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Considering a homogeneous, isotropically scattering target slab of optical thickness x, containing an internal plane emitting source, the aim of the present paper is to determine the distribution of the internal emitting source by measuring the angular distribution of the intensity of radiation emergent from the slab. First, a system of differential recurrence relations for the intensity of finite order scattered radiation emergent from the top is deduced. In other words, a Cauchy system for determining the finite order emergent intensity is expressed in terms of a source function in the diffuse radiation field and the Fredholm resolvent that were computed in a preceding paper [cf. Bellman, Ueno, and Vasudevan, Math. Biosci. 15, 195–203 (1972)]. Then, the quasilinearization technique is used to solve the inverse problem of determining characteristics of the internal source by measuring the radiation field emergent from the top. This method will be employed in future computations relating to realistic situations in the radiation diagnosis and therapy.

This paper reports on the results of a computer experiment demonstrating some of the capabilities of computerized versions of a stochastic model of population growth [4] and a stochastic model of human reproduction [5] in family planning evaluation. The paper is divided into five sections, with Sec. 1 being devoted to a discussion of some of the motivations underlying the paper and the limitations of the results. Section 2 contains the numerical specifications of the component distributions of the model, and in Sec. 3 an experimental design is defined whereby the interactions of two alternative family building schemes, late versus early marriage, and low versus high desired family size may be studied with respect to population growth. The experimental design consists of eight reproductive regimes, and in Sec. 4 graphs of the calculated net maternity functions of these regimes together with the corresponding Malthusian parameter, the crude birth rate, and the mean number of female offspring in a completed family are presented. Section 5 is devoted to two population projections designed to measure the impact on population growth of two experimental schemes of transition from a high reproductive regime to a lower one. The indicators of population change and distributions used to measure the impact of these two experimental transitions in reproductive regimes as a function of time in these projections were crude birth rate, annual rate of population growth, age-specific birth rates, age distribution, and mean total population size.

A terminating nonhomogeneous renewal process is formulated and studied in relation to the development of quantitative methods in family planning evaluation. The motivation underlying the results presented here was to satisfy a need, which arose during computer studies, to complete and extend a stochastic model of human reproduction reported previously. The paper is divided into seven sections detailing the nature of changes made in previous work. Two novel developments in the paper are the formulation of the waiting time between live births in terms of an absorbing semi-Markov process and the waiting time to conception in populations utilizing several methods of contraception as an absorbing Markov chain. These formulations not only permit the accommodation of many family planning variables but also lead to greater conceptual simplification than formulations used in previous work.

In Mathematical Biosciences5(1969), 495–509, I introduced a new theory of message transport in nerves based on an interpretation of the role played by the sodium ion diffusion in the transport process. In this paper I report the results of a quantum statistical mechanical theory of the fundamental message transport step. The theory is developed using the assumption that the message charge transfer step and the sodium ion diffusion from the Donnan nonequilibrium state are coupled through modes of the environmental dielectric continuum. The analysis is carried out by means of second-order, time-dependent, quantum mechanical perturbation theory. The analysis yields an expression for the message current which is of the form of a second-order kinetic process. The requirement of at least a second-order rate for the message transport mechanism was pointed out in the above mentioned paper.

It is known that the electric polarization and associated sodium ion concentration gradient on either side of a nerve membrane undergo great changes during the passage of a nerve impulse. Why this happens and what triggers the change is not known. In this paper I present a theory which accounts for the role played by the sodium ion diffusion in impulse transmission. The theory presented accounts for the following: the nature and use of the sodium diffusion; the necessity of a critical current; and, finally, the reason for the lack of attenuation of the impulse current during its passage. The action of the sodium pump is to store energy in the nerve system. This stored energy, when released, is used in the activation of microscopic charge transfer of message charges between molecular sites in the nerve. The message current depends on the concentration of message charges and on the concentration of sodium ions. It is argued in this paper that the rate of message charge transfer owes its “all-or-none” character to a condition analogous to a thermal explosion.

