ALBERT

Notes: Abstract A simple one-dimensional model of single-species populations is studied by means of computer simulations. Although the model has a rich spectrum of dynamics including chaotic behavior, the introduction of survival thresholds makes the chaotic region so small that it can be hardly observed. Stochastic fluctuations further reduce the chaotic region because they accidentally lead populations to extinction. The model thus naturally explains the observation that the majority of natural populations do not show chaotic behavior but a monotonic return to a stable equilibrium point following a disturbance.

Notes: Abstract Current understanding of the pattern of proliferation within intestinal crypts involves the notion of a cutoff region introduced by Cairnieet al. (Exp. Cell. Res. 39, 539–553, 1965b). (Cells produced above the cutoff are non-cycling, whereas cells produced below the cutoff are cycling.) They contrasted the predicted distribution of proliferation in the extreme cases of a cutoff of width 0 (a sharp cutoff) with one eight cells wide (a slow cutoff) and concluded that the data were better explained by the latter. We have shown that crypt size variation artificially broadens the apparent distribution of proliferating cells in the crypt (Totafurnoet al., Biophys. J. 54, 845–858, 1988). Here we show that the measurement and analysis of crypts of a specified height reduces this artifact. This work introduces the use of distance from the crypt base (in microns) to specify the location of cells within the crypt as an improvement over the cell position ordering traditionally used in the determination of the distribution of proliferating cells. We also show how to explicitly correct for several artifacts in the measurement of the labelling index. We conclude that cell proliferation within the crypt is more localized than previously realized; in fact, a cutoff as slow as eight cells wide is rejected.

Notes: Abstract Multiple string (sequence) alignment is a difficult and important problem in computational biology, where it is central in two related tasks: finding highly conserved subregions or embedded patterns of a set of biological sequences (strings of DNA, RNA or amino acids), and inferring the evolutionary history of a set of taxa from their associated biological sequences. Several precise measures have been proposed for evaluating the goodness of a multiple alignment, but no efficient methods are known which compute the optimal alignment for any of these measures in any but small cases. In this paper, we consider two previously proposed measures, and given two computationaly efficient multiple alignment methods (one for each measure) whose deviation from the optimal value isguaranteed to be less than a factor of two. This is the novel feature of these methods, but the methods have additional virtues as well. For both methods, the guaranteed bounds are much smaller than two when the number of strings is small (1.33 for three strings of any length); for one of the methods we give a related randomized method which is much faster and which gives, with high probability, multiple alignments with fairly small error bounds; and for the other measure, the method given yields a non-obviouslower bound on the value of the optimal alignment.

Notes: Abstract The kinematics of helical motion are descirbed for an organism treated as a rigid body with six degrees of freedom relative to the organism's frame of reference, i.e. the organism can translate in the direction of, or rotate around any of, three orthogonal axes fixed to its body. Equations are derived that express the unit vectors of the Frenet trihedron and the torsion and curvature of the trajectory in terms of the organism's translational and rotational velocities. These equations permit description of the radius, pitch, angular velocity and axis of a helical trajectory in terms of the translational and rotational velocities of the organism swimming along that trajectory. The results of this analysis are then used in two later papers that describe how organisms can orient to an external stimulus.

Notes: Abstract Organisms that move along helical trajectories change their net direction of motion largely by changing the direction, with respect to the body of the organism, of their rotational velocity (Crenshaw and Edelstein-Keshet, 1993,Bull. math. Biol. 55, 213–230). This paper demonstrates that an organism orients to a stimulus field, such as a chemical concentration gradient or a ray of light, if the components of its rotational velocity, with respect to the, body of the organism, are simple functions of the stimulus intensity encountered by the organism. For example, an organism can orient to a chemical concentration gradient if the rate at which it rotates around its anterior-posterior axis is proportional to the chemical concentration it encounters. Such an orientation can be either positive or negative. Furthermore, it is true taxis—orientation of the axis of helical motion is direct. It is neither a kinesis nor a phobic response—there is no random component to this mechanism of orientation.

Notes: Abstract A basic but neglected property of neuronal trees is their finite length. This finite length restricts the length of a segment to a certain maximum. The implications of the finite length of the tree with respect to the segment length distributions of terminal and intermediate segments are shown by means of a stochastic model. In the model it is assumed that branching is governed by a Poisson process. The model shows that terminal segments are expected to be longer than intermediate segments. Terminal and intermediate segments are expected to decrease in length with incrasing centrifugal order. The results are compared with data fromin vivo pyramidal cells from rat brain and tissue cultured ganglion cells from chicken. A good agreement between data and model was found.

