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  • 1
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    Bulletin of mathematical biology 45 (1983), S. 287-293 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We postulate that the biomass distribution function for an ecological population may be derived from the condition that the biomas diversity functional is maximal subject to an energetic constraint on the total biomass. This leads to a biomass distribution of the form $$p(m) = \bar m^{ - 1} \exp ( - m/\bar m)$$ , where $$\bar m$$ is the mean biomass per individual. The same condition yields a unique value for the biomass diversity functional. These predictions are tested against fishery data and found to be in good agreement. It is argued that the existence of a unique value for biomass diversity may provide a preliminary theoretical foundation for the observed upper limit to species diversity.
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    Bulletin of mathematical biology 45 (1983), S. 311-321 
    ISSN: 1522-9602
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    Notes: Abstract Pigment distribution presages hydranth regeneration in the marine hydroidTubularia. We suggest that such a distribution could result from a reaction-diffusion system. A model system based on a practical reaction scheme is studied and spatial structures found which closely resemble this pigment distribution. Finite-amplitude spatial structures in reaction-diffusion systems are considered. Whereas in one spatial dimension the final structures are normally very similar to the transient patterns which emerge from a linear analysis, it is shown that in more than one dimension this is not necessarily the case. The reasons for this are discussed.
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    Bulletin of mathematical biology 45 (1983), S. 409-424 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract An analytical model is used to described the behavior of inhaled particulate matter in the human respiratory tract. Three different geometries, symmetric and asymmetric, are utilized to simultate the tracheobronchial (TB) tree. The suitability of each geometry for representing the human is evaluated by comparing calculated aerosol deposition probabilities with experimental data from inhalation exposure tests. A symmetric, dichotomously branching pattern is found to be a reliable description of the TB tree for studies of factors affecting aerosol deposition in the human lung. Calculations with the theoretical model are in excellent agreement with measured aerosol deposition efficiencies. Furthermore, the model accurately predicts experimentally observed features of inhalation exposure data, such as effects of inter-subject lung morphology differences and relative efficiencies of specific deposition mechanisms, on aerosol deposition patterns in the TB tree.
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    Bulletin of mathematical biology 45 (1983), S. 436-436 
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    Bulletin of mathematical biology 45 (1983), S. 437-437 
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    Bulletin of mathematical biology 45 (1983), S. 579-590 
    ISSN: 1522-9602
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    Notes: Abstract In this paper we are concerned with problems of the long-term behavior for nonlinear systems in random environment. The general model is assumed to be given by an ordinary differential equation with random parameters or random input. The disturbance process can be taken from a fairly general class of Markov processes having a bounded state space. In terms of the system’s dynamics we give sufficient conditions for the existence and uniqueness of invariant probabilities. Finally, we apply these results to the two-dimensional biochemical model which is known as the Brusselator.
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    Bulletin of mathematical biology 45 (1983), S. 571-577 
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    Notes: Abstract In various applications one faces the problem of estimating a signal from discontinuous observations. For example, in biomedical applications the signal may be the ‘state’ of a given organ and one observes through an external counter the amount of radioactivity sequestered by the organ after injection of a radioactive tracer. Here the problem is studied in the context of nonlinear filtering when the signal can be modelled as either a random variable or a diffusion process, and the observations have a continuous and a purely discontinuous component; both components may be affected by the signal. When the signal is a random variable an explicitly computable solution is obtained; for the diffusion case the solution is given as a sequence of approximating filters that can be computed recursively.
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    Bulletin of mathematical biology 45 (1983), S. 627-634 
    ISSN: 1522-9602
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    Notes: Abstract Eigenvalue problems arise in various biological models. We outline a useful comparison method and a technique using Lyapunov functions that can be applied in many cases. An application to lateral diffusion is discussed.
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    Bulletin of mathematical biology 45 (1983), S. 605-616 
    ISSN: 1522-9602
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    Notes: Abstract This paper reviews, up to their recent developments, two types of models of the cell cycle: those considering the size controls over the cycle events and the transition probability models. The distribution of inter-mitotic time and the sister-sister and motherdaughter correlations implied by the two approaches are discussed in view of some relevant experimental data.
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    Bulletin of mathematical biology 45 (1983), S. 617-626 
    ISSN: 1522-9602
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    Notes: Abstract The development of a blood cell line originating from a pluripotent stem cell pool is modelled by a chain of multidimensional branching processes in which the sojourn times of the cells in certain resting states depend on the size of the following subpopulation. The stability of such a model is discussed qualitatively and some considerations concerning a possible malignant degeneration are presented. The behaviour of models for normal and malignant cell production are illustrated by stochastic stimulations. The model presented here describes the development of a certain line of blood cells (e.g. erythrocytes, monocytes or granulocytes) originating from the pluripotent stem cell up to the functional cell in the blood (for related models see, e.g., Rubinow and Lebowitz,J. math. Biol. 1, 87–225;Biophys. J. 16, 897–910).
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    Bulletin of mathematical biology 45 (1983), S. 635-641 
    ISSN: 1522-9602
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    Notes: Abstract This paper reviews some recent advances in single population stochastic differential equation growth models. They are a natural way to model population growth in a randomly varying environment. The question of which calculus, Itô or Stratonovich, is preferable is addressed. The two calculi coincide when the noise term is linear, if we take into account the differences in the interpretation of the parameters. This clarifies, among other things, the controversy on the theory of niche limiting similarity proposed by May and MacArthur. The effects of correlations in the environmental fluctuations and statistical methods for estimating parameters and for prediction based on a single population trajectory are mentioned. Applications to fisheries, wildlife management and particularly to environmental impact assessment are now becoming possible and are proposed in this paper.
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    Bulletin of mathematical biology 45 (1983), S. 643-658 
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    Notes: Abstract A survey is given of the application of (functions of) continuous-time Markov chains in the statistical analysis of behavioural time series.
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    Bulletin of mathematical biology 45 (1983), S. 659-659 
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    Bulletin of mathematical biology 45 (1983), S. 661-664 
    ISSN: 1522-9602
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    Notes: Abstract This paper demonstrates that there is one and only one solution to a non-linear singular two-point boundary-value problem which describes oxygen diffusion in a spherical cell. Previous authors have calculated numerical results that differ substantially. Numerical computations using the multiple shooting method support the results of McElwain.
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    Bulletin of mathematical biology 45 (1983), S. 665-720 
    ISSN: 1522-9602
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    Notes: Abstract The mathematics of distance geometry constitutes the basis of a group of algorithms for revealing the structural consequences of diverse forms of information about a macromolecule's conformation. These algorithms are of proven utility in the analysis of experimental conformational data. This paper presents the basic theorems of distance geometry in Euclidean space and gives formal proofs of the correctness and, where possible, of the complexity of these algorithms. The implications of distance geometry for the energy minimization of macromolecules are also discussed.
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    Bulletin of mathematical biology 45 (1983), S. 721-737 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A fully developed pulsatile flow in a circular rigid tube is analysed by a microcontinuum approach. Solutions for radial variation of axial velocity and cell rotational velocity across the tube are obtained using the momentum integral method. Simplified forms of the solutions are presented for the relevant physiological data. Marked deviations in the results are observed when compared to a Newtonian fluid model. It is interesting to see that there is sufficient reduction in the mass flow rate, phase lag and friction due to the micropolar character of the fluid.
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    Bulletin of mathematical biology 45 (1983), S. 749-758 
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    Notes: Abstract A mathematical model of the transport of fluorescein across the blood-retina barrier in the transient state and the subsequent diffusion of fluorescein in the vitreous body is presented. The function of the barrier is lumped in a single parameter—the permeability. The sensitivity of this parameter due to changes in the other parameters of the model is given. This establishes the foundation for the quantitative assessment of the barrier function through vitreous fluorophotometry.
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    Bulletin of mathematical biology 45 (1983), S. 739-748 
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    Notes: Abstract The objective of this preliminary study was to develop a new quantitative method of setting the initial insulin infusion patterns in treatment of diabetic patients. The method is based upon the mathematical estimation of the insulin profile required to maintain the glucose level within the normal range after glucose loading in diabetic patients. Using our previously developed equivalent circuit model of glucose kinetics and the reported data of an intravenous glucose tolerance test (IVGTT) in two groups of normal and diabetic patients, two important physiological parameters of the model (the peripheral tissue's insulin resistivity and the hepatic sensitivity to glucose level) were computed for two clinical groups. Then the insulin profile was obtained by computing the plasma insulin concentrations required to keep the total glucose utilization rate of the tissue and the liver in the diabetic group equal to that of the normal group. The simulation result indicated that the computed insulin profile produced a plasma glucose profile which was more closely matched to the normal group's glucose profile than with the case of emulating the normal group's insulin profile in the diabetic group.
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    Bulletin of mathematical biology 45 (1983), S. 759-780 
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    Notes: Abstract This paper shows that the Na conductance changes can be explained quantitatively, based on the following assumptions: (1) there exist in nerve membranes the electron transfer (ET) complexes and traps, (2) there is energy migration among them. The gating mechanism is explained in physical terms. Its mathematical expression differs from the Hodgkin-Huxley equations, but resembles the Hoyt formulation. In the present model, the physical parameters for the squid axon can be estimated from currently available experimental data. The density of the ET complexes is on the order of 105/μm2, and the density of the traps is 103/μm2. The magnitude of the energy transfer rate between ET complexes is about 106/sec at large depolarization and decreases with decreasing depolarizations, as does the Na inactivation rate. The energy gap between the two stable states of the transfer electron in the ET complex is estimated to be around 0.1 eV, which is approximately the same as that for the photosynthetic systems.
