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  • Springer  (122,265)
  • 1980-1984  (48,999)
  • 1970-1974  (73,266)
  • 1925-1929
  • 1984  (48,999)
  • 1973  (37,150)
  • 1972  (36,116)
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  • 1980-1984  (48,999)
  • 1970-1974  (73,266)
  • 1925-1929
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  • 1
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    Springer
    In:  EPIC3Naturwissenschaften, Springer, 71(12), pp. 599-608, ISSN: 0028-1042
    Publication Date: 2014-06-04
    Repository Name: EPIC Alfred Wegener Institut
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  • 2
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    Bulletin of mathematical biology 34 (1972), S. i 
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  • 3
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    Bulletin of mathematical biology 34 (1972), S. 53-63 
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    Notes: Abstract A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.
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  • 4
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    Bulletin of mathematical biology 34 (1972), S. 439-441 
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    Notes: Abstract It is shown that from the definition of organismic sets (Rashevsky,Organismic Sets. Some Reflections on the Nature of Life and Society, Holland, Michigan, Mathematical Biology, Inc. and Grosse Pointe, Michigan, J. M. Richards Laboratory) a complete sensory deprivation of an organismic set of ordern=2 should result in malfunctioning of the set. A generalization to higher order sets is suggested.
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    Bulletin of mathematical biology 34 (1972), S. 431-438 
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    Notes: Abstract Optimality in branching structure of the vascular tree was studied. Analysis on its physiological roles as the duct system for blood supply to the capillaries predicted that the vascular tree should be constructed with minimum volume under restriction of determinant pressure, flow and location at the origin and the terminals. Mathematical derivations of this conditional extremum problem yielded some equations expressing the relations between the radii of the branches and their branching angles, which provided numerical solutions for branching points of bi- and poli-terminal minimum volume trees. Comparison of the peritoneal vascular tree in a dog with the minimum volume one computed under the same restrictive conditions showed good agreement in their branching structure.
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    Bulletin of mathematical biology 34 (1972), S. 443-456 
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    Notes: Abstract In order to determine the kinetics of passage of a substance through an organ containing a tangle of vessels, we study the response of a tube to various inflows (perfusion, brief injection, ...). The introduction of the catabolic terms and of the spatial dependence between bulk concentration and surface concentration allows one to account for the difference of arteriovenous concentrations observed experimentally for many metabolites. The relationships between the physico-chemical parameters of the organ and the operational parameters of the model demonstrate the importance of the transit time through the considered vessels. If one considers the different pathways as independent, the introduction of the transit time distribution for an inert substance enables one to compute the response of the organ analytically or by recurrence, using convolution. The parameters of the model can be obtained by the moments method.
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  • 7
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    Bulletin of mathematical biology 34 (1972), S. 457-466 
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    Notes: Abstract The solution of the diffusion equation in the gas phase of the human lung is very difficult because of the structure of the bronchial tree. It is shown by means of physical arguments, how one can reduce the diffusion equation to a simple one-dimensional form. The solution is then obtained by a stochastic simulation, which is easily realized on a digital computer.
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  • 8
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    Bulletin of mathematical biology 34 (1972), S. 467-481 
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    Notes: Abstract A new mathematical model of the oscillatory behavior of the respiratory center has been developed based upon published records of neuronal activity during respiration in the pons and medulla. In contrast with a previous model, four, rather than two, networks are assumed to interact in the respiratory center so as to produce the respiratory oscillation. A mathematical description of this interaction, in the form of a set of four first-order, nonlinear, coupled differential equations, is derived; the behavior of the solutions of this system is studied qualitatively, and expressions for the durations of the inspiratory and expiratory phases are obtained in terms of some parameters. It is found that central and chemical influences drive the medullar neurons to a position somewhere between saturation and full cutoff, and the pontine neurons deeply into cutoff. The control of the duration of the different phases by these chemical and central means is discussed. In order to effect a decrease in the magnitude of the various times, the neurons have to be driven towards operating points of higher central facilitation.
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    Bulletin of mathematical biology 34 (1972), S. 483-502 
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    Notes: Abstract In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.
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    Bulletin of mathematical biology 34 (1972), S. 521-532 
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    Notes: Abstract Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.
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    Notes: Abstract Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.
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    Bulletin of mathematical biology 34 (1972), S. 547-558 
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    Notes: Abstract Two chemicals,A andB, are allowed to diffuse together and a reaction described by $$A + B\mathop \rightleftharpoons \limits_{K_{ - 1} }^{K_1 } C$$ is allowed to proceed. This system is described mathematically by a system of partial differential equations. A numerical procedure is presented to find the rate constants ofK 1 andK −1. A systematic analysis of the effects of errors is also presented.
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    Bulletin of mathematical biology 34 (1972), S. 533-546 
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    Notes: Abstract Equations are developed to describe the energy expenditure of the human heart. As well as the external potential and kinetic energy terms, general consideration is given to other possible avenues of energy consumption. Emphasis is placed upon using mathematical variables which are readily available for experimental verification. The errors involved in assuming that mean values for the physiological parameters give reasonable estimations for the external mechanical performance are examined, and a theoretical estimation for the discrepancy in the kinetic component is presented. Logical extension of the mathematical derivation leads to a determination of cardiac external mechanical efficiency and clearly demonstrates the significance of the ventricular pressure-volume loop in this context. Finally, experimental procedures are suggested to clarify further some of the conclusions reached through the theoretical analysis.
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    Bulletin of mathematical biology 34 (1972), S. 559-563 
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    Notes: Abstract The question is discussed as to the reason why some animal societies, such as bees or ants, are sexually differentiated, that is, onlysome of its members are exhibiting reproducing activities. It is indicated that human society may be on its way to such a sexual differentiation which may eventually come.
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    Bulletin of mathematical biology 34 (1972), S. 565-565 
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    Bulletin of mathematical biology 34 (1972), S. 567-567 
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    Bulletin of mathematical biology 35 (1973), S. 301-311 
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    Notes: Abstract X-ray diffraction patterns obtained experimentally for fibers, together with their chemical structures, can be analyzed theoretically in terms of an integral equation. The partially unknown electron density function can be solved by iteration. This mathematical technique has been applied with success to study the secondary structures of DNA fibers.
