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  • 1
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    Bulletin of mathematical biology 52 (1990), S. 153-197 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
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  • 2
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    Bulletin of mathematical biology 52 (1990), S. 319-334 
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  • 3
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    Bulletin of mathematical biology 52 (1990), S. 335-337 
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  • 4
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    Bulletin of mathematical biology 52 (1990), S. I 
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    Bulletin of mathematical biology 52 (1990), S. 335-348 
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    Notes: Abstract The classical metaphor of the genetic program written in the DNA nucleotidic sequences is reconsidered. Recent works on algorithmic complexity and logical properties of computer programs and data are used to question the explanatory value of that metaphor. Structural properties of strings are looked for which would be necessary to apply to DNA sequences if the metaphor is to be taken literally. The notion of sophistication is used to quantify meaningful complexity and to distinguish it from classical computational complexity. In this context, the distinction between program and data becomes relevant and an alternative metaphor of DNA as data to a parallel computing network embedded in the global geometrical and biochemical structure of the cell is discussed. An intermediate picture of an evolving network emerges as the most likely where the output of the cellular computing network can produce, at a different time scale, changes in the structure of the network itself by means of changes in the DNA activity patterns.
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  • 6
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    Bulletin of mathematical biology 52 (1990), S. 349-358 
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    Notes: Abstract When two strings of symbols are aligned it is important to know whether the observed number of matches is better than that expected between two independent sequences with the same frequency of symbols. When strings are of different lengths, nulls need to be inserted in order to align the sequences. One approach is to use simple approximations of sampling for replacement. We describe an algorithm for exactly determining the frequencies of given numbers of matches, sampling without replacement. This does not lead to a simple closed form expression. However we show examples where sampling with, or without, replacement give very similar results and the simple approach may be adequate for all but the smallest cases.
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  • 7
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    Bulletin of mathematical biology 52 (1990), S. 509-525 
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    Notes: Abstract Pairwise optimal alignments between three or more sequences are not necessarily consistent as a whole, but consistent and inconsistent residues are usually distributed in clusters. An efficient method has been developed for locating consistent regions when each pairwise alignment is given in the form of a “skeletal representation” (Bull. math. Biol. 52, 359–373). This method is further extended so that the combination of pairwise alignments that gives the greatest consistency is found when possibly many alignments are equally optimal for each pairwise comparison. A method for acceleration of simultaneous multiple sequence alignment is proposed in which consistent regions serve as “anchor points” limiting application of direct multi-way alignment to the rest of “inconsistent” regions.
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  • 8
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    Bulletin of mathematical biology 52 (1990), S. 527-534 
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    Notes: Abstract Sufficient conditions are given for the unlimited growth or otherwise in multitype population size dependent Galton-Watson processes. These conditions are given in terms of moments of offspring distributions and extend known conditions for processes with one type.
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  • 9
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    Bulletin of mathematical biology 52 (1990), S. 535-547 
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    Notes: Abstract The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.
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  • 10
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    Bulletin of mathematical biology 52 (1990), S. 583-596 
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  • 11
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    Bulletin of mathematical biology 52 (1990), S. I 
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  • 12
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    Bulletin of mathematical biology 52 (1990), S. 549-582 
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    Notes: Abstract Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated (S. A. Maggelakis and J. A. Adam,Math. Comput. Modeling, in press). In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters:Q, γ, b, andδ, which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place. The integro-differential growth equation is solved for the outer spheroid radiusR 0(t) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics ofin vitro multicellular spheroids. This will be one objective of future studies.
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  • 13
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    Notes: Abstract Workers of the speciesLeptothorax acervorum show age-polyethism, they start their life as broodworkers and later on they become nestworkers and foragers. Nestworkers and foragers of this ant species are inactive for 72% and 15% of the total time respectively. The short bursts of activity within the nest do not occur randomly but are synchronized so that the whole nest population exhibits nonperiodic pulses of activity: the ants were seen to wake each other actively. In addition starvation experiments were done to assess whether ants react upon food availability. In appeared that during a longlasting period of starvation the proportion of active ants in the nest is at a higher approximately constant level.
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  • 14
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    Bulletin of mathematical biology 52 (1990), S. 455-475 
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    Notes: Abstract Random clone mapping of genomic DNA is a subject of great interest in molecular biology.E. coli has just been mapped and work is progressing on some human chromosomes. In this paper we give estimates of the fraction of genomic DNA which is not clonable by partial digest with a restriction enzyme.
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  • 15
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    Bulletin of mathematical biology 52 (1990), S. 483-484 
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  • 16
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    Bulletin of mathematical biology 52 (1990), S. I 
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  • 17
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    Bulletin of mathematical biology 52 (1990), S. 1-1 
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  • 18
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    Bulletin of mathematical biology 52 (1990), S. 3-23 
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  • 19
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    Bulletin of mathematical biology 52 (1990), S. 25-71 
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    Notes: Abstract This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al., 1952,J. Physiol. 116, 424–448; Hodgkin and Huxley, 1952,J. Physiol. 116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6).
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  • 20
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    Bulletin of mathematical biology 52 (1990), S. 73-97 
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    Notes: Abstract The McCulloch-Pitts paper “A Logical Calculus of the Ideas Immanent in Nervous Activity” was published in theBulletin of Mathematical Biophysics in 1943, a decade before the work of Hodgkin, Huxley, Katz and Eccles. The McCulloch-Pitts neuron is an extremely simplified representation of neural properties, based simply on the existence of a threshold for the activation of an action potential.