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    Bulletin of mathematical biology 45 (1983), S. 781-792 
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    Notes: Abstract The role of symmetry in simplifying the theory of complex neural systems is argued. When the structural symmetries of a network are expressed as an ismorphism group, implications emerge for the dynamics. Various qualitative possibilities concerning stability of uniform motion in homogeneous nets are discussed and an approach to neural hierarchies is outlined.
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    Bulletin of mathematical biology 45 (1983), S. 793-805 
    ISSN: 1522-9602
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    Notes: Abstract By constructing appropriate Liapunov functionals, asymptotic behaviour of the solutions of various delay differential systems describing prey-predator, competition and symbiosis models has been studied. It has been shown that equilibrium states of these models are globally stable, provided certain conditions in terms of instantaneous and delay interaction coefficients are satisfied.
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    Bulletin of mathematical biology 45 (1983), S. 807-826 
    ISSN: 1522-9602
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    Notes: Abstract Sensitivity analyses have been used to examine the flow structure of two hypothetical ecosystem models. These analyses have results which relate to important aspects of ecosystem theory. Cycles are shown to increase the sensitivity of the network, while increased throughflow is shown to decrease the sensitivity. Such results indicate that several factors can be modified to decrease the sensitivity of ecosystems to environmental stress.
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    Bulletin of mathematical biology 45 (1983), S. 827-836 
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    Notes: Abstract A continous, deterministic mathematical model is used to predict population distributions by age at any time, given the initial distribution and the variation of birth and death rates with age and time. Solutions are obtained on a computer using a semi-discretization algorithm in which time derivatives in the partial differential equations are replaced by finite-difference expressions. The resulting sets of ordinary differential equations are solved by a predictor-corrector method. Graphical results are shown for some examples.
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    Bulletin of mathematical biology 45 (1983), S. 849-855 
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    Notes: Abstract A new formula for the complexity of graphs is proposed and applied to the points lines and ‘connections’ of some chemically relevant graphs.
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    Bulletin of mathematical biology 45 (1983), S. 837-847 
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    Notes: Abstract This paper reports general and specialized results on analytical solutions to the governing phenomenological equations for chemotactic redistribution and population growth of motile bacteria. It is shown that the number of bacteria cells per unit volume,b, is proportional to a certain prescribed function ofs, the concentration of the critical substrate chemotactic agent, for steady-state solutions through an arbitrary spatial region with a boundary that is impermeable to bacteria cell transport. Moreover, it is demonstrated that the steady-state solution forb ands is unique for a prescribed total number of bacteria cells in the spatial region and a generic Robin boundary condition ons. The latter solution can be approximated to desired accuracy in terms of the Poisson-Green's function associated with the spatial region. Also, as shown by example, closed-form exact steady-state solutions are obtainable for certain consumption rate functions and geometrically symmetric spatial regions. A solutional procedure is formulated for the initialvalue problem in cases for which significant population growth is present and bacteria cell redistribution due to motility and chemotactic flow proceeds slowly relative to the diffusion of the chemoattractant substrate. Finally, a remarkably simple exact analytical solution is reported for a stradily propagating plane-wave which features motility, chemotactic motion and bacteria population growth regulated by substrate diffusion.
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    Bulletin of mathematical biology 45 (1983), S. 857-867 
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    Notes: Abstract This paper discusses the flow of blood in large artries under the influence of linear periodic acceleration. The governing equations and boundary conditions are established and analytical solutions for the velocity, fluid acceleration, bulk flow and shear stress are obtained. The results for these physical quantitites are computed for the case of an artery the size of a normal human aorta. It is found that the flow field variables are directly proportional to the external accelerating force. The behaviour of the velocity profile along the radial distance at different stages of times at fixed applied acceleration is also shown.
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    Bulletin of mathematical biology 45 (1983), S. 931-968 
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    Notes: Abstract The evolutionary selection circuits model of learning has been specified algorithmically. The basic structural components of the selection circuits model are enzymatic neurons, that is, neurons whose firing behavior is controlled by membrane-bound macromolecules called excitases. Learning involves changes in the excitase contents of neurons through a process of variation and selection. In this paper we report on the behavior of a basic version of the learning algorithm which has been developed through extensive interactive experiments with the model. This algorithm is effective in that it enables single neurons or networks of neurons to learn simple pattern classification tasks in a number of time steps which appears experimentally to be a linear function of problem size, as measured by the number of patterns of presynaptic input. The experimental behavior of the algorithm establishes that evolutionary mechanisms of learning are competent to serve as major mechanisms of neuronal adaptation. As an example, we show how the evolutionary learning algorithm can contribute to adaptive motor control processes in which the learning system develops the ability to reach a target in the presence of randomly imposed disturbances.
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    Bulletin of mathematical biology 45 (1983), S. 981-990 
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    Notes: Abstract In the present paper we discuss the behaviour of solutions of a dynamical system describing the growth of cells in a well-mixed continuous culture where the supply of the growth-limiting nutrient depends on the activity of an enzyme outside the cell membrane. It turns out that for positive dilution rates there exists an exponentially attractive two-dimensional simplex. Furthermore, the reversed system restricted to this simplex is quasimonotone. In every case all trajectories tend to an equilibrium state.
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    Bulletin of mathematical biology 45 (1983), S. 991-1004 
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    Notes: Abstract We present a Gause predator-prey model incorporating mutual interference among predators, a density-dependent predator death rate and a time lag due to gestation. It is well known that mutual interference is stabilizing, whereas time delays are destabilizing. We show that in combining the two, a long time-lag usually, but not always, destabilizes the system. We also show that increasing delays can cause a bifurcation into periodic solutions.
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    Bulletin of mathematical biology 45 (1983), S. 969-980 
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    Notes: Abstract The cycle structure of enzymatic neural networks may be characterized in terms of number of cycles exhibited, size of cycle state sets and cycle lengths. Simulation experiments show that the stability properties of these networks have some unusual features which are not exhibited by networks of two-state switching elements or by randomly constructed ecosystem models. The behavioral and structural stability of these systems decreases with their structural complexity, as measured by the number of components. The behavioral and structural stability of enzymatic neural networks also decreases with structural complexity, as measured by the number of excitase types, but only up to the middle level of excitases per neuron. This is the point of highest potential responsiveness of the system to environmental stimuli. Beyond this point the behavioral and structural stability increase. This is due to the fact that the number of possible states increases up to this point and decreases beyond it. The number of possible states, not the number of components, serves as the useful measure of complexity in these types of systems. The selection circuits learning algorithm has been used to evolve networks whose cycle structures have desired features.
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    Bulletin of mathematical biology 45 (1983), S. 1005-1011 
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    Notes: Abstract Similarity criteria of the functional design of the mammalian cardiovascular system are scant. For the analysis of mammalian cardiac energetics physiological parameters such as mean arterial blood pressure, stroke volume, heart rate, metabolic rate and heart and body weights are considered pertinent. Based on these parameters, a new similarity principle is established via allometric equations, dimensional analysis and Buckingham's pi-theorem. The principle states that the ratio of left ventricular external work to metabolic rate is inversely proportional to resting heart rates of mammals. The proportionality constant is dimensionless and is invariant of mammalian body weights.
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    Bulletin of mathematical biology 45 (1983), S. 1029-1045 
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    Notes: Abstract The mathematical theory of categories is used as a tool in the description of the structure and function of natural systems. The connections between the category of natural systems, with observables and dynamics, and the phenomenological calculus of response tensors, duality- and adjoint-invariance diagrams are established. The unified theory is applied to the analysis of hierarchies, pattern generation and the structure and dynamics of proteins.
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    Bulletin of mathematical biology 45 (1983), S. 1047-1072 
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    Notes: Abstract This is an investigation of natural systems from the standpoint of the mathematical theory of categories. It examines the relationships which exist between different descriptions through measurement of observables and dynamical interactions. We begin with a category theory of formal systems with observables, and then proceed to a category theory of dynamical systems. The two categories are then combined to represent natural systems. Topological considerations enter in the study of stability and bifurcation phenomena. Special emphasis is placed on natural systems which model biological processes. The categorical system theory developed is applied to the analysis of several biological problems and biological system theories.
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    Notes: Abstract Tetanic hyperpolarization for theXenopus node is simulated by means of iterative solutions of the Frankenhaeuser-Huxley excitation equations together with an active transport current density term which is dependent on sodium and potassium levels as well as the ADP/ATP ratio. All time-dependent variables at the end of one interspike interval are introduced as initial conditions for the next response, whereupon all time-dependent changes in voltage and permeability factors appear identical for the third and fourth responses of a sequence. Net change in internal sodium concentration is zero throughout the third and fourth intervals if sodium loading of the system is initially adjusted to a critical level. Extent of tetanic hyperpolarization is a function of the pump conductance.
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    Bulletin of mathematical biology 45 (1983), S. 1097-1097 
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    Bulletin of mathematical biology 45 (1983), S. 1073-1096 
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    Notes: Abstract The properties of nonlinear equations describing the solute and solvent transport across a simplified Patlak-Goldstein-Hoffman model (two membranes in series without unstirred layers) are investigated both analytically and numerically. The analysis shows that the principal coefficients measured in transport experiments in the presence of active transport are dependent on the experimental conditions. These ‘apparent’ system parameters are extensions of the corresponding parameters determined both in passive systems and in the linear Kedem-Katchalsky theory. Moreover, they are related to the local phenomenological coefficients of the single membranes of the array. Several relationships between measurable quantities and the local system parameters are indicated, allowing the planning of experiments aimed at the measurement of the latter. Data in the literature have been used to check the proposed volume flow equation.
