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  • Springer  (39,521)
  • Frontiers Media
  • 2015-2019
  • 1975-1979  (39,521)
  • 1975  (39,521)
  • 1
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    Bulletin of mathematical biology 37 (1975), S. 37-49 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    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 
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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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    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 
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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 
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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 
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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 
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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 
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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 
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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 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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    Topics: Agriculture, Forestry, Horticulture, Fishery, Domestic Science, Nutrition
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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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    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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    Topics: Computer Science , Mathematics
    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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    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.
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    Computing 14 (1975), S. 153-166 
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    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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    Computing 14 (1975), S. 195-203 
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    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.
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    Computing 14 (1975), S. 225-234 
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    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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    Computing 14 (1975), S. 261-270 
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    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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    Computing 14 (1975), S. 285-312 
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    Description / Table of Contents: Zusammenfassung Eine interpretierte zweisortige Logik wird so definiert, daß sie die Repräsentation einer Unterklasse von Flußdiagrammschemata erlaubt. Ein Entscheidungsverfahren wird entwickelt für die Äquivalenz einfacher Darstellungsformeln innerhalb der Logik und infolgedessen wird ein Entscheidungsverfahren für Flußdiagrammschemata innerhalb der Unterklasse demonstriert. Anwendungen der in der Arbeit entwickelten Techniken haben weitere Entscheidungsverfahren zur Folge. Ein Entscheidungsverfahren für die Äquivalenz von Programmen in einer einfachen Programmiersprache wird auch angegeben.
    Notes: Abstract An interpreted two-sorted logic is defined to allow the representation of a subclass of flowchart schemata. A decision procedure for the equivalence of simple representing formulae within the logic is developed and as a consequence a decision procedure for flowchart schemata within the subclass is demonstrated. Applications of the techniques developed within the paper result in further decision procedures for the equivalence of flowchart schemata in other subclasses. In addition, a decision procedure for the equivalence of programs in a simple programming language is given.
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    Computing 14 (1975), S. 389-396 
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    Description / Table of Contents: Abstract This paper contains an ALGOL-Procedure for solving mixed integer programming problems with convex objective function and constraints according to a method of Burkard [2].
    Notes: Zusammenfassung Diese Arbeit enthält ein ALGOL-Programm zur Lösung gemischt-ganzzahliger Optimierungsaufgaben mit konvexer Zielfunktion und konvexen Restriktionen nach einer Methode von Burkard [2].
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    Computing 15 (1975), S. 33-39 
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    Description / Table of Contents: Zusammenfassung Folgende Algorithmen wurden untersucht: 1. Warshall's Algorithm [17], 2. Purdom's Algorithm [13], 3. der modifizierte Algorithmus von Yen [14], 4. der Algorithmus von Dzikiewicz [3, 4]. Die getesteten Digraphen wurden durch einen Zufallsgenerator erzeugt. Das Literaturverzeichnis enthält alle Veröffentlichungen über Algorithmen zur Bildung der transitiven Hülle, welche den Verfassern bekannt sind.
    Notes: Abstract The paper contains results of computational experiences with the following algorithms for finding the transitive closure of a digraph: (i) Warshall's algorithm [17], (ii) Purdom's algorithm [13], (iii) the modification of Yen's algorithm [14], and (iv) the new algorithms for finding the transitive closure [3, 4]. The tested digraphs were generated at random. The enclosed references contain all papers known to the authors concerning transitive closure algorithms.
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  • 89
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    Description / Table of Contents: Abstract In this paper Aireys converging factors of the exponential integral are transformed into an asymptotic expansion with remainder. This expansion is obtained by an Euler transformation of the remaining divergent series in analogy to the formal procedure due to Airey. The remainder termR N (z) is developed by Abels asymptotic principle. Transforming the terms of this series we get Aireys converging factors. Finally we improve the estimation of the remainder term which has been given by J. B. Rosser.
    Notes: Zusammenfassung In der vorliegenden Abhandlung werden die Aireyschen Konvergenzfaktoren für das Exponentialintegral zu einer asymptotischen Entwicklung mit Restglied umgeformt. Diese Entwicklung wird in Anlehnung an das Aireysche formale Herleitungsverfahren durch Eulersche Transformation der divergenten Restreihe gewonnen. Das RestgliedR N (z) seinerseits wird nach dem Prinzip der Abelschen Asymptotik entwickelt. Durch Umformen dieser Reihe erhält man die Aireyschen Konvergenzfaktoren; deren Restabschätzung nach J. B. Rosser wird in der vorliegenden Arbeit verbessert.