Notes: Abstract Shape and size of elongating cells were examined in three plant tissues: the adaxial epidermis of the petiole ofZebrina pendula L., the abaxial epidermis ofAnacharis densa L. leaves and the abaxial epidermis of the scale leaf ofAllium cepa L. Based on a few simple assumptions, the expected probability distribution frequencies (pdf) for cell length and number of adjacent walls were calculated. Actual data of cell lengths closely approximated those expected with the pdfs being asymmetrical since there are more younger, shorter cells than older, longer cells. Data for number of lateral walls of real cells were similar to that expected and these walls increase in compensating mechanism exists to maintain a constant range of cell lengths through many cell generations. It is expressed by longer than average new daughter cells dividing relatively soon while shorter than average new daughter cells divide after a relatively long cycle.

Notes: Abstract Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.

Notes: Abstract The particular dynamics of the previously proposed model of a catalytic network formed byn error-prone self-replicative species without and with superimposed competition is analysed. In the first case, two situations are studied in detail: a uniform network in which all the species are inter-coordinated in the same way, and a network with a species differentiated in its catalytic relation with the remaining elements. In the second case, the superimposed competition is introduced at two levels: first, as an asymmetry in one of the network species amplification factor considering a null self-catalytic vector, and secondly, as a non-null self-catalytic vector with no asymmetry in the other propertics of the species. This kind of system does not present complex behaviour and can be adequately deseribed by performing a standard linear analysis, which gives direct information on the asymptotic behaviour of the sytem. Finally, the biological implications of this analysis within the framework of biological evolution are discussed.

Notes: Abstract A theoretical model is proposed for the formation of cell distribution patterns in the slug stage of the cellular slime moldDictyostelium discoideum. The equilibrium distribution of two types of cells, prestalk and prespore, is obtained by minimizing the free energy, which is defined in terms of differential chemotaxis, differential cell adhesion and randomness of cell movement. Resulting distributions show various segregation patterns of cell types. The condition for cell sorting is obtained from stability analysis of the set of diffusion equations governing the evolution of cell type distribution and the concentration of chemoattractant. The intensities of differential chemotaxis and random cell movement are quantitatively evaluated from experimental data to show that two cell types can sort themselves completely by these forces.

Notes: Abstract Multicell spheroids, small spherical clusters of cancer cells, have become an importantin vitro model for studying tumour development given the diffusion limited geometry associated with many solid tumour growths. Spheroids expand until they reach a dormant state where they exhibit a grossly static three-layered structure. However, at a cellular level, the spheroid is demonstrably dynamic with constituent cells migrating from the outer well-nourished region of the spheroid toward the necrotic central core. The mechanism that drives the migrating cells in the spheroid is not well understood. In this paper we demonstrate that recent experiments on internationalization can be adequately described by implicating pressure gradients caused by differential cell proliferation and cell death as the primary mechanism. Although chemotaxis plays a role in cell movement, we argue that it acts against the passive movement caused by pressure differences.

Notes: Abstract In recent years, methods of consensus, developed for the solution of problems in the social sciences, have become widely used in molecular biology. Westudy a method of consensus originally due to Watermanet al. (Waterman, Galas and Arratia. 1984. Pattern recognition in several sequences: consensus and alignment.Bull. math. Biol. 46, 515–527) which is used to identify patterns or features in a molecular sequence where a pattern can vary in position within a given window. We show that some well-known consensus methods of the social sciences, the median and the mean, are special cases of this method for certain choices of the parameters used in it and give a precise account of the parameters for which these special cases arise. We also show that the specific parameters used in the method of Watermanet al. make their method equivalent to the median procedure which is widely used in the social sciences.

Notes: Abstract We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensinalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-tye equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.

Notes: Abstract Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.

Notes: Abstract We show that the existence of diffusional resistance due to the presence of a solid phase can have a positive effect on the metabolic reactions of plant cells. In this case the efficiency of metabolic reactions, defined as the ratio of rate of production of biomass of aggregated cells/rate of production of biomass of dispersed cells, can be greater than unity for a certain range of aggregate sizes for both solid spheres (common plant cell aggregates) and hollow spheres (e.g.Volvox aggregates). This means that, under appropriate conditions, plant cells tend to stay in the aggregated form to improve the efficiency of their metabolic reactions. The result of the present analysis provides an explanation as to why aggregates of plant cells are observed under typical culture conditions.