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    Bulletin of mathematical biology 45 (1983), S. 139-142 
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    Notes: Abstract As an alternative to optimum-processor models in which sensors attempt to circumvent internal and external noise, a mechanism-independent argument is presented for Weber's law in vision and hearing. In vision, the argument is that categories of objects should be independent of the light intensity on these objects. In hearing, sound categorization should be independent of the distance from the sound source. An analogous desideratum for computer-based image segmentation is also presented.
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    Bulletin of mathematical biology 45 (1983), S. 193-207 
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    Notes: Abstract Rashevsky's treatment of general binary relations between sets of biological elements is extended using the novel mathematical concept of lattice-valued relation (l.v.r.). This yields a quantitative measure of the strength of the relations between components of a biological organism, and some illustrative examples are given. Specific l.v.r.'s are used to define (more precisely than in Rashevsky's preliminary theory of binary relations) the biologically important relationships amongst hormones, metabolism and energy exchange involved in metabolic reactions. The ‘strongest link’ between the set of hormones and the set of metabolic reactions is quantified using a special l.v.r., and other specific biological realisations of lattice-valued relations in abstract-relational biology are presented. L.v.r.'s may also be regarded as a form ofG-relation in relational biology, or as a particular case of generating diagrams. Further possible developments of this approach, using more complex tools of the newly developed mathematical theory of lattice-valued relations, such as function space l.v.r., group l.v.r., l.v.r. morphisms, l.v.r. homology andn-ary l.v.r.'s are suggested.
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    Bulletin of mathematical biology 45 (1983), S. 259-267 
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    Notes: Abstract The effects of the viscosity-concentration dependence and of the concentration profile on blood flow through a vessel with stenosis have been studied. The flow resistance and the wall shear stress have been found to be smaller than in the two-fluid model with constant viscosities.
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    Bulletin of mathematical biology 45 (1983), S. 507-519 
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    Notes: Abstract A survey is given of branching process type methods in cell kinetics. Some results are given that allow circadian rhythm and do not require complete independence between cells. Some more classical results on balanced exponential growth are given and some comments are made on flow microfluorometry.
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    Bulletin of mathematical biology 45 (1983), S. 439-442 
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    Bulletin of mathematical biology 45 (1983), S. 443-465 
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    Notes: Abstract The spike train activity of neurones is considered as a point process, and methods of analysing and interpreting recorded spike trains are considered. The generation of a continuous process (membrane noise) from interacting point processes is described.
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    Bulletin of mathematical biology 45 (1983), S. 521-554 
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    Notes: Abstract Stochastic models of population genetics are studied with special reference to the biological interest. Mathematical methods are described for treating some simple models and their modifications aimed at the problems of the molecular evolution. Unified theory for treating different quantities is extensively developed and applied to some typical problems of current interest in genetics. Mathematical methods for treating geographically structured populations are given. Approximation formulae and their accuracy are discussed. Some criteria are given for a structured population to behave almost like a panmictic population of the same total size. Some quantities are shown to be independent of the geographical structure and their dynamics are described.
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    Bulletin of mathematical biology 45 (1983), S. 11-20 
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    Notes: Abstract This paper is concerned with a generalization of the simple epidemic model in which the infective population is partitioned intom classes, each of specific infectiousness. Attention is restricted, however, to the case where all the meeting rates between two individuals are equal to each other. Both deterministic and stochastic versions are examined. In either case the development in time of the epidemic process is investigated by exploiting a connection with the standard simple epidemic model. Finally, it is shown that the technique used also applies to a similar model for the spread of information.
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    Bulletin of mathematical biology 45 (1983), S. 33-40 
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    Notes: Abstract For each rooted binary tree witht labeled terminal vertices (leaves) a natural number can be assigned uniquely. Unrooted trees witht labeled terminal vertices andt-2 unlabeled internal vertices of degree 3 can also be numbered uniquely using the same convention. Rooted trees in which the hights of the internal vertices are rank ordered are also considered. Applications to problems in taxonomy are discussed.
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    Bulletin of mathematical biology 45 (1983), S. 41-50 
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    Notes: Abstract Expressions and numerical values for hematocrit reduction are calculated as blood flows from a cylindrical feeding tube into a cylindrical capillary at a right-angle branch. Blood is considered to consist of two Newtonian fluids, plasma and red cell suspension, which have equal densities but different viscosities. The concentration profile of the red cells is concluded to depend on the size of the feeding tube. An estimate for the thickness of the plasma layer adjacent to the wall is obtained.
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    Bulletin of mathematical biology 45 (1983), S. 51-67 
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    Notes: Abstract Asimple model system of two self-reproducing objects is considered. A set of equations, similar to Eigen's equation, describing competition of these objects is derived and analyzed under the effect of an ‘ecological constraint’. The relation with other constraints used in the literature is discussed.
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    Bulletin of mathematical biology 45 (1983), S. 69-90 
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    Notes: Abstract A membrane with an arbitrary distribution of fixed charges inside and on its surfaces is considered. A procedure for calculating the local electrostatic potential at an arbitrary point of the system is described and its validity discussed. This procedure is based on the linearization of the 3-dimensional Poisson-Boltzmann equation around an exact 1-dimensional solution.
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    Bulletin of mathematical biology 45 (1983), S. 103-116 
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    Notes: Abstract In this, the first of a series of papers on stochastic and deterministic non-linear allometric growth models, a deterministic model is proposed which generalizes the widely applicable classical linear model of Huxley and Needham. There aren types of producers, each type depositing a product which accumulates monotonically in the environment. Producers interact via a mass action law satisfying an optimality condition. Coefficients may be interpreted as competition between the various producer types in the usual Volterra sense. An ideal coral reef is studied in which then species of coral polyps lay down aragonite calcium carbonate in building the reef framework. This deterministic model predicts that younger reefs are strongly unstable relative to initial species abundance, while older reefs grow in the classical sense of Huxley and Needham, asymptotically, as time goes to infinity.
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    Bulletin of mathematical biology 45 (1983), S. 151-153 
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    Notes: Abstract First-order spatial gradients are reliquished in the Schrödinger-Bloch equation for bacterial chemotaxis if and only if the flux coefficient-motility ratio equals 2, the precise value measured in recent experiments onEscherichia coli attracted by oxygen. Moreover, for δ/μ=2 the Schrödinger-Bloch function Ψ is simply equal to the number of bacteria cells per unit volume divided by the chemoattractant concentration.
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    Bulletin of mathematical biology 45 (1983), S. 169-192 
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    Notes: Abstract A mathematical formulation useful in characterizing the effects of light on the pupal eclosion rhythm of the fruit flyDrosophila pseudoobscura is presented. It is based upon the premise that the underlying pacemaker oscillator behaves in a manner analogous to a customized version of a phase-locked loop circuit. Theoretical analyses supplemented with numerical simulations reveal that this phase-locked loop approach yields a concise mathematical characterization which is quite comprehensive in its scope, yet surprisingly accurate in the detail with which it can be used to successfully predict observed experimental results.
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    Bulletin of mathematical biology 45 (1983), S. 155-168 
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    Notes: Abstract Pointwise upper and lower bounds for the solution of a class of nonlinear problems arising in the steady-state finite cable model of cell membranes are presented. Simple analytical bounding curves are obtained for an illustrative example in the theory of nerve membranes.
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    Bulletin of mathematical biology 45 (1983), S. 209-227 
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    Notes: Abstract Single-species reaction-diffusion models are analyzed to determine the effect of various diffusion mechanisms on species persistence or extinction.
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    Bulletin of mathematical biology 45 (1983), S. 401-408 
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    Notes: Abstract A restriction is imposed on the number of particles that can possibly move at any time from a compartment, so that any other particles present in the compartment must wait until such particles have moved out. The equations for such a system are formulated and the solution is given for a single compartment system; increased variability of the compartmental particle count is one effect of this restriction.
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    Bulletin of mathematical biology 45 (1983), S. 323-345 
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    Notes: Abstract Two classes of dose-effect models for cell killing (additive damage and independent effects) are developed under alternative hypotheses about the damage that leads to cell death. Generalized models, along with specific models for cell killing after exposure to a specific cytotoxicant, are used to make predictions about the effects of sequential or simultaneous exposure to different cytotoxicants. It is demonstrated that with the additive damage models developed one can adequately account for the combined effects of the cytotoxicants considered. Theoretical results are presented which suggest that after simultaneous exposure of cells to low total doses (〈0.1 Gy) of different ionizing radiations, use of the conventional relative biological effectiveness approach to predict cell killing risks is unnecessary; cell killing risks can adequately be determined by assuming the effects of the different radiations to be independent. Also, for simultaneous exposure of cells to total doses of different radiations much larger than 0.1 Gy, use of the conventional RBE approach to arrive at cell killing risk could lead to overestimation of the risk.