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    Bulletin of mathematical biology 46 (1984), S. 967-969 
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    Notes: Abstract It is observed that a dynamical continuity equation for biomass distribution yields the asymptotic steady-state exponential dependencen=A exp( $$ - m/\bar m$$ ) exhibited by certain fishery data, wherem is the biomass of an individual,n is the number of individuals per unit biomass interval, andA, $$\bar m$$ are positive constants. This dynamical approach to biomass distribution is an alternative to the global maximization principle proposed recently by Lurié and Wagensberg.
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    Bulletin of mathematical biology 46 (1984), S. 971-972 
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    Bulletin of mathematical biology 46 (1984), S. 973-974 
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    Bulletin of mathematical biology 35 (1973), S. 663-688 
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    Notes: Abstract The paper demonstrates that it is possible to construct memory models where the information inserted is stored in disseminated form, using sequential coding, the changes in the units forming the models being determined by their geometrical connections and by the incoming stream of information. The models are shown to have large storage capacity and their efficiency can be made insensitive to loss of or damage to a large fraction of their units. The satisfactory verification by computer simulation of the analysis and results described in the present paper will be the subject of a future paper.
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    Bulletin of mathematical biology 46 (1984), S. 11-17 
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    Notes: Abstract Linear birth and death processes are used to derive simple expressions for sequential extinction times and gene fixation probabilities in asexual populations.
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    Bulletin of mathematical biology 46 (1984), S. 1-10 
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    Notes: Abstract We are here concerned with the functionf which assigns to each pointP of an object the numberf(P) which is the shortest distance fromP to the border. This function appears in various guises in diverse biological studies. The functionf(P) is itself a measure of shape—or more precisely, an infinite set of measures, one for each point (and hence, in view of its geometric definition, usually in a form inconvenient for use). Thus in this paper we sought a reasonable representative of this infinite set of measures, namely themean of the numbersf(P) asP ranges over all points of the entity. Computability studies are developed for various classes of shapes. For example, (1) the mean for a lamina bounded by a polygon circumscribable about a circle of radiusr isr/3; (2) the mean for a domain bounded by a polyhedron circumscribable about a sphere of radiusr isr/4. The transition from pointwise to piecewisef(P), especially in the non-convex case, requires working with inequalities.
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    Bulletin of mathematical biology 46 (1984), S. 19-40 
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    Notes: Abstract A mathematical model for traveling bands of motile and chemotactic bacteria in the presence of cell growth and death is examined. It is found that asymptotic traveling wave solutions exist in the absence of chemotaxis, due to the balance of growth, death and random motility. Thus random motility confers the ecological advantage of population propagation through migration into nutrient-rich regions. The presence of chemotaxis amplifies this advantage by moving more cells into higher nutrient concentration regions, resulting in larger and faster bands. Therefore there seem to be two types of traveling bands that can be attained by chemotactic bacteria in the presence of growth and death: (1) these growth/death/motility bands; and (2) pure chemotactic ‘Keller-Segel'-type bands. Comparison to experimental observations by Chapman in 1973 indicate that the latter seem to be formed. The relationship between these two types of solution is at present uncertain. The growth/death/motility bands may have relevance on longer time or distance scales characteristic of microbial ecological systems.
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    Bulletin of mathematical biology 46 (1984), S. 115-125 
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    Notes: Abstract Based on the principle of minimum power, a mathematical model of the pathological functional state of the oxygen transport system is presented. The model is used to determine the optimal functional parameters of the oxygen transport system in hyperthyroidism, anemia and hypertension. Theoretical results are compared with clinical data.
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    Bulletin of mathematical biology 46 (1984), S. 139-153 
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    Notes: Abstract A modified SIRS model is developed as a framework for the study of epizootiological dynamics in an insect-pathogen system. Linearized stability analysis reveals that the system with one immune and one susceptible host class can exhibit stable, periodic or unstable behavior depending on model parameters. In general, high pathogenicity, short pathogen propagule lifespan and high host reproductive rate are stabilizing influences. Pathogen transmissibility and propagule production/host do not influence local stability. The effect of seasonal host reproduction is studied because most insect hosts are seasonal in temperate climates. The basic stability dependence on model parameters holds except as modified by the length of the reproduction interval. The results of this study are compared with the recent work of Anderson and May.
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    Bulletin of mathematical biology 46 (1984), S. 175-184 
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    Notes: Abstract The random predator-prey type interactions of the population species in a random varying environment have been investigated. The Fokker-Planck equation for the transition probability, corresponding to the stochastic differential equations established from Lotka-Volterra equations by the introduction of randomness and variability, has been integrated in the form of a path integral. The transition probabilities for extinction or survival of one or several species have been approximately evaluated and investigated.
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    Bulletin of mathematical biology 46 (1984), S. 155-174 
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    Notes: Abstract If two microbial populations compete for a single resource in a homogeneous environment with time invariant inputs they cannot coexist indefinitely if the resource competed for is not renewed by biological activity within the system. Mathematical studies have shown that in a predator-prey system, where the resource (prey) is self-renewing, the two competitors (predators) can coexist in a limit cycle. This suggests that if the resource competed for is renewed by biological activity within the system coexistence can occur in any microbial system provided that it exhibits the same features as, but without being, a predator-prey one. A food chain involving commensalism, competition and amensalism is presented here. Two subcases are considered. It is only when maintenance effects are taken into account that coexistence, in limit cycles, can occur for this system. Limit cycle solutions for the system are demonstrated with the help of computer simulations. Some necessary conditions for coexistence are presented, as are some speculations regarding the possible physical explanations of the results.
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    Bulletin of mathematical biology 46 (1984), S. 127-137 
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    Notes: Abstract The branching structure of the mammalian arterial tree has been known to be close to that of an optimal conduit system of the minimum work model characterized as the branch system of constant wall shear rate. The physiological mechanism producing such construction was considered to be based on the local response of arterial caliber induced by the wall shear stress (shear rate × blood viscosity) and thereby maintaining this stress constant, which was previously observed at the canine common carotid artery shunted to the external jugular vein. The stress levels at various parts of the arterial system estimated from available data fell within ±50% of the mean (15 dyn/cm2), which was consistent with the value predicted from the model. Theoretical analyses on the cost function of the model indicated that the suspected variation of shear rate levels in the arterial tree due to the anomalous changes in blood viscosity which might bring about 3- to 4-fold differences between the minimum and maximum shear rates would cause less than 10% increase in the total energy cost. It was concluded that a local adaptive response to wall shear stress is the mechanism which effectively optimizes the design of the arterial tree.