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    Bulletin of mathematical biology 52 (1990), S. 99-115 
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    Notes: Abstract Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.
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  • 22
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    Bulletin of mathematical biology 52 (1990), S. 199-207 
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    Bulletin of mathematical biology 52 (1990), S. 117-152 
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    Notes: Abstract The evolution of spatial pattern is a central issue in developmental biology. Turing's (Phil. Trans. R. Soc. Lond. B237, 37–72, 1952) chemical theory of morphogenesis is a seminal contribution. In this talk I give a personal and necessarily limited view of its impact on mathematical and developmental biology. I briefly describe some of the interesting mathematical aspects of Turing's reaction-diffusion mechanism and discuss some of the different models which Turing's vision inspired. The emphasis throughout is on the practical biological applications of the various theories.
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  • 24
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    Bulletin of mathematical biology 52 (1990), S. 209-240 
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    Notes: Abstract Mathematical expressions are found for the effect of selection on simple Mendelian populations mating at random. Selection of a given intensity is most effective when amphimixis does not affect the character selected, e.g. in complete inbreeding or homogamy. Selection is very ineffective on autosomal recessive characters so long as they are rare.
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  • 25
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    Bulletin of mathematical biology 52 (1990), S. 241-295 
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  • 26
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    Bulletin of mathematical biology 52 (1990), S. 297-318 
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    Bulletin of mathematical biology 52 (1990), S. 359-373 
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    Notes: Abstract A new algorithm for optimal sequence alignment allowing for long insertions and deletions is developed. The algorithm requires O((L+C)MN) computational steps, O(LN) primary memory and O(MN) secondary memory storage, whereM andN(M≥N) are sequence lengths,L (typicallyL≤3) is the number of segment specifying the gap weighting function, andC is a constant. We have also modified our earlier traceback algorithm so that it finds all and only the optimal alignments in a compact form of a directed graph. The current versions accept a set of aligned sequences as input, which facilitates multiple sequence alignment by some iterative procedures.
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    Bulletin of mathematical biology 52 (1990), S. 397-429 
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    Notes: Abstract The kinetic model of cortical-like neural systems is amplified with elements related to a gross lamination in three parts of the basic system. A reduced version of the model, aiming to study only short-range, random aspects of activity propagation, was subjected to an intensive computational experimentation, and the results are briefly presented. The comparison of these results with those of a previous paper, in which aspects related mainly to long-distance effects were described, suggested the possibility of extending the model by considering new features related to phenomena of attention. The results of experimentations with the whole model revealed the possibility of laying the bases for a description of the cognitive activity in a neural frame.
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    Bulletin of mathematical biology 52 (1990), S. 431-453 
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    Notes: Abstract A method of inductive inference known asminimum message length encoding is applied to string comparison in molecular biology. The question of whether or not two strings are related and, if so, of how they are related and the problem of finding a good theory of string mutation are treated as inductive inference problems. The method allows the posterior odds-ratio of two string alignments or of two models of string mutation to be computed. The connection between models of mutation and existing string alignment algorithms is made explicit. A fast minimum message length alignment algorithm is also described.
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    Bulletin of mathematical biology 52 (1990), S. 477-482 
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    Bulletin of mathematical biology 52 (1990), S. 629-641 
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    Notes: Abstract The geometrical properties of neurons are important for the way they function within neural circuits. The arborescent processes of neurons that are necessary for the transmission of the information are formed by branching and elongation of segments. In studies that model the outgrowth the tree structures have generally been considered as binary. However, multifurcations do occur. It will be shown that if the multifurcations can be considered as aggregates of bifurcations they may be included in the topological analysis of neuronal branching patterns.
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  • 32
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    Notes: Abstract Observations on activity of ants of the speciesLeptothorax acervorum show that ants within the nest are inactive for about 72% of their time (Frankset al., 1990.Bull. math. Biol.,52, 597–612). By examination of the activity of individual ants it is demonstrated that activity bouts of individuals are highly synchronized. The bursts of activity detected by Frankset al. occurred three to four times per hour. In this paper we develop a model to describe the phenomenon. As a result of the interdependence of the number of active ants within the nest and the high level of community activity some predictions are made, which are supported by experimental data in a quantitative way. In case of starvation the number of active ants will increase and no rhythms should occur. When proportionally more brood is present the rhythms should occur with a higher frequency. Eventually the rhythm breaks down and a stable equilibrium is reached.
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    Bulletin of mathematical biology 52 (1990), S. 643-655 
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    Notes: Abstract A computer model of body mass and composition in relation to gross energy balance is constructed. The model is built using conventional empirical physiological formulae rather than statistical or analytical mathematical techniques. The model is applied to the Minnesota and other experiments and produces as good or better simulations of observed values of changes in body weight than reported for other formulae or models. Alternative physiological mechanisms concerning metabolic adaptions to starvation, changes in time activity budgets and the energy equivalents of weight loss offer equally good simulations of experimental results. The present analysis highlights the survival value of a basal metabolic depression during starvation and indicates an optimal body composition of 10% mobilizable fat for starvation survival for a 70 kg man. Proper quantification of the effects of the physiological mechanisms involved depends on new experimental data, however. Long term continuous monitoring of time activity budgets are a necessary part of such experiments.