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    Bulletin of mathematical biology 45 (1983), S. 347-400 
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    Notes: Abstract A predator which is preying on a model-mimic system can choose either the single-trial strategy or a multi-trial strategy as its behavior in learning to prudently harvest such a prey system. In this learning behavior, an important and often-posed problem is to determine which among these two strategies is better suited for the predator and why one is preferable over the other. We present in this article, using Markov chain methods, an extensive analysis of these strategies (and also of eat-everything, strategy). We conclude that the multi-trial strategy is the one that the predator should adopt (but we will also describe the situations when the single-trial strategy seems to be better). Our conclusions are based on the comparisons of quantities such as the mean benefit to the predator, energy derived by a predator from the model-mimic system and (a newly introduced notion of) contagion in eating mimics and models (these quantities are computed for different strategies). The first two quantities are functions of the abundancep and noxiousnessb of models. The contagion is a function of onlyp; and, though independent ofb, it is also in support of multi-trial, strategy. We introduce, in the present context, a biological analog of the d'Alembert principle and also derive functions describing the influences of eating a specified type of prey at a given time on eating any type of prey at a later time. Various results of Estabrook-Jespersen (single-trial strategy) and Bobisud-Potratz (multi-trial strategy) are re-derived as special cases of our more general results. A central limit theorem under the eat-everything strategy is given.
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    Bulletin of mathematical biology 45 (1983), S. 431-435 
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    Bulletin of mathematical biology 45 (1983), S. 425-430 
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    Notes: Abstract We consider the one-dimension (one-compartment) exponential model using a diffusion process approach. In particular, we summarize the known results in the case where the stochastic component of the model is a Gaussian white noise process with mean zero and variance σ2. Finally, we briefly illustrate a number of cases where similar forms of model arise.
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    Bulletin of mathematical biology 45 (1983), S. 467-482 
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    Notes: Abstract The surfaces of many cells are viscous fluids; consequently, most membrane proteins are able to diffuse laterally, in a more or less random fashion, with diffusion coefficients typically of order 10−10 cm2/sec. If a molecule (ligand) in solution outside the cell and a protein molecule on the surface (receptor) each have two or more sites at which they can interact with one another, large, branched receptor-ligand networks can form on the cell surface by virtue of the chemical interactions that surface fluidity permits. Evidence from a variety of systems indicates that such receptor clustering plays a role in the sequence of events leading to cellular activity. This paper describes a number of mathematical problems that arise in the analysis of experiments in which clustering occurs. I begin by reviewing methods for finding the time evolution of the cluster size distribution function in terms of reaction rate constants. The methods solve an essentially infinite system of coupled nonlinear differential equations. Next, the rate constants are analyzed, the Brownian motion problems that arise in attempting to understand ligand recognition are described and relevant experimental systems are discussed. Finally the notion of ligand as a signal amplifier is introduced—an idea that emerges naturally from the requirement that receptors be clustered for a finite amount of time before a signal can be transmitted.
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    Bulletin of mathematical biology 45 (1983), S. 555-569 
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    Notes: Abstract The covariances among the populations of distinct compartments are studied for the multi-compartmental birthless death-migration-immigration process. The important role played by the transition matrix of the deterministic process is underlined. Several examples are discussed.
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    Bulletin of mathematical biology 45 (1983), S. 483-506 
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    Notes: Abstract Qualitative theory for multidimensional stochastic dynamical models $$\dot x = f(x, \xi )$$ is presented where the random influences ξ may be white or colored, i.e. a (possibly bounded) diffusion process. We concentrate on transience, stationary solutions and boundary behavior and discuss a set-up for reliable simulations. The method consists in associating a deterministic control system where the (approximate) controllability properties determine the qualitative behavior of the stochastic system. Applications to some biological systems indicate the usefulness of qualitative theory in life sciences.
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    Bulletin of mathematical biology 45 (1983), S. 599-603 
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    Notes: Abstract We study some linear stochastic differential equations in Hilbert spaces.
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    Bulletin of mathematical biology 45 (1983), S. 591-597 
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    Notes: Abstract In this article we show how linearity with respect to the output of a stochastic dynamic model can be exploited in order to simplify the computation of moments or conditional moments. The results are presented for two examples, one of which includes delays. This feature is often encountered in biological models.
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    Bulletin of mathematical biology 45 (1983), S. 873-874 
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    Bulletin of mathematical biology 45 (1983), S. 869-872 
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    Notes: Abstract In the derivation of the biomass distribution function for an ecological population critical use is made of an energetic constraint on the maximization of biomass diversity. The nature of this constraint is explored in detail using Kleiber's relation σ(m)=cm γ between animal metabolic rate σ(m) and body weightm in conjuction with the Prigogine-Wiame thermodynamic paradigm for specific entropy production in biological stationary states. These two inputs fix the energetic constraint on the maximization of biomass diversity to be the constancy of the mean metabolic rate of the ecosystem. The resulting biomass distribution function is tested against observational data.
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    Bulletin of mathematical biology 45 (1983), S. 875-875 
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    Bulletin of mathematical biology 45 (1983), S. 901-915 
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    Notes: Abstract A system of mixed integrodifferential and partial differential equations for an agestructured predator-prey system is studied here. The predator eats all ages of prey, but more of the very young and very old than of the intermediate ages. The existence of periodic solutions corresponding to stable coexistence is proved for a suitable range of parameters by bifurcation theory.
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    Bulletin of mathematical biology 45 (1983), S. 877-900 
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    Notes: Abstract In this paper, stability of two-prey, one-predator communities is investigated by Lyapunov's direct method and Hopf's bifurcation theory. Three patterns of three-species coexistence are possible. A globally stable non-negative equilibrium exists for the system even if two competing prey species without a predator cannot coexist. The stable equilibrium bifurcates to a periodic motion with a small amplitude when the predation rate increases. It is also shown that a chaotic motion emerges from the periodic motion when one of two prey has greater competitive abilities than the other. This predator-mediated coexistence can be realized by the intimate relationship between preferences of a predator and competitive abilities of two prey.
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    Bulletin of mathematical biology 45 (1983), S. 917-930 
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    Notes: Abstract A non-linear stability analysis using a multi-scale perturbation procedure is carried out on the practical Thomas reaction-diffusion mechanism which exhibits bifurcation to non-uniform states. The analytical results compare favourably with the numerical solutions. The sequential patterns generated by this model by variations in a parameter related to the reaction-diffusion domain indicate its capacity to represent certain key morphogenetic features required in a recent model by Kauffman for pattern formation in theDrosophila embryo.
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    Bulletin of mathematical biology 45 (1983), S. 1-9 
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    Notes: Abstract The urn model discussed in part is generalized so that the random depletion of balls from the urn in any cycle is not necessarily the same as the reinforcement in that cycle. This model is applied to an urn containing balls of three colors (white, red, black) for which the black balls always receive reinforcements, whereas there is only one cycle in which red balls are added. Experimental data are considered in which red balls correspond to radioactive iodine atoms, black balls to stable iodine atoms and white balls to empty space, all relating to the thyroid gland. Half-life periods for the radioactive iodine in relation to the time of uptake (ten years, fifteen years, etc.) are considered.
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    Bulletin of mathematical biology 45 (1983), S. 21-31 
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    Notes: Abstract Kinetic equations for the action of a reagent on a population which is increasing by Gompertzian growth are solved by a combination of analytical and numerical methods for both dilute distributions and cellular aggregates. For the aggregates an empirical function controlling extent of diffusion is used with external reagent concentration held constant, and the effective diminution of reagent activity caused by aggregation is related to the extent of penetration by diffusion. Diffusion from the centre is considered. Solutions with time-dependent reagent concentration are obtained for a distribution, and the effect of successive inputs of reagent studied.
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    Notes: Abstract Bilayer lipid membranes consist of an inner hydrocarbon tail region with the hydrophilic polar heads on either side. The order-disorder transition in the hydrocarbon tail reigon, from liquid crystalline (fluid) to gel state, is characterised in terms of a Landau-de Gennes description, in which the effect of an external electric field is incorporated through its description, in which the effect of an external electric field is incorporated through its interaction with the surface charges on the bilayer (placed as it is in an ionic medium) or with the polar heads. Biological implications of such a phase transition, for excitable membranes, resides in a model wherein ion channels (taken to be composed of protein bundles) are postulated to be surrounded by lipid molecules in the fluid phase when the membrane is in its resting state, while surface charges and/or the polar heads of the lipid molecules responding to an electric stimulus, if of adequate magnitude, induces a transition in the hydrocarbon tail region of the (boundary) lipid surrounding the ion channels from the liquid crystalline (fluid) to the crystalline (gel) phase which, in turn, through coupling with the relevant modes of the protein bundles, results in the opening of the ion channels, provinding thereby a mechanism for the desired response.
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    Bulletin of mathematical biology 45 (1983), S. 117-138 
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    Notes: Abstract Concepts of the uniqueness of the amino acid sequences of proteins were defined in a prior report (Saroff, H. A. and F. A. Kutyna. 1981. “The Uniqueness of Protein Sequences: A Monte Carlo Analysis.”Bull. math. Biol. 43, 619–639), which presented a detailed discussion ofi-uniqueness, i.e. the tendency of small peptides to be repeated within an amino acid sequence of a protein. We now report on the quantitative analysis ofo-uniqueness, which evaluates the tendency of small peptides to be repeated amongst different proteins, usually of a single species. A detailed analysis of theo-uniqueness of several proteins is presented to illustrate the method and the range of values encountered. Uniqueness data on sequences of human proteins in a data bank of sequences containing about 32,500 amino acids are made available in the form of a microfiche. Analysis of biologically active subsequences such as the angiotensins and the enkephalins suggest a tendency of the subsequences contributing to the property ofo-uniqueness to cluster in portions of the parent protein sequence which are biologically active. This property may provide a general method for predicting biologically active areas of proteins. Current data may already be adequate to permit useful predictions, and the rapidly accumulating and interrelated new data on nucleic acid and protein sequences will further enhance the power ofo-uniqueness analysis.