This paper presents an alternative model to Wiener's system of coupled nonlinear oscillators to account for the peak-dip shape of an EEG power spectrum he reported. The model also offers an alternative to the qualitative description of this peak-dip trimodal shape as the fourth derivative of a Gaussian distribution. The Gaussian distribution is not a necessary feature of the reported model. The model considered here is a system of entrainable oscillators being driven by a single frequency. The oscillators are considered to be at most only indirectly coupled with their natural periods of oscillation probabilistically distributed. The main result of the paper is that the distribution of entrained oscillator frequencies has the peak-dip shape characterized by a high central peak rising out of a dip on either side. Both peak height and depth and distance of the dips from the central peak are related to the sharpness of the entrainment curve and the entrainment range for an oscillator.

In a series of preceding papers, we have solved with the aid of invariant imbedding technique a number of diffuse reflection and transmission problems for parallel rays incident on the top of the target slab. In this paper, again using the principles of invariant imbedding, we obtain the intensity of the internal radiation field due to rays which have suffered a finite number of scatterings during passage through the slab where energy-dependent and anisotropic scattering is assumed. Thus, the internal radiation field is expressed in terms of the finite order scattering and transmission functions, which have been determined in our preceding paper. It is then shown that the internal radiation field permits us to evaluate the required absorbed dose of x rays and gamma rays in a target slab.

A simplified mathematical model of a nerve fiber is described by a semilinear partial differential equation. In our previous paper the existence of a threshold was suggested by numerical calculation. An excitation approaches a peculiar pulselike waveform if the value is above the threshold, and on the other hand vanishes during propagation if it is below the threshold. In this paper the existence of the threshold is analytically investigated: It is shown that every solution of the equation in some neighborhood of zero converges to zero as time increases.

A stochastic model was constructed by the author in an earlier paper to describe the spread of an epidemic in a finite homogeneous population subject to a non-Markovian type of epidemic. Based on this model asymptotically optimal tests were derived to test for contagion of the disease. The purpose of the present paper is to illustrate the use of such optimal tests by applying them first to some observed data of Mastocytoma (mast cell tumour) and then to the simulated Mastocytoma data generated by using Monte Carlo method. The stochastic model, from which the test criteria were derived, describes the temporal development of the epidemic by utilizing the occurence times T 1 ,…, T n of the infective cases. The transition phenomenon of the process is characterized by mutation and contagion. The essential feature of the model is that the infectiousness of a diseased individual depends not only on the length of time he has been sick but also on the time at which he caught the disease. These characteristics are described in the model by a contagion function f . In the present paper, we have used contagion functions of the indicator type to perform the test for contagion of Mastocytoma. The simulated data are used to study the convergence rate of the distribution of several test statistics to the limiting normal distribution; these results are compared numerically. It is found that one particular test statistic W 3 always gives better normal approximation that the others. The asymptotic power is calculated, its sensitiveness with respect to the choice of the contagion function is also discussed. The test result of the observed data, which consists of 174 cases recorded in two counties over two years, does not indicate the presence of contagion.

In a series of our papers invariant imbedding was used to derive the integro-differential and integral recurrence relations for the finite order scattering and transmission functions, the finite order X and Y functions, by which the diffusely reflected and transmitted intensities of finite order scattered radiation can be numerically computed. In the present paper, extending the Bellman-Krein-like recurrence relation for the Fredholm resolvent to the derivation of a modified decomposition formula, we show how to express the internal intensities of finite order radiation in terms of the finite order X , Y , f , and g functions. Then, once the finite order X and Y functions have been determined by the use of the preceding methods, making use of the integral recurrence relations for the finite order f and g functions, we can obtain the numerical solution of the finite order internal radiation field. Hence the absorbed dose distribution of finite order gamma radiation and neutron beams can be directly computed.

An exact solution for the problem of detection of extrasystoles was obtained in a previous paper assuming an additive stochastic signal and noise model for the electrocardiogram. As the exact solution is not easy to implement in terms of present day hardware, an approximation procedure for the detection criterion based on minimum variance estimation procedure on a restricted class of estimators is presented in this paper. The relative performance of the approximation with respect to the exact solution is discussed.

In connection with the estimation of the average fecundability of a group of women, it has been assumed in the past that the waiting time for a conception follows a type I geometric distribution, and its two parameters have been estimated by the method of moments and the method of maximum likelihood. In a follow-up study, starting with a group of women, there is always the possibility that some women will remain nonpregnant in the limited period of time they are observed, and the problem arises as to how to estimate the same parameters if the follow-up experiment is terminated after a fixed period of time. The present paper deals with this truncation problem, and provides three different methods for estimating the parameters, which are applied to data from the Menstrual and Reproductive History Study at the University of Minnesota. In the last section, an extended model is presented which incorporates the possibility of dropout cases.