Notes: Abstract The Hodgkin and Huxley equations model action potentials in squid giant axons. Variants of these equations are used in most models for electrial activity of excitable membranes. Computational tools based upon the theory of nonlinear dynamical systems are used here to illustrate how the dynamical behavior of the Hodgkin Huxley model changes as functions of two of the system parameters.

Notes: Abstract The description of the “microbial loop” has led to some major changes in our understanding of nutrient cycling within aquatic ecosystems. It now appears that in many settings it is not uncommon for some 50% of phytoplankton production to be diverted into microbial pathways rather than passing up to higher trophic levels. As a result the microbial loop is responsible for enhanced and rapid nutrient cycling at the very base of the food web. Since tight recycling is often associated with unstable positive feedback, we use a model to examine the possible repercussions in more detail. The model simulates the dynamics of the microbial loop and finds it to greatly affect the way in which aquatic primary production responds to nutrient pulses.

Notes: Abstract The maintenance activity of plants is investigated in terms of a simple model. Maximization of a certain biomass fraction we refer to asnonactive biomass is postulated. Optimal behaviour of plants according to this principle is explicitly derived and expressed depending on environmental conditions. Several interesting hypotheses result, e.g. a quadratic law relating specific growth rate and gross rate of photosynthesis. A qualitative comparison with data from the literature is performed, with a special emphasis on the question whether plants stressed by air pollutants repair optimally. Regarding long-term constant environmental conditions, no data were found that contradict optimal behaviour. Exact quantitative testing of the theory is desirable, appropriate experiments are suggested.

Notes: Abstract In an earlier work a model of the autocrine and paracrine pathways of tumor growth control was developed (Michelson and Leith. 1991. Autocrine and paracrine growth factors in tumor growth.Bull. math. Biol. 53, 639–656). The target population, a generic tumor, was modeled as a single, homogeneous population using the standard Verhulst equation of logistic growth. Mitogenic signals were represented by modifications to the Malthusian growth parameter and adaptational signals were represented by modifications to the carrying capacity. Three growth scenarios were described: (1) normal tissue wound healing, (2) unperturbed tumor growth, and (3) tumor growth in a radiation damaged environment, a phenomenon termed the Tumor Bed Effect (TBE). In this paper, we extend those results to include a “triad” of growth factor controls (autocrine, paracrine and endocrine) and heterogeneity of the target population. The heterogeneous factors in the model represent either intrinsic, epigenetic or environmental differences in both normally differentiating tissues and tumors. Three types of growth are modeled: (1) normal tissue differentiation or wound healing, assuming no communication between differentiated and undifferentiated cell compartments; (2) normal wound healing with feedback inhibition, due to signalling from the differentiated compartment; and (3) the development of hypoxia in a spherical tumor. The signal processing within the triad is discussed for each model and biologically reasonable constraints are defined for limits on growth control.

Notes: Abstract A transient multispecies model for quantifying microbial space competition in biofilm is derived from existing models, introducing a new approach to biomass detachment modelling. This model includes inert biomass, substrate diffusion and utilization rate within the biofilm and diffusional layers. It predicts the evolution of biofilm thickness, bulk substrate concentration, species distribution and substrate concentration within the biofilm. A zero-dimensional transient model is described. Its steady-state solution is used to set up initial conditions of the one-dimensional model and case computation towards steady-state solution. Some numerical tools have been developed, enabling fast computation on microcomputers. Simulations show the validity of a zero-dimensional model and perturbated systems are also simulated. Simulations with experimental data give acceptable results.

Notes: Abstract Recently, we proposed a new model of DNA sequence evolution (Arquès and Michel. 1990b.Bull. math. Biol. 52, 741–772) according to which actual genes on the purine/pyrimidine (R/Y) alphabet (R=purine=adenine or guanine, Y=pyrimidine=cytosine or thymine) are the result of two successive evolutionary genetic processes: (i) a mixing (independent) process of non-random oligonucleotides (words of base length less than 10: YRY(N)6, YRYRYR and YRYYRY are so far identified; N=R or Y) leading to primitive genes (words of several hundreds of base length) and followed by (ii) a random mutation process, i.e. transformations of a base R (respectively Y) into the base Y (respectively R) at random sites in these primitive genes. Following this model the problem investigated here is the study of the variation of the 8 R/Y codon probabilities RRR,..., YYY under random mutations. Two analytical expressions solved here allow analysis of this variation in the classical evolutionary sense (from the past to the present, i.e. after random mutations), but also in the inverted evolutionary sense (from the present to the past, i.e. before random mutations). Different properties are also derived from these formulae. Finally, a few applications of these formulae are presented. They prove the proposition in Arquès and Michel (1990b.Bull. math. Biol. 52, 741–772), Section 3.3.2, with the existence of a miximal mean number of random mutations per base of the order 0.3 in the protein coding genes. They also confirm the mixing process of oligonucleotides by excluding the purine/pyrimidine contiguous and alternating tracts from the formation process of primitive genes.