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    Bulletin of mathematical biology 46 (1984), S. 185-185 
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    Bulletin of mathematical biology 46 (1984), S. 187-203 
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    Notes: Abstract The structure of the genetic code is related to a Gray code, which is a plausible theoretical model for an amino acid code. The proposed model implies that the most important factor in shaping the code was the effects of mistakes in translation, not effects of mutations. Another possible implication is that the preservation of stiffness and flexibility at appropriate places in a protein chain is as important in protein structure as the appropriate placement of hydrophilic (external) and hydrophobic (internal) residues. Other results are a simple conceptualization of the relationships among the 20 amino acids and their relations to their codons. The detailed relationships are summarized in the following ‘similarity alphabet’: ala, thr, gly, pro, ser; asp, asn, glu, gln, lys; his, arg, trp, tyr, phe; leu, met, ile, val, cys; (ATGPS DNEQK HRWYF LMIVC in the one-letter code). This alphabet falls into four groups of amino acids: small, external, large, internal. The approximate relation of the groups to their codons is expressed as: the first base of a codon controls size—a purine means a small amino acid, a pyrimidine means large; the middle base controls cloisterednes—purine means external, pyrimidine means internal. These relationships express the minimum change principle upon which the code appears to be founded.
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    Bulletin of mathematical biology 46 (1984), S. 269-282 
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    Notes: Abstract A theoretical study of the Brusselator model with non-uniform distribution of component A and a concentration-dependent diffusion coefficient has been performed. Numerical simulation reveals that a variable diffusion coefficient alters the bifurcation pattern and the stability properties of the steady-state as well as periodic solutions. A simple approximate method, based on one-point collocation, has been proposed to analyze the bifurcation phenomena for the case of fixed boundary conditions and low system size.
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    Bulletin of mathematical biology 46 (1984), S. 283-294 
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    Notes: Abstract In this paper the effects of the occurrence of cut trees in the topological analysis of branching patterns have been studied. It is assumed that branches are removed at random from the trees. We prove that, for both the segmental and terminal growth models, the probability distributions of the cut trees are identical to those of complete trees.
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    Bulletin of mathematical biology 46 (1984), S. 247-268 
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    Notes: Abstract The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL p[0,∞], (1〈-p〈-∞). A discussion is included of theL p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O K (relating extravascular concentration to intravascular) and K a, k (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O K ‖ p 〈-1, ‖K a, k ‖ p 〈-1), isotone, linear operators onL p . The uniform convergence relation $$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$ (as operators onL p) is derived, whereP n (a) is the Poisson probabilitye −a a n /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖Ok‖=‖Ka,k‖p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.
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    Bulletin of mathematical biology 46 (1984), S. 295-326 
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    Notes: Abstract One particular kind of structure offers possible explanations, for long-term memory, efficient consolidation of stored information from the environment, clustering of data strings and multimodal functioning. It is a possible model for pieces of neural structure and its use offers a uniform method for both studying and constructing an extensive class of mechanisms.
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    Bulletin of mathematical biology 46 (1984), S. 327-332 
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    Notes: Abstract Levenshtein dissimilarity measures are used to compare sequences in application areas including coding theory, computer science and macromolecular biology. In general, they measure sequence dissimilarity by the length of a shortest weighted sequence of insertions, deletions and substitutions required, to transform one sequence into another. Those Levenshtein dissimilarity measures based on insertions and deletions are analyzed by a model involving valuations on a partially ordered set. The model reveals structural relationships among poset, valuation and dissimilarity measure. As a consequence, certain Levenshtein dissimilarity measures are shown to be metrics characterized by betweenness properties and computable in terms of well-known measures of sequence similarity.
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    Bulletin of mathematical biology 46 (1984), S. 337-337 
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    Bulletin of mathematical biology 46 (1984), S. 333-336 
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    Notes: Abstract It is pointed out that the methane flux measured experimentally for certain ponds and swamps is quantitatively consistent with a commensal dependence of Methanobacteria on O2-chemotactic motile aerobic bacteria. The Methano species is thereby shielded from oxygen and provided with carbon dioxide for the anaerobic production of methane.
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    Bulletin of mathematical biology 46 (1984), S. 357-370 
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    Notes: Abstract A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra-type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of the random fluctuations are not too large.
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    Bulletin of mathematical biology 46 (1984), S. 371-377 
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    Notes: Abstract One of Bobisud's models for the evolution of cannibalism is reanalyzed by applying the method of finding evolutionarily stable strategies (or ESS's). It is demonstrated that ‘no cannibalism’ never will be an ESS if the initial rate of cannibalism is too large. It is further demonstrated that individual selection may even result in the evolution of cannibalism during food abundance. Some empirical case studies are briefly discussed in relation to this model.
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    Bulletin of mathematical biology 46 (1984), S. 379-387 
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    Notes: Abstract A computer algorithm is presented which equiprobably generates any member of the set of all directed trees withk labeled terminal nodes and unlabeled interior nodes. The algorithm requires roughlyk 2 /2 storage locations. The one-time initialization requiresO(k 2 ) time, while generating each tree requiresO(k) time.
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    Bulletin of mathematical biology 46 (1984), S. 515-527 
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    Notes: Abstract The comparison of several sequences is central to many problems of molecular biology. Finding consensus patterns that define genetic control regions or that determine structural or functional themes are examples of these problems. Previously proposed methods, such as dynamic programming, are not adequate for solving problems of realistic size. This paper gives a new and practical solution for finding unknown patterns that occur imperfectly above a preset frequency. Algorithms for finding the patterns are given as well as estimates of statistical significance.