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    Bulletin of mathematical biology 45 (1983), S. 143-149 
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    Notes: Abstract In this note we examine Eigen's nonlinear rate equations for the relation of coupled biomacromolecules. We obtain an exact solution to the equations for constant overall population densities and constant rate parameters. We conclude that there is only one stationary solution where all molecular species coexist when they are coupled by mutation.
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    Bulletin of mathematical biology 45 (1983), S. 229-237 
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    Notes: Abstract Previously developed light absorption models have treated the effective quantity of light-absorbing material within the experimental environment as a constant, i.e. a parameter. These models are, however, probabilistic in nature and are properly applied and interpreted only in a statistical sense. Thus, it is clearly logical to regard the effective quantity of light-absorbing material to be a random variable. In this paper an asymptotic distribution is derived for this random quantity, and it is shown how this distribution may be incorporated into present models. These results may be applied to light absorption by plant and crop canopies as well as to liquid or solid media. Furthermore, previous models are based upon the assumption that light is parallel, or effectively so, as for solar light. Such models may be inadequate for an artificial (laboratory) environment which utilizes point source light. Present models for estimating light interception and radiation intensity are modified so as to accommodate a proximate point source of light. Numerical examples are included to illustrate the theory.
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    Bulletin of mathematical biology 45 (1983), S. 239-257 
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    Notes: Abstract This paper presents an exact analysis of a mutually coupled relay oscillator based on a method orignated by Tsypkin. Limit-cycle frequencies and phases can be determined exactly using this method, unlike other approximate methods based on describing functions and harmonic balance techniques. A new method of exact determination of limit-cycle stability is also shown to give excellent agreement with simulation studies. Different types of intercoupling are shown to give different stability conditions, and these are discussed in relation to gastrointestinal (GI) smooth muscle modelling. GI tract electrical activity has previously been modelled using bidirectionally coupled nonlinear oscillators, and the results of the present analysis of relay oscillators is compared with other studies using van der Pol dynamics.
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    Bulletin of mathematical biology 45 (1983), S. 269-285 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Information about the way of branching of dendritic arborizations may be obtained by comparing the frequency distributions of observed branching patterns with theoretical distributions based on well-defined growth models. Two models usually get much attention in geomorphological and (neuro)biological studies, viz. terminal growth and segmental growth. Formulae to construct the exact probability distributions for both growth models are presented. It is shown that ranking and lumping of the individual branching patterns enable the analysis of very large arborizations with relatively few data. The application of the Kolmogorov goodness-of-fit test for discrete distributions to the analysis is discussed.
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    Bulletin of mathematical biology 45 (1983), S. 295-309 
    ISSN: 1522-9602
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    Notes: Abstract An autonomous system of two differential equations with an asymptotically stable steady state is considered. It is shown that there exists a class of time delays which preserves the stability of such a steady state. The charactristic return time (or decay rate) for the solutions to approach the steady state is estimated using the variational system for the perturbations.
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    Meteorology and atmospheric physics 32 (1983), S. 53-57 
    ISSN: 1436-5065
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    Description / Table of Contents: Zusammenfassung Etliche jüngst erhaltene, empirische Resultate und theoretische Untersuchungen der Abschätzung der thermischen Strahlung der Atmosphäre aus wolkenlosem Himmel werden diskutiert. Es wird gezeigt, daß des strahlungsmäßige Verhalten des Wasserdampfes bei Lufttemperaturen über dem Gefrierpunkt so gut als möglich und nötig parameterisiert worden ist, doch bei Temperaturen unterhalb des Gefrierpunktes sind noch Verbesserungen möglich. Zusätzlich wird gezeigt, daß Vorschriften zur genauen Berechnung der atmosphärischen Thermalstrahlung in der Lage sein müssen, die Effekte der zeitlichen und räumlichen Variabilität der Staubverteilung in Betracht zu ziehen.
    Notes: Summary Several recent empirical findings and theoretical developments related to the estimation of clear-sky atmospheric thermal radiation are discussed. It is demonstrated that the radiative behavior of water vapor at air temperatures above freezing has been parameterized as well as can or need be done, but that there may be room for some improvement at air temperatures below freezing. It is additionally shown that any procedure designed to accurately calculate atmospheric thermal radiation must be able to handle effects of airborne dust variability of both a temporal and spatial nature.
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    Meteorology and atmospheric physics 32 (1983), S. 1-22 
    ISSN: 1436-5065
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    Description / Table of Contents: Zusammenfassung Die interannuale Veränderlichkeit in der Ausdehnung des antarktischen Meereseises wird während der winterlichen Zuwachsperiode (Juni bis September) für die fünf Jahre 1973–77 untersucht. Statistische Korrelationen zwischen den synoptischen atmosphärischen Zirkulationsverhältnissen, gegeben durch die Frequenz der von Satelliten beobachteten außertropischen Tiefdruckgebiete, und der breitenmäßigen. Ausdehnung des Meereseises werden auf hemisphärischer und regionaler Basis für jeden Winter ausgewertet. Markante Unterschiede in der winterlichen Zyklonentätigkeit begleiten die jährlichen Unterschiede im Wachstum und in der Ausdehnung des Eises, gemessen im zonalen Mittel, und sind besonders in den extremen Eiswintern 1973 und 1977 evident. Ein Zusammenhang zwischen der Ausdehnung des Meereseisgebietes während des Winters und der Frequenz von Zyklogenese ist besonders für die geographischen Breiten unmittelbar nördlich der Eisgrenze angedeutet. Drei Schlüsselgebiete starker interannualer Veränderlichkeit in den Eisbedingungen werden identifiziert und statistisch auf Rückwirkungen zwischen Eis und Zyklonen hin untersucht. Korrelationen zwischen der geographischen Breite des Eisrandes und der Zyklonenfrequenz zeigen die geringste interannuale Veränderlichkeit im ostantarktischen Sektor. Durch diese Gegebenheit wird die Rolle des semi-permaneten Tiefdruckzentrums im Wintereisregime dieser Region betont. Andererseits variiert die Korrelation zwischen Eis und Zyklonentätigkeit von Jahr zu Jahr bedeutend in der Weddellsee, wo die meeresbedingten Advektionsverhältnisse von zusätzlicher Bedeutung sind. Einzeluntersuchungen aus dem Zeitraum der fünf Winter dienen als Unterstützung der für jeden Sektor gewonnenen, statistischen Resultate.
    Notes: Summary Interannual variations in the extent of Antarctic sea ice are investigated for the winter growth season (June through September) for the five years 1973–77. Statistical correlations between the synoptic-scale atmospheric circulation, given by the frequencies of satellite-observed extratropical depressions, and the latitudinal extent of the ice are evaluated on hemispheric and regional scales for each winter. Marked differences in winter cyclone activity accompany the year-to-year variations in ice growth and extent on a zonally-averaged basis, and are particularly evident for the extreme ice winters of 1973 and 1977. A link between the expansion of the sea ice zone during winter and the frequencies of new cyclonic developments (cyclogenesis) is evident for latitudes immediately north of the ice-ocean boundary. Three key regions of high interannual variability in ice conditions are identified and examined statistically for ice-cyclone interactions. Correlations between ice-edge latitude and cyclone frequencies show least variation between years in the East Antarctic sector, emphasizing the role of the semi-permanent low pressure center in the winter ice regime of this region. Conversely, ice-cyclone correlations vary markedly from year-to-year in the Weddell Sea, where ocean-related ice advection patterns are of additional significance. Case studies from the five-winter period are used to support the statistical results obtained for each sector.
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    Meteorology and atmospheric physics 32 (1983), S. 99-107 
    ISSN: 1436-5065
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    Description / Table of Contents: Zusammenfassung Die Ergebnisse der Untersuchung zeigen, daß in Jodhpur die Feuchtigkeit in kahlem sandigen Boden größer ist als in einem Boden unter einer Vegetationsdecke. Das mittlere Temperaturmaximum war in 5 cm Tiefe im allgemeinen um 1 bis 2°C höher als im Boden unter einer Vegetationsdecke, ausgenommen im Juh, wo eine beträchtliche Temperatur-abnahme im kahlen Boden eintrat. Die Tagestemperatur in 5 cm Tiefe war im Sommer höher als in tieferen Schichten, was zu einem abwärtsgerichteten Wärmefluß führte. Im Winter waren die Temperaturen in tieferen Schichten höher, was einen aufwärtsgerichteten Fluß von Wärme und Wasserdampf bewirkte. Es wurde eine große Veränderlichkeit des Wärmeflusses mit im allgemeinen positivem Fluß im Sommer und negativem Fluß im Winter beobachtet.
    Notes: Summary The results of the studies conducted during 1977–78 show higher moisture regime of a bare sandy soil of Jodhpur than the soil with vegetative cover. Mean maximum temperature of a bare soil at 5 cm depth, in general, was 1 to 2°C higher than the soil with vegetative cover except during July when there was considerable fall in temperature of a bare soil. Daytime soil temperatures at 5 cm depth during summer were higher than the lower depths leading thereby to downward movement of heat. In winter, however, the temperatures of lower layers were higher facilitating the upward flow of heat and the vapour flux. A high variability in heat fluxes with generally positive daytime heat fluxes in summer and negative in winter were observed.