A discrete time stochastic growth process for human populations accommodating marriages is introduced. In the introduction the motivations underlying the formulation of the model are discussed and the basic random functions of the process are defined in Sec. 2. The basic random functions of the model change because of deaths, marriages, the dissolutions of couples, and births, and Sects. 3, 4, 5, and 6 are devoted to stochastic processes describing these aspects of population dynamics. Finally, Sec. 7 is devoted to the derivation of a system of nonlinear renewal type equations satisfied by the mean functions of the process. The techniques used in the construction of the population process introduced in this paper are related to those used in the construction of multitype age-dependent branching processes.

A new method of finding the roots of the quadratic matrix equation A + BR + RC + RDR = 0 , where A , B , C , and D are n × n matrices is described in this paper. The method gives all roots of the equation when the eigenvalues of the characteristic matrix are distinct and a subset of the roots where the eigenvalues are repeated. The sign and Walsh functions are utilized in the root extraction algorithm.

This paper describes a new computational method for calculating the asymptotic solution to the matrix Riccati equation. This method is fast, efficient and gives all possible solutions to the matrix quadratic form. Matrix sign functions are used to find the asymptotic solutions.

This paper discusses some properties of a nerve-axon-like transmission line made up from random networks of threshold elements. A random network contains a large number of threshold elements of which the threshold values are Gaussian random variables, and it can act as a monostable or a bistable multivibrator. The signal wave propagation in a transmission line is analyzed by a statistical method. The results show that a signal wave can propagate along the line and that its waveform is shaped, during propagation, into a specific form peculiar to the line. A self-oscillatory system that consists of two random networks is also analyzed. Even in this simple system, various modes of oscillation can exist, with periods varying over a wide range according to the values of system parameters. Examples of self-oscillations obtained by a graphical method are presented, and also some results of computer experiments are shown.

This paper presents an application of a dynamic optimization procedure to the problem of finding the most economical use of active and passive immunization to control an infectious disease in a closed population. The mathematical model used is suitable and accurate for representing most such diseases, taking into consideration the latent and infectious periods and their statistical variations. Variables representing active and passive immunization controls are incorporated in the model and the resulting model includes functions of present and past values of both state and control variables. Conditions for cost optimization are obtained from the Minimum Principle and numerical solutions of the problem, obtained by the use of parallel tangent gradient techniques, are presented.

In this paper, we outline a new method for numerical solution of the nonlinear partial differential equation of the Hodgkin-Huxley model. Preliminary results of numerical experimentation are presented.

This paper continues the study of the asymptotic properties of a unified class of biological models which are represented by positive homogeneous operators on ordered Banach spaces. Ergodic properties of these models are analyzed in terms of the spectral properties of the operators.

Following an introductory discussion on the implications of stochastic models of carcinogenesis on and of deterministic models for the growth kinetics of solid tumors, and a brief review of the experimental evidence, it is proposed that solid tumor growth kinetics should be represented mathematically by a nonmultiplicative time-homogeneous stochastic process of the birth and death type. Since nutrients have to enter the solid tumor through its surface, it is in a first approximation to a more adequate representation assumed that the birth rate is proportional to the 2 3 power of the tumor cell population size, and the death rate proportional to the population size; both rates are assumed to be age-independent. A solution for the forward Kolmogorov equation for this process could not be obtained, but bounds are given for the probability of extinction, which is shown to tend to unity asymptotically, for the waiting time to extinction, and for the mean and the variance of the population size. Bounds for the asymptotical conditional mean and mode of the tumor size, conditioned on no extinction, are also given, and the results of computer simulations of the process are discussed. It is shown that the (singular) Fokker-Planck equation for the continuous (diffusion) analog of the process has a regular boundary at the origin, which would require the imposition of an arbitrary boundary condition. Since this is at least biologically unfeasible, it is concluded the the surface proportionality assumption for the birth rate should be replaced by a proposition resulting in a better behavior at the origin, and it is indicated that a better formulation has been obtained and will be presented in a subsequent paper.

In this paper we consider a generalization of the system of differential equations proposed by S.N. Levine [1] to describe the kinerics of enzyme amplifier systems, based on the presence of a negative physiological feedback in the systems. In particular, we conjecture the identity between the models of Levine and Seegers for blood coagulation.