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    Bulletin of mathematical biology 52 (1990), S. 657-675 
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    Notes: Abstract Irradiation affects numerous physiological processes within cells and tissues and can lead to damage or death. If the damage is not too severe, cells have the ability to repair and regenerate. Many small injuries are repaired more easily than ones causing extensive damage and, consequently, tissue typically respond differently to one large dose of radiation than to many small doses, separated in time. In the radiotherapy of tumors, the choice of the fractionation regimen of dose over time is therefore as crucial as the total radiation dose. The interdependence between total dose, fractionation regimen, and radiation effect has been described mathematically with variousisoeffect relationships. These relationships appear to be fundamentally distinct and have been considered unrelated; some even claim that one class of isoeffect relationship is appropriate whereas other relationships are rather useless. We examine how alternative isoeffect models relate to each other and test the reliability of estimating parameter values of one model from the other.
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    Bulletin of mathematical biology 52 (1990), S. 697-708 
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    Bulletin of mathematical biology 52 (1990), S. 677-696 
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    Notes: Abstract A stability analysis is given for a model of plankton dynamics introduced by Wroblewskiet al. (Global Biogeochem. Cycles 2, 199–218, 1988). The detailed dependence of the steady-states and their stability on the various model parameters is explicitly presented and analysed. It is shown that under certain conditions the coexistence of phytoplankton and zooplankton occurs in an orbitally stable oscillatory mode. A distinguished parameter is varied and the steady-states computed. The significance of the lack of stable steady-states leading to periodic population levels is investigated and related to certain oceanographic data.
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    Bulletin of mathematical biology 52 (1990), S. I 
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    Bulletin of mathematical biology 52 (1990), S. 709-709 
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    Bulletin of mathematical biology 52 (1990), S. 733-740 
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    Notes: Abstract An ecosystem can be visualized as a graph of certain preassigned trophic compartments; these nodes are then mutually connected through the internal exchanges of material and energy. The mathematical theory of information can be applied to such a graph in order to define two relevant indices: a measure of connectivity (the entropy H of the connections) and a measure of the degree of the “energetic” specialization (the internal transfer of informationI). The computation of these indices in stationary real cases suggests that the observed complexity of ecosystems is conditioned by two competing effects. The first can be interpreted as a “thermodynamical” principle related to the unavoidable irreversibility taking place inside the system, whereas the second can be taken as a “biological” principle concerned with the selection of some particular interactions: those which maximize the information circulating between the compartments.
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    Bulletin of mathematical biology 52 (1990), S. 773-784 
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    Notes: Abstract An accurate approximation is derived to the distribution of the length of the longest matching word present between two random DNA sequences of finite length, using only elementary probability arguments. The distribution is shown to be consistent with previous asymptotic results for the mean and variance of longest common words. The application of the distribution to assessing the statistical significance of sequence similarities is considered. It is shown how the distribution can be modified to take account of non-independence of neighbouring bases in real sequences.
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    Notes: Abstract Experimental data of the radial incorporation of labeled cholesterol [14C-4] into the artery wall is regressed against a mathematical model that predicts macromolecular transport in this biological system. Data is obtained using excised canine carotid arteries that are perfusedin vitro under pulsatile hemodynamic conditions for 2 hr. Vessels are exposed to either normotensive hemodynamics, hypertensive hemodynamics, or simulations in which the rate of flow or vessel compliance is deliberately altered. Several arteries are studied under normotensive conditions following balloon catheter deendothelialization. Transmural concentration profiles of [14C-4] activity are determined by microcryotomy of longitudinal sections of perfused vessels. Nonlinear Marquardt regression on 12 experimental cases yields parameter estimates of effective diffusivity,D and solute filtration velocity,V. Results of this experimental investigation support our hypothesis that hemodynamics and the endothelial lining influence wall flux in intact vessels. Exposure to altered (vs normotensive) hemodynamics is associated with increased incorporation of labeled cholesterol. A similar observation is made for deendothelialized vessels (e.g. a greater accumulation of label and a rise in convective flux). Based upon our companion measurements of vessel wall forces and endothelial cellular morphology accompanying hemodynamic simulations, we suggest that hemodynamically induced alterations to endothelial structures lead to the increased permeability, convection and incorporation that we observe in this work.
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    Bulletin of mathematical biology 52 (1990), S. 797-805 
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    Bulletin of mathematical biology 52 (1990), S. 741-772 
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    Notes: Abstract Statistical studies of gene populations on the purine/pyrimidine alphabet have shown that the mean occurrence probability of thei-motif YRY(N) i YRY (R=purine, Y=pyrimidine, N=R or Y) is not uniform by varyingi in the range [1,99], but presents a maximum ati=6 in the following populations: protein coding genes of eukaryotes, prokaryotes, chloroplasts and mitrochondria, and also viral introns, ribosomal RNA genes and transfer RNA genes (Arquès and Michel, 1987b,J. theor. Biol. 128, 457–461). From the “universality” of this observation, we suggested that the oligonucleotide YRY(N)6 is a primitive one and that it has a central function in DNA sequence evolution (Arquès and Michel, 1987b,J. theor. Biol. 128, 457–461). Following this idea, we introduce a concept of a model of DNA sequence evolution which will be validated according to a shema presented in three parts. In the first part, using the last version of the gene database, the YRY(N)6YRY preferential occurrence (maximum ati=6) is confirmed for the populations mentioned above and is extended to some newly analysed populations: chloroplast introns, chloroplast 5′ regions, mitochondrial 5′ regions and small nuclear RNA genes. On the other hand, the YRY(N)6YRY preferential occurrence and periodicities are used in order to classify 18 gene populations. In the second part, we will demonstrate that several statistical features characterizing different gene populations (in particular the YRY(N)6YRY preferential occurrence and the periodicities) can be retrieved from a simple Markov model based on the mixing of the two oligonucleotides YRY(N)6 and YRY(N)3 and based on the percentages of RYR and YRY in the unspecified trinucleotides (N)3 of YRY(N)6 and YRY(N)3. Several properties are identified and prove in particular that the oligonucleotide mixing is an independent process and that several different features are functions of a unique parameter. In the third part, the return of the model to the reality shows a strong correlation between reality and simulation concerning the presence of large alternating purine/pyrimidine stretches and of periodicities. It also contributes to a greater understanding of biological reality, e.g. the presence or the absence of large alternating purine/pyrimidine stretches can be explained as being a simple consequence of the mixing of two particular oligonucleotides. Finally, we believe that such an approach is the first step toward a unified model of DNA sequence evolution allowing the molecular understanding of both the origin of life and the actual biological reality.
