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

Natural Boundaries for Solutions to a Certain Class of Functional Differential Equations

AbstractThis paper is concerned with a generalization of a functional differential equation known as the pantograph equation. The pantograph equation contains a linear functional argument. In this paper we generalize this functional argument to include nonlinear polynomials. In contrast to the entire solutions generated by the pantograph equation for the retarded case, we show that in the nonlinear case natural boundaries occur. These boundaries are linked to the Julia set of the polynomial functional argument

Generalized transversality conditions for the Hahn quantum variational calculus

We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory

The Genetic Analysis Workshop 16 Problem 3: simulation of heritable longitudinal cardiovascular phenotypes based on actual genome-wide single-nucleotide polymorphisms in the Framingham Heart Study

The Genetic Analysis Workshop (GAW) 16 Problem 3 comprises simulated phenotypes emulating the lipid domain and its contribution to cardiovascular disease risk. For each replication there were 6,476 subjects in families from the Framingham Heart Study (FHS), with their actual genotypes for Affymetrix 550 k single-nucleotide polymorphisms (SNPs) and simulated phenotypes. Phenotypes are simulated at three visits, 10 years apart. There are up to 6 "major" genes influencing variation in high- and low-density lipoprotein cholesterol (HDL, LDL), and triglycerides (TG), and 1,000 "polygenes" simulated for each trait. Some polygenes have pleiotropic effects. The locus-specific heritabilities of the major genes range from 0.1 to 1.0%, under additive, dominant, or overdominant modes of inheritance. The locus-specific effects of the polygenes ranged from 0.002 to 0.15%, with effect sizes selected from negative exponential distributions. All polygenes act independently and have additive effects. Individuals in the LDL upper tail were designated medicated. Subjects medicated increased across visits at 2%, 5%, and 15%. Coronary artery calcification (CAC) was simulated using age, lipid levels, and CAC-specific polymorphisms. The risk of myocardial infarction before each visit was determined by CAC and its interactions with smoking and two genetic loci. Smoking was simulated to be commensurate with rates reported by the Centers for Disease Control. Two hundred replications were simulated

Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-201

Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850

Fractional Euler-Lagrange differential equations via Caputo derivatives

We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form will appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu et al. (eds.), Springer New York, 2012, DOI:10.1007/978-1-4614-0457-6_9, in pres

A retrospective claims analysis of combination therapy in the treatment of adult attention-deficit/hyperactivity disorder (ADHD)

<p>Abstract</p> <p>Background</p> <p>Combination therapy in managing psychiatric disorders is not uncommon. While combination therapy has been documented for depression and schizophrenia, little is known about combination therapy practices in managing attention-deficit/hyperactivity disorder (ADHD). This study seeks to quantify the combination use of ADHD medications and to understand predictors of combination therapy.</p> <p>Methods</p> <p>Prescription dispensing events were drawn from a U.S. national claims database including over 80 managed-care plans. Patients studied were age 18 or over with at least 1 medical claim with a diagnosis of ADHD (International Classification of Diseases, Ninth Revision, Clinical Modification [ICD-9-CM] code 314.0), a pharmacy claim for ADHD medication during the study period July2003 to June2004, and continuous enrollment 6 months prior to and throughout the study period. Dispensing events were grouped into 6 categories: atomoxetine (ATX), long-acting stimulants (LAS), intermediate-acting stimulants (IAS), short-acting stimulants (SAS), bupropion (BUP), and Alpha-2 Adrenergic Agonists (A2A). Events were assigned to calendar months, and months with combined use from multiple categories within patient were identified. Predictors of combination therapy for LAS and for ATX were modeled for patients covered by commercial plans using logistic regression in a generalized estimating equations framework to adjust for within-patient correlation between months of observation. Factors included age, gender, presence of the hyperactive component of ADHD, prior diagnoses for psychiatric disorders, claims history of recent psychiatric visit, insurance plan type, and geographic region.</p> <p>Results</p> <p>There were 18,609 patients identified representing a total of 11,886 months of therapy with ATX; 40,949 months with LAS; 13,622 months with IAS; 38,141 months with SAS; 22,087 months with BUP; and 1,916 months with A2A. Combination therapy was present in 19.7% of continuing months (months after the first month of therapy) for ATX, 21.0% for LAS, 27.4% for IAS, 23.1% for SAS, 36.9% for BUP, and 53.0% for A2A.</p> <p>For patients receiving LAS, being age 25–44 or age 45 and older versus being 18–24 years old, seeing a psychiatrist, having comorbid depression, or having point-of-service coverage versus a Health Maintenance Organization (HMO) resulted in odds ratios significantly greater than 1, representing increased likelihood for combination therapy in managing adult ADHD.</p> <p>For patients receiving ATX, being age 25–44 or age 45 and older versus being 18–24 years old, seeing a psychiatrist, having a hyperactive component to ADHD, or having comorbid depression resulted in odds ratios significantly greater than 1, representing increased likelihood for combination therapy in managing adult ADHD.</p> <p>Conclusion</p> <p>ATX and LAS are the most likely drugs to be used as monotherapy. Factors predicting combination use were similar for months in which ATX was used and for months in which LAS was used except that a hyperactive component to ADHD predicted increased combination use for ATX but not for LAS.</p

Collisional kinetics of non-uniform electric field, low-pressure, direct-current discharges in H2_{2}

A model of the collisional kinetics of energetic hydrogen atoms, molecules, and ions in pure H$_2$ discharges is used to predict H$_\alpha$ emission profiles and spatial distributions of emission from the cathode regions of low-pressure, weakly-ionized discharges for comparison with a wide variety of experiments. Positive and negative ion energy distributions are also predicted. The model developed for spatially uniform electric fields and current densities less than $10^{-3}$ A/m$^2$ is extended to non-uniform electric fields, current densities of $10^{3}$ A/m$^2$, and electric field to gas density ratios $E/N = 1.3$ MTd at 0.002 to 5 Torr pressure. (1 Td = $10^{-21}$ V m$^2$ and 1 Torr = 133 Pa) The observed far-wing Doppler broadening and spatial distribution of the H$_\alpha$ emission is consistent with reactions among H$^+$, H$_2^+$, H$_3^+$, and H$^-H$ ions, fast H atoms, and fast H$_2$ molecules, and with reflection, excitation, and attachment to fast H atoms at surfaces. The H$_\alpha$ excitation and H$^-$ formation occur principally by collisions of fast H, fast H$_2$, and H$^+$ with H$_2$. Simplifications include using a one-dimensional geometry, a multi-beam transport model, and the average cathode-fall electric field. The H$_\alpha$ emission is linear with current density over eight orders of magnitude. The calculated ion energy distributions agree satisfactorily with experiment for H$_2^+$ and H$_3^+$, but are only in qualitative agreement for H$^+$ and H$^-$. The experiments successfully modeled range from short-gap, parallel-plane glow discharges to beam-like, electrostatic-confinement discharges.Comment: Submitted to Plasmas Sources Science and Technology 8/18/201