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  • 1
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    Bulletin of mathematical biology 20 (1958), S. 71-93 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A somewhat different approach to the principle of biotopological mapping, discussed in previous publications, is given. The organism is considered as a set of properties, each of which is in its turn a set of numerous subproperties which are logically included in the corresponding properties. Topology is introduced by an appropriate definition of neighborhoods, and four postulates are stated which concern the mapping of the spaces corresponding to higher organisms on those of lower ones. A number of conclusions are drawn from the postulates. Some of them correspond to well-known facts. For example, in man and some higher organisms appropriate emotional stimuli should produce gastrointestinal or cardiovascular disturbances; or some microorganisms should produce substances harmful to other microorganisms (antibiotics). Some other conclusions are still awaiting verification. One of them is, for example, that there must exist unicellular organisms which produce antibodies to appropriate antigens.
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  • 2
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    Bulletin of mathematical biology 20 (1958), S. 25-32 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Zusammenfassung Für die Praxis der Pflanzenernährung ist es wichtig, zu wissen, in welcher Weise die Ertragsbildung von der Konzentration eines mineralischen Nährstoffes in der Umgebung der Pflanze abhängt. Da nur diejenigen Nährstoffmengen das physiologische Geschehen in der Pflanze unmittelbar zu beeinflussen vermögen, die sich in der Pflanze befinden, wird angenommen, dass das Wachstum zum Zeitpunktt, d.h. die Geschwindigkeit der Trockensubstanzzunahme zu diesem Zeitpunkt, eine Funktion der zur Zeitt in der Pflanze enthaltenen Nährstoffmenge ist. Diese Nährstoffmenge wird natürlich im Intervall vor dem Zeitpunktt aufgenommen. Deshalb und auch noch aus anderen Gründen hängt das Wachstum zur Zeitt davon ab, wie die in der Umgebung der Pflanze herrschende Konzentration des betrachteten Nährstoffes in demjenigen Zeitintervall verläuft, das sich von der Aussaat bis zum Zeitpunktt erstreckt. Die angegebene Annahme fürhrt zusammen mit einigen weiteren naheliegenden Annahmen zu einem Ansatz, der Ergebnisse liefert, die in verschiedener Hinsicht gut mit der Erfahrung übereinstimmen. Jedoch gibt es auch noch Widersprüche zwischen Theorie und Erfahrung. Durch weitere Ausgestaltung der Theorie lassen sich diese Widersprüche beseitigen. Es wird angeregt, Versuche durchzuführen, deren Resultate Hinweise für die weitere Ausgestaltung der Theorie liefern.
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    Bulletin of mathematical biology 20 (1958), S. 33-70 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The dynamics of cell multiplication and differentiation in tissues in asteady state and the kinetics of isotope incorporation into the DNA have been theoretically analyzed. Equations have been derived, with the aid of which thegeneration time, thelife span, and the distribution or rate of death of the cells can be obtained if the tissue is in asteady state, i.e., if the number of cells is maintained constant by constant, equal rates of cell division and cell death and if the mean DNA content per cell is also constant. An equation has also been derived which gives thegeneration time in the case of logarithmic multiplication of cells. Two special cases have been analyzed: InCase 1, the isotope is considered as being introduced into the metabolic system at zero time only; inCase 2, the specific activity of the DNA precursor is considered as being maintained constant. The use of the method has been illustrated by an example in which thegeneration time and themean, themedian, and themode life span, as well as the curve of the rate of death of leukocytes in a patient with chronic leukemic granulocytic leukemia, have been obtained from the rate of P32 incorporation into the DNA. The merits and the limitations of the method are discussed.
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    Bulletin of mathematical biology 20 (1958), S. 95-95 
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    Bulletin of mathematical biology 37 (1975), S. 37-49 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The chromosomal theory of inbreeding based on a gametic interaction system lead us to define a depression coefficientD. Comparison of random, sib and half-sib matings (with inbreeding coefficientF=0, 1/4 and 1/8) shows thatD depends on the structure of the starting population and on values of the model parameters. This result accounts for responses of lines whose depression does not depend directly on the inbreeding coefficient and which theories of inbreeding based on increasing homozygosity fail to explain.
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    Bulletin of mathematical biology 37 (1975), S. 59-69 
    ISSN: 1522-9602
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    Notes: Abstract An idealization of chemical combination is formulated as a model of computability, and it is shown that this model has universal computational power just in case assembly has at least two-dimensional space in which to occur. It is also shown that this model, under reinterpretation, corresponds to a cellular automaton in which growth occurs by differentiation only (i.e., the state into which any cell is born is thereadfter fixed). Hence this latter model of growth is also computationally universal.
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  • 7
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    Notes: Abstract Kinetics of biological light emission processes do not mean what they seem to mean, because measured light intensity is not proportional to reactant concentration but to reaction rate. Therefore, the differential equation for light decay is usually different from that of concentration decay, so that mass action interpretations cannot be applied directly to light intensity decay. An observed second order light decay for Chlorella at 6.5°C, implies Elovich solid state reaction kinetics, which agrees with other evidence for solid state processes in photosynthesis. An observed 1.5 order light decay for Cholorella at 28°C implies second order liquid or solid state reaction kinetics. First ordere light decay implies first order reaction kinetics.
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    Bulletin of mathematical biology 37 (1975), S. 71-78 
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    Notes: Abstract Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ R θ(x 1,x 2,t) dx 1 dx 2 of a biological species with an area-density distribution function θ=θ(x 1,x 2,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇2θ +fθ −gθ n+1 whereD (〉0),n (〉0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(x 1,x 2, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn〉1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ R θ(x 1,x 2, 0)2 dx 1 dx 2 sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.
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    Bulletin of mathematical biology 37 (1975), S. 127-138 
    ISSN: 1522-9602
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    Notes: Abstract The equilibrium probability distribution of the process level is studied for a general class of reversible stochastic reactions. A calculationally convenient approximation for equilibrium probabilities is derived and its accuracy is investigated over a range of values of the equilibrium constant. A method of approximating the equilibrium means and variance is developed and illustrated forQ th-order processes.
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    Bulletin of mathematical biology 37 (1975), S. 565-572 
    ISSN: 1522-9602
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    Notes: Abstract Beside the concept of material inputs and outputs of components of the representation of biological systems given to us by Rosen, the concept of energy is incorporated. The interaction of material and energy is represented by a cartesian product; and separate material and energetical mappings are considered as the new representation of components. These developments generate aMα category, and it is shown thatMα is isomorphic to theM category of previous developments.
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    Bulletin of mathematical biology 37 (1975), S. 555-564 
    ISSN: 1522-9602
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    Notes: Abstract This paper discusses the solution of a generaln-compartment system with time dependent transition probabilities utilizing the technique described by Cardenas and Matis (1975) (hereafter abbreviated (CM)). In addition, the cumulant generating function is derived for a special class of reversiblen-compartment systems where the time-dependent intensity coefficients corresponding to the migration and death rates are some multiple of each other. The immigration rates can be any integrable function of time. The moments are also obtained and the solution to the two-compartment system is presented explicitly. The solution is illustrated with a linear and a periodic function which forms have been widely reported in the literature.
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    Bulletin of mathematical biology 37 (1975), S. 573-588 
    ISSN: 1522-9602
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    Notes: Abstract The relations (inflow) = (dose)/(area under indicator curve), and (volume of distribution) = (throughflow) × (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem, is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution.