The paper deals with dynamical behaviors of the solutions of a single neuronic equation: x(t+1) = 1[ ∑ k=0 n−1 akx(t−k)θ] , where 1[u] = 1 for u>0 and = 0 for u≦0, in order to have a systematic insight into a certain family of transition phenomena among 2n state configurations. For this purpose we are concerned with the specific operators of the six kinds: L ω, L omega;, L αl, L alpha;l, L βl and L beta;l (l=0, 1, 2,…, n−1). In Sections 5 and 6 we discuss reverberation phenomena for L alpha;0. A regular articular representation of each state configuration is used to find out all the types and the total number of reverberation cycles under the applications of the operator L alpha;0. Section 7 deals with the applications of L ālpha;n−1, while Sections 8 and 9 deal with L beta;l. Section 10 gives some ideas on the use of our family of operators for the representation of the linear threshold functions associated with the single neuronic equation.

This paper treats the two component radiation approximation for the calculation of absorbed dose when γ radiation passes through irradiated material. We describe a method of solving in a successive fashion the vector transfer equation for the two component radiation. We use a two-by-two matrix to represent the differential attenuation coefficient. Employing the imbedding principle, we obtain a set of integro-differential recurrence relations for the two-by-two matrices of finite order, scattering and transmission functions. Making use of these functions, the two component radiation field within the homogeneous, anisotropically and noncoherently scattering target slab (Bellman, Ueno, and Vasudevan, to be published), can be computed. This enables us to determine the absorbed dose.

Scan-rescan processes are assuming a more and more important role in applications of nuclear medicine to cancer detection as well as in detection of other medical abnormalities, especially when real time processing is desirable. In this paper it is shown that dynamic programming can both improve accuracy and decrease the time required for examination. Very interesting analytic questions arise in this fashion. The numerical resolution of these questions and the methods developed will have extensive application in other areas where pattern recognition and search are involved. In particular, a hierarchal application of scan-rescan processes, suitably generalized, could have extensive use in the national Medicare program.

The primary purpose of this paper is to explain some principal aspects of various approaches appealing to certain mathematical formulations based upon cell spaces and to illustrate their implications to biomathematics. In Sects. 2 and 3 we explain our specific investigations on cell space in connection with a family of local transformations satisfying the principle of local majority. In Sect. 4 general considerations and a set of five interpretations of basic notions of cell space approaches are given to explain how and why an abstract and purely mathematical formulation of cell space can have any connection with biological problems and hence can deserve to be a theoretical framework in biological sciences. The general attitude given in Sect. 4 is illustrated in Sect. 5 in which various biological problems, including birth, growth, formation of forms, pattern disintegration, and death, as well as some ecosystem problems are discussed with reference to various works done by many authors. Suggestions are given regarding the problem of what notions will be indispensable to further investigation of the cell space approach. In Sect. 6 we summarize our view on the role of biomathematics which is to be expected to have a certain role different from biophysical consideration. It is the standpoint of the author to build up a branch of biomathematics which is closely and directly connected with the information science approach, as we explain it in Sect. 7, with specific reference to the automation and language problem in the cell space approach.

Following an interpretation of nonlinear birth rates as the product of a linear birth rate and a growth fraction, it is concluded that the assumption of a surface-proportional birth rate used in a previous paper for a stochastic process for solid tumor growth is rather untenable from a biological viewpoint. Here, a nonlinear birth rate is derived from a simple model formulating the diffusion and consumption of a nutrient in a solid tumor, assuming that the diffusion rate is proportional to the Laplacian of the concentration and that nutrient consumption is proportional to concentration. The local birth rate at a point in the tumor is then assumed to be proportional to the nutrient concentration, resulting in a total birth rate which is (for proper choice of the value range of the parameters) proportional to tumor size for small tumors and becomes proportional to tumor surface with increasing tumor sizes. Assuming a linear death rate, the resulting spatially discrete birth and death process permits early exponential and later decelerated and asymptotic mean growth. The process is ergodic with absorption at zero certain; it is shown that for biologically reasonable parameter values the expected absorption time becomes so large as to make this process almost ergodic null for all practical considerations. Bounds are derived for the time course of the absorption probability, the unconditional mean and variance, and the conditional mean. It is shown that the conditional equilibrium distribution is reversed- J -shaped or unimodal depending on the ratio of the birth and death rate parameters, and bounds for the mode of the conditional equilibrium distribution are derived. The boundary classification for the spatially continuous formulation of the process yields an exit boundary at zero and thus eliminates the problem encountered with the surface-proportional birth rate of imposing an arbitrary boundary condition at the origin.