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    Bulletin of mathematical biology 52 (1990), S. I 
    ISSN: 1522-9602
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    Notes: Abstract A non-linear three-compartment model is proposed to describe a new strategy for the administration of 2′,3′-dideoxycytidine (ddCyd) in the treatment of HIV infections. The drug is injected after having been encapsulated in a non-diffusible form (ddCMP) into erythrocytes. Nummerical solutions show that by this treatment the highest ddCyd blood concentration is strongly reduced and in turn its toxicity, while long-lasting therapeutic effect is assured. The model is compared with experimental data in vitro.
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    Bulletin of mathematical biology 52 (1990), S. 375-396 
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    Notes: Abstract Some insect populations exhibit cycles in which successive population peaks may correspond to effectively discrete generations. Motivated by this observation, we investigate the structure of matriarchal generations in five simple, continuous-time, stage structure models in order to determine the proportion of individuals in one population peak who are the offspring of individuals in the pervious peak. We conclude that in certain models (including a model of Nicholson's blowflies) successive population peaks do not correspond to discrete generations, whereas in others (including some models of uniform larval competition) successive peaks may well approximate discrete generations. In all models, however, there is eventually significant overlap of generations.
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    Bulletin of mathematical biology 52 (1990), S. 485-508 
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    Notes: Abstract Catalysis in replication networks has become an important issue in biophysics and other areas of biology. Examples are RNA catalysis, idiotype recognition in the immune response and dynamical models of Maynard-Smith games in sociobiology. Chemical reaction networks describing catalysed, template-induced reproduction of three species are analysed in full generality. The nine-dimensional parameter space is reduced to three relevant angular coordinates which determine completely the phase portraits (PPs) and the bifurcation patterns. All cases are classified and all generic as well as most of the nongeneric transitions are listed and described.
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    Mathematical programming 49 (1990), S. 163-187 
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    Notes: Abstract A commonly studied relaxation of the travelling salesman problem is obtained by adding subtour elimination constraints to the constraints of a 2-factor problem and removing the integrality requirement. We investigate the problem of solving this relaxation for a special type of objective function. We also discuss some ways in which this relates to the concept of rank introduced by Chvátal.
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    Mathematical programming 46 (1990), S. 1-29 
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    Keywords: Global optimization ; concurrent ; parallel ; stochastic ; network of computers
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    Notes: Abstract The global optimization problem, finding the lowest minimizer of a nonlinear function of several variables that has multiple local minimizers, appears well suited to concurrent computation. This paper presents a new parallel algorithm for the global optimization problem. The algorithm is a stochastic method related to the multi-level single-linkage methods of Rinnooy Kan and Timmer for sequential computers. Concurrency is achieved by partitioning the work of each of the three main parts of the algorithm, sampling, local minimization start point selection, and multiple local minimizations, among the processors. This parallelism is of a coarse grain type and is especially well suited to a local memory multiprocessing environment. The paper presents test results of a distributed implementation of this algorithm on a local area network of computer workstations. It also summarizes the theoretical properties of the algorithm.
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    Mathematical programming 46 (1990), S. 105-122 
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    Keywords: Nondifferentiable minimization ; convex programming ; numerical methods ; descent methods
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    Notes: Abstract Proximal bundle methods for minimizing a convex functionf generate a sequence {x k } by takingx k+1 to be the minimizer of $$\hat f^k (x) + u^k |x - x^k |^2 /2$$ , where $$\hat f^k $$ is a sufficiently accurate polyhedral approximation tof andu k 〉 0. The usual choice ofu k = 1 may yield very slow convergence. A technique is given for choosing {u k } adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.
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    Mathematical programming 46 (1990), S. 127-151 
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    Keywords: Dual descent ; monotropic program ; Tucker tableau ; elementary vector
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    Notes: Abstract We propose a dual descent method for the problem of minimizing a convex, possibly nondifferentiable, separable cost subject to linear constraints. The method has properties reminiscent of the Gauss-Seidel method in numerical analysis and uses theε-complementary slackness mechanism introduced in Bertsekas, Hosein and Tseng (1987) to ensure finite convergence to near optimality. As special cases we obtain the methods in Bertsekas, Hosein and Tseng (1987) for network flow programs and the methods in Tseng and Bertsekas (1987) for linear programs.
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    Mathematical programming 46 (1990), S. 219-224 
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    Keywords: Algebraic optimization ; location theory ; ellipsoid algorithms
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    Notes: Abstract The Fermat-Weber location problem is to find a point in ℝ n that minimizes the sum of the (weighted) Euclidean distances fromm given points in ℝ n . In this work we discuss some relevant complexity and algorithmic issues. First, using Tarski's theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem, where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one-dimensional polynomial. Secondly, we exhibit an explicit solution to the strong separation problem associated with the Fermat-Weber model. This separation result shows that anε-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.
