A Theory of Fluid Flow in Compliant Tubes

Starting with the Navier-Stokes equations, a system of equations is obtained to describe quasi-one-dimensional behavior of fluid in a compliant tube. The nonlinear terms which cannot be shown to be small in the original equations are retained, and the resulting equations are nonlinear. A functional pressure-area relationship is postulated and the final set of equations are quasi-linear and hyperbolic, with two independent and two dependent variables. A method of numerical solution of the set of equations is indicated, and the application to cases of interest is discussed.     

Analysis of the transient behavior of kidney models

Non-steady-state equations for kidney models are stated. General conservation relations for these equations are derived. Transient equations for the central core model of the renal medulla are developed. Solution of the equations by Laplace transform methods for time invariant volume flows is discussed. The general theory of solving models with time dependent flows by finite difference methods is developed.     

Outline of a mathematical theory of the removal of malarial parasites from the blood stream

By making some plausible assumptions, a set of differential equations is proposed which describes the kinetics of interaction between periodically multiplying parasites and continuously produced phagocytes. These equations establish some general conditions which are necessary in order that the number of parasites should increase or decrease. Some consequences of the equations are discussed, and some experimental procedures for the determination of the parameters entering those equations are suggested.     

种间竞争的一个新的数学模型—对经典的Lotka—Vol—terra竞争方程的扩充

本文根据营养动力学理论,建立了一类种间竞争的新的数学模型:它是单种群增长的Cui-Lawson模型,在种间竞争上的推广。新的种间竞争模型克服了经典的种间竞争的Lotka-Volteira方程的局限与不足,具有更广泛和复杂的行为,并在特殊条件下以Lotka-Volterra竞争方程为其特例。因此,新的种间竞争的数学模型是更一般的解释性模型,是对经典的Lotka-Voterra竞争方程的扩充。     

The appropriate use of regression equations for the estimation of segmental inertia parameters

Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base. OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. A set of non-linear equations is provided.     

Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses.

Generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response. These equations are to be solved jointly with score equations from a generalized linear model for mean parameters. A class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion, and proposals for the use of such equations are developed. Comments on the relative merits of the ad hoc and model-based approaches to estimation are given and a regression illustration with a bivariate response is provided.     

Equations and calculations of product yields and preferred pathways for butanediol and mixed-acid fermentations

Using the available information of fermentation biochemistry, fermentation (stoichiometric) equations are derived for anaerobic saccharolytic fermentations of butanediol and mixed acids. The equations describe the interrelations among the fermentation products, biomass, and consumed substrate (glucose). The validity of the equations is tested using a variety of batch data from the literature. The validity of the equations is expected to extend to steady-state and transient fermentations, as well. Uses, improvements, and extensions of the equations are also discussed in detail. Among others, it is shown that the equations are useful for checking the consistency of experimental data, for calculating maximal yields and selectivities for the fermentation products, and calculating the extent of utilization of the Embden-Meyerhof-Parnas pathway versus the Hexose Monophosphate pathway of glucose utilization.     

Quantum-mechanical theory of CD for infinite helical polymers

Quantum-mechanical equations are derived that are particularly well suited to actual computations of the CD for helical polymers. They make use of cyclic boundary conditions and helical symmetry, so that only two matrices with a size equal to the number of transitions considered need be diagonalized. The final equations are expressed directly in terms of monomer properties and helical parameters to invite the same input as earlier calculations, and are given as a rotational strength times a shape function for ease of comparison with the earlier work. The shape of the helix term is expressed as a derivative with respect to ω and depends on the distance between monomers along the helix axis. Other terms involving two electric transition dipoles depend on the distance from the helix axis to the transition center. These equations are directly comparable to the classical equations derived for cyclic boundary conditions and helical symmetry. We present an outline of the derivation and enough intermediate steps to clarify how the equations arise.     

Photosynthesis in Stands of Green Peppers. An Application of Empirical and Mechanistic Models to Controlled-environment Data

The photosynthetic response of stands of green peppers to lightand CO² is examined by means of various mathematical models.Several single leaf response equations are considered, fromthe simple and ubiquitous rectangular hyperbola, to more sophisticatedforms incorporating photorespiration and the oxygen effect.By making the usual assumptions about stand structure and lightpenetration, mechanistic crop response equations arc derivedfor each of the different leaf response equations. First, asa purely empirical approach, the leaf response equations areapplied directly to crop data, and it is concluded that therectangular hyperbola with a term for dark respiration [eqn(2)] gives an adequate summary of crop response to light andCO₂. Second, four mechanistic equations of crop response areapplied to the data, and, although the results are equivocal,it is suggested that the simpler crop equations [eqns (15) and(17)] are satisfactory at the present time, and it is not yetpossible to detect the results of photorespiration and the oxygeneffect directly in the crop data.     

Mathematical model for transient isoelectric focusing of simple ampholytes

A mathematical model describing transient processes in isoelectric focusing (IEF) of L biprotic ampholytes is presented. The model is a generalization of our previous research on steady slate in IEF and consists of L nonlinear partial differential equations coupled with 2L+2 algebraic equations. Constraints imposed by the mode of operation, viz., constant current. voltage or power, are described. Due to the nonlinearity of the equations, analysis of the model requires computer simulation. Model equations suitable for computer implementation are derived.     