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    Bulletin of mathematical biology 46 (1984), S. 501-514 
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    Notes: Abstract A new development is introduced here in the use of dynamic programming in finding pattern similarities in genetic sequences, as was first done by Needleman and Wunsch (1969). A condition of pattern similarity is defined and an algorithm is given which scans any set of similarities and screens out those which fail to meet the condition. When the set to be scanned contains every pair of segments, one from each of two given sequences of lengthsm andn (i.e. every possible location for a pattern similarity), then it completes the scan in a number of computational steps proportional tom·n, leaving those pairs of segments which satisfy the similarity condition. The algorithm is based on the concept of match density, as suggested by Goad and Kanehisa (1982).
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    Bulletin of mathematical biology 46 (1984), S. 529-543 
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    Notes: Abstract This paper concerns sequences of letters in which certain “distinguished” words are of interest. Such sequences arise as data in numerous fields including genetics and neuroscience. A probability distribution is given for the number of occurrences of a chosen word in a randomized sequence of letters. Such words are considered “favored” if they occur more than expected at random. Favored words have been discovered in nerve impulse trains and may reflect a neural coding scheme.
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    Bulletin of mathematical biology 46 (1984), S. 545-552 
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    Notes: Abstract As the volume of protein sequence data grows, rapid methods for searching the protein sequence database become of primary importance. Rigorous comparison of sequences is obtained with the well-known dynamic programming algorithms. However, these algorithms are not rapid enough to use for routinely searching the entire database. In this paper we discuss some methods that can be used for rapid searches.
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    Bulletin of mathematical biology 46 (1984), S. 553-566 
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    Notes: Abstract We give algorithms for computing the extent of similarity between two or three sequences of letters. The similarity measures we consider include a penalty for inserting gaps within the sequence in order to enhance similarity. The magnitude of the penalty for gaps is assumed to be independent of their size in order to accommodate certain biological applications. Our algorithm for three sequence comparisons, which is based on solving a system of recursive equations, improves upon the efficiency of existing methods. Although the system of recursive equations utilized by the algorithm is quite complicated as it stands, it has none the less been simplified by appeal to combinatorial considerations.
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    Bulletin of mathematical biology 46 (1984), S. 567-577 
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    Notes: Abstract Well-known dynamic programming algorithms exist for comparing two finite sequences inO(N 2) time and storage, whereN is the common sequence length. Extensions to the comparison ofM finite sequences requireO((2N) M) time and storage, making such algorithms difficult even forM=3. A simple generalization of the sequences makes it possible to obtain some results about the geometry of sequence alignments. These ideas suggest heuristic approaches to problems of comparing several sequences. IfM sequences are known to be related by a binary tree, they can be aligned inO(MN 2) time andO(N 2+NM) storage.
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    Bulletin of mathematical biology 46 (1984), S. 661-672 
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    Notes: Abstract Protein sequences of the Dayhoff databank of 1984 have been analyzed to evaluate the occurrences of the 400 dipeptides and 8000 tripeptides. Expected values and standard deviations for the di- and tripeptides were determined by Monte Carlo and binomial approximation. A condensed format containing this information, labeled a uniqueness diagram, is presented and made available in the form of a microfiche.
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    Bulletin of mathematical biology 46 (1984), S. 827-844 
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    Notes: Abstract In two-state sliding filament models of muscle contraction a partial differential equation must be solved to find the cross-bridge distribution functionn(x, t). In this paper the analytical form of this function is obtained by integration along the characteristic line and special cases are presented in which the explicit expression forn(x, t) can be completely determined. These analytical solutions provide a direct mathematical connection between the microscopic contraction parameters contained in the kinetic theories and macroscopic muscle dynamics and are thus used to investigate what parameters influence the transient contractile tension in typical experimental conditions. The results of this investigation are consistent with relevant aspects of muscle physiology.
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    Bulletin of mathematical biology 46 (1984), S. 845-857 
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    Notes: Abstract Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP〉2 nets is examined briefly.
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    Bulletin of mathematical biology 46 (1984), S. 869-877 
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    Notes: Abstract The influence of topographical situation on the spread of infection is studied. The investigation is based on a multigroup model. The population under consideration is thought to be divided into subpopulations living in regions that are separated from each other by natural barriers (mountains). Infection is carried from one region to another by migrating infectives. Migration is possible only along the river system so that the structure of the epidemiological network is that of a symmetric tree. The results allow comparison of the velocity of propagation of the epidemic for different geographical situations and allow quantification of the “channel-effect”, according to which mountainous regions are channels rather than barriers to the spread of an epidemic.
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    Bulletin of mathematical biology 34 (1972), S. 173-204 
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    Notes: Abstract As a further attempt to determine the stresses and strains of the individual myocardial fibers, the heart muscle is considered as an orthotropic material. A theory is presented which leads to the expression of the equilibrium conditions for the left ventricle in the form of three simultaneous differential equations. Solution of these equations would give the changes in shape of the left ventricle throughout the cardiac cycle, and, in addition, the stresses and strains of the individual myocardial fibers. It is pointed out, however, that meaningful solutions of the equations cannot be obtained at the present time because of difficulties in experimental determination of certain parameters.
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    Bulletin of mathematical biology 34 (1972), S. 213-222 
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    Notes: Abstract This paper compares two previously published neural models for epilepsies (Bull. Math. Biophysics,33, 539–553, 1971;34, 71–78, 1972). The second model is developed in more detail and an attempt is made to bring it more in line with established neurological findings. The question of classification of some epilepsies is briefly discussed.
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    Bulletin of mathematical biology 34 (1972), S. 223-230 
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    Notes: Abstract A formal mathematical model is proposed for a spontaneously repetitively firing neuron. It is based on the assumption that an excitatory and inhibitory substance, possibly different from those involved in synaptic transmissions, is formed in the soma of everynormal neuron. Furthermore, the decay of the substances is ascribed to their combination with some other substances, present in healthy individuals. A generalized two factor system of differential equations is used. It is shown that when the normally present substances are absent, possibly due to genetic defects so that the decay constants become zero, the equations lead to undamped sinusoidal solutions of the difference between excitatory and inhibitory factors, thus producing a trulyspontaneous repetitive discharge, in the absence of external currents or other stimulation. It is suggested that convulsants may act by destroying the substances present in healthy individuals. It is further suggested that by administering to epileptics those substances, which are present in normal healthy persons, perhaps by using brain extracts fromhealthy higher animals which sometimes suffer from epilepsy, an actual cure rather than symptomatic treatment by anticonvulsants may be obtained.