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    Mathematical programming 46 (1990), S. 255-256 
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    Keywords: Linear complementarity problem ; P-matrix
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    Notes: Abstract We give a short proof of the finiteness of Murty's principal pivoting algorithm for solving the linear complementarity problemy = Mz + q, y T z = 0,y ≥ 0,z ≥ 0 withP-matrixM.
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    Mathematical programming 47 (1990), S. 305-336 
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    Keywords: Trust region ; linear constraints ; convex constraints ; global convergence ; local convergence ; degeneracy ; rate of convergence ; identification of active constraints ; Newton's method ; sequential quadratic programming ; gradient projection
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    Notes: Abstract We develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem. Global convergence is established for general convex constraints while the local analysis is for linearly constrained problems. The main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations. As a consequence of the identification properties, we develop rate of convergence results by assuming that the step is a truncated Newton method. Our development is mainly geometrical; this approach allows the development of a convergence theory without any linear independence assumptions.
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    Mathematical programming 47 (1990), S. 425-439 
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    Keywords: Isotonic regression ; active sets
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    Notes: Abstract In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms. In this paper we consider the isotonic regression problem with respect to a complete order $$\begin{gathered} minimize\sum\limits_{i = 1}^n {w_i } (y_i - x_i )^2 \hfill \\ subject tox_1 \leqslant x_2 \leqslant \cdot \cdot \cdot \leqslant x_n \hfill \\ \end{gathered} $$ where eachw i is strictly positive and eachy i is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n 2). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eeden's method and show that it is of worst-case exponential time complexity.
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    Mathematical programming 47 (1990), S. 463-463 
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    Mathematical programming 48 (1990), S. 1-17 
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    Keywords: Coercivity ; quadratic programming ; linear programming ; aggregation ; relaxation ; multigrid methods
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    Notes: Abstract This paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces. The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described. The auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program. Coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates. Equivalents of this condition are presented in special cases.
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    Mathematical programming 49 (1990), S. 91-111 
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    Keywords: Sparse matrices ; linear programming ; bipartite matching
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    Notes: Abstract Many optimization algorithms involve repeated processing of a fixed set of linear constraints. If we pre-process the constraint matrixA to be sparser, then algebraic operations onA will become faster. We consider the problem of making a given matrix as sparse as possible, theSparsity Problem (SP). In a companion paper with S. Frank Chang, we developed some theoretical algorithms for SP under a non-degeneracy assumption (McCormick and Chang, 1988). Here we investigate what must be done to make those algorithms applicable in practice. We report encouraging computational results in making linear programming constraint matrices sparser. We also find that the Simplex Algorithm can solve the reduced LPs faster. Comparisons are made to a heuristic algorithm for SP of Adler et al. (1989).
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  • 59
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    Keywords: Quasi-Newton methods ; collinear scalings ; conic approximations ; local and q-superlinear convergence
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    Notes: Abstract This paper is concerned with collinear scaling algorithms for unconstrained minimization where the underlying local approximants are forced to interpolate the objective function value and gradient at only the two most recent iterates. By suitably modifying the procedure of Sorensen (1980) for deriving such algorithms, we show that two members of the algorithm class derived related to the DFP and BFGS methods respectively are locally and q-superlinearly convergent. This local analysis as well as the results they yield exhibit the same sort of “duality” exhibited by those of Broyden, Dennis and Moré (1973) and Dennis and Moré (1974) for the DFP and BFGS methods. The results in this paper also imply the local and q-superlinear convergence of collinear scaling algorithms of Sorensen (1982, pp. 154–156) related to the DFP and BFGS methods.
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    Mathematical programming 49 (1990), S. 139-141 
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    Keywords: Linear complementarity problem ; copositive ; strictly copositive ; Q-matrix
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    Notes: Abstract Recently, Jeter and Pye gave an example to show that Pang's conjecture, thatL 1 ⋂Q $$ \subseteq R_0 $$ , is false. We show in this article that the above conjecture is true for symmetric matrices. Specifically, we show that a symmetric copositive matrix is inQ if and only if it is strictly copositive.
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    Mathematical programming 49 (1990), S. 145-162 
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    Keywords: Linear programming ; interior-point methods ; primal—dual algorithms ; feasibility
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    Notes: Abstract A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a “big-M”, and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.
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    Mathematical programming 49 (1990), S. 231-251 
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    Keywords: Nonsmooth optimization ; subgradient ; ε-subgradient ; exact penalty function ; successive quadratic programming
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    Notes: Abstract In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.
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    Mathematical programming 49 (1990), S. 325-340 
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    Keywords: Convex quadratic programming ; interior point method ; Karmarkar's method ; logarithmic barrier function method
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    Notes: Abstract We present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach. This approach generates a sequence of problems, each of which is approximately solved by taking a single Newton step. It is shown that the method requires $$O(\sqrt n L)$$ iterations and O(n 3.5 L) arithmetic operations. By using modified Newton steps the number of arithmetic operations required by the algorithm can be reduced to O(n 3 L).