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    Computing 15 (1975), S. 71-74 
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    Description / Table of Contents: Zusammenfassung Es werden einige Sätze über die Erhaltung der Eindeutigkeit von n-tupel-Sprachen präsentiert. Weiters wird der Begriff der „n-stratified” Mengen als Verallgemeinerung der von Ginsburg definierten „stratified” Mengen eingeführt und damit eine notwendige und hinreichende Charakterisierung der beschränkten eindeutigen n-tupel-Sprachen — in Analogie zu den beschränkten eindeutigen umgebungsunabhängigen Sprachen — angegeben.
    Notes: Abstract Several results concerning the preservation of unambiguity in n-tuple languages are presented here. Furthermore, the notion of n-stratified sets — as appropriate generalization of the straified sets defined by Ginsburg — is introduced and a necessary and sufficient characterisation result for bounded unambiguous n-tuple languages analogous to bounded unambiguous context free languages is given.
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    Computing 15 (1975), S. 137-146 
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    Description / Table of Contents: Abstract If an Integral Equation is solved by Hermite Collocation we get essentially the same error bound as by using a more complicate method, supposed the collocation-points are suitably chosen. How to do this is one essential part of this paper. The discussion of numerical problems is confined to the case of linear problems.
    Notes: Zusammenfassung Gegenstand der Arbeit ist die Lösung nichtlinearer Integralgleichungen durch hermitesche Kollokation. Bei geeigneter Wahl der Kollokationsstellen —hierzu benötigen wir die Theorie der Gaußschen Quadraturformeln — ergibt sich praktisch dieselbe Genauigkeit wie bei komplizierteren Verfahren. Da nichtlineare Probleme bei der praktischen Rechnung meist linearisiert werden, beschränkt sich die Diskussion der numerischen Probleme auf den linearen Fall.
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    Computing 15 (1975), S. 147-156 
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    Description / Table of Contents: Abstract Effective procedures for the decision whether two finite graphs are isomorphic or not make use of some information about the automorphism groups of these graphs. The paper deals with a systematic representation of this information and proposes an algorithm scheme for the determination of the automorphisms of a graph.
    Notes: Zusammenfassung Bekannte Verfahren zum Nachweis der Isomorphie zwischen zwei endlichen GraphenG undG′ benutzen Informationen über die Automorphismengruppen beider Graphen. Die vorliegende Arbeit befaßt sich mit einer systematischen Darstellung dieser Information und enthält ein Algorithmenschema zur Bestimmung der Automorphismen eines Graphen.
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    Computing 15 (1975), S. 173-179 
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    Description / Table of Contents: Zusammenfassung Dieser Aufsatz fängt mit einer Untersuchung zweier besonderer Gestalten der Gauß-Turán-Quadratur des Chebyshev-Typs der Präzision 6n−1 an. Dann werden die Restformeln dieser Quadraturen entwickelt und scharfe Irrtumgrenzen für die Funktionen inC q [−1, 1] gezeigt, woq eine positive ganze Zahl ist. Am wichtigsten beweist diese Studie, daß diese Ergebnisse verlängert werden können, um die Resformeln und scharfe Irrtumsabschätzungen für alle solchen Quadraturen höherer Präzision zu erhalten.
    Notes: Abstract This paper begins with an investigation of two special forms of the Gauss-Turán quadrature of Chebyshev-type of precision 6n−1. Then the remainder formulas of these quadratures are developed and sharp error bounds for the functions inC q [−1, 1] are shown, whereq is a positive integer. Most importantly this study proves that these reslts can be extended in order to yield sharp error estimates for all such quadratures of higher precision.
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    Computing 15 (1975), S. 205-216 
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    Description / Table of Contents: Abstract By Mac Leod there was presented an improved algorithm for the computation of cubic natural Splines with equi-spaced knots. In this paper his algorithm is extended to the case of type I and II L-Splines corresponding to operators of second order with constant coefficients and applied to a problem concerning the static deflection of beams.