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    Bulletin of mathematical biology 34 (1972), S. 277-291 
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    Notes: Abstract The general kinetic behavior of a multicompartment system is shown to depend upon certain general structural features, including its connectivity, whether it is open, and whether it contains cyclic pathways. Structural influences are clarified by putting the system matrix in a certain form. For systems not strongly connected, a distinction is drawn between partially and completely open systems. Necessary and sufficient conditions are given for non-singularity of the system matrix and for asymptotic stability of the system. Sufficient conditions are given for non-overshooting and monotonic transitions. A system is demonstrated whose solution may contain a prolonged series of damped oscillations; but the oscillations are very slow and small; and it seems unlikely that oscillations could be detected experimentally in any biological system.
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    Bulletin of mathematical biology 34 (1972), S. 243-275 
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    Notes: Abstract It is shown how the fundamental laws of chemical kinetics for either open or closed systems with an arbitrarily large number of reactants can be represented as a system of Riccati-like differential equations. Through the use of a concise tensor notation, it is shown when and how the differential system is exactly reducible to linear form, a reduction without approximation that parallels the well-known similar reduction of a single simle Riccati equation. An example is worked out to show how open kinetics can lead to oscillatory chemical concentrations of the Change-Higgins type. The biologically central problem of great chemical speciation is discussed from the viewpoint of Gibbs ensemble theory within the linearized kinetics and, approximately, within the starting nonlinear kinetics where it is shown roughly how to estimate, from an overall temperature-like parameter characterizing the whole system, mean chemical levels and mean frequencies of oscillation, and where a gross oscillation of the total mass is estimated in terms of an anharmonic oscillator whose general structure is fixed from the structure of the chemical kinetic laws.
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    Bulletin of mathematical biology 34 (1972), S. 293-296 
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    Notes: Abstract A closed chain of compartments in which there is unidirectional transport between adjacent members can exhibit damped oscillations. For a system ofn equivalent compartments, the value ofn which gives the greatest difference between the first maximum and first minimum isn=11, the difference being 1.57%. The greatest difference between the first maximum value and the steady state value is 4% and is obtained whenn=25. The results are illustrated graphically forn equal to 5, 10, 25 and 100.
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    Bulletin of mathematical biology 34 (1972), S. 297-304 
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    Notes: Abstract This paper is a concrete approach to the problem of the number of the sexes. We try to imagine—on the example of three sexes—the mechanisms which would have to accompany a reproduction with several sexes. We have limited our study to the monohybridism, dihybridism and determinism of the sex.
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    Bulletin of mathematical biology 34 (1972), S. 325-335 
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    Notes: Abstract An analysis of the effect of cilia on fluid transport in tubules is presented. The applicability of the results for the flow rates observed in the ductus efferentes of the male tract is discussed.
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    Bulletin of mathematical biology 34 (1972), S. 305-324 
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    Notes: Abstract The dipole models for steady-state currents in excitable membranes of Arndt, Bond and Roper and of Hamel and Zimmerman are compared by fitting the equations to the data of Gilbert and Ehrenstein. The more complex Hammel and Zimmerman model does not fit the data as well as does the simpler Arndt, Bond and Reper model. When fitting the data, the Hammel and Zimmerman current equation reduces to the Arndt, Bond and Roper current equation because of the values assumed by the parameters. An interpretation is given for the parameter values obtained with the Arndt, Bond and Roper model.
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    Bulletin of mathematical biology 34 (1972), S. 337-341 
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    Notes: Abstract It is shown that, under rather general conditions, it is possible to formally decompose the dynamics of ann-dimensional dynamical system into a number of non-interacting subsystems. It is shown that these decompositions are in general not simply related to the kinds of observational procedures in terms of which the original state variables of the system are defined. Some consequences of this construction for reductionism in biology are discussed.
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    Bulletin of mathematical biology 34 (1972), S. 343-353 
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    Notes: Abstract For a certain class of physical machines, termed “structure-determined,” the problem of self-reproduction can be reduced to the problem of serial message reproduction. Serial message reproduction however presupposes a sort of “open system” constraint. This leads to the principle of pseudo, or exogenously standardized, respectively, self-reproduction. It seems to be consistent with both chemical and biological self-reproduction. It thus may reflect a general principle of biological design. The proposed principle is a physico chemical analog to Robert Rosen's abstract relational self-reproduction constraint.
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    Bulletin of mathematical biology 34 (1972), S. 355-377 
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    Notes: Abstract Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.
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    Bulletin of mathematical biology 34 (1972), S. 413-418 
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    Notes: Abstract Analytical solutions are presented for transient heat conduction in biological media. General boundary conditions and internal sources varied in both spatial and time variables are considered, thus, solutions for many special cases can be obtained with ease from the general solutions presented in this analysis.
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    Bulletin of mathematical biology 34 (1972), S. 393-412 
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    Notes: Abstract Two mathematical models of pulmonary single breath gas washout (one analytic, one numerical) are developed and their predictions compared with experimental data on human subjects. Weibel's 23 generation symmetric anatomical model is used as a guide to bronchial tree geometry. Experimental plots of nitrogen concentration versus volume expired, dead space versus breath holding time, and dead space versus tidal volume are compared with plots predicted by the models. Agreement is good. A plot of nitrogen concentration in the airways as predicted by the numerical model at different times during inhalation and exhalation of a single breath of oxygen is shown. Model predictions for changes in dead space with changes in washout gas and expiratory flow rate are discussed. Use of the analytic model for obtaining average values of the path length from mouth to alveoli in a given subject is discussed. To the extent of their agreement with experiment, the models provide a sound physical basis for the correlation of airway structure and function.
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    Bulletin of mathematical biology 34 (1972), S. 429-429 
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    Bulletin of mathematical biology 34 (1972), S. 419-427 
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    Notes: Abstract The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m 1 exp (n 1x) −m 2 exp (n 2 x), wherem 1,m 2,n 1 andn 2 are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.
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    Bulletin of mathematical biology 35 (1973), S. 313-317 
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    Notes: Abstract Various philosophers have repeatedly denounced knowledge as a source of unpleasantness, thereby criticizing education. This paper presents a set theoretical approach to knowledge and education and tries to explain how they could lead occasionally to a feeling of unpleasantness.