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    Bulletin of mathematical biology 37 (1975), S. 219-219 
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    Bulletin of mathematical biology 37 (1975), S. 291-299 
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    Notes: Abstract Perturbation methods are applied to a differential equation predator-prey model to find the approximate amplitudes and period of limit cycle solutions. In the model the feeding rate per unit predator per unit prey decreases as the prey become scare. The rigorous applicability of the perturbation technique depends on the assumptions that the limit cycle amplitude is relatively small and that near the equilibrium point the growth rate of each species is most sensitive to changes in the density of the other species. The second assumption is usually roughly satisfied in practice and examples are considered which suggest that the first assumption can be greatly relaxed.
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    Bulletin of mathematical biology 37 (1975), S. 367-387 
    ISSN: 1522-9602
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    Notes: Abstract Signal Detection Theory can be used to provide a mathematical model describing the choice of a predator trying to distinguish between a model and a Batesian mimic. The mathematical model yields a number of a deductions, in particular that it may or may not assist the mimic population if mimics more closely resemble their models. The assumptions underlying the analysis are discussed in some detail.
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    Bulletin of mathematical biology 37 (1975), S. 419-425 
    ISSN: 1522-9602
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    Notes: Abstract A new type of physical transition, denotedS→S *, has been detected in irradiated organic molecules (λ=546 nm) through their interaction with specific biological macromolecules. In a specific enzyme-substrate interaction, a clear enhancement of the reaction rate is observed, when the substrate is irradiated with sharply well defined times. These “efficient irradiation times” are always of the 5k sec type (k=1, 2, 3, …). They have been consistently revealed in a great number of specific biological interactions. The present note demonstrates an important property, i.e. that forevery irradiation time aS→S * transition is induced in organic molecules. It is shown that for any irradiation times different from the 5k sec type (k=1, 2, 3, …) states of theS * type may occur, but the biological macromolecules may “detect” only theS * states induced by irradiations of the 5k sec type.
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    Bulletin of mathematical biology 37 (1975), S. 459-470 
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    Notes: Abstract A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. It envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. It has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematocrit. The wall layer thickness is found to decrease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the case of blood may be attributed to the deformability of the erythrocytes.
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    Bulletin of mathematical biology 37 (1975), S. 489-504 
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    Notes: Abstract Three dimensional laminar, viscid flow is developed for Newtonian fluids which provides absolute values for axial, radial and tangential velocity fields everywhere if the dimensions of the vessel are known and two simultaneous axial velocities e.g. on and off the central axis in the same plane, and the central axis axial velocity gradient are measured. In addition, normal and shear stresses are determinable. The equation set satisfies geometric and other known flow limiting conditions such as no slip at surfaces etc. and are amenable for inclusion in general, dynamic flow expressions. Alternatively they may be used alone for certain problems involving gradients and secondary flows. A range of illustrations are shown for a distorting vessel with elliptic cross-section and small axial taper (analogous to the pulmonary trunk during ejection).
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    Bulletin of mathematical biology 37 (1975), S. 521-553 
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    Notes: Abstract A regulated left ventricular dynamics model is presented which involves interaction of the dynamics of the left ventricular and circulatory systems and their regulation by the central nervous system. On-line human parametric simulation (parameter estimation) and consequential prognostic implications (based on parametric values) are demonstrated. Model responses to simulated physiologic stresses help delineate tolerances of subjects. In order to have an estimate of the reliability of the model, the sensitivity of the model's responses to changes in the values of its intrinsic parameters is assessed. Also determined is the extent to which errors in measuring the pressure affect the calculated values of the model's simulation parameters and subsequently influence the values of other diagnostically useful variables (such as contractility, oxygen consumption rate, heart rate), when the model is used to determine the limiting physiological stress sustainable by the subject. A comparison of the model's composition with those of other similar cardio-circulatory models is included.
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    Bulletin of mathematical biology 37 (1975), S. 659-673 
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    Notes: Abstract Then-stage harvesting strategy of Elizarov and Svirezhev is examined. As a result, some important new features appear. A discussion is presented on whether or not one should harvest a species at one time stage or wait until a later time. The paper is concerned with contributions which are primarily mathematical formulations and results for continuous, as well as discrete time, logistic growth of a single species being harvested. Age class structure is ignored.
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    Bulletin of mathematical biology 37 (1975), S. 97-100 
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    Bulletin of mathematical biology 37 (1975), S. 1-9 
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    Notes: Abstract The study of systems exhibiting a band-pass function is completed for systems whose parameters are time-dependent. In the case of periodic parametric excitations, it is demonstrated that some systems can get into “resonance” for a particular frequency. By studying this problem, a new and probably fruitful approach of some rhythmic behaviours can be made.
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    Bulletin of mathematical biology 37 (1975), S. 19-35 
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    Notes: Abstract Analytical techniques are developed which permit objective control of asiist device driving systems. In addition to being objective, the techniques described in this paper are optimal in the sense of minimizing a performance index which consists of a term involving left ventricular power and a term involving deviations of aorta hemodynamic parameters from normal values. Comparisons are included of off-line computations and measurements on dogs with experimentally induced myocardial infarctions undergoing intraaortic balloon pumping.
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    Bulletin of mathematical biology 37 (1975), S. 427-458 
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    Notes: Abstract A mathematical model simulating a cell growing in a culture medium is obtained. Using this model, various behavioral patterns of the cell are obtained under different types of disturbances, in particular when (i) a Mg2+ deficiency experiment and, (ii) a split-dose ionizing radiation experiment are carried out, (iii) when disturbances on the rate constants of the biochemical reactions taking place in the nucleus of the cell are applied, and (iv) when the cell's interior components are perturbed. The cell model results obtained agree well with experimental results for the Mg2+ and split dose experiments, and explain the mechanism of the split dose radiation experiment without the need to introduce additional axioms (e.g. healing processes) into the dynamics of the cell. Conditions are obtained which cause the cell to behave in a rapidly growing ‘tumor-like’ mode; it is shown that once the cell moves into this ‘tumor-like’ mode, its behavior is irreversible, i.e. if a disturbance of opposite type is then applied to the ‘tumor’ cell, the cell will not revert back to its original normal behavior.
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    Bulletin of mathematical biology 37 (1975), S. 85-90 
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    Notes: Abstract Solution of the equation that describes simulatenous liquid flow and diffusion in a spherical model of the vitreous body of the eye shows that a small dissolved specie can move both anteriorly and posteriorly from a source behind the lens even though there is a slow liquid movement almost entirely in the posterior direction. This results explains why tracer studies using large particles (dyes or colloids) show only a posterior flow, whereas studies using sodium ion show anterior movement as well.
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    Bulletin of mathematical biology 37 (1975), S. 101-107 
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    Bulletin of mathematical biology 37 (1975), S. 111-111 
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    Bulletin of mathematical biology 37 (1975), S. 221-221 
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    Bulletin of mathematical biology 37 (1975), S. 255-268 
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    Notes: Abstract The equations relating hybridized RNA to free RNA, in the case of simple hybridization, or to ratio of labelled and unlabelled RNA in competitive hybridization, are derived. Analysis of the equations shows how hybridization data may be used to infer properties of the distribution of components in an RNA mixture, or the relation between two distributions in competitive hybridization. A critical examination of the assumptions underlying the equations indicates that some of then may be violated in certain cases, or have no current support, evidential or theoretical. The consequences of such qualifications for the interpretation of hybridization data are indicated.
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    Bulletin of mathematical biology 37 (1975), S. 589-636 
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    Notes: Abstract The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.