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    Meteorology and atmospheric physics 32 (1983), S. 23-34 
    ISSN: 1436-5065
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    Description / Table of Contents: Zusammenfassung Sorgfältig ausgewählte AMTEX-Daten wurden zur Berechnung von ähnlichkeitsfunktionen für die Parametrisierung des Oberflächen-Stress und des Wärmetransportes in der konvektiven atmosphärischen Grundschicht verwendet. Die ähnlichkeitsfunktionen für den Stress, die aus den über die Schicht gemittelten mittleren Windkomponenten bestimmt wurden, stimmen für den baroklinen Fall nicht mit ihren aus numerischen Modellen für den barotropen Fall bestimmten Gegenstücken überein, im Widerspruch zu dem, was wir von der Hypothese von Arya und Wyngaard [ 10] erwarten. Vielleicht waren die verwendeten Rawinsonden-Daten für die Verifikation dieser Hypothese nicht gut genug. Es wurde jedoch gefunden, daß die konvektive PBL-Höhe in übereinstimmung mit der Hypothese von Kazanski und Monin [6] mit dem Stabilitätsparameter gut korreliert ist.
    Notes: Summary Carefully selected AMTEX data were used to compute the similarity functions for the parameterization of the surface stress and heat flux for the convective atmospheric boundary layer. The similarity functions for the stress determined from the layer-averaged mean wind components for the baroclinic case don't agree with their counterparts predicted by numerical models for the barotropic case, contrary to what we expect from the hypothesis of Arya and Wyngaard [10]. Perhaps the rawinsonde data used in this study were not good enough for the verification of this hypothesis. However, the non-dimensionalized convective PBL height is found to be well correlated with the stability parameter in accordance with the Kazanski-Monin hypothesis [6].
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    Meteorology and atmospheric physics 32 (1983), S. 55-69 
    ISSN: 1436-5065
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    Description / Table of Contents: Zusammenfassung Eine lineare, barokline Stabilitätsanalyse wird auf das vertikale, zonale Windprofil der Grundströmung vor der Bildung einer Depression während der MONEX-79-Phase in der Bucht von Bengalen angewendet, wobei ein quasi-geostrophisches, mehrschichtiges, numerisches Modell verwendet wird. Die aus der Zuwachsrate des Spektrums abgeleitete, bevorzugte Wellenlänge beträgt 1400 km für eine Kanalbreite von 1500 km und eine e-fache Vergrößerungszeit von 5.2 Tagen. Die vertikale Wellenstruktur wurde berechnet und mit der beobachteten Struktur der Depression verglichen. Beide besitzen mehrere gemeinsame Grundzüge. Die vertikale Struktur der bevorzugten Welle, aus einem modifizierten Windprofil abgeleitet, in dem die Windscherung unterhalb des Westwindmaximums entfernt wurde, zeigt bessere Übereinstimmung mit der Depressionsstruktur, als die aus dem vollen Windprofil erhaltene Struktur. Der Effekt der vertikalen Veränderlichkeit der statischen Stabilität in den unteren Schichten auf das Wachstum der kurzen, instabilen Wellen wird untersucht. Es zeigt sich, daß eine Abnahme der statischen Stabilität in den unteren Schichten zu einer Zunahme der Wachstumsrate und einer Abnahme der bevorzugten Wellenlänge führt.
    Notes: Summary A linear baroclinic stability analysis is performed for the basic vertical zonal wind profile prior to the formation of the depression during Bay of Bengal phase of MONEX-79 by utilizing a quasi- geostrophic, multilayer, numerical model. The preferred wavelength obtained from the growth rate spectrum is 1400 km for the channel width 1500 km, with an e-folding time 5.2 days. The wave structure in the vertical is computed and compared with the observed structure of the depression. Many common features are noticed between them. Further, the vertical structure of the preferred wave obtained from the modified wind profile, removing wind shear below the westerly maximum level, shows closer agreement to the depression structure than that obtained from the full profile. The effect of vertical variation of static stability in the lower layers on the growth of short unstable waves is studied. It is found that a decrease in static stability in the lower layers resulted in the increase in growth rate and decrease in the preferred wavelength.
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    Meteorology and atmospheric physics 32 (1983), S. 97-101 
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    Description / Table of Contents: Zusammenfassung In der Arbeit wird der Hintergrund des äquivalent barotropen Modells, wie es von Mahanti [6, 7, 8] eingeführt worden ist, im Detail besprochen.
    Notes: Summary The physical background of the equivalent barotropec model, introduced by Mahanti [6, 7, 8], has been discussed in details in the present study.
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    Meteorology and atmospheric physics 32 (1983), S. 161-185 
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    Description / Table of Contents: Zusammenfassung Bei der Modellierung mesoskaliger Küstenklimaeffete sind wegen des Übergangs Land-Meer, repräsentiert durch die Rauhigkeitslangez 0 =z 0 (x, y) und die WärmekapazitätC 0 =C 0 (x,y), und des verstärkten Einflusses der Feuchte ausgeprägte Inhomogenitaten zu berücksichtigen, welche die meteorologischen Verhältnisse signifikant mitgestalten. Mit Hilfe eines dreidimensionalen numerischen Modells werden in der vorliegenden Arbeit die Auswirkungen solcher horizontaler Inhomogenitaten auf Struktur und Variation der Grenzschicht in der Küstenregion studiert, Bowie die Entstehung von Advektionsnebel unter dem Einfluss unterschiedlicher geostrophischer Winde untersucht. Die Modifizierung der Windstärke und Windrichtung in ihrer Abhängigkeit von der Rauhigkeitslange und der unterschiedlichen thermischen Schichtung Bowie zwei Beispiele von Advektionsnebelbildung bei unterschiedlichem geostrophischem Wind werden als Ergebnisse vorgestellt.
    Notes: Summary When modelling mesoscale effects in the coastal climate we have to account for the significant influence of humidity and for the land-sea contrast which is represented by the roughness lengthz 0 =z 0(x, y) and the heat capacityC 0 =C 0 (x, y). This brings about strong horizontal inhomogenities which affect the meteorological situation significantly. The effects of these inhomogenities on the structure and variation of the boundary layer in the coastal region are studied by use of a three-dimensional numerical model; the formation process of advective fog in connection with different geostrophic winds is also investigated. As main results we present 1) the modifications of wind velocity and wind direction as a function of the roughness distribution and different thermal stratifications and 2) two simulations of advective fog formation assuming different geostrophic wind values in each example.
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    Meteorology and atmospheric physics 32 (1983), S. 201-218 
    ISSN: 1436-5065
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    Description / Table of Contents: Zusammenfassung Statistische auf Andauer und Klimatologie begründete Modelle werden zur Vorhersage der Bahnen tropischer Zyklonen von der Nordwestküste Australiens her angewendet. Ein auf die Zeit 1980–81 bezogener Vergleich zeigt, daß eine Anpassung erster Ordnung nach kleinsten Quadraten von acht primären Vorhersagegrössen eine hohe Stufe statistischer Signifikanz hat und vergleichbare mittlere Fehler bei Vorhersagen auf Grund von Regressionsgleichungen höherer Ordnung gibt.
    Notes: Summary Statistical models based on persistence and climatology are used to forecast tropical cyclone tracks off the northwest coast of Australia. A comparison based on the 1980–81 season shows that a first order least squares fit of the eight primary predictors has a higher level of statistical significance and comparable mean forecast errors to predictions based on higher order regression equations.
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    Meteorology and atmospheric physics 32 (1983), S. 297-308 
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    Description / Table of Contents: Zusammenfassung Die asymptotische Form des Diffusionstensors in einer Scherungsströmung (im besonderen in einer Ekman-Strömung) wird auf eine zweidimensionale vertikal gemittelte Diffusionsgleichung mit einem Schwächungsterm erster Ordnung angewendet. Einige klassische Lösungen für punktförmige Quellen werden besprochen und auf ein Modell einer Scherungsströmung angewendet. Lösungen für nichtkonstante Quellen werden diskutiert, wobei der Effekt des Abschneidens des Diffusionsprozesses im hochfrequenten Bereich dargelegt wird. Eine einfache Anwendung auf experimentelle Daten in mesoskalem Abstand wird auch gezeigt.
    Notes: Summary The asymptotic form of the diffusivity tensor in shear flow (in particular Ekman flow) is applied to a two-dimensional, vertically averaged diffusion equation with a decay term of the first order. Some classical solutions for point sources are reviewed and applied to the shear flow model; solutions for non-constant sources are discussed, putting into evidence the cut-off effect of the diffusive process in the high frequency range. A simple application to experimental data on mesoscale distance is also shown.
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    Meteorology and atmospheric physics 32 (1983), S. 301-307 
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    Description / Table of Contents: Zusammenfassung Die vorliegende Arbeit gibt die Bedingung der Instabilität von Wellen mit unbegrenzter seitlicher Ausdehnung an. Es wird auch gezeigt, daß die obere atmosphärische kurze Welle der zonalen Wellenzahl 8 und die niedrige atmosphärische Zyklonenwelle beim Fehlen diabatischer Erwärmung instabil werden und zusammenbrechen.
    Notes: Summary The present paper gives a condition of instability of waves of infinite lateral extent. Also, it is shown that the upper atmospheric short wave of zonal wave number 8 and lower atmospheric cyclone scale wave become unstable and break down in the absence of diabatic heating.