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    Bulletin of mathematical biology 35 (1973), S. 275-286 
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    Notes: Abstract An analysis of countercurrent exchange in a U-tube is presented for a single-solute, constant-volume flow rate system with spatially varying source fluxes and permeabilities. Analytical solutions are given for the steady-state equations and numerical solutions for the unsteady-state equations. The solutions indicate that an external source of solute delivered to the stream flowing away from the U-tube bend can be distributed by the exchanger so that the concentration in both limbs increases toward the bend. In particular, there exist source fluxes whose magnitude decreases monotonically toward the bend for which the maximum solute concentration occurs at the bend. The point at which a concentration maximum occurs is governed principally by the solute permeability of the barrier separating the two limbs and by the volume flow rate through the exchanger. The system dynamics depend strongly on the relative cross-sectional areas of the two limbs or, equivalently, on the flow velocities within them. The model is used as a basis for discussion of various functional aspects of the renal vasa recta system.
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    Bulletin of mathematical biology 35 (1973), S. 287-300 
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    Notes: Abstract The circulatory mixing process was analyzed as the time course of the dispersion of indicator after its injection into the heart. In simplified models, which had one or two lumped mixing chambers and circulatory pathways connected with them, it was suggested that the extent of dispersion could be evaluated by the variance of indicator distribution in the total circulating blood when the circulation time distributions between the chambers and the concentration curves in the chambers were known. The method of determining the circulation time distributions through the pulmonary, systemic and total circulations was derived and the actual distributions were obtained in dogs by indicator dilution techniques. With the use of these distributions, the time course of the circulatory mixing process was numerically calculated. The results showed that there was considerable difference in velocities of the process between the case of the right heart injection and the left heart injection of the indicator.
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    Bulletin of mathematical biology 35 (1973), S. 319-337 
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    Notes: Abstract An investigation is made as to whether or not the existence of a band-pass filter function, analogous to that in electronics, can be proved from the fundamental laws of chemical kinetics. The problem is important for better understanding of the preference of certain biological rhythms to others. It is shown with simple examples that such behavior is possible for a number of systems of coupled chemical reactions far enough from the thermodynamic equilibrium. It is of interest to generalize this behavior since it could conceivably play a role in the transmission of “usable information” in biology.
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    Bulletin of mathematical biology 35 (1973), S. 339-344 
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    Notes: Abstract The phenomenon of mental creativity is considered from the standpoint of the theory of organismic sets, developed by the author in a series of previous publications. It is shown how the differences in creativity between different individuals may be interpreted on this basis, and why extreme creativity is rare. A parallel interpretation for facility in observation is given, and it is shown why facility in creativity and observation is much rarer than either individual facility. A further conclusion is drawn regarding the deducibility of the laws of nature by purely logical means.
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    Bulletin of mathematical biology 35 (1973), S. 359-374 
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    Notes: Abstract The equation for the quantum transitions (spontaneous and stimulated) of membrane dipoles is solved for the various forms of time-varying stimulation in nerve. From the condition of ever-increasing dipole population in the upper state, the threshold for excitation is determined in each case. The results obtained are in agreement with the established facts. The optimum frequency for stimulation is given asv 0=0.0615/T 2 whereT 2 is the dipole relaxation time. The feature of the theory is that the mathematical formulation is based upon a physical mechanism and the results can thus provide some understanding in the observed phenomena.
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    Bulletin of mathematical biology 35 (1973), S. 415-415 
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    Bulletin of mathematical biology 35 (1973), S. 417-418 
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    Bulletin of mathematical biology 35 (1973), S. 419-419 
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    Bulletin of mathematical biology 35 (1973), S. 565-575 
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    Notes: Abstract Evaluation of the Van der Waals energy per filament suggests that molecular dispersion forces should not be very important in determining the stability of the myofilament lattice in resting muscle. In order to explain the lattice stability and other important properties of the striated muscle, it is suggested that a balance between electrostatic forces and forces developed by some interfibrillar structures is mainly responsible.
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    Bulletin of mathematical biology 35 (1973), S. 549-563 
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    Notes: Abstract A method is presented for the simultaneous determination of (i) the blood flow to the organs and (ii) the cardiac output. Part I of the paper deals with the analysis of ann compartment (organ) vascular system model. The data, employed in the analysis, consists of continuous monitoring of the amounts of indicatorM i in the organs (or compartments). An analysis for determination of the cardiac output and the absolute flows to the organs is presented. Since it is difficult to isolate certain organ systems and measure the amounts of indicator in them exclusively, a more realistic model of then compartment vascular system is presented in Part II. Herein, the analysis has accounted for the finite transit time, of the indicator, from one organ system to another. Further, estimation theory is employed to make estimates of blood flow to different organs by taking note of (i) the measurement errors due to the detectors' monitoring (for an organ system) some combination ofM i 's instead of theM i for the particulari th organ and (ii) noise uncertainties introduced by the measuring instruments.
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    Bulletin of mathematical biology 35 (1973), S. 577-589 
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    Notes: Abstract Analytic expressions for the velocity profile and particle distribution of a dilute suspension in flow were obtained as functions of radial distance. Einstein's linear viscosity model and the hypothesis of “minimum energy dissipation” were used. The methods of variational calculus were applied during the mathematical development. A parabolic velocity profile, which is a modified form of that for Hagen-Poiseuille flow, and a uniform particle distribution were obtained. An attempt is made to explain the results in light of some of the widely held theories on suspension flow and the rather severe limitations of Einstein's viscosity model. A suggestion for future work is made for improving the results of the present.
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    Bulletin of mathematical biology 35 (1973), S. 591-605 
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    Notes: Abstract A possible mechanism for microwave-neuron interaction, when the nerve is irradiated by a thermally insignificant electromagnetic field, is described. The radiation field is treated classically, but the atomic system which interacts with this field is treated quantum mechanically using the density matrix approach. Attention is given to both homogeneous and inhomogeneous broadening effects, and the degrading influence of inhomogeneous broadening upon the neural membrane's ability to interact with the electromagnetic field is shown.