Notes: Abstract It is believed that the native folded three-dimensional conformation of a protein is its lowest free energy state, or one of its lowest. It is shown here that both a two-and three-dimensional mathematical model describing the folding process as a free energy minimization problems is NP-hard. This means that the problem belongs to a large set of computational problems, assumed to be very hard (“conditionally intractable”). Some of the possible ramifications of this results are speculated upon.

Notes: Abstract A model employing separate dose-dependent response functions for proliferation and differentiation of idiotypically interacting B cell clones is presented. For each clone the population dynamics of proliferating B cells, non-proliferating B cells and free antibodies are considered. An effective response function, which contains the total impact of proliferation and differentiation at the fixed points, is defined in order to enable an exact analysis. The analysis of the memory states is restricted in this paper to a two-species system. The conditions for the existence of locally stable steady states with expanded B cell and antibody populations are established for various combinations of different field-response functions (e.g. linear, saturation, log-bell functions). The stable fixed points are interpreted as memory states in terms of immunity and tolerance. It is proven that a combination of linear response functions for both proliferation and differentiation does not give rise to stable fixed points. However, due to competition between proliferation and differentiation saturation response functions are sufficient to obtain two memory states, provided proliferation preceeds differentiation and also saturates earlier. The use of log-bell-shaped response functions for both proliferation and differentiation gives rise to a “mexican-hat” effective response function and allows for multiple (four to six) memory states. Both a primary response and a much more pronounced secondary response are observed. The stability of the memory states is studied as a function of the parameters of the model. The attractors lose their stability when the mean residence time of antibodies in the system is much longer than the B cells' lifetime. Neither the stability results nor the dynamics are qualitatively chanbed by the existence of non-proliferating B cells: memory states can exist and be stable without non-proliferating B cells. Nevertheless, the activation of non-proliferating B cells and the competition between proliferation and differentiation enlarge the parameter regime for which stable attractors are found. In addition, it is shown that a separate activation step from virgin to active B cells renders the virgin state stable for any choice of biologically reasonable parameters.

Notes: Abstract More than 20 years after its proposal, Keller and Segel's model (1971,J. theor. Biol.,30, 235–248) remains by far the most popular model for chemical control of cell movement. However, before the Keller-Segel equations can be applied to a particular system, appropriate functional forms must be specified for the dependence on chemical concentration of the cell transport coefficients and the chemical degradation rate. In the vast majority of applications, these functional forms have been chosen using simple intuitive criteria. We focus on the particular case of eukaryotic cell movement, and derive an approximation to the detailed model of Sherrattet al. (1993,J. theor. Biol.,162, 23–40). The approximation consists of the Keller-Segel equations, with specific forms predicted for the cell transport coefficients and chemical degradation rate. Moreover, the parameter values in these functional forms can be directly measured experimentally. In the case of the much studied neutrophil-peptide system, we test our approximation using both the Boyden chamber and under-agarose assays. Finally, we show that for other cell-chemical interactions, a simple comparison of time scales provides a rapid check on the validity of our Keller-Segel approximation.