Square-root models for the Volterra equations and the explicit solution of these models

Volterra's (1926) equations for competition and predator-prey interactions are modified by introduction of root terms. A critical comparison with the original equations shows that the dynamic properties of the systems remain essentially alike, while the modification allows for explicit solution of the differential equations. Detailed solutions and numerical examples are given.     

The attractiveness of the Droop equations.

The Droop equations are a system of three coupled, nonlinear ordinary differential equations describing the growth of a microorganism in a chemostat. The growth rate of the organism is limited by the availability of a single nutrient. In contrast to the better known Monod equations, the nutrient is divided into external and internal cellular pools. Only the internal pool can catalyze growth. This paper proves that the Droop equations are globally stable. Based on a single combination of parameters, either the chemostat organism goes extinct or it tends to a fixed, positive concentration.     

Nonlinear theory of the collective Cherenkov interaction of a dense relativistic electron beam with a dense plasma in a waveguide

A nonlinear theory of the instability of a straight relativistic dense electron beam in a plasma waveguide is derived for conditions of the stimulated collective Cherenkov effect. A study is made of a waveguide with a dense plasma such that the plasma wave excited by the beam during the instability can be escribed, with a good degree of accuracy, as a potential wave. General relativistic nonlinear equations are btained that describe the temporal dynamics of beam-plasma instabilities with allowance for plasma nonlinearity and the generation of harmonics of the initial perturbation. Under the assumption that the resonant interaction between the beam waves and the plasma waves is weak, the general equations are reduced to relativistic equations with cubic nonlinearities by using the method of expansion in small perturbations of the trajectories and momenta of the beam and plasma electrons. The reduced equations are solved analytically, the time scales on which the instability saturates are determined, and the nonlinear saturation amplitudes are obtained. A comparison between analytical solutions to the reduced equations and numerical solutions to the general nonlinear equations shows them to be in good agreement. Nonlinear processes caused by the relativistic nature of the beam are found to prevent stochastization of the system in the nonlinear stage of the well-developed instability. In contrast, a nonrelativistic electron beam is found to be subject to significant anomalous nonlinear stochastization.     

Spectral properties of Eigen evolution matrices

Eigen [7] has employed deterministic kinetic-like equations to describe macromolecular replication and mutation leading to selection. The solutions to these equations and their physically interesting properties depend upon the spectrum of a type of matrix appearing in those equations. Below we explicitly solve for the spectrum of a fairly general class of such matrices. These solutions are obtained recursively for equations describing macromolecules of any length v.     

Application of theoretical models in ecology.

Certain general prey-predator models are compared with a particular ecosystem simulation. The adequacy of the models is considered in relation to the structure of the equations; to the representation of ecological features by terms in these equations; and to the need to incorporate resource equations in such models.     

Solution of a set of Maxwell-Lorentz equations for a ring relativistic electron beam

Nonlinear solutions to a set of Maxwell’s equations and the relativistic equations of electron motion are obtained that describe the equilibrium of a high-power ring relativistic electron beam against the background of immobile ions. By transforming the basic equations, a set of equations for a three-component vortex vector field is derived that describes ring beam configurations for plasma confinement. An example of a numerical calculation of the steady state of a compact beam torus of immobile ions and relativistic electrons is presented.     

一类生物数学模型参数反演的Picard压缩映射方法

微分方程（组）系统是常见的一类生物数学模型．我们利用模型对应的Picard算子的压缩性和Collage定理,提出一类模型参数的反演算法,并通过数值例子说明了方法的有效性．     

Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations.

Modifications based on experimental results reported in the literature are made to the Hodgkin-Huxley equations to describe the electrophysiological behavior of the Aplysia abdominal ganglion R15 cell. The system is then further modified to describe the effects with the application of the drug tetrodotoxin (TTX) to the cells' bathing medium. Methods of the qualitative theory of differential equations are used to determine the conditions necessary for such a system of equations to have an oscillatory solution. A model satisfying these conditions is shown to preduct many experimental observations of R15 cell behavior. Numerical solutions are obtained for differential equations satisfying the conditions of the model. These solutions are shown to have a form similar to that of the bursting which is characteristic of this cell, and to preduct many results of experiments conducted on this cell. The physiological implications of the model are discussed.     

Regression analysis of four physical characteristics—stature,length of thigh,length of leg and length of foot of the male population in Sichuan Province

Using four physical characteristics of 394 adult male individuals in the Sichuan Province of China — stature, length of thigh, length of leg and length of foot, a series of linear regression equations and ternary regression equations have been established. Meanwhile three stepwise regression equations also have been established including a quarternary regression equation. Generally speaking, the established multiple equations in this study should be applied as much as possible, where there are two or three independent variables, because they predict more effectively in the individual identification of forensic practice.     

Differential equations for moments of an interacting particle process on a lattice

Differential equations are derived for a class of interacting lattice particle processes first introduced in [11]. When there is one type of particle, the equations are in terms of the moments of boundary counts and particle counts. When there are two types of particles, the equations are in terms of the moments of certain neighbourhood statistics as well.This work has been supported by grants from the Natural Sciences and Engineering Research Council of Canada. WJB also acknowledges support of an Ontario Graduate Scholarship