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    Bulletin of mathematical biology 20 (1958), S. 1-23 
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    Notes: Abstract To account for some of the more important aspects of drug interaction we shall consider a model which can also account for certain general properties of the action of a single drug. A simple model in which there may be enzymatic detoxification of a drug is studied theoretically. The relation between time for appearance of an effect due to the drug and the size of the dose is found to contain the same parameters as the relation between the effectiveness of paired doses and the interval of time between doses. A similar situation holds when the drug is given at a constant rate. When two drugs are administered together, their effect will depend on the manner of interaction, how much of each drug is given, which is given first, and on the interval of time between each administration. A number of plausible types of interaction is considered theoretically in terms of the model, analytical expressions being given for a number of cases. The interaction may be synergistic or antagonistic. In the former case the potentiation may be more than or less than additive depending on the order of delivery and on the time between injections. Methods for the estimation of the parameters from data are discussed.
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    Bulletin of mathematical biology 37 (1975), S. 109-109 
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    Bulletin of mathematical biology 37 (1975), S. i 
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    Bulletin of mathematical biology 37 (1975), S. 139-146 
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    Notes: Abstract Krylov-Bogoliubov-Mitropolsky perturbation method was used to study the effect of nonlinearity in the Volterra-gause-Witt (VGW) model for a two species prey-predator system. The first order corrections to both the frequency of oscillation and the amplitude of the linearized system were computed. It was found that the basic qualitative features of the nonlinearity are exhibited by the first order result. We have also discussed the Lotka-Volterra problem which is a special case of VGW model.
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    Bulletin of mathematical biology 37 (1975), S. 147-160 
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    Notes: Abstract This paper presents a mathematical analysis of a tumor model first proposed by Skipper and Zubrod. The tumor model is comprised of three compartments, a proliferative compartment, a nonproliferative but viable compartment, and a dead compartment. By the suitable selection of functions describing loss of cells from the proliferative and nonproliferative compartments, the model is capable of describing tumor behavior during periods of growth and drug treatment. The loss functions during treatment are related to pharmacokinetic functions and may be chosen according to known drug properties. Tumor properties may be simulated by the appropriate choice of cell cycle parameters. It therefore seems feasible to simulate tumor behavior for scheduled treatment with chemotherapeutic agents. Another important result of this analysis is the derivation of a fraction labelled mitosis function which incorporates the nonproliferative compartments.
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    Bulletin of mathematical biology 37 (1975), S. 161-180 
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    Notes: Abstract Techniques of modelling and simulation are discussed as they relate to bioengineering systems. The advantages and disadvantages of different analytical engineering methods utilized to gather information concerning the behavior of complex physiological and neuromuscular control mechanisms are explained. An Inners Criterion is developed to determine if the roots of a model lie within a certain “biologically realistic region” ΓB, in the complex plane which contains the roots of linearized models for a large variety of neuromuscular systems. Several algorithmic methods based on the Jury Inners Test are described which specify whether the model roots lie within the desired region, thereby providing an indication as to the validity of the proposed model. This technique can help to eliminate tedious simulation on an unrealistic model with roots lying far outside this region. An exemplary model for control of vergence eye movements is presented and shown to satisfy the ΓB criterion; several counter-examples are also discussed. The Inners approach can be adapted to other classes of bioengineering systems by specifying the region based on models that are contained in the class of interest.
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    Bulletin of mathematical biology 37 (1975), S. 215-218 
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    Bulletin of mathematical biology 37 (1975), S. 220-220 
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    Bulletin of mathematical biology 37 (1975), S. 223-254 
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    Notes: Abstract In this paper I consider how information is required to specify various systems. It is shown that the transitive information of any physical system, is distributed among three distinct components. One of these, the selective component, is required to specify the elemental parts of the system. Another, the connective component, is required to specify the macrostructure of the system; that is the way the parts are put together. And a third, the conformative component, is required to specify the intrinsic complexion or microstructure of the system. An interesting method for analyzing branched systems which takes account of connective ambiguity is described in some detail. The relationship between information and entropy, known as the Clausius-Shannon Identity, is then discussed with reference to selected thermodynamic models: and that aspect of information which is often overlooked, namely the distinction, between transitive and intransitive information is highlighted. The applications (or perhaps more correctly, the limitations of applying this treatment) to problems of biological interest are also indicated.
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    Bulletin of mathematical biology 37 (1975), S. 269-275 
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    Notes: Abstract This paper discusses two compartment models with interaction allowed between the compartments. The total number of particles in the system at any time is discussed along with the number to the found in each separate compartment. An interesting result is that the number of particles in each of the two compartments areindependent random variables. Some asymptotic results are also given. The paper is a continuation of some earlier work by the author.
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    Bulletin of mathematical biology 37 (1975), S. 277-289 
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    Notes: Abstract General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ∂c i /∂t =D i ∇2 c i +Q i (c) subject to boundary conditions of practical interest, where the enumerator indexi runsl ton, c i =c i (x,t) denotes the concentration or density of theith participating molecular or biological species,D i is the diffusivity constant for theith species, andQ i (c), an algebraic function of then-tuplec=(c 1,...,c n ), expresses the local rate of production of theith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals ofc which decrease monotonically with time can often be found, as exemplified here for Volterra and Verhulst-Volterran-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reaction-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, $$\mathop {\lim }\limits_{t \to \infty } c_i (x, t) = \hat c_i (x)$$ , if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl ofQ(c) inc-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of “weak diffusion” with the reaction rate terms dominant in the governing equations.
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    Bulletin of mathematical biology 37 (1975), S. 323-365 
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    Notes: Abstract A model nonlinear network involving chemical reactions and diffusion is studied. The time evolution and bounds on the steady state solutions are analyzed. Spatially ordered solutions of the equations of the dissipative structure type are found by bifurcation theory. These solutions are calculated analytically and their qualitative properties are discussed.
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    Bulletin of mathematical biology 37 (1975), S. 637-657 
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    Notes: Abstract Models based on molecular mechanisms are presented for pattern formation in developing organisms. It is assumed that there exists a diffusion governed gradient in the morphogenetic field. It is shown that cellular differentiation and the subsequent pattern formation result from the interaction of the diffusing morphogen with the genetic regulatory mechanism of cells. In a second stage it is shown that starting from a homogeneous distribution of morphogen, polarity can be generated spontaneously in the morphogenetic field giving rise to the establishment of a gradient. The stability of these gradients is demonstrated. The onset of a morphogenetic gradient and pattern formation are combined in a single coherent model. Size invariance and its biological implications are discussed.
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    Bulletin of mathematical biology 37 (1975), S. 11-17 
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    Notes: Abstract A simple population model consisting of one adult and two larval stages with cannibalism or competition among the larval stages is presented. The solutions are found to be either periodic or of a steady state nature depending on the ratios of fertility and cannibalism among the larvae. Two similar cannibalism pressure functions are compared and the conditions that lead to steady or periodic solutions, or to extinction, are examined.
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    Bulletin of mathematical biology 37 (1975), S. 301-321 
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    Notes: Abstract Thyroid hormone synthesis shows two important characteristics at each iodine organification step: iodine fixation to thyroglobulin tyrosyl residues in the vicinity of the thyroid cell microvilli, and the storage of thyroglobulin in the follicular lumen. In order to study the influence of kinetic parameters (chemical reactions, diffusion coefficient) and, of the structure (follicular radius and reactional space width) we have determined the impulsional response of a diffusion sphere exchanging matter with a compartment where the chemical reactions are taking place. This study gives the starting point of a mathematical model of hormonal secretion by a thyroid follicle. Moreover it suggests a simple way of estimating the thyroglobulin diffusion coefficient by the mean transit time determination. Finally, we discuss respectively the validity limits of a compartmental description of the model, and of a continuous description by an infinite sum of exponentials of a system where chemical reactions interfere with a storage process by diffusion.