Notes: Abstract The formal structure of evolutionary theory is based upon the dynamics of alleles, individuals and populations. As such, the theory must assume the prior existence of these entities. This existence problem was recognized nearly a century ago, when DeVries (1904,Species and Varieties: Their Origin by Mutation) stated. “Natural selection may explain the survival of the fittest, but it cannot explain the arrival of the fittest.” At the heart of the existence problem is determining how biological organizations arise in ontogeny and in phylogeny. We develop a minimal theory of biological organization based on two abstractions from chemistry. The theory is formulated using λ-calculus, which provides a natural framework capturing (i) the constructive feature of chemistry, that the collision of molecules generates specific new molecules, and (ii) chemistry's diversity of equivalence classes, that many different reactants can yield the same stable product. We employ a well-stirred and constrained stochastic flow reactor to explore the generic behavior of large numbers of applicatively interacting λ-expressions. This constructive dynamical system generates fixed systems of transformation characterized by syntactical and functional invariances. Organizations are recognized and defined by these syntactical and functional regularities. Objects retained within an organization realize and algebraic structure and possess a grammar which is invariant under the interaction between objects. An organization is self-maintaining, and is characterized by (i) boundaries established by the invariances, (ii) strong self-repair capabilities responsible for a robustness to perturbation, and (iii) a center, defined as the smallest kinetically persistent and self-maintaining generator set of the algebra. Imposition of different boundary conditions on the stochastic flow reactor generates different levels of organization, and a diversity of organizations within each level. Level 0 is defined by selfcopying objects or simple ensembles of copying objects. Level 1 denotes a new object class, whose objects are self-maintaining organizations made of Level 0 objects, and Level 2 is defined by self-maintaining metaorganizations composed of Level 1 organizations. These results invite analogy to the history of life, that is, to the progression from self-replication to self-maintaining procaryotic organizations to ultimately yield self-maintaining eucaryotic organizations. In our system self-maintaining organizations arise as a generic consequence of two features of chemistry, without appeal to natural selection. We hold these findings as calling for increased attention to the structural basis of biological order.

Notes: Abstract A model is developed to describe neuronal elongation as a result of the polymerization of microtubules and elastic stretching of the neurites by force produced by the growth cone. The model for a single segment with a single growth cone revealed a constant elongation rate, while the concentration of tubulin in the soma rises, and the concentration of tubulin becomes constant in the growth cone. Extending the model to a neurite with a single branch point and two growth cones revealed the same results. When the assembly or the disassembly rate of microtubules is unequal in both growth cones, transient retraction of one of the terminal segments occurs, which results in complete retraction of the segment when the difference in (dis)assembly rate between the two growth cones is large enough. When the model is applied to large trees, a maximal sustainable number of terminal segments as a function of the production rate of tubulin appears. Mechanisms to stop outgrowth are discussed in relation to the establishment of synaptical contacts between cells.

Notes: Abstract We have developed a new model describing the relationship between plasma and red cell tracers flowing through the lung. The model is the result of an analysis of the transport of radiolabeled plasma albumin between two flowing phases and shows that differences between red cell and plasma tracer curves are related to microvascular hematocrit. The model was tested in an isolated, blood-perfused dog lung preparation in which we injected51Cr-labeled red cells and125I-labeled plasma albumin into the pulmonary artery. From the tracer concentration-time curves at the venous outflow, we calculatedh r, the ratio of microvascular hematocrit to large-vessel hematocrit. In 18 baseline experiments,h r=0.92±0.01 (mn±sem) at a blood flow rate of 10.7±0.3 ml s−1. We determined the effects of (a) glass bead embolization, (b) alloxan, and (c) lobe ligation onh r. Embolization attenuated the separation between plasma and red cells (increasedh r), probably as a consequence of passive vasodilation. Alloxan enhanced separation of plasma and red cells (decreasedh r), possibly as a result of arteriolar vasoconstriction. Ligation of a fraction of the perfused tissue at constant flow did not cause significant change inh r in the remaining perfused tissue. The model assumes that large-vessel transit times are uniform and that all dispersion occurs in the microvasculature. A theoretical analysis apportioning dispersion between large and small vessels disclosed that the error associated with these assumptions is likely to be less than 15% of the measuredh r. We conclude from this study that the microvascular hematocrit model describes experimental plasma and red cell curves. The results imply thath r can be readily deduced from tagged red cells and plasma and can be accounted for in calculating permeability-surface area in diffusing tracer experiments.

Notes: Abstract We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seenin vivo, including immunostimulation of tumor growth, “sneaking through” of the tumor, and formation of a tumor “dormant state”. The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL1 in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measuredin vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.

Notes: Abstract We present a new, practical algorithm to resolve the experimental data in restriction site analysis, which is a common technique for mapping DNA. Specifically, we assert that multiple digestions with a single restriction enzyme can provide sufficient information to identify the positions of the restriction sites with high probability. The motivation for the new approach comes from combinatorial results on the number of mutually homeometric sets in one dimension, where two sets ofn points are homeometric if the multiset ofn(n−1)/2 distances they determine are the same. Since experimental data contain errors, we propose algorithms for reconstructing sets from noisy interpoint distances, including the possibility of missing fragments. We analyse the performance of these algorithms under a reasonable probability distribution, establishing a relative error limit ofr=Θ(1/n 2) beyond which our technique becomes infeasible. Through simulations, we establish that our technique is robust enough to reconstruct data with relative errors of up to 7.0% in the measured fragment lengths for typical problems, which appears sufficient for certain biological applications.