The Lotka-Volterra nonlinear differential equations for two competing species P and Q contain six independent parameters. Their general analytic solutions, valid for arbitrary values of the parameters, are at present unknown. However, when two or more of these parameters are interrelated, it is possible to obtain the exact solutions in the P , Q phase plane, and six cases of solvability are given in this paper. The dependence of the solutions on the parameters and the initial conditions can thus be readily investigated.

The dose-response relationship in a pharmacologic system is analyzed using basic concepts of drug-receptor interaction and the theory of stochastic processes. Response is viewed as resulting from additive recruitment of tissue receptor-effector units. A Fokker-Planck (continuity) equation is written relating response to drug dosage in time. The solution to this equation is recalled from a previous paper. An expression relating response at equilibrium to drug dose follows, and easy methods for evaluation of experimental parameters are developed. Another mode of solution of the Fokker-Planck equation using Laplace transforms is presented, and simple equations are derived permitting straightforward evaluation of experimental parameters in transient states. These equations are adapted to “relaxation” type experiments.

A discrete time version of a generalized one-type age-dependent branching process is considered in relation to stable population theory in demography. The motivation underlying the discrete time version of the theory is to make it amenable to computations involving demographic data. After giving a brief discussion of the foundations underlying the process, discrete type renewal equations for the mean and covariance functions of the process are derived. It is then shown how those renewal type equations may be used for making population projections with respect to age-specific birth rates, rates of population growth, and the number of live individuals in each age group, given an initial population with an arbitrary age distribution. A novel feature of the method of population projection introduced in this paper is that it is possible to derive confidence bounds for the projected number of individuals in any age group by utilizing the covariance functions of the process and the central limit theorem.

The paper presents the fundamentals of the theory of neural nets with loops based on functional matrices, which are a generalization of the state transition matrices for a net. Problems of analysis and synthesis; given a net, find its functional matrix and vice versa, are treated for probabilistic and deterministic nets. Questions about universal nets, oscillations, and stability are studied for deterministic nets. Reduction of probabilistic nets with loops is considered. It is shown that any probabilistic net with loops can be duplicated by a deterministic net with loops plus a probabilistic loop-free encoder. The motivation for the work is a search for formulation of the general theory of neural nets that could be tied to the theory of triadic intensional relations, as suggested by Warren S. McCulloch.

This paper will prove the following result which was stated without proof by Dr. John Jacquez: A linear compartmental system has a trap if and only if the associated system of differential equations has a zero eigenvalue. It will then use this result to prove an approximation theorem which says roughly that a linear compartmental system has an approximate trap if and only if the associated system of differential equations has an eigenvalue which is approximately zero. Let S represent a linear compartmental system consisting of compartments C1, C2 . . . . C, and qj be the amount of material in Cj. Let f~,j. be the fractional exchange coefficient so that the rate of flow of material from Cj to Ci isf~,jqj; and letf0,iqj be the rate of flow of material from C s. to the environment. The total outflow from Cj is (fo,j + Xl,jf~,j)qj which we will write as fs,iqJ. This leads us to consider the system of differential equations

Graphical tools are presented to assist in the prediction of future achievements and in the early recognition of breakthroughs in leukemia research and for the comparative evaluation of results. Survival and hazard function curves are shown, both for past experience and extrapolated for estimating future achievements. This paper discusses methods for early recognition of substantial changes in the rate of progress (or “breakthroughs”). Methods are presented for comparing the performance of treatment protocols, or for evaluating the performance of treatment centers, by comparing with broad-based results to derive a value of time lead or lag. The objective is to provide feedback in the design of clinical trials and to identify the most promising approaches for rapid exploitation.

The paper discusses the extension of the well-known stability theorem for digital filters to two-dimensions. Several independent proofs of the stability theorem, which were first obtained by Shanks and Justice in the context of Banach algebras and z-transforms, are also discussed.