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    Bulletin of mathematical biology 35 (1973), S. 709-714 
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    Notes: Abstract Mathematical models of neurons are studied which exhibit spontaneous and repetitive firings in the absence of normally occurring substances. The activity of such neurons could result in the symptoms of epilepsy. The identification of such substances would be important in the treatment of the disease as well as in the understanding of their mode of action. The importance of the geometrical and physical parameters for the generation of spontaneous repetitive discharges is brought out.
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    Circuits, systems and signal processing 3 (1984), S. 267-294 
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    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract Pipeline techniques have been successfully applied to speeding up processing in both general- and special-purpose digital computers. Application of these techniques to nonrecursive (FIR) filters has been suggested and is quite straightforward. Application to recursive (IIR) filters has not previously been shown. In this paper, the technique for applying pipeline techniques to recursive filters is shown and the advantages and disadvantages of the technique are discussed. Using these techniques, recursive digital filters operating at hitherto impossibly high rates can be designed.
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    Circuits, systems and signal processing 3 (1984), S. 295-314 
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    Notes: Abstract The long-standing problem of reconstructing a function from its samples is considered again. Assuming a sequence of oversampled values, a set of appropriate idealized reconstruction filters can be defined, which do not suffer from instability or slow convergence. The realization — a cascade of a nonrecursive digital filter, D/A-converter, and a fixed/analog smoothing filter — demands the design of the digital filter for the increase of the sampling rate. The design of this nonrecursive filter is the purpose of this paper. Approximations in the frequency as well as in the time domain are presented.
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    Circuits, systems and signal processing 3 (1984), S. 105-122 
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    Notes: Abstract In this paper we formulate power systems as nonlinear nearly Hamiltonian systems. Using the invariance principle for ordinary differential equations, necessary and sufficient conditions for asymptotic stability are established and a new method of estimating the domain of attraction of the stable equilibrium point is developed. The present results constitute a novel approach to stability analysis and involve the following three steps: a. Given a system with dissipation, the stability of its equilibrium is ascertained by determining the stability of the associated conservative system. b. Attractivity of the stable equilibrium of the entire system (with dissipation) is determined from the system topology. c. An estimate of the domain of attraction of the asymptotically stable equilibrium is obtained by making use of results obtained in (a) and (b). The stability criterion developed in this paper sheds new light on the mechanism of instability in power systems and it provides analytical verification to the concept of the potential-energy boundary surface (PEBS). The PEBS is a hypersurface which makes up a part of the boundary of the domain of attraction of the stable equilibrium in a power system. The existence and properties of the PEBS have thus far been deduced primarily via simulations and heuristic methods.
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    Circuits, systems and signal processing 3 (1984), S. 161-176 
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    Notes: Abstract In conventional television systems, picture scanning in vertical and temporal directions is usually very defective with regard to the sampling theorem. In this paper some deficiencies such as aliasing, line-structure distortion, line flickering, and large-area flickering are investigated with regard to their dependence on the inter-lacedpicture-scanning process. The three-dimensional reconstruction filtering of the sampled picture is especially analyzed with respect of the viewer's perception. Furthermore, it will be shown that in connection with a new concept of picture scanning published earlier [1], [2], a flat-field reproduction without any 25/30-Hz flicker can be achieved by vertical filtering only. This is true even though the final reproduction by the monitor is performed with interlace. The vertical filtering can then be optimized in the sense of maximum picture sharpness and resolution with negligible ringing as well. Practical results are given in this paper.
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    Circuits, systems and signal processing 3 (1984), S. 177-191 
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    Notes: Abstract In this paper a new type of “velocity-selecting/rejecting filter” which passes or stops a particular event in a seismic signal is proposed. The velocity-selecting filter is based on a time-space band-pass filter with sharp passband for a particular direction, and similarly, the velocity-rejecting filter is based on a time-space band-stop filter. A technique for designing such filters, in terms of an infinite-impulse-response (IIR) filter, is presented, in which a rotated version of separable filter is used. Finally, numerical examples are included to illustrate the design theory.
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    Circuits, systems and signal processing 3 (1984), S. 347-359 
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    Circuits, systems and signal processing 3 (1984), S. 315-325 
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    Notes: Abstract A special type of factorization for operators defined on partially ordered Hilbert resolution spaces is considered. The main result includes, as a particular case, the classical Schur-Coleski triangular factorization. Connections with stochastic optimization and image-processing problems are established.
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    Circuits, systems and signal processing 3 (1984), S. 409-417 
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    Notes: Abstract In a recent paper on nonlocal expansions necessary and sufficient conditions are given under whichf −1 has a generalized power series expansion, whenf is an invertible locally Lipschitz map between certain general subsets of a complex Banach space. Here we establish the validity of a conceptually interesting algorithm for obtaining the expansion. Basically, we show that a certain contraction mapping iteration generates iteratesℐ 1,ℐ 2,... such that eachℐ k yields all of the terms of the generalized power series expansion off −1 up to order (k + 1), assuming merely that the expansion off −1 exists. An earlier different result along related lines is mentioned.
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    Circuits, systems and signal processing 3 (1984), S. 447-475 
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    Notes: Abstract The representation of functions in a basis function expansionz(t)= ∑k=1/∞=,a k〉 x k (t) is straightforward when the basis functionsx k (t) are orthogonal. There has been very little work up to this time in determining how to use nonorthogonal bases in signal representation. On the other hand, applications in data compression and signal synthesis often require using specific tailor-made bases. Presented here is a method for constructing very general nonorthogonal bases. Orthogonality has often been used to show that a basis spans the set of functions of interest and to calculate the coefficients of the representation. In this paper, both of these fundamental aspects are addressed for nonorthogonal bases. A new basis {y k (t)} is obtained by performing a linear transformation on a known existing basis {x k (t)}. This transformation is constructed such that the coefficients of signal representation on the new basis are readily found. Then, a useful and sufficient condition is placed upon the new basis such that representations converge. The fundamental methods are applied to the standard examples of signal representation. The complex sinusoids, the Rademacher functions, the orthogonal polynomials, and the decaying exponentials are used as the original basis {x k (t)} from which a new basis {y k (t)} is generated. Two examples are given to illustrate general applications: one in signal synthesis and one in signal analysis.