Notes: Abstract A simple chemical model of the idiotypic network of immune systems, namely the AB model, has been developed by De Boeret al. The complexity of the system, such as the steady states, periodic oscillations and chaotic motions, has been examined by the authors mentioned above. In the present paper, the periodic motions and chaotic behaviours exhibited by the system are intuitively described. To clarify in which parameter domains concerned the system exhibits periodic oscillations and in which parameter domains the system demonstrates chaotic behaviours the Lyapounov exponent is explored. To characterize the strangeness of the attractors, the fractal dimension problem is worked out.

Notes: Abstract We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.

Notes: Abstract Dextran has been the most commonly employed test molecule for probing the selectivity of glomerular filtration to macromolecules of varying size. The usual theories for hindered transport of solid spheres through pores have limited utility in interpreting clearance data for dextran or other linear polymers because such polymers in solution more closely resemble random, solvent-filled coils than solid spheres. To provide a model for glomerular filtration of random-coil macromolecules, the equilibrium partitioning of random coils between cylindrical pores and bulk solution was simulated using Monte Carlo calculations, and those results were combined with a hydrodynamic theory for restricted motion of solvent-filled polymer coils in pores. The rates of transport predicted for either neutral random coils or for solid spheres of the same Stokes-Einstein radius were significantly lower than observed transport rates of dextran through the glomerular capillary wall or across synthetic porous membranes. This facilitation of dextran transport was modeled by postulating weak, attractive interactions between dextran monomers and the pore wall. The random-coil model with attractive interactions, modeled using a short-range, square-well potential, was found to adequately represent dextran sieving data in normal rats. Various limitations of this approach are discussed.

Notes: Abstract Method-dependent mechanisms that may affect dynamic numerical solutions of a hyperbolic partial differential equation that models concentration profiles in renal tubules are described. Some numerical methods that have been applied to the equation are summarized, and ways by which the methods may misrepresent true solutions are analysed. Comparison of these methods demonstrates the need for thoughtful application of computational mathematics when simulating complicated time-dependent phenomena.

Notes: Abstract The regulation of the interactions between the actin binding proteins and the actin filaments are known to affect the cytoskeletal structure of F-actin. We develop a model depicting the formation of actin cytoskeleton, bundles and orthogonal networks, via activation or inactivation of different types of actin binding proteins. It is found that as the actin filament density increases in the cell, a spontaneous tendency to organize into bundles or networks occurs depending on the active actin binding protein concentration. Also, a minute change in the relative binding affinity of the actin binding proteins in the cell may lead to a major change in the actin cytoskeleton. Both the linear stability analysis and the numerical results indicate that the structures formed are highly sensitive to changes in the parameters, in particular to changes in the parameter ϕ, denoting the relative binding affinity and concentration of the actin binding proteins.

Notes: Abstract To investigate morphogenesis and in particular circularization mechanisms in young mycelia, we observe cultures of the zygomyceteMucor spinosus and develop discrete models of two-dimensional filamental branching growth. The models are based on the hypothesis that the fungus secretes a regulatory substance that diffuses into the surrounding medium and is detected by the growing hyphae. We also present a simple Markovian growth model without such a feedback, but yielding to analytical computations.

Notes: Abstract In vivo volume growth of two murine tumor cell lines was compared by mathematical modeling to their volume growth as multicellular spheroids. Fourteen deterministic mathematical models were studied. For one cell line, spheroid growth could be described by a model simpler than needed for description of growthin vivo. A model that explicitly included the stimulatory role for cell-cell interactions in regulation of growth was always superior to a model that did not include such a role. The von Bertalanffy model and the logistic model could not fit the data; this result contradicted some previous literature and was found to depend on the applied least squares fitting method. By the use of a particularly designed mathematical method, qualitative differences were discriminated from quantitative differences in growth dynamics of the same cells cultivated in two different three-dimensional systems.

Notes: Abstract This paper develops and applies a dynamic mathematical model for optimal scheduling of nitrogen fertilization and irrigation that minimizes nitrogen leaching subject to a target level of yield. The analysis assumes a single crop grown during a single growing season of a given length. It is shown that substitution of water for nitrogen along a given plant growth path decreases nitrogen leaching and, therefore, groundwater contamination. It is proved that a minimum leaching solution to the optimization problem is obtained with a single nitrogen application at the beginning of the season and irrigation scheduling that maintains a wet soil throughout the growing period. A numerical example utilizing experimental data for an irrigated summer corn in Israel confirms and quantifies the analytical findings.