    Notes: Zusammenfassung Mac Leod hat ein numerisch stabiles Verfahren zur Berechnung von kubischen natürlichen Polynom-Splines mit äquidistanten Knoten hergeleitet, das lediglich einen Rechenaufwand in der Größenordnung der Knotenanzahl erfordert. Sein Ansatz wird in dieser Arbeit auf den Fall von L-Splines vom Typ I und II zu Operatoren zweiter Ordnung mit konstanten Koeffizienten erweitert und auf eine dem Spline-Problem adäquate Fragestellung bei der statischen Biegung von Balken angewendet.
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    Computing 15 (1975), S. 217-234 
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    Description / Table of Contents: Zusammenfassung Um einen natürlichen, von Kodierungen unabhängigen Vergleich zwischen primitiv-rekursiven Funktionen und Automatentransduktionen zu ermöglichen, werden die Grzegorczyk-Hierarchie, die Rekursionszahl-Hierarchie und die Loop-Hierarchie von arithmetischen auf Wortfunktionen verallgemeinert. Dabei ergeben sich einige Unterschiede zum arithmetischen Fall. Unter Benutzung von Turingmaschinen und verallgemeinerten endlichen Automaten werden alle Inklusionsprobleme der Funktionenklassen dieser Hierarchien gelöst. Von Automaten definierte Funktions- und Sprachklassen werden innerhalb dieser Hierarchien klassifiziert. Außerdem werden primitiv-rekursive Transformationen zwischen verschiedenen Monoiden behandelt.
    Notes: Abstract In order to compare primitive recursive functions and transductions defined by automata in a natural way independent of encodings, we generalize the Grzegorczyk hierarchy, the recursion number hierarchy and the loop hierarchy from arithmetical to wordfunctions. We observe several differences between the arithmetical and the non-arithmetical theory. By means of turingmachines and generalized sequential machines all inclusion problems for the function classes of these hierarchies are solved. Transductions and languages defined by automata are classified within these hierarchies. Moreover, we introduce and study primitive recursive transformations between different monoids.
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    Computing 15 (1975), S. 275-286 
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    Description / Table of Contents: Abstract A method for the determination of Eigenvalues and Eigenvectors of symmetric non-hermitian matrices has been developed. This method is characterized by a “complex unitarity” of the transformation matrix. Thereby it is possible to extend the Jacobi procedure to symmetric non-hermitian matrices and to guarantee convergency.
    Notes: Zusammenfassung Es wird ein Verfahren zur Bestimmung der Eigenwerte und Eigenrichtungen von symmetrisch nicht-hermiteschen Matrizen entwickelt, das dadurch charakterisiert ist, daß für die auf die Matrix angewandte Transformation eine „Komplexe Unitarität” gefordert wird. So gelingt es, das Jacobi-Verfahren auf symmetrische nicht-hermitesche Matrizen zu erweitern und die Konvergenz zu gewährleisten.
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    Computing 15 (1975), S. 311-328 
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    Description / Table of Contents: Abstract We derive explicit bounds for the termination index and for the error bound of algorithms as they were studied in part I. Furthermore, we analyze a relation between the theory of part I and the theory of Dahlquist.
    Notes: Zusammenfassung Es werden explizite Schranken angegeben für den Abbrechindex und für die Fehlerschranke von Algorithmen, wie sie in Teil I untersucht wurden. Weiterhin wird ein Zusammenhang zwischen der Theorie von Teil I und der Theorie von Dahlquist untersucht.
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    Computing 14 (1975), S. 1-3 
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    Computing 14 (1975), S. 5-14 
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    Description / Table of Contents: Abstract Wir geben eine neue Methode für die Reduktion des Grades eines Intervallpolynoms. Es wird gezeigt, daß diese Methode besser ist als eine frühere Methode ([4] und [5]), wenn der Grad der Reduktion klein ist. Einige numerische Vergleiche und eine Anwendung auf Grenzwertprobleme werden angegeben.
    Notes: Abstract A new Method for reducing the degree of an interval polynomial is presented. The method is shown to be better than a previously suggested method ([4] and [5]) when the degree of the reduction is small. Some numerical comparisons and an application to a boundary value problem is given.
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