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    Mathematical programming 49 (1990), S. 371-379 
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    Keywords: Z-function ; strong solvability with nonnegativity ; strictly semimonotone function
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    Notes: Abstract This paper focuses on the relationship between the ‘strong’ solvability of a certain system involving a Z-function, and the strict semimonotonicity of such a function. Our main result shows that, for a system defined by a continuous, superhomogeneous Z-function, the additional condition of strict semimonotonicity is both necessary and sufficient for strong solvability.
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    Mathematical programming 49 (1990), S. 397-411 
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    Keywords: Concave functions ; knapsack problems ; strict minimizers ; NP-hard ; nonconvex ; local minimizers
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    Notes: Abstract We consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint. We characterize strict local miniimizers of concave minimization problems subject to linear constraints, and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is NP-hard, it is possible to determine a local minimizer of this problem with at most O(n logn) operations and 1+[logn] evaluations of the function. If the function is quadratic this algorithm requires at most O(n logn) operations.
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    Monatshefte für Mathematik 110 (1990), S. 115-116 
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    Notes: Abstract We point out that a construction due to J. Vaughan yields an example of a regularT 1-space that does not admit a uniformly regular quasi-uniformity.
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    Monatshefte für Mathematik 110 (1990), S. 101-114 
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    Notes: Abstract This paper establishes a heat semigroup version of Bernstein's theorem, applicable to any unimodular Lie group. The result has an intrinsic geometric content, involving estimates for the norms of the heat kernels for small time and large time. The theorem is stated in terms of certain Lipschitz spaces whose definition incorporates these two geometric features of the group in question. The geometric content is further underlined by showing that, in a certain sence, the theorem is best-possible.
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    Monatshefte für Mathematik 110 (1990), S. 117-136 
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    Notes: Abstract We investigate monotone semilinear equations in the setting of generalized functions. We derive existence and uniqueness results even in situations where no solution exists in the sense of distributions; next we show it is consistent with continuous solutions whenever they exist. Furthermore we prove that those transient generalized solutions stabilize toward steady state solutions the way weak solutions do. Lastly we point out an example how these techniques can be applied to different type of nonlinearities.
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    Monatshefte für Mathematik 110 (1990), S. 189-206 
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    Notes: Abstract Selection is often viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to relate these stability notions with asymptotic stability in the so-called “replicator dynamics”, by generalizing results, which are well-known for elementary situations, to a fairly general setting applicable, e.g. to complex populations. Moreover, a purely dynamical characterization of evolutionary stability and uninvadability is presented.
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    Monatshefte für Mathematik 110 (1990), S. 207-216 
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    Notes: Abstract We use the notion of self-similar sequences, introduced by the author in [1], to obtain in a natural way some sequences of points in the interval [0, 1] with the property: $$0〈 \ell \left( U \right)〈 L\left( U \right)〈 \infty $$ whereL(U) and ℓ (U) stand respectively for the lim sup and the lim inf of the ratioN·D N * (U)/LogN. HereD N * (U) is the star discrepancy of the sequenceU.
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    Monatshefte für Mathematik 110 (1990), S. 231-256 
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    Notes: Abstract Letn be the conductor of an abelian number fieldK. The numbersicot (πk/n), (k, n)=1, belong to, then-th cyclotomic field; theirK-traces form an additive group whose index in the “imaginary part” of the ringO K involves the relative class numberh K − ofK. This was shown previously. In the present paperh K − is decomposed into “branch factors”, each of which is shown to be the index of an additive group of modified cotangent numbers. Put together in the right way, the said numbers yield formulas forh K − simpler than the previous ones. The different types of cotangent numbers are mutually connected by Dirichlet convolution, whose meaning in the construction of cyclotomic numbers is studied. Finally, our results are rephrased in terms of Stickelberger ideals.
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    Monatshefte für Mathematik 110 (1990), S. 261-278 
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    Notes: Abstract The Convergence of Types Theorem on ℝ d is wellknown as an important tool for investigations on the limit behaviour of normalized sums or r.v. It is natural to look for a generalization for group-valued r.v. While for simply connected nilpotent Lie groups the Theorem is valid in general the existence of non-trivial compact subgroups causes problems. For compact extensions of nilpotent groups we prove restricted versions of the Convergence of Types Theorem.
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    Monatshefte für Mathematik 110 (1990), S. 283-295 
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    Notes: Abstract LetG be a locally compact Vilenkin group. We give a maximal function characterization of the weightedH p spaces overG, and give a Hörmander type multiplier theorem for these spaces.
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    Monatshefte für Mathematik 110 (1990), S. 297-314 
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    Notes: Abstract We consider the Robbins—Monro process with constant step-size in ℝ k and design confidence regions (via a comparison result) and a stopping time which is based on the inner products of consecutive gradients. We show that this stopping time leads to confidence regions which are uniform in the starting value.
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    Monatshefte für Mathematik 110 (1990), S. 315-319 
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    Notes: Abstract In this paper, the famous Lebesgue covering theorem and a useful completeness theorem ofFletcher andLindgren are extended to relator spaces, which are straightforward generalizations of Weil's uniform spaces.
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    Monatshefte für Mathematik 110 (1990), S. 321-327 
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    Notes: Abstract We explicitly compute the outer automorphism group Out π1 M of the fundamental group of the Hantzsche — Wendt manifoldM. It is an extension 1→(ℤ2)3→Outπ1 M→S 3⊕ℤ2→1, but not the semidirect product (ℤ2)3⋉(S 3⊕ℤ2) as claimed in [3] (see also [4]). As a consequence, we get a quick algebraic computation of the symmetry groups of the Borromean rings and the figure-8-knot.