Notes: Abstract We analyse the stochastic properties of dynamical systems with finite populations of a few differentreplicator species. Our main interest is to evaluate the typicallifetime, i.e. the time for the extinction of the first species in the network, for different catalytic structures, as a function of the population size.

Notes: Abstract The capacity of a model immune network in terms of the number of different antigens that can be vaccinated against without any memory lost is computed and tested by numerical simulations. We also investigate memory loss and failure to vaccinate due to overcrowding the network with too many antigens. The computations are done for two different strategies for proliferation, one implying all the antigen specific clones and the second one being more thrifty.

Notes: Abstract The technique of model-building a protein of known sequence but unknown tertiary structure from the structures of homologous proteins is probably so far the most reliable means of mapping from primary to tertiary structure. A key step towards the realization of the aim is to develop ways of aligning three-dimensional structures of homologus proteins, thereby deriving the rules useful for protein modelling. We have developed a generalized differential-geometric representation of protein local conformation for use in a protein comparison program which aligns protein sequences on the basis of their sequence and conformational knowledge. Because the differetial-geometric distance measure between local conformations is independent of the coordinate frame and remains chirality information, the comparison program is easily implemented, relatively rational and reasonably fast. The utility of this program for aligning closely and distantly related homologous proteins is demonstrated by multiple alignment of globins, serine proteinases and aspartic proteinase domains. Particularly, the method has reached the rational alignment between the mammalian and microbial serine proteinases as compared with many published alignment programs.

Notes: Abstract New formulas for deriving the sensitivities of stable stage structures and reproductive values to changes in vital rates are presented. They enable comparison of the sensities to changes of different elements in the projection matrix; in other words, comparison of partial derivatives of the eigenvectors. These kinds of sensitivities can be used in applied problems such as an analysis of the effect of harvesting on the population structure. However, in this paper, we examine the application of the sensitivities in a more general ecological context. We investigate why the stable stage structure of the mustard aphid,Lipaphis erysimi, changes very little in the temperature interval 10–30°C. The sensitivities of the stable stage structure at 15°C and 25°C were derived. The character of the sensitivites were the same in both temperatures although the stage structure was more sensitive to changes at 15°C than at 25°C. The sensitivity analysis also revealed that the temperature variation results in changes in fecundity and developmental rate that have a counteractive effect on the population structure.

Notes: Abstract Plankton populations undergo dramatic surges. Rapid increases and decreases by a factor of 10 or more are observed, often separated by relatively stable interludes. We propose a description of plankton communities as excitable systems. In particular, we present a model for the evolution of phytoplankton and zooplankton populations which resembles models for the behaviour of excitable media. The parameter dependency of the various “excitable” phenomena, trigger mechanism, threshold, and slow recovery, is clear, and permits ready investigation of the influence of properties of the physical environment, including variations in nutrient fluxes, temperature or pollution levels.

Notes: Abstract Analysis schemes for the classification of synergism and antagonism for mixed agents operate on the discrepancies between observed and calculated results. As such they cannot be confirmed by experiments and therefore have to be tested in terms of mathematical and logical self-consistency. The concept of independent action is close to the literal meaning of the term “non-interaction”. Since this concept does not depend on the mechanisms of actions nor on the type of effect scale used, it is suitable as one of the basic criterion for the definition of synergism and antagonism. A general mathematical framework of independent action is presented in this paper based on the concept of “relative effect” as used in the literature. The, different equations for independent action currently used in various areas are shown to be manifestations, of a general formula under different sets of boundary conditions, which are the natural limiting values of the effects of the corresponding system observed at low and at high doses of the agents. The framework can, be generalized to the combined action ofn-agents as well as to the interaction of an agent with itself. In addition, the differential form of the formula for independent action is derived. This framework of systematic definitions and derived equations enable a more in-depth study of the implications of the concept of independent action and its relation to other concepts of non-interaction.