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    Bulletin of mathematical biology 37 (1975), S. 389-405 
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    Notes: Abstract In order to represent the biological evolution of a predator-prey ecology it is necessary to add to the equations of population dynamics terms corresponding to spontaneous mutation. Using a Volterra-Lotka ecology as an example, a model is developed for this. It is based on the assumption of two levels of description; a local one containing mutation probabilities, and the other the macroscopic average equations for the whole system. Diffusion processes link the two. The “evolutionary state” of a species is interpreted as an average effectiveness in terms of a genetic parameter space and it is shown that as a result of random mutations the ecosystem drifts irreversibly through this space.
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    Bulletin of mathematical biology 37 (1975), S. 407-417 
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    Notes: Abstract A number of recent experiments have revealed the existence of mutants with different free run periods in their circadian rhythms. Parameter variations in mathematical models can be used to simulate such changes. In addition, phase response curves (PRC) are derived and the effect of parameter variation in their shape is studied. It is shown that changes in global parameters can also distort their shape. Therefore one cannot conclude that genetic experiments provide evidence in favor of “chronon” models since “kinetic” models can also simulate their outcome.
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    Bulletin of mathematical biology 37 (1975), S. 51-57 
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    Notes: Abstract A methematical description of a coupling between a chemical reaction, the diffusion through a cellular membrane and a fluid flow is presented. This coupling may occur at the membrane levels of the cells bathed by fluid flows (e.g. endothelial cells). By such a coupling, the fluid flow and the diffusion can act as drivers of some intracellular endergonic reactions.
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    Bulletin of mathematical biology 37 (1975), S. 91-95 
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    Notes: Abstract The classical epidemic equations of Kermack and McKendrick are cast into dimensionless form. This allows discussion of the assumptions underlying the standard approximate solutions.
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    Bulletin of mathematical biology 37 (1975), S. 471-488 
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    Notes: Abstract A study is made of blood flow by assuming that the blood constitutes a suspension of cells in plasma instead of a simple homogeneous fluid. A macroscopic theory governing the motion of plasma in a plasma-cell system is derived from the local volume averaging method for a system without mass transfer between the phases, and its characteristic length is much larger than the size of the cells. The equations governing the motion of the local averaged fluid quantities include one additional term in the equation of motion and two additional terms in the energy equation. These terms represent, respectively, the force exerted upon the fluid by the particles, and the rate of heat transfer and work done upon the fluid by the particles. The theory is applied to obtain the effective viscosity as the explicit function of the volume concentration of the cells by assuming that the cells behave like rigid spherical particles with slip-collision, and the plasma is an incompressible Newtonian fluid. Comparison with existing experimental results shows a good agreement. The theory is also used to obtain the effects of cell distribution upon the overall effective viscosity in a circular tube. The quantitative result shows that there is a decrease in overall effective viscosity as the concentration of cells increases toward the center of the tube, and the overall effective viscosity is smaller than the flow with evenly distributed cells.
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    Bulletin of mathematical biology 37 (1975), S. 505-519 
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    Notes: Abstract The distribution of the two-compartment, reversible system with time-dependent transitions is proposed and verified. Inasmuch as the required probabilities cannot, in general, be expressed in closed form, a method of approximating these probabilities is described. An example with specific inverse functions of time is presented.
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    Bulletin of mathematical biology 37 (1975), S. 113-126 
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    Notes: Abstract The airway system of the lung from the mouth to the pulmonary membrane is modelled by matching a cylindrical model of a pathway through the respiratory region of the lung onto a one-dimensional trumpet model for the conducting airways. The concentration of O2 in gas expired from this model airway system is investigated following an inspiration of air at two different flow rates (10 litres/min and 85 litres/min). In each case, expiration occurs at the same constant flow rate as that during the previous inspiration. The inspirations, which are studied in an earlier paper, are each of 2 sec duration and begin at a lung volume of 2300 ml and a lung oxygen tension of 98 mm Hg. The equations are solved numerically and plots of expired O2 concentration against time and against expired volume are shown. It is found that at 85 litres/min, gas mixing in the lung is complete after about 0.7 sec of expiration whereas at 10 litres/min, about 2.6 sec of expiration is required for complete equilibration. It is suggested that the experimental alveolar plateau slope is not in general caused by a slow approach to equilibrium of gas concentrations; except at very low flow rates in the early part of the concentration/time plateau.
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    Bulletin of mathematical biology 37 (1975), S. 181-192 
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    Notes: Abstract On the basis of an abstract, simple bistable reaction system (‘homogeneous Eccles-Jordan trigger’) used as anRS flip-flop, an abstract homogeneousastable flip-flop is devised. It can be run also as amonostable flip-flop and as aT flip-flop. The qualitative behavior of the three systems can be understood, in the limiting case, with the aid of Poincaré's notion of bifurcation of steady states. The reaction system is proposed as a paradigm for a specific class of ‘decomposable’ chemical and dynamical systems (so-called DC-type dynamical automata). Two possible biological applications are mentioned.
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    Bulletin of mathematical biology 37 (1975), S. 193-213 
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    Notes: Abstract The stochastic theory of a nonlinear game is presented which incorporates some of the essential properties of living systems: metabolism, reproduction and mutability. The steady state distribution function as well as the complete time development are given explicitly. The second law of thermodynamics is generalized to a certain class of nonequilibrium systems. An order parameter is introduced as a measure of the system's internal organization. From the point of view of phase transition theory, the model exhibits a transition at the absolute zero of temperature, with critical behaviour showing up in the low temperature region.
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    Bulletin of mathematical biology 37 (1975), S. 675-689 
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    Notes: Abstract A theory for environmental systems is defined on the basis of two elements, termed ‘environmental unity’ and ‘behavior’. Environmental systems are regarded as non-living systems, each one related with only one biological system. We construct a material-energetic environmental diagram, which is introduced in terms of the theory of categories, thereby giving rise to a new categoryE. By means of two biological conditions, and the definition of static property of the biological system (related to its own environment), a set of theorems is obtained, exhibiting mathematical consequences for the represented theory.
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    European journal of wildlife research 21 (1975), S. 15-34 
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    European journal of wildlife research 21 (1975), S. 81-81 
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    European journal of wildlife research 21 (1975), S. 82-82 
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    European journal of wildlife research 21 (1975), S. 139-144 
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    European journal of wildlife research 21 (1975), S. 192-194 
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    European journal of wildlife research 21 (1975), S. 194-194 
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    European journal of wildlife research 21 (1975), S. 182-190 
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    Description / Table of Contents: Summary The results are presented in a survey of peeling by mouflon in spruce stands in the former mouflon research district Padberg, county of Hochsauerland, of the Research Station for Hunting Lore and Wildlife Damage Prevention. Observations were made from 1958 to 1970. Peeling in winter reached a peak in the extreme winter of 1962/63 when 8% of the trees were peeled. Summer peeling was considerably less, usually about 10% of the total peeling. Based on total peeling, root peeling was about 15% to 25% in winter, while the value for the dry summer of 1959 was 60%, based on the summer peeling per trunk. Mouflon peeled spruce stands over 30 years old more than those under 30. Further, little peeling of beech by Mouflon was proven. Of the tested anti-peeling preparations, chemical ones were effective on beech and spruce and mechanical protection in the form of a dry band for spruce. The protection was effective over a period of five or more years. The mechanical-biological protection, a sort of smoothing, reduced peeling up to 80% and more, compared to untreated trees, as far as the experiments are evaluated in which unprotected trees were peeled 3% and more. One stand of spruce, 18 years old, was badly damaged by the Flammigersche protective method; 12 trees died. The smoothing method caused only minor bark damage, which can be tolerated since this method is quite inexpensive.