This paper discusses strategies for storing information in a memory, subject to the kinds of constraints that would correspond to some of the known restrictions on the anatomy and physiology of the mammalian nervous system. Thus it is assumed that individual memory “traces” (the elementary unit of memory storage) correspond to spatial patterns of activity in very large groups of cooperating neurons. It is also assumed that traces interact so that the stored information is the sum of all the previously stored traces. Although a great deal of information is lost in the storage process, it is still possible to answer many simple questions about the stored information. Two models are proposed which use such storage systems. The first model, the “recognition model,” assumes that we are interested in knowing whether or not an incoming trace is identical to one that was stored previously. It is shown that this question can be answered to a high degree of accuracy. Two variations of the recovery procedure are suggested. The second model, the “association model,” assumes that two traces have become associated in the past. We then present the system with the first trace and we wish to generate, at the output of the storage system, the second trace. A simple model for accomplishing this is developed and the degree of noise added by the system is evaluated. It is shown that this model can generate indefinitely long strings of associations from a single trace while the noise added by the system approaches a finite limit. The general features of the model are discussed and some possible experimental approaches to proof or disproof of the model are suggested. The models are discussed in light of some of the relevant psychological data.

Random drift causes fixation in a finite population not subject to recurrent mutation. The paper shows that, with respect to fixation at two loci in a monoecious population, the introduction of monogamous matings, rather than random unions, has little effect on the rate of fixation, or on the probabilities of which gamete type will be fixed. The differences are minimal when the number of gametes passed on to the next generation by any individual has a variance approximately equal to its mean.

This work points out certain inaccuracies in the paper by Temple F. Smith which first described the maximum in the entropy function of protein within the base composition range of vertebrates. The significance of this maximum is discussed.

The aim of the present paper is concerned with the formulation of a general stochastic bivariate model without specifying the nature of the relationship between two species. The stochastic model is equally well adopted to either the predator-prey type of interaction between two species or to mutual competition. From the bivariate distribution function estimators for the parameters in the stochastic model can be easily derived and a realistic insight into the behavior of the system can be gained.

This paper is concerned with the dynamic programing approach to the design of optimal pattern classification systems when the costs of feature measurements describing the patterns are of considerable importance. A multistage or sequential classifier that requires, on the average, a substantially smaller number of feature measurements than equally reliable nonsequential classifiers based on a fixed number of measurements is defined and constructed through the principle of recursive optimization. Two methods of reducing the dimensionality in computation are presented for the cases where the observed feature measurements are (1) statistically independent and (2) Markov dependent. Both models, in general, provide a ready numerical solution to the optimal sequential classification problem. Computersimulated experiments are used to illustrate the feasibility of this approach.

The paper describes the bases of the new approach in the synthesis of the artificial legged and anthropomorphic gait, in particular. In detail there has been described the concept of prescribed synergy which in the authors' opinion gives great possibilities for treating realistically the legged systems locomotion. There is also given a new algorithm that makes it possible to form automatically (by a computer) the mathematical models of anthropomorphic mechanisms of general configurations. The method is a general one and can be applied with equal success to both the analysis and synthesis of mechanical configurations that correspond to the manipulators.

For the past thirty years Green's table in Glasser's Medical Physics [6] has been used as the common quantitative source for data on the arterial tree in mammals. The present paper updates that table by coordinating additional anatomical data from man and dog into a unified model of branching levels. The model and data appear to be consistent as to geometry, topology, and fluid resistance, as well as in number, size, and volume, of tubes. Application of such quantitative modeling to two problems is sketchily presented. The first application concerns the way the quantitative modeling information is involved in the treatment of pulsatile flow in the arterial system. The second application adduces some evidence that the wide resistive range of real arterial systems (as opposed to the average model developed) should be interpreted as mean power regulation by the system, rather than mean pressure regulation.

Organ transplantations from a donor animal D to a recipient animal R will be successful only when R possesses all antigens possessed by D. When there is no direct method to determine the antigenic structure of R and D, one procedure which can be used to determine whether D possesses an antigen R does not (in which case, the transplant will fail) is the third-man test. In the third-man test, an indifferent recipient I receives a skin graft from both R and D. Unfortunately, the results of this procedure may be deceptive and difficult to interpret. In this paper, the author considers an algorithm that provides a means of avoiding the pitfalls associated with the interpretation of the third-man test provided the number of indifferent animals used is sufficiently large. Proper use of the algorithm can allow researchers to determine the compatibility of donor-recipient pairs with certainty; also, information may be obtained concerning the number of antigens possessed by D, the number of antigens possessed by Rs compatible with D, and the total number of antigens affecting the results of the third-man test. Finally, a number of examples illustrating the use of the algorithm are given.