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    Meteorology and atmospheric physics 32 (1983), S. 281-288 
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    Description / Table of Contents: Zusammenfassung Die täglichen Höhenwindbeobachtungsdaten, die von Schiffsbeobachtungen im MONEX-79 erhalten wurden, sind einer 2-D Filterung unterworfen worden, deren Reaktion hinsichtlich der Wellenzahl so ist, daß Wellen mit Wellenlängen kleiner als 760 km ausgefiltert wurden. Diese gefilterten Windkomponenten wurden „Wirbeln” zugeordnet. Die kinetische Energie dieser „Wirbel” in 3 Phasen des Monsuns, nämlich vor dem Beginn des Monsuns (Mai), zum Monsuneinsatz (Juni) und im vollentwickelten Monsun (Juli), wird untersucht. Es wurde festgestellt, dass der Beitrag dieser „Wirbel” zur kinetischen Energie des Gesamtwindes von einem Minimum von 15,5% in der Vormonsunzeit bis 40% in der Phase des voll entwickelten Monsuns variiert. Der Energieaustausch zwischen den „Wirbeln” und den grosskaligen Windkomponenten zeigt an, daß während der Phase des entwickelten Monsuns die untere Troposphäre bis 700 mb eine baroklin instabile Grundströmung zu haben scheint, während in den anderen Phasen die Grundströmung im allgemeinen barotrop instabil ist.
    Notes: Summary The daily upper-air wind data, obtained from the MONEX-79 ship observing platforms, are subjected to a 2-D filter, whose wavenumber response is such as to filter out waves having wavelengths less than 760 km. The filtered wind components are referred to as “eddies”. The kinetic energies of these eddies during the three phases of the monsoon, viz. (a) pre-onset (May), (b) onset (June), (c) established (July) are studied. It is found that the contribution to the kinetic energies of the total wind, by the eddies, vary from a minimum of about 15.5% during the pre-onset phase to a maximum of 40% during the established phase. The energy exchanges between the eddy and large-scale wind components, indicate that during the established phase the lower troposphere, upto 700 mb, appears to have a baroclinically unstable basic flow, while during the other phases, the basic flow is, in general, barotropically unstable.
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    Meteorology and atmospheric physics 32 (1983), S. 310-310 
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    Meteorology and atmospheric physics 32 (1983), S. 339-352 
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    Description / Table of Contents: Zusammenfassung Ein zeitabhängiges, auf den Grundgleichungen basierendes, numerisches Modell wurde verwendet, um die Hypothese zu prüfen, daß solare Veränderlichkeit eine Änderung in den Grundvariablen der oberen Atmosphäre und somit in der planetaren Langwellenstruktur der unteren Atmosphäre hervorrufen kann. Dieser Mechanismus wurde zur Erklärung einer offensichtlichen, atmosphärischen Reaktion auf solare Schwankungen herangezogen. Im Modell wird diese Veränderlichkeit durch eine diabatische Wärmequelle hervorgerufen, welche als eine einfache Repräsentation der Jouleschen Dissipationserwärmung angenommen wird. Die Langwellenstruktur der unteren Atmosphäre zeigte sich als unbeeinflußbar durch sonneninduzierte Vorgänge in der oberen Atmosphäre. Tatsächlich auftretende Veränderungen sind auf eine Distanz von 25 bis 40 km unterhalb des Niveaus maximaler Erwärmung beschränkt und sind von nur kurzer Lebensdauer.
    Notes: Summary A time-dependent, primitive-equation numerical model is used to test the hypothesis that solar variations induce changes in the distributions of basic state variables at high levels in the atmosphere, and thus induce changes in planetary-scale wave structure at lower atmospheric levels. This mechanism was proposed to explain apparent atmospheric responses to solar activity. The changes are brought about in the model by a diabatic heat source, which is taken to be a simple representation of Joule dissipative heating. Lower atmospheric wave structure is found to be insensitive to solar-induced changes in the upper atmosphere. Such changes as do occur are limited to within 25 to 40 km below the level of maximum heating, and are also quite short-lived.
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    Meteorology and atmospheric physics 32 (1983), S. 389-397 
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    Description / Table of Contents: Zusammenfassung Wir betrachten die Dynamik langer Wellen mit begrenzter Amplitude in einer quasigeostrophischen Atmosphäre. Es wird gezeigt, daß sie den von Charney und Drazin formulierten Theoremen für vertikale Reflexion und Verbindung mit der zonalen Strömung genügen. Es wird ferner gezeigt, daß der typische Effekt des kritischen Niveaus auch für solche Wellen gilt.
    Notes: Summary We consider the dynamics of finite amplitude long waves in a quasi-geostrophic atmosphere. It is shown that they satisfy the well known theorems, formulated by Charney and Drazin, for vertical reflection and coupling with zonal flow. It is further shown that the typical critical level effect holds for such waves as well.
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    Meteorology and atmospheric physics 32 (1983), S. 405-406 
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    Meteorology and atmospheric physics 32 (1983), S. 337-360 
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    Description / Table of Contents: Zusammenfassung Die bodennahen nordhemisphärischen Temperaturvariationen 1881–1980 und einige in Frage kommende externe Einflußparameter werden statistisch untersucht. Die statistischen Maßzahlen und Tests jährlicher und jahreszeitlicher Daten zeigen, daß es sinnvoll ist, den tiefpaßgefilterten langfristigen Trend (T 〉 30 Jahre) vom verbleibenden hochpaßgefilterten Anteil zu trennen, wobei dieser weitgehend die Charakteristika einer stationären und vielleicht auch stochastischen Zeitreihe erfüllt. Der langfristige Trend muß jedoch als die nicht-stationäre Komponente angesehen werden, die vermutlich von externen Einflußmechanismen des Klimasystems gesteuert wird. Die spektral-statistischen Untersuchungen stützen dieses Konzept, weisen jedoch auch, wie die nicht-spektralen Untersuchungen, auf jahreszeitliche Besonderheiten hin. 84–91% des langfristigen (aber nur 38–65% des jährlichen) Anteils der Varianz lassen sich durch vulkanisch-solare Parameter sowie den beobachteten anthropogenen CO2-Anstieg erklären, wobei die Erfassung des solaren Einflusses sehr problematisch ist. Multiple lineare Korrelations- und Regressionsanalysen führen zu Abschätzungen der Signal-Amplituden und des Signal-Rausch-Verhältnisses. Neben dem vulkanischen und solaren Signal ist das CO2-Signal in den Jahresdaten, weniger signifikant in den Sommerdaten, auffindbar.
    Notes: Summary This is a statistical study on the northern hemisphere temperature variations near surface, 1881–1980, and some possible external forcing parameters. The statistical measures and tests of yearly and seasonal data indicate that it is reasonable to separate the low-pass filtered long-term trend (T 〉 30 years) from the remaining high-pass filtered quantity, where the latter fulfills closely the characteristics of a stationary may be stochastic time series. The long-term trend, however, has to be seen as the non-stationary component presumably driven by external forcing mechanisms of the climatic system. The spectral statistics maintain this concept; but there are, as in case of the non-spectral statistics, distinct seasonal peculiarities. 84–91% of the long-term (but only 38–65% of the yearly data's) variance can be explained by volcanic/solar parameters and the observed anthropogenic CO2 increase, where the solar influence, however, is very problematical to realize. A multiple linear correlation and regression analysis leads to assessments of the signal magnitudes and the signal to noise ratios. Besides the volcanic and solar signal, the CO2 signal is detectable in yearly and — less significant — in summer data.
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    Meteorology and atmospheric physics 33 (1983), S. 19-30 
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    Description / Table of Contents: Zusammenfassung Auf der Grundlage von ausgeglichenen Energieflüssen (einschließlich Abfuhr der Nutzungsenergie) innerhalb einer flachen Sonnenenergie-Kollektorplatte und von Energieflüssen zwischen Kollektor und seiner Umgebung wird mit Verwendung von bekannten Klimadaten ein Módell für die Simulation von Tagesgängen des Energieaustausches und -gewinns entwickelt. Numerische Experimente mit diesem Modell zeigen auch eine interessante Abhängigkeit der nutzbaren Energie von einigen meteorologischen Parametern wie Nebel, Niederschläge und Wind.
    Notes: Summary On the basis of balanced energy fluxes (including the removal of useful energy) inside the flat-plate solar-energy collector, and the energy fluxes between the collector and its environment, a model for simulation of daily courses of energy exchanges and gain is constructed, using the known climatic data. Numerical experiments with this model show also an interesting dependence of useful energy on some meteorological parameters like: fog, precipitations and wind.
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    Meteorology and atmospheric physics 33 (1983), S. 11-18 
    ISSN: 1436-5065
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geography , Physics
    Description / Table of Contents: Zusammenfassung Zur Untersúchung der Ausbreitungs- und Konzentrationsformen von Autoabgasen von einer Straße in einem V-förmigen Tal wird ein numerisches Modell angewendet. Diese Formen sind mit dem Windzirkulationssystem der bei Tag und bei Nacht vorherrschenden Strömung quer zum Tal eng verbunden. An einem typischen Tag mit einer relativ starken Oberflächenerwärmung war die Abgasfahne zum Leehang gegen den Kamm hin geneigt und bog dann zurück zur Abwindrichtung mit der vorherrschenden Strömung. In einer typischen Nacht mit stabiler Schichtung zeigte die Ausbreitung nicht die Form einer Abgasfahne, sondern eher eine diffuse Ausbreitung aufwärts und eine mehr oder weniger horizontale Verbreitung.