    Abstract: Résumé Il est rendu compte de relevés d'écorcements causés par le Mouflon dans l'ancien territoire de chasse expérimental (Station de Recherches Cynégétiques et de Prévention des Dégâts de Gibier) de Padberg (Haut Sauerland). La période d'observation s'étendit de 1958 à 1970. L'écorcement d'hiver atteignit sa valeur la plus élevée au cours de l'hiver exceptionnel de 1962–63 avec un taux de 8% de pieds touchés. L'écorcement d'été, par contre, ne présenta généralement pas un caractère aigu en se situant aux environs ou en-dessous de 10% de la totalité des pieds écorcés au niveau du tronc. Par rapport à l'ensemble des écorcements causés aux troncs, l'écorcement causé aux racines traçantes au cours de l'hiver s'est situé entre 15 et 25% tandis qu'au cours de l'été sec de 1959, ce type d'écorcement s'est élevé à 60% des écorcements d'été causés aux troncs. Le Mouflon écorça plus intensivement les peuplements d'Epicéa de plus de trente ans que ceux de moins de 30 ans. Quelques écorcements d'été au Hêtre furent également signalés. Parmi les répulsifs expérimentés, les produits chimiques (pour l'Epicéa et le Hêtre) et la protection mécanique au moyen d'un manchon de ramilles sèches (pour l'Epicéa) se sont avérés pleinement efficaces tout au long d'une période de cinq ans et plus. Le traitement mécanique-biologique au moyen d'un rabot réduisit l'écorcement des pieds traités par rapport aux pieds non-traités dans une proportion de 80% et plus, l'écorcement concernant 3% des pieds non-traités. Un peuplement d'Epicéa traité à l'âge de 18 ans au moyen du grattoir de Flammig fut fortement endommagé par le traitement en question et 12 arbres succombèrent. Le traitement au rabot ne causa que des dommages faibles à modérés à l'écorce; ils ne portèrent pas à conséquence en raison du coût avantageux du traitement mécanique-biologique.
    Notes: Zusammenfassung Mitgeteilt werden Auszählergebnisse für das Schälen durch Muffelwild in Fichtenbeständen in dem ehemaligen Muffelwildversuchsrevier Padberg, Hochsauerlandkreis, der Forschungsstelle für Jagdkunde und Wildschadenverhütung. Der Beobachtungszeitraum erstreckte sich von 1958 bis 1970. Die Winterschäle erreichte mit 8% geschälter Stämme in dem Extremwinter 1962/63 den Höchstwert. Die Sommerschäle trat demgegenüber meist zurück und bewegte sich um oder unter 10% der Gesamtstammschäle. Bezogen auf die Gesamtstammschäle lag die Wurzelschäle im Winter im Bereich von 15% bis 25%, der Wert für den trockenen Sommer 1959 lag bei 60% bezogen auf die Sommerschäle am Stamm. Das Muffelwild schälte über 30 Jahre alte Fichtenbestände stärker als unter 30 Jahre alte. Nachgewiesen wurden weiterhin geringe Sommerschälschäden durch Muffelwild an Buche. Von den erprobten Schälschutzmitteln zeigten die chemischen Präparate bei der Buche und Fichte und der mechanische Schälschutz in Form des Trockeneinbandes bei der Fichte über einen Zeitraum von fünf und mehr Jahren eine volle Abwehrwirkung. Der mechanisch-biologische Schälschutz, als Hobelverfahren durchgeführt, minderte die Muffelwildschäle gegenüber unbehandelt um 80% und mehr, soweit die Versuche gewertet werden, bei denen an ungeschützten Stämmen 3% und mehr Schäle auftrat. Ein im Alter von 18 Jahren mit Flammigerschen Schutzkratzer behandelter Fichtenbestand wurde durch das Schutzverfahren stark geschädigt, 12 Stämme starben ab. Das Hobelverfahren brachte nur geringe bis mäßige Rindenschäden, die zum Teil wegen der Billigkeit des mechanisch-biologischen Schutzes in Kauf genommen werden könnten.
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    European journal of wildlife research 3 (1957), S. 32-39 
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    Description / Table of Contents: Summary Investigations proved that damages caused in the fields by red deer, fallow deer, and wild boars reach their culmination during certain summer months. Winter damages are caused only by wild boars in the potatoe fields of the year before now sown with rye. The question of the number of nights of damaging following one another could be cleared for the area of investigation. With red and fallow deer likewise a quick decrease of field damages could be observed with the beginning of ruttishness. A noticeable decrease of winter field damages caused by wild boars could not be observed during ruttishness. The opinion stated up to now that wandering fallow deer would not cause concentric field damages did not prove true within the district of observation.
    Abstract: Résumé Les recherches ont prouvé que les dégâts causés dans les champs par les bêtes fauves, les daims er les bêtes noires arrivent à leur point culminant pendant certain mois d'été. Dégâts causés dans les champs hibernales sont uniquement produits par les bêtes noires sur soles de pommes de terre de l'année précédente dont la semaille fut faite avec seigle. La question après le nombre des nuits se succédant pouvait être éclaircie pour la domaine d'observation. Une diminution rapide des dégâts causés dans les champs et produits par les bêtes fauves et les daims pouvait être observé également pour tous deux. Par contre il était impossible de constater une diminution sensible des dégâts causés dans les champs hibernales et produits par les bêtes noires pendant le temps de leur ivresse. L'opinion soutenue jusqu'ici que le daim vagabond ne causera pas de dégâts concentrés dans les champs ne se trouvera pas justifié pour le district d'observation.
    Notes: Zusammenfassung Durch Untersuchungen wurde bewiesen, daß die Feldschäden von Rot-, Dam- und Schwarzwild in bestimmten Sommermonaten ihren Höhepunkt erreichen. Winterfeldschäden werden nur durch Schwarzwild auf den mit Roggen bestellten vorjährigen Kartoffelschlägen verursacht. Die Frage nach der Anzahl der aufeinanderfolgenden Schadensnächte konnte für das Untersuchungsgebiet geklärt werden. Bei Rot- und Damwild wurde übereinstimmend mit Beginn der Brunft ein schnelles Absinken der Feldschäden beobachtet. Ein merkbarer Rückgang der Winterfeldschäden durch Schwarzwild während der Rauschzeit konnte nicht festgestellt werden. Die bisher vertretene Ansicht, daß durch das unstete Damwild keine konzentrierten Feldschäden verursacht werden, erwies sich für das Beobachtungsrevier als nicht zutreffend.
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    European journal of wildlife research 3 (1957), S. 154-159 
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    Description / Table of Contents: Summary Although Mr.Gruschwitz chooses a percentage of increase of 70%, he reaches in his examples only 54%, because he relates the percentage to the does on March 31 st and not on June 1 st. The sexual ratio is not the ratio in numbers of the pubescent animals in summer, but the ratio of all male and female animals. It is necessary to make a difference between this ratio in March and in June. A large sexual ratio has to be reduced essentially to the lack of males, the cause being a much too heavy killing off of males. The insufficient killing of females is of importance, too. Measure and means of each interference with the population has to be decided by the slate pencil. It is not to be recommended to kill half of all the does within one year, because most of the does have young ones and choice is difficult.