The pulmonary clearance delay (PCD) is the most commonly used index of uneven pulmonary ventilation. However, the only method previously available for its calculation involves the fitting of a mathematical model to which several investigators have raised objections. This paper presents two new methods. One involves a different mathematical model and illustrates the way in which the PCD can be derived from a variety of models. The second method does not utilize the parameters of any fitted model but calculates the PCD by simple arithmetic procedures from the original data observations.

This three-part study investigates the application of optimization techniques, especially dynamic programming, to decision making in agricultural pest management. In Part III, a specific model is developed to find an optimal combination of control techniques including biological control (a parasite, in this case) and chemical control (an insecticide). The insecticide is toxic to the parasite and thus chemical control interferes with biological control. Calculations are also made of the effect on production costs of possible governmental actions (such as insecticide taxation or prohibition) aimed at reducing insecticide use. The last section of this paper discusses the way in which additional factors can be included in the optimization model without overtaxing computer storage requirements.

Membranes are often assumed to be impermeable barriers penetrated by a repeating pattern of discrete pores. When such membranes are enclosed by concentration boundary layers with nonnegligible diffusion resistances, analyses of transport through the membrane must take them into account. In this paper a mathematical model of this membrane system is chosen that includes the resistance of the boundary layers and takes specific account of the heterogeneous nature of the membrane. The solution for the diffusion flux is shown to correspond in the appropriate parametric limits to the idealized models of a one- dimensional series of resistances or of a collection of single pores opening into semi-infinite media. The solution is used to establish the range of validity of these idealized models. Moreover, the unique characteristics of the dependence of the overall diffusion flux on boundary layer thickness when the membrane contains pores suggests ways of experimentally determining whether a membrane has pore structure and, in certain cases, of estimating the average pore diameter.

All possible configurations of relative fitnesses can be obtained by drawing an arbitrary triangle and marking a point on two sides. The lines of equal fitness become circles for hyperbolas and the point of maximum fitness (if any), and the ultimate destiny of the population can be seen at a glance. The new method may be helpful for purposes of analysis and to avoid calculations, but the paper otherwise contains no new results.

Stochastic models of neuronal firing have been proposed previously, in which neuronal firings are determined by the interaction between two renewal processes. This paper proposes a stochastic model of neuronal firing taking account of a threshold level and derives the interval density between two successive firing events.

The paper is concerned with the probability of noninfectivity of virus particles to which antibodies become attached. The particular case considered is one where detachment of antibodies is also possible. A solution is found for the deterministic model, in particular in the case where the attachment and detachment parameters take a linear form. Probabilities of noninfectivity are calculated in two simple cases.

In a previous study of human locomotion, an important aspect of our mathematical modelling was the decomposition of the complex body dynamics into two parts: one describes the lower extremity motion; the other pertains to the motion of the upper trunk. The generation of gait patterns via optimal programming was studied in that paper. In the present study the torso motion is considered. This involves as investigation of the unstable equilibrium of the torso about its upright position. A multidimensional “inverted pendulum” model is derived to study this behavior. To stabilize the motion, a linear feedback law coupled with an on-off perturbation is proposed. A Liapunov function is then constructed to show that such a control mechanism provides effective stabilization of the torso for all initial configurations of motion. Significance of the control law is discussed and a possible EMG-photographic experiment is suggested on the basis of this theoretical study.

It is shown for a supercritical Galton-Watson process with immigration, {Xn}, that there exists a sequence of constants {Cn} such that the process {Xn/Cn} converges almost surely to a random variable with continous distribution concentrated on (0, ∞), or diverges almost surely to infinity according as the log moment σ ∞ 1 b j log j of the immigration distribution is finite or not. These results include the suggested refinements proposed in a previous paper of the author [6].

In this paper a new statistical test is proposed for detecting major gene differences in quantitative inheritance. It is based on the Smirnov distribution free test and may be used to detect single gene differences between two inbred lines using data from parental, F1, F2 and backcross generations. An application of the test is demonstrated using data from previously published experiments on the genetic control of specific antibody production.