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    Monatshefte für Mathematik 110 (1990), S. 3-13 
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    Notes: Abstract In this paper, we show that an entire function of a complex variable of slow growth assuming values very close to rational integers at the points of a geometric sequence is a polynomial.
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    Monatshefte für Mathematik 110 (1990), S. 15-21 
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    Notes: Abstract We give a formula for the Hausdorff dimension of fractals which are the support of certain Riesz-product type measures.
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    Monatshefte für Mathematik 110 (1990), S. 23-32 
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    Notes: Abstract We establish a criterion for the applicability of the finite section method to Toeplitz operators with continuous matrix-symbols on the Bergman space of the polydisk.
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    Monatshefte für Mathematik 110 (1990), S. 33-46 
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    Monatshefte für Mathematik 110 (1990), S. 47-61 
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    Notes: Abstract LetN(x, n, α) denote the number of integer lattice points inside then-dimensional sphere of radius (an)1/2 with center at x. This numberN(x,n, α) is studied for α fixed,n → ∞, andx varying. The average value (asx varies) ofN(x,n, α) is just the volume of the sphere, which is roughly of the form (2 βe, α) n/2. it is shown that the maximal and minimal values ofN (x,n, α) differ from the everage by factors exponential inn, which is in contrast to the usual lattice point problems in bounded dimensions. This lattice point problem arose separately in universal quantization and in low density subset sum problems.
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    Monatshefte für Mathematik 110 (1990), S. 79-88 
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    Monatshefte für Mathematik 110 (1990), S. 63-72 
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    Notes: Abstract Let $$T(x) = \sum\limits_{ord(G) \leqq x} {t(G),} $$ , wheret(G) define the number of direct factors of a finite Abelian group.E. Krätzel ([5]) defined a remainderΔ 1(x) in the asymptotic ofT(x) and proved $$\Delta _1 (x)〈〈 x^{{5 \mathord{\left/ {\vphantom {5 {12}}} \right. \kern-\nulldelimiterspace} {12}}} \log ^4 x.$$ Using two different methods to estimate a special three-dimensional exponential sum we get the better results $$\Delta _1 (x)〈〈 x^{{{282} \mathord{\left/ {\vphantom {{282} {683}}} \right. \kern-\nulldelimiterspace} {683}}} \log ^4 x$$ and $$\Delta _1 (x)〈〈 x^{{{45} \mathord{\left/ {\vphantom {{45} {109}}} \right. \kern-\nulldelimiterspace} {109}} + \varepsilon } (\varepsilon 〉 0).$$
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    Monatshefte für Mathematik 110 (1990), S. 73-78 
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    Topics: Mathematics
    Notes: Abstract It is shown by analytical means that, if one assumes the Riemann hypothesis, the asymptotic formula $$\sum\limits_{n \leqslant x} {\omega (n) = x 1n1n } x + B - x\int_l^{x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } {\frac{{\{ t\} }}{{t^2 (1n x - 1n t)}}dt + O(x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon } )} $$ holds. This improves a result ofB. Saffari, who got a weaker error term by using the Dirichlet “hyperbola method”. The above formula, in turn, implies the Riemann hypothesis.
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    Monatshefte für Mathematik 110 (1990), S. 89-100 
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    Topics: Mathematics
    Notes: Abstract Improved estimates are established regarding the accuracy which can be achieved by a suitable choice of generator in a single-generator lattice quadrature rule (as used in the “method of good lattice points”) in the general case wherem, the number of quadrature points, is not necessarily prime. The result obtained for the general case is asymptotically the same as the best currently-known result for the prime case. However, it is also shown that when these rules are applied to some customary test functions the mean error (over different rules with the same number of points) can be arbitrarily large compared to the corresponding mean value for rules with a comparable but prime value ofm. These mean values are of interest in relation to computerised searches for good generators.
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    Monatshefte für Mathematik 110 (1990), S. 137-144 
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    Notes: Abstract We study the behaviour under uncountable products of some structures of interest in the Frölicher-Kriegel calculus of differentiation theory: the Mackey closure topology, the compactly and sequentially generated topologies and bornologies. Some properties turn out to be independent from ZFC+(⌝CH).
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    Monatshefte für Mathematik 110 (1990), S. 145-166 
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    Topics: Mathematics
    Notes: Abstract We develop differentiation theory based on a new definition of derivative: in terms of integrals of curves. It applies to a very wide class of domains. Some ‘elementary’ maps between Fréchet spaces, excluded from calculus by previous theories, now emerge as smooth maps.
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    Monatshefte für Mathematik 110 (1990), S. 167-176 
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    Monatshefte für Mathematik 110 (1990), S. 177-188 
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    Topics: Mathematics
    Notes: Abstract LetM be a compact Riemannian manifold. The discrepancy of two measures onM can be defined by means of geodesic balls onM. It is shown that a sequence of positive measures converges weakly to an absolutely continuous measure (w.r.t. the volume measure ofM) if and only if the discrepancy converges to 0, and the geodesic balls with vanishingv-measure of the boundary constitute a convergence determining class of sets for weak convergence of a sequence of positive measures tov. Estimates for the distance of probability measures with respect to the Kantorovich metric in terms of the discrepancy are obtained.
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    Monatshefte für Mathematik 110 (1990), S. 217-230 
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    Topics: Mathematics
    Notes: Abstract We considers-number sequences {s n (T)} n=1 ∞ of linear and continuous operatorsT on Banach spaces and prove product formulas for the operator ideals ℒ p,u (s) . Furthermore, we investigate the relationship between the eigenvalues of a Riesz operator and its Hilbert numbers.