    Abstract: Résumé Quoique le pourcentage d'accroissement queM. Gguschwitz avait choisi était 70% il atteint dans ces exemples seulement 54%, puisqu'il se rapporte chez les chevrettes sur le pourcentage du 31 mars et point sur celui du 1 er juin. La proportion entre les sexes n'est point la proportion numérique des pièces en état de puberté pendant l'été; mais la proportion de tous les mâles envers toutes les femelles. Il est donc nécessaire de faire une différence entre la proportion du mois de mars et celle du mois de juin. Une grande proportion entre les sexes doit être attribuée essentiellement à un manque en boucs; la raison en est un abattement en boucs de beaucoup trôp élevé. A part de cela, un abattement insuffisant en chevrettes n'a presque pas d'importance. C'est le crayon d'ardoise qui fixe l'étendue et la façon de chaque empiètement dans l'état. Mais il nést point applicable de faire abattre la moitiè de toutes les chevrettes au cours d'une année, puisque la plupart des chevrettes conduisent et en cela même un abattement sélectif devient de plus difficile.
    Notes: Zusammenfassung ObwohlGruschwitz einen Zuwachsprozent von 70 wählt, kommt er in seinen Beispielen nur auf 54%, weil er den Hundertsatz auf die Geißen vom 31. 3. bezieht und nicht auf die vom 1. 6. Das Geschlechterverhältnis ist nicht das Zahlenverhältnis der geschlechtsreifen Stücke im Sommer zueinander, sondern das Verhältnis aller männlichen zu allen weiblichen Stücken. Nötig ist, dieses Verhältnis im März von dem im Juni zu unterscheiden. Ein weites Geschlechterverhältnis ist im wesentlichen auf Mangel an Böcken zurückzuführen, die Ursache ist ein viel zu hoher Bockabschuß. Ungenügender Rickenabschuß ist daneben kaum von Bedeutung. Das Maß und die Art jedes Eingriffes in den Bestand bestimmt der Rechenstift. Die Hälfte aller Ricken innerhalb eines Jahres abzuschießen, ist nicht angebracht, weil die meisten Ricken führen und der Wahlabschuß erschwert wird.
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    European journal of wildlife research 3 (1957), S. 172-180 
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    European journal of wildlife research 3 (1957), S. 92-92 
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    European journal of wildlife research 3 (1957), S. 96-100 
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    European journal of wildlife research 3 (1957), S. 122-123 
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    European journal of wildlife research 3 (1957), S. 107-114 
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    Description / Table of Contents: Summary Description of rather a number of methods of catching birds with ring nets, nets, and snares. Dates concerning the catching of quails with throwing sticks, the hunt of snowcock, the catching of ducks by means of pumpkin masks, and hawking. Geese are being shot with shooting weapons out of a rifle pit or after having been rounded in with the field lorry. Rock partridges are approached behind movable screens, pheasant cocks are attracted by the flattering noise of house fowl. — This survey includes also ancient means of trapping birds, possibly no longer in use, and is restricted to the Southern marginal regions of the Caspic Sea (including the Eastern Caucasus), according to literature and to personal observations in 1956. In the beginning questions of the imperilment of birds and the necessity of educational measures are being discussed.
    Abstract: Résumé On donne ici la description de quelques methodes pour attraper les oiseaux, soit à I'aide d'un panneau ou à I'aide des rets ou lacets. En plus sont donnés des indications sur la chasse aux cailles avec javelot, la chasse à courre sur Tetraogallus, la chasse aux canards à I'aide d'une masque faite d'un potiron et la chasse au vol. Les oies sont abattus à I'aide des fusils d'une tranchée ou après les avoir cerné d'une voiture allant en terrain. Les bartavelles sont abordés derrière un paravent portable; les coqs de faisan pipé avec le son ballottant des poules domestique. — Ce précis comprend aussi des methodes anciens pour attraper les oiseaux partiellement éteindus aujourd'hui, et se restreind sur les régions méridionals et marginals de la mer caspienne (les régions caucasiennes orientales inclus), selon la literature et des observations propres en 1956. Au commencement les questions des dangers aux oiseaux et la des mesures pédagogiques son traités.
    Notes: Zusammenfassung Beschreibung einer größeren Zahl von Methoden des Fanges von Vögeln mit Käschern, Netzen und mit Schlingen. Angaben über die Jagd auf Wachteln mit Wurfstöcken, die Hetzjagd auf Königshühner, den Entenfang mit Hilfe der Kürbismaske und die Beizjagd. Mit Schußwaffen werden Gänse vom Schützenloch oder nach Einkreisen mit dem Geländewagen erlegt. Steinhühner werden hinter einem tragbaren Schirm angegangen, Fasanenhähne mit dem Flatterlaut der Haushennen angelockt. — Diese Übersicht bezieht auch alte, vielleicht zum Tiel jetzt erloschene Fangweisen ein und beschränkt sich auf die südlichen Randgebiete des Kaspischen Meeres (einschließlich Ost-Kaukasus), nach der Literatur und nach eigenen Beobachtungen 1956. Eingangs werden die Fragen der Gefährdung des Federwildes und die Dringlichkeit erzieherischer Maßnahmen behandelt.
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    European journal of wildlife research 3 (1957), S. 168-172 
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    European journal of wildlife research 4 (1958), S. 48-56 
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    European journal of wildlife research 4 (1958), S. 93-99 
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    Description / Table of Contents: Summary The fortunate circumstance that the world of ancient Mesopotamia from the third up to the first millenium b. C. has preserved not only proofs of plastic art and reliefs, but also thousands of seal pictures, which often show animals, enables us to study the fauna of the ancient Orient much better than that of many other periods and territories. Especially about 2800, 2300, and 1300 b. C. people were very diligent with animal portraits, and naturally they chose in the first place to illustrate chase or chaseable animals. Starting with the lion the bear, wild oxen, wild sheep and goat, the oriental kinds of antelopes and stags up to the ostrich and the wild boar we find here almost complete documentary evidence of the animals to be hunted, which will be continued once more in the later Assyrian reliefs — here not treated. These testimonies are the more valuable because they are generally very animated and prove a surprising capacity of observation of these creatures.
    Abstract: Résumé C'est un fait heureux que du monde de l'ancienne Mésopotamie du troisième jusqu'au premier millénaire a. J.-C. nous sommes remis à coté des pièces justificatives de l'art plastique et de l'art du relief aussi des images de sceaux présentant très souvent des animaux, ce que nous donne la possibilité d'étudier de beaucoup mieux la faune de l'ancien Orient que celle de beaucoup d'autres époques et régions. C'est surtout vers le temps de 2800, 2300 et 1300 a. J.-C. qu'on soignait très bien le portrait d'animaux et ce n'est que trôp compréhensible que se sont surrout les représentations de chasse ou au moins des bêtes bonne à chasser qui y sont en vedette. Commancé par le lion en passant par l'ours au boeuf sauvage, le mouton et la chèvre sauvage, les espèces d'antilopes se trouvant en Asie antérieur, jusqu'au cerf, l'autruche et le sanglier; on y trouve une documentation presque complète sur tous les bêtes bonne à chasser; documentation qui sera encore continuée dans les reliefs de l'ancien Assyrie, quoique ces derniers ne soient pas traités dans cet ouvrage. Ces témoignages sont d'autant plus précieux, qu'ils font le plus souvent épreuve d'une grande vivacité et qu'ils accusent un talent d'observations étonnant chez leurs créateurs.
