of contacts that each resident requires from HCWs increases; an introduction of a colonized individual into an MRSA-free nursing home has a much higher probability of leading to a major outbreak taking off than an introduction of a contaminated HCW

Estimating the within-household infection rate in emerging SIR epidemics among a community of households

This paper is concerned with estimation of the within household infection rate λL for a susceptible → infective → recovered epidemic among a population of households, from observation of the early, exponentially growing phase of an epidemic. Specifically, it is assumed that an estimate of the exponential growth rate is available from general data on an emerging epidemic and more-detailed, household-level data are available in a sample of households. Estimates of λL obtained using the final size distribution of single-household epidemics are usually biased owing to the emerging nature of the epidemic. A new method, which accounts correctly for the emerging nature of the epidemic, is developed by exploiting the asymptotic theory of supercritical branching processes and proved to yield a strongly consistent estimator of λL as the population and sampled households both tend to infinity in an appropriate fashion. The theory is illustrated by simulations which demonstrate that the new method is feasible for finite populations and numerical studies are used to explore how changes to the parameters governing the spread of an epidemic affect the bias of estimates based on single-household final size distributions