    Notes: Summary A numerical model is utilized to investigate the dispersion and concentration pattern of vehicular exhaust gas from a v-shaped valley highway. These patterns are closely associated with the wind circulation regimes in the prevailing cross-valley flow during typical day and night times. In the typical day with relative strong surface heating the exhaust plume first inclined to the lee slope toward to the ridge then curved back to the downwind direction with the prevailing flow; in the typical night with stable condition the dispersion showed no plume pattern but rather diffused upward and spread out more or less horizontally.
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  • 97
    ISSN: 1436-5065
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geography , Physics
    Description / Table of Contents: Zusammenfassung Es wird eine räumliche and zeitliche Verteilung einiger Komponenten des Zyklus der Energetik der allgemeinen Zirkulation auf der Südhalbkugel im IGY analysiert und besprochen. Versuchsweise werden Diagramme für den Energiezyklus für Sommer, Winter und für den Jahresdurchschnitt auf Grund einer Annahme über die Zerstreuung der kinetischen Turbulezenergie angegeben. Die allgemeinen Verhältnisse des Energiezyklus auf der südlichen Halbkugel sind den auf der nördlichen Halbkugel beobachteten ähnlich; die Intensität der allgemeinen Zirkulation ist aber im Winter größer. Die äquator überschreitenden Wechselwirkungen zwischen den beiden Hemisphären sind speziell im Sommer der Südhemisphäre bedeutend. Die Bildung zonal verfügbarer potentieller Energie ist im Sommer stärker und sie ist eine der bedeutendsten Energiequellen für die Erhaltung der allgemeinen Zirkulation. Diabatische Erwärmung durch Auslösung von latenter Wärme beim Phasenübergang von Wasserdampf liefert den vorherrschenden Beitrag für die Bildung zonal verfügbarer potentieller Energie. Der Anteil der Bildung von verfügbarer potentieller Turbulenzenergie ist nach indirekter Schätzung im Sommer größer, während der barokline Prozeß der Umwandlung von zonal verfügbarer potentieller Energie in verfügbare potentielle Turbulenzenergie im Winter stärker ist. Der antarktische Kontinent scheint für die Energetik der südlichen Hemisphäre eine bedeutende Rolle zu spielen; seine Beiträge zur Bildung von zonal verfügbarer potentieller Energie und zur Umwandlung dieser Energieform in verfügbare potentielle Turbulenzenergie sind beachtenswert.
    Notes: Summary The spatio-temporal distribution in the mixed space-time domain of some components of the cycle of the energetics of the general circulation in southern hemisphere during the IGY are analyzed and discussed. Tentative box diagrams of the energy cycle are given for summer, winter and annual mean conditions, on the basis of an assumption concerning the dissipation of eddy kinetic energy. The general features of the energy cycle in southern hemisphere are similar to those observed in the northern hemisphere; the intensity of the general circulation is higher in winter. The transequatorial interactions between the two hemispheres are important specially during southern hemisphere summer. The generation of zonal available potential energy is stronger in summer, and is one of the most important sources of energy to maintain the general circulation; diabatic heating due to the release of latent heat in the phase transitions of water vapour gives the dominant contribution to the generation of zonal available potential energy. The rate of generation of eddy available potential energy, indirectly estimated, is higher in summer while the baroclinic process of transformation of zonal available potential energy into eddy available potential energy is more intense in winter. The antarctic continent seems to play an important role in the energetics of southern hemisphere; its contributions to the generation of zonal available potential energy and to the conversion of this form of energy into eddy available potential energy are noteworthy.
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  • 98
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    Meteorology and atmospheric physics 32 (1983), S. 135-143 
    ISSN: 1436-5065
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geography , Physics
    Description / Table of Contents: Zusammenfassung In dieser Arbeit wird der Tages- und Jahresgang der Konzentration der positiven und negativen großen Ionen in der Luft über Athen (ϑ = 37°58′N, λ = 23°43'E,h = 107 m) für den Zeitraum 1969 bis 1977 untersucht. Der Tagesgang der Konzentration der grossen Ionen zeigt eine Doppelwelle. Der Jahresgang verläuft parallel zum Jahresgang der Luftverunreinigung und weist den größten Wert im Winter auf, während der kleinste Wert auf den Sommer fällt. Der Tagesgang dieser Ionen wird auch einer Fourier-Analyse unterworfen. Es wird auch der Einfluß der Temperatur und der relativen Feuchtigkeit der Luft auf die Konzentration der positiven und der negativen großen Ionen untersucht.
    Notes: Summary In the present study we examine the daily and annual variation of the concentration of positive and negative large ions in the air above Athens (ϑ = 37°58′N, λ = 23°43′E,h = 107 m), for the 9-year period 1969–1977. The diurnal variation of the large ion concentration follows a double oscillation. Their annual variation is parallel to that of smoke, presenting a maximum in winter and a minimum in summer. A Fourier analysis of their diurnal variation was also made. In addition we have examined the influence of air temperature and relative humidity on the concentration of positive and negative largeions.
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  • 99
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    Meteorology and atmospheric physics 32 (1983), S. 231-252 
    ISSN: 1436-5065
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geography , Physics
    Description / Table of Contents: Zusammenfassung Die mittleren monatlichen und jährlichen Regenfallsreihen (1930–1977) für ein 30×30 Bogenminuten-Netz im östlichen Transvaal und Swaziland wurden analysiert. Diese Reihen zeigen eine charakteristische zeitliche und räumliche Veränderlichkeit des Niederschlagsregimes, deren Signifikanz auf quasi-periodische Anregung mit Perioden von ungefähr 16–20 Jahren und 13–14 Monaten, im gesamten Gebiet evident, zurückgeführt werden kann. Schwingungen im Bereich eines quasi-biennalen Zykluses wurden ebenfalls beobachtet, jedoch mit geringerer Signifikanz. Eine Analyse der Hauptkomponenten ergibt eine Anzahl von Eigenvektoren, welche mit den Änderungen im jahreszeitlichen Charakter des regionalen Niederschlages identifiziert werden können. Der Jahresgang dominiert eindeutig in der gesamten Region, wobei Januar (Mittsommer) der nasseste Monat ist. Der zweite Eigenvektor besitzt hohe Amplituden im Frühling und Frühsommer (November) und zeigt eine räumliche Verteilung, die auf topographisch induzierte Unterschiede in den Konvektionsprozessen dieses Gebietes zurückzuführen ist. Die Hauptkomponenten höherer Ordnung sind in ihrer zeitlichen Struktur nicht eindeutig definiert. Der dritte Eigenvektor kann jedoch zum Teil auf synoptische Vorgänge bezogen werden, welche in den Küstenebenen und im Plateau des Hinterlandes verschiedene Niederschlagsvertefungen bewirken.
    Notes: Summary The mean monthly and annual rainfall series (1930–1977) for 30×30 blocks in the eastern Transvaal-Swaziland region ara analyzed. These series show temporal and spatial characteristics in the regional precipitation regime. Significance is attributed to oscillatory forcings with quasi-periodicities of about 16–20 years and 13–14 months that operate throughout the area. Oscillations near the quasi-biennial cycle are also observed but with less significance. Principal component analysis provides a number of eigenvectors which are identified with changes in the seasonal characteristics of the regional rainfall. The annual cycle clearly dominates over the entire region with January (mid-summer) being the wettest month. The second eigenvector has pronounced loadings in the spring/early summer (November) period and clear spatial responses that are attributed to topographically induced differences in the deep convective processes of the area. Higher principal components are not as clearly defined in their temporal structure. The third eigenvector, however, is attributed partially to synoptic scale forcings which produce different rainfall characteristics over the coastal plains relative to the high inland plateau.
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  • 100
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    Meteorology and atmospheric physics 32 (1983), S. 279-286 
    ISSN: 1436-5065
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geography , Physics
    Description / Table of Contents: Zusammenfassung Wenn grosse Werte unabhängig verteilt sind und wenn ihre Verteilungsfunktion bekannt ist, kann die Verteilungsfunktion des grössten Wertes (des Maximums) von dieser Verteilungsfunktion abgeleitet werden. Als Konsequenz dieses Prinzips ist es möglich, für das Jahresmaximum meteorologischer Variabler gute Schätzungen für Grössen mit langer Wiederholungsperiode, wie z.B. ein säkulares Maximum, auf Grund von relativ kurzen Beobachtungsreihen zu geben. Es werden zwei Beispiele mit Verwendung fünfjähriger Beobachtungen für das Jahresmaximum der Windgeschwindigkeit in Uccle und das Jahresmaximum der Tagesregenmenge in Uccle gebracht.
    Abstract: Résumé Lorsque les grandes valeurs sont réparties de façon indépendante let lorsque leur fonction de répartition est connue, la fonction de répartition de la plus grande de ces valeurs, c'est-à-dire le maximum peut être déduite de la première. Il en résulte que pour le maximum annuel d'une variable météorologique il est possible de dormer de bonnes estimations des fractiles à grande période de retour tels que le maximum séculaire à partir de séries d'observations relativement courtes. On donne deux exemples basés sur une série de cinq années d'observations: a) le maximum annuel de la vitesse instantanée du vent à Uccle, b) le maximum annuel de la cote pluviométrique journalière à Uccle.
    Notes: Summary When large values are independently distributed and when their distribution function is known, the distribution function of the largest (i.e. maximum) value may be derived from the former distribution function. As a consequence of this principle, it is possible to give for the annual maximum of a meteorological variate good estimations for fractiles with large return period such as the secular maximum with using relatively short series of observations. Two examples are given with the use of five years of observations: a) the annual maximum windspeed at Uccle, b) the annual maximum of the daily rainfall at Uccle.
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