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    Monatshefte für Mathematik 110 (1990), S. 257-259 
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    Topics: Mathematics
    Notes: Abstract LetN 1,N ∞ be thel 1 andl ∞ norms on ℝ n . We denote by $$\mathcal{N}$$ the set of all normsN on ℝ n such thatN ∞≤N≤N 1. The aim of the paper is to present a characterization of the extreme points of $$\mathcal{N}$$ .
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    Monatshefte für Mathematik 110 (1990), S. 279-282 
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    Topics: Mathematics
    Notes: Abstract The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Based on this theorem we will prove a series of inequalities connecting the product of certain successive minima with certain intrinsic volumes.
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    Monatshefte für Mathematik 110 (1990), S. 348-348 
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    OR spectrum 12 (1990), S. 24-24 
    ISSN: 1436-6304
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    Topics: Mathematics , Economics
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    OR spectrum 12 (1990), S. 42-42 
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    OR spectrum 12 (1990), S. 25-34 
    ISSN: 1436-6304
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    Topics: Mathematics , Economics
    Description / Table of Contents: Summary The Complex-Method of linear programming is a tool in computer aided planning and Decision-Support-Systems. This method qualifies the management for finding bottlenecks and disproportions and for computing optimal productivity by means of optimal variation of the production-, prices- and salesconditions. The paper describes its philosophy of application and the algorithms. The Complex-Method demands an effective Human-Computer-Dialogue, its software KOLIOP at personal computer is discussed. Finally experiences and an example are given.
    Notes: Zusammenfassung Die Komplexmethode der linearen Optimierungist ein Instrument der interaktiven computergestützten Planung und Entscheidungsvorbereitung. Mit ihr kann in den Unternehmen die Erfüllbarkeit betriebswirtschaftlicher Zielvorgaben durch eine schrittweise Berechnung optimaler Modellverbesserungen analysiert und somit eine mathematisch begründete Entscheidungsvorbereitung gegeben werden. Die Komplexmethode befähigt die Unternehmensführung, Engpässe und Disproportionen aufzudecken und mit gezielten Veränderungen der Produktions-, Preis- und Absatzbedingungen den größtmöglichen Produktionseffekt zu ermitteln. In dieser Arbeit wird ihre Anwendungsphilosophie und Algorithmierung erläutert. Die Komplexmethode nutzt effektiv den Mensch-Computer-Dialog im Planungsprozeß, ihre rechentechnische Realisierung mit der Software KOLIOP auf Personal-computer wird vorgestellt. Anwendungserfahrungen und ein Anwendungsbeispiel beschließen die Arbeit.
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    OR spectrum 12 (1990), S. 1-23 
    ISSN: 1436-6304
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    Topics: Mathematics , Economics
    Description / Table of Contents: Summary This paper gives a review of some new developments in the field of economic quality control achieved since 1980. The main results are a generally applicable approach to optimal control of production processes together with the derivation of powerful and at the same time simple methods to solve the problem of determining the optimal control procedure for a given situation, and the development of a new generation of economic single sampling plans for product control, which take into consideration the nowadays tightened process control and consequently improved quality of incoming lots.
    Notes: Zusammenfassung Dieser Artikel gibt eine Übersicht über einige neue Entwicklungen auf dem Gebiet der wirtschaftlichen Qualitätskontrolle. Die Hauptresultate sind ein allgemein anwendbares Konzept zur optimalen Prozeßkontrolle zusammen mit der Herleitung leistungsstarker und zugleich einfacher Methoden zur Bestimmung optimaler Kontrollpläne für eine gegebene Situation und die Entwicklung einer neuen Generation von einfachen Stichprobenplänen für die Produktkontrolle, die die heutige, schärfere Prozeßkontrolle und damit die verbesserte Qualität der angelieferten Warenpartien berücksichtigt.
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    OR spectrum 12 (1990), S. 35-41 
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    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Es wird ein stochastisches Modell für die Zirkulation eines wiederverwendbaren Materials untersucht. Dabei wird angenommen, daß zu festen Zeitpunktenn=1,2,... NachfragenX n auftreten, von denen am Ende der jeweiligen Periode ein fester Prozentsatz zum Verteiler zurückkehrt. Ein Beispiel ist das Pfandflaschensystem. WennX 1,X 2,... unabhängige, identisch verteilte Zufallsvariablen sind undY n die Menge des bereits verwendeten, aber wieder einsetzbaren Materials zur Zeitn bezeichnet, bildet (Y n)n ⩾ 1 eine Markovkette, die im einzelnen untersucht wird. Für gleichmäßig oder exponentiell verteilteX n wird die Grenzverteilung vonY n explizit bestimmt; im exponentiellen Fall werden auch geschlossene Formeln für die Verteilungsfunktion vonY n hergeleitet.
    Notes: Summary A stochastic model for the circulation of restorable materials is presented. A fixed portion of the material which is distributed at then-th time instant is assumed to be returned at the end of then-th period and can thereafter be redistributed. LetX n be the quantity needed at timen. If theX n are i.i.d. random variables, the accumulated amountY n of returned material forms a Markov chain which is analyzed in detail. For uniform and exponentialX n the asymptotic distribution ofY n is found explicitly. In the exponential case also closed formulas for the distribution function ofY n are given.
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    OR spectrum 12 (1990), S. 50-50 
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    Topics: Mathematics , Economics
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