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    Bulletin of mathematical biology 34 (1972), S. 1-12 
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    Notes: Abstract A method is described for estimating cell-cycle parameters from experimental fraction-of-labeled-mitoses measurements. The method is closely related to that of J. C. Barrett (1970) but is based on the analysis of Brockwell and Trucco (1970) which takes into account population growth in the calculation of theoreticalFLM-functions. Several sets of experimental data are analyzed, among them the data for the Marshall tumor considered by Barrett. It is found that population growth has a small but nevertheless detectable effect on the estimates of the cell parameters.
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    Bulletin of mathematical biology 34 (1972), S. 33-44 
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    Notes: Abstract The radioactivity disappearance curves of glucose-6-14C albumin-I131 after a single injection of tracer into a human subject have been determined in detail, particularly at early time intervals. The curves, expressed as sums of exponentials, have been analyzed as the infinite sum of convolutions of single passage time densities. The resultant transfer time distribution of a single circulatory pass allows examination of all delays in the system no matter how long they take. The structural detail evident by this means and the long mean time of a single pass of glucose (〉5 min) supports the thesis that factors other than rapid and uniform diffusion play a role in the extravascular movements of glucose molecules.
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    Bulletin of mathematical biology 34 (1972), S. 13-31 
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    Notes: Abstract A bisexual multiple branching process is studied. Consider a population with respect to three genotypes in both the female and male populations and let $$X(n) = \left\langle {X_1 (n), X_2 (n), X_3 (n)} \right\rangle and Y(n) = \left\langle {Y_1 (n), Y_2 (n), Y_3 (n)} \right\rangle$$ be random vectors giving the number of females and males (respectively) of each genotype in generationn. The mating of females and males is accommodated in the model withZ ij (n) representing the number of females of theith genotype mated with a male of thejth genotype in generationn. The mating system is such that a female may be mated to only one male but a male may be mated with more than one female. By arranging the nine random variablesZ ij (n),i, j=1, 2, 3, in a 1×9, vectorZ(n) it is shown that under certain conditions there is a positive constant ϱ such that when ϱ〉1 the vectorsZ n /ρn,X n /ρn andY n /ρn converge almost surely asn→∞ to random vectors with fixed directions. The paper is divided into four sections. In section 1 the model is described in detail and its potential applications to population genetics are discussed. In section 2, the generating function of the transition probabilities of theZ-process are derived. Section3 is devoted to the study of the limiting behavior of the first and second moments of theZ-process, and in section4 the results of section3 are utilized to study the behavior of the random vectorsZ(n),X(n) andY(n) asn→∞.
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    Bulletin of mathematical biology 34 (1972), S. 45-52 
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    Notes: Abstract Herein we show that the voltage-clamp current density at zero time calculated from electrodiffusion equations is linear in the clamping voltage for a simple membrane (no charge structure) and for a membrae with fixed charges. Such membranes are nonexcitable. Excitable membranes can be represented by a homogeneous membrane with dipole layers at the surface. In this case the initial current density will be linear in the clamping voltage if a critical field for a dipole layer reorientation is not passed through in changing from holding to clamping potential. Otherwise, deviation from nonlinearity may occur. This is in agreement with experimental data for the squid giant axon.
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    Bulletin of mathematical biology 34 (1972), S. 65-69 
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    Notes: Abstract By generalizing a previous paper on periodicities in the endocrine system (Bull. Math. Biophysics,30, 735–749, 1968), it is shown that nonperiodic, sporadic oscillations in the system are also possible. A procedure of describing the feedback mechanism between the endocrine system and the central nervous system is suggested. It is shown that the combined system: endocrine—CNS, also may show sporadic fluctuations.
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    Bulletin of mathematical biology 34 (1972), S. 79-86 
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    Notes: Abstract This paper deals with a mathematical attempt to determine the wall shear during normal flow of blood in the ascending and the descending thoracic aorta. A simple model is used, but the results obtained are in agreement with published experimental results for the descending thoracic aorta. It is suggested that the degree of fluctuation in the pressure gradient at a given station is the major factor in determining the level of wall shear at that point.
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    Bulletin of mathematical biology 34 (1972), S. 71-78 
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    Notes: Abstract A neural model based on a generalization of a model proposed in 1938 in the first edition of the author'sMathematical Biophysics (Chicago: Univ. of Chicago Press) is described. It possesses the property that due to some endogenous or exogenous stimulus, which may be of random nature, a pathway may suddenly begin to fire spontaneously. This spontaneous firing may either gradually spread over other pathways and eventually cease, or it may remain localized within one or a few pathways and then cease. Which of the two types of events occurs depends on the values of a number of parameters. The case of spreading reminds one of Jacksonian epilepsy.
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    Bulletin of mathematical biology 34 (1972), S. 93-102 
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    Notes: Abstract The diffusion equation is solved for a membrane-bounded sphere situated in an infinite medium with different diffusion properties. The formal solution is obtained through Laplace transformation in the time variable. It is not possible to find a closed form solution in terms of analytical functions, and therefore a numerical inversion technique is applied to obtain the final solution. The application on a biological problem is discussed.
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    Bulletin of mathematical biology 34 (1972), S. 87-92 
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    Notes: Abstract It is shown that the individual rate constants can be determined for the composite chemical system: $$A + B_i \rightleftarrows C_i ; i = 1...N$$ with only measurements of the unbound species,A(t), required. The dissociation rate constants can be determined by direct analysis of a single steady state tracer study. The association constants then follow from the analysis of stable equilibrium determinations reported earlier (Hart, 1965). An approximate solution when tracer methods are in-applicable is also given.
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    Bulletin of mathematical biology 34 (1972), S. 103-112 
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    Notes: Abstract Certain arrangements of enzymatic (bimolecular) subsystems lead to characteristic threshold-type response. Two simple cases of such systems are studied here in terms of steady state behavior and explicit relationships between system and curve parameters. It is found that the curvature of the threshold curve is directly related to the equivalent Michaelis constant and, in the case of saturated threshold curve, the slope of the curve at the idealized threshold is limited by the ratio of saturation to threshold. This slope may be appreciably increased up to a stepwise response at the threshold if a multisubstrate complex of the enzyme is the only species which affects the enzyme mediated transport.
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