    Notes: Zusammenfassung Der glückliche Umstand, daß die Welt Altmesopotamiens vom 3. bis zum 1. Jahrtausend v. Chr. uns neben Belegen der Vollplastik und Reliefkunst Tausende von Siegelbildern aufbewahrt hat und auf diesen häufig Tiere abgebildet sind, ermöglicht es uns, die Fauna des Alten Orients weit besser studieren zu können als die vieler anderer Zeiten und Gebiete. Besonders um 2800, um 2300 und um 1300 v. Chr. hat man sich um Tierporträts sehr bemüht, und hierbei steht verständlicherweise das Jagdbild oder zum mindesten das jagdbare Wild im Vordergrund. Vom Löwen angefangen über Bär, Wildrind, Wildschaf und-ziege, die vorderasiatischen Antilopenarten und den Hirsch bis hin zum Strauß und das Wildschwein wird hier eine nahezu vollständige Beurkundung der jagdbaren Tiere geboten, die dann in den — hier nicht behandelten —spätassyrischen Reliefs noch einmal fortgesetzt wird. Diese Zeugnisse sind um so kostbarer, als sie meist von großer Lebendigkeit sind und eine erstaunliche Beobachtungsgabe ihrer Schöpfer beweisen.
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    Mathematical programming 8 (1975), S. 91-103 
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    Notes: Abstract This paper describes what is termed the “generalized assignment problem”. It is a generalization of the ordinary assignment problem of linear programming in which multiple assignments of tasks to agents are limited by some resource available to the agents. A branch and bound algorithm is developed that solves the generalized assignment problem by solving a series of binary knapsack problems to determine the bounds. Computational results are cited for problems with up to 4 000 0–1 variables, and comparisons are made with other algorithms.
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    Mathematical programming 8 (1975), S. 43-53 
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    Notes: Abstract A break in a {0, 1}-matrix is defined as a 0 with at least one 1 to its left and at least one 1 to its right in the same row. This paper is concerned with {0, 1}-matrices with given column sums and an upper limit for the row sums. In addition, there are limits on the distance from the first to the last 1 in a row. The problem that is considered is to find a {0, 1}-matrix satisfying the conditions such that the total number of breaks is minimum. An algorithm for solving this problem is presented. Computational results illustrate the effectiveness of the algorithm. The investigation originated in a problem of crew rostering.
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    Mathematical programming 8 (1975), S. 104-116 
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    Notes: Abstract This paper presents a new algorithm for the solution of multi-state dynamic programming problems, referred to as the Progressive Optimality Algorithm. It is a method of successive approximation using a general two-stage solution. The algorithm is computationally efficient and has minimal storage requirements. A description of the algorithm is given including a proof of convergence. Performance characteristics for a trial problem are summarized.
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    Mathematical programming 8 (1975), S. 250-250 
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    Mathematical programming 8 (1975), S. 272-307 
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    Notes: Abstract A method of solving the 0–1 knapsack problem which derives from the “shrinking boundary method” is described and compared to other methods through extensive computational experimentation.
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    Mathematical programming 8 (1975), S. 332-344 
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    Notes: Abstract Consequences of a general formulation of the theorem of the alternative are exploited.
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    Mathematical programming 8 (1975), S. 369-374 
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    Notes: Abstract A perturbation method is introduced which transforms any fixed cost transportation problem F into a degeneracy-free equivalent F′. If a basic optimal solution to F′ is known, an optimal solution to F can be obtained by means of simple rounding.
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    Mathematical programming 8 (1975), S. 345-368 
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    Notes: Abstract Consider the relaxation of an integer programming (IP) problem in which the feasible region is replaced by the intersection of the linear programming (LP) feasible region and the corner polyhedron for a particular LP basis. Recently a primal-dual ascent algorithm has been given for solving this relaxation. Given an optimal solution of this relaxation, we state criteria for selecting a new LP basis for which the associated relaxation is stronger. These criteria may be successively applied to obtain either an optimal IP solution or a lower bound on the cost of such a solution. Conditions are given for equality of the convex hull of feasible IP solutions and the intersection of all corner polyhedra.
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    Mathematical programming 8 (1975), S. 378-381 
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    Notes: Abstract We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield a sharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e., the convex hull of tours. In fact, some of the new inequalities as well as some of the well-known subtour elimination constraints are indeed facets of the travelling salesman polytope, i.e., belong to the class of inequalities that uniquely characterize the convex hull of tours to an-city problem.
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    Mathematical programming 9 (1975), S. 57-68 
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    Notes: Abstract A family of test-problems is described which is designed to investigate the relative efficiencies of general optimisation algorithms and specialised algorithms for the solution of nonlinear sums-of-squares problems. Five algorithms are tested on three members of the family, and it is shown that the best choice of algorithms is critically affected by the value of one parameter in the test functions.
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    Mathematical programming 9 (1975), S. 87-99 
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    Notes: Abstract This paper identifies necessary and sufficient conditions for a penalty method to yield an optimal solution or a Lagrange multiplier of a convex programming problem by means of a single unconstrained minimization. The conditions are given in terms of properties of the objective and constraint functions of the problem as well as the penalty function adopted. It is shown among other things that all linear programs with finite optimal value satisfy such conditions when the penalty function is quadratic.
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    Mathematical programming 9 (1975), S. 130-130 
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    Mathematical programming 9 (1975), S. 137-137 
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    Mathematical programming 9 (1975), S. 139-160 
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    Notes: Abstract Supposez ∈ E n is a solution to the optimization problem minimizeF(x) s.t.x ∈ E n and an algorithm is available which iteratively constructs a sequence of search directions {s j } and points {x j } with the property thatx j →z. A method is presented to accelerate the rate of convergence of {x j } toz provided that n consecutive search directions are linearly independent. The accelerating method uses n iterations of the underlying optimization algorithm. This is followed by a special step and then another n iterations of the underlying algorithm followed by a second special step. This pattern is then repeated. It is shown that a superlinear rate of convergence applies to the points determined by the special step. The special step which uses only first derivative information consists of the computation of a search direction and a step size. After a certain number of iterations a step size of one will always be used. The acceleration method is applied to the projection method of conjugate directions and the resulting algorithm is shown to have an (n + 1)-step cubic rate of convergence. The acceleration method is based on the work of Best and Ritter [2].
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    Mathematical programming 9 (1975), S. 255-255 
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    Mathematical programming 9 (1975), S. 247-254 
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    Mathematical programming 9 (1975), S. 257-277 
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    Notes: Abstract The nonlinear complementarity problem is the problem of finding a point x in the n-dimensional Euclidean space,R n , such that x ⩾ 0, f(x) ⩾ 0 and 〈x,f(x)∼ = 0, where f is a nonlinear continuous function fromR n into itself. Many existence theorems for the problem have been established in various ways. The aim of the present paper is to treat them in a unified manner. Eaves's basic theorem of complementarity is generalized, and the generalized theorem is used as a unified framework for several typical existence theorems.
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    Mathematical programming 9 (1975), S. 313-335 
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    Notes: Abstract This paper presents a local convergence analysis of Broyden's class of rank-2 algorithms for solving unconstrained minimization problems, $$h(\bar x) = \min h(x)$$ ,h ∈ C1(R n ), assuming that the step-size ai in each iterationx i+1 =x i -α i H i ▽h(x i ) is determined by approximate line searches only. Many of these methods including the ones most often used in practice, converge locally at least with R-order, $$\tau \geqslant \sqrt[n]{2}$$ .
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    Mathematical programming 9 (1975), S. 350-357 
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    Notes: Abstract This paper presents an algorithm for ranking the vertices of a directed graph. Its space and time requirements are bounded byc 1 n 2 +c 2, wheren is the number of vertices of the graph andc 1,c 2 are positive constants which are independent of the size or other properties of the graph. The algorithm can be easily modified to solve the problem of determining longest distances from a vertex to all other vertices in a positive real valued graph with at mostc 1 n 2 +c 2 elementary operations; the same result holds for shortest distances in negative real valued graphs.
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    Mathematical programming 9 (1975), S. 371-376 
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    Notes: Abstract This paper reports an empirical discovery in integer programming. A version of the branch-and-bound approach is used as a control and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree. It is shown that the imbedded linear search locates solutions within 1%, and solutions within 5% of the theoretical optimum, which in fact can be seen to have this proximity to the theoretical optimum at the time of termination of computation, over ten times faster than the control.
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    Computing 14 (1975), S. 51-65 
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    Description / Table of Contents: Abstract In this paper the mathematical structures occuring in rounded computations with complex numbers and intervals are studied. By means of the special representation of the complex numbers and the properties of their operations one can show, that the same structures as in the real case occur again. Nevertheless we can prove formulae for the interval arithmetic wellknown for complex intervals ([1]) which are easy to calculate.
    Notes: Zusammenfassung In der vorliegenden Arbeit werden die beim numerischen Rechnen mit komplexen Zahlen und Intervallen auftretenden mathematischen Räume untersucht. Unter Verwendung der speziellen Darstellung komplexer Zahlen und der Eigenschaften ihrer Verknüpfungen läßt sich zeigen, daß wieder dieselben Strukturen wie im reellen Fall vorliegen. Trotzdem lassen sich für die Intervallarithmetik leicht auswertbare Formeln herleiten, wie sie für komplexe Intervalle bekannt sind ([1]).
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    Computing 14 (1975), S. 107-117 
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    Description / Table of Contents: Zusammenfassung Es handelt sich um eine nomographische approximative Lösungsmethode der folgenden Minimierungsaufgabe: $$g(t_0 ,t_1 ) + g(t_1 ,t_2 ) + ... + g(t_{n - 1} ,t_n ) = \min !$$ unter der Bedingung $$t_0 \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 0〈 t_1〈 t_2〈 ...〈 t_{n - 1}〈 T\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } t_n ,$$ wobein die Anzahl der Variablen ist und auch unbekannt sein kann. Der Nomograph besteht aus den Höhenlinien vong(.,.) und anderen Hilfslinien. Die Erstellung der Nomographen durch Computer wird auch gezeigt.
    Notes: Zusammenfassung A graphical procedure is presented to obtain an approximate solution to the minimization problem of the follwing form: Minimize the function $$\varphi (t_0 ,t_1 ,...,t_{n - 1} ,t_n ) = g(t_0 ,t_1 ) + g(t_1 ,t_2 ) + ... + g(t_{n - 1} ,t_n )$$ subject to the constraints $$t_0 \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 0〈 t_1〈 t_2〈 ...〈 t_{n - 1}〈 T\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } t_n $$ wheren the number of the variables is not predetermined. The nomograph for the procedure is constructed of contour lines of the functiong(.,.) as well as two other auxiliary curves. Procedures to prepare such nomographs by computer are also presented.
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    Computing 14 (1975), S. 141-147 
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    Topics: Computer Science
    Description / Table of Contents: Abstract A theorem is proved by means of Brouwer's theorem which allows to decide whether a given interval matrix [S] has the inclusion property relative to the unknown inverseA −1 of a known matrixA. It is demonstrated in which cases the theorem may be used and how to choose [S] in these cases.
    Notes: Zusammenfassung Mittels des Satzes von Brouwer wird ein Satz bewiesen, nach dem man entscheiden kann, ob eine gegebene Intervallmatrix [S] die Einschließungseigenschaft bezüglich der unbekannten InversenA −1 einer bekannten MatrixA hat. Es wird vorgeführt, in welchen Fällen der Satz anwendbar ist und wie [S] in diesen Fällen gewählt werden kann.
    Type of Medium: Electronic Resource
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  • 97
    Electronic Resource
    Electronic Resource
    Springer
    Computing 14 (1975), S. 153-166 
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Zur Konstruktion von Algorithmen zur numerischen Lösung einer allgemeinen Volterraschen Integralgleichung zweiter Art werden Splinefunktionen vom Gradem und der DefizienzJ−1, d. h. inC m−J , zusammen mit Gaußschen Quadraturformeln benutzt. Die Methode ist, für gegebenesm, von der Ordnung (m+1), und sie erfordert im allgemeinen 0(N) Auswertungen des Kerns. Die bisher bekannten Methoden erfordern 0(N 2) Auswertungen. Es wird gezeigt, daß die Methode für Splinefunktionen mit voller Stetigkeit (J=1) numerisch instabil ist für allem〉2. Dagegen wird die Stabilität bewiesen fürJ=m, m −1 undm beiliebig. Weiter wird fürm=3,J=1 gezeigt, daß bei geeigneter Modifikation der ursprünglichen Methode eine ganze Familie stabiler Methoden gewonnen werden kann.
    Notes: Abstract Spline function of degreem, deficiencyJ−1, i. e. inC m−J , are used in conjunction with (Gaussian) quadrature rules to construct algorthms for the numerical solution of a general Volterra integral equation of the second kind. For a givenm, the method is of order (m+1) and, in general, requires 0(N) evaluations of the kernel. This is in sharp contrast to the 0(N 2) evaluations required by hitherto known methods. It is shown that the method for spline functions with full continuity (J=1) is numerically unstable for allm〉2. However, stability is established forJ=m, m−1, for allm. Furthermore, form=3,J=1, it is demonstrated that by appropriately modifying the original method, a whole family of stable methods is obtained.
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  • 98
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    Electronic Resource
    Springer
    Computing 14 (1975), S. 195-203 
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird ein Algorithmus beschrieben, der das chromatische Polynom eines Graphen in Koeffizientenform liefert. Ein FORTRAN-Programm wird angegeben und einige praktische Erfahrungen werden kurz zusammengefaßt.
    Notes: Abstract An algorithm for deriving the chromatic polynomial of a graph in coefficient form is described. A FORTRAN implementation is given, and some computational experience is summarized.
    Type of Medium: Electronic Resource
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  • 99
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    Springer
    Computing 14 (1975), S. 225-234 
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract In the present paper a new method is given for the numerical treatment of the initial problemsy (n)=f(x,y,y′, ...,y (n−1),y (i) (x o )=y o (i) , i=0, 1, ...,n−1. This method is an one-step process of order four. For a class of linear differential equations the exact solution is obtained. Moreover some numerical results are presented.
    Notes: Zusammenfassung In der vorliegenden Arbeit wird ein neues Verfahren zur numerischen Berechnung des Anfangswertproblemsy (n)=f(x,y,y′, ...,y (n−1),y (i)(x o )=y o (i) , i=0, 1, ..., n−1 gegeben. Es handelt sich hierbei um ein Einschrittverfahren der Konsistenzordnung 4. Das Verfahren liefert für eine Klasse linearer Differentialgleichungen die exakte Lösung. Abschließend geben wir einige numerische Beispiele.
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  • 100
    Electronic Resource
    Electronic Resource
    Springer
    Computing 14 (1975), S. 261-270 
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract It is not possible to support by numerical tests the common opinion that the filter method is better than the additive algorithm. The reasons for the limited efficiency of the filter method are stated together with results of numerical tests.
    Notes: Zusammenfassung Die in der Literatur verbreitete Meinung, die Filtermethode sei eine Verbesserung des additiven Algorithmus, kann durch numerische Tests nicht gestützt werden. Zusammen mit den Ergebnissen der numerischen Tests werden Ursachen für die beobachtete geringe Leistungsfähigkeit der Filtermethode angegeben.
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