Notes: Abstract We model how auto-reactiveB cells are kept under control by an idiotypic network. Autoimmunity occurs when the control is broken by an infection or not achieved through an abnormal ontogenetic evolution. We describe the idiotypic network, viz., the central immune system, by idiotype-anti-idiotype pairs which are coupled to a set of highly connected clones, which interact with each clone of the network. Some clones of the central immune system recognize self-antigen. We find a huge variety of fixed points which can be classified as tolerant, autoimmune, and neutral states according to the concentration of the auto-reactive antibody. Most significant are auto-reactive clones which are a member of an idiotype-anti-idiotype pair. In a healthy individual, an autoimmune disease is induced by an antigen infection which triggers a transition from a tolerant to an autoimmune state. Autoimmunity is induced more readily by an antigen coupling to theanti-idiotype than by one interacting with the auto-reactive clone itself. We indicate a possible therapy which simply reverses the processes that have lead to the autoimmune disease. In the early development of the central immune system its highly connected, core part serves to draw the more specific clones of idiotype-anti-idiotype pairs into the network. In order to avoid autoimmunity in ontogenetic evolution the anti-idiotype of an auto-reactive clone must be formed in advance by a sufficiently long period of time. Thus, a well ordered succession of the appearance of the more specific clones is required.

Notes: Abstract A method of dimensionless time-scaling based on extrinsic expectation of life at birth but intrinsic to a system generating a survival distribution is introduced. Such scaling allows the survival fraction function and its associated mortality function to serve as Green's functions for their generalized equivalents. i.e. a “population” function and a “death” function. The analytical mechanics of utilizing these concepts are formulated, applied to the classical Gompertz and Weibull survival models, and discussed with respect to biological relevance.

Notes: Abstract It is now widely accepted that localized high concentrations of Ca2+ (Ca2+ domains) play a major role in controlling the time course of neurotransmitter release. In the present work we calculate the magnitude and the time course of Ca2+ domains that evolve in the vicinity of a Ca2+ channel and an adjacent release site. In the calculations we consider a accurately dimensioned Ca2+ channel. Moreover, the Ca2+ current is continuously adjusted with regard to the accumulated intracellular Ca2+ and, in addition, endogenous buffers are considered. The calculations, carried out by the software FIDAP, based on finite element method, show that the Ca2+ concentrations achieved near the release sites are significantly lower than claimed by other investigators. Furthermore, we present arguments indicating that the Ca2+ domains, regardless of their magnitude, do not play a role in controlling the time course of release of neurotransmitter.

Notes: Abstract Mammalian white blood cells are known to bias the direction of their movement along concentration gradients of specific chemical stimuli, a phenomenon called chemotaxis. Chemotaxis of leukocyte cells is central to the acute inflammatory response in living organisms and other critical physiological functions. On a molecular level, these cells sense the stimuli termed chemotactic factor (CF) through specific cell surface receptors that bind CF molecules. This triggers a complex signal transduction process involving intracellular biochemical pathways and biophysical events, eventually leading to the observable chemotactic response. Several investigators have shown theoretically that statistical fluctuations in receptor binding lead to “noisy” intracellular signals, which may explain the observed imperfect chemotactic response to a CF gradient. The most recent dynamic model (Tranquillo and Lauffenburger,J. Math. Biol. 25, 229–262. 1987) couples a scheme for intracellular signal transduction and cell motility response with fluctuations in receptor binding. However, this model employs several assumptions regarding receptor dynamics that are now known to be oversimplifications. We extend the earlier model by accounting for several known and speculated chemotactic receptor dynamics, namely, transient G-protein signaling, cytoskeletal association, and receptor internalization and recycling, including statistical fluctuations in the numbers of receptors among the various states. Published studies are used to estimate associated constants and ensure the predicted receptor distribution is accurate. Model analysis indicates that directional persistence in uniform CF concentrations is enhanced by increasing rate constants for receptor cytoskeletal inactivation, ternary complex dissociation, and binary complex dissociation, and by decreasing rate constants for receptor internalization and recycling. For most rate constants, we have detected an optimal range that maximizes orientation bias in CF gradients. We have also examined different desensitization and receptor recycling mechanisms that yield experimentally documented orientation behavior. These yield novel insights into the relationship between receptor dynamics and leukocyte chemosensory movement behavior.

Notes: Abstract Given two independent sequences of letters, we seek the probability distribution of the length of the longest matching word. This word can be in different positions in the two sequences and we consider both perfect and nearly perfect matching. We derive bounds and approximations for the probability and compare them with other bounds and approximations. The results can be applied to DNA sequences in molecular biology and generalized matching between two independent random sequences.

Notes: Abstract Once designed, implementation of an optimal mean-square binary morphological filter is extremely fast, especially when the erosions are implemented on a suitable parallel processor. On the other hand, optimal filter design involes a computationally burdensome search procedure that can, in practice, be intractable. The present paper provides an algorithm for filter design that is based on the relationship between the optimal morphological filter and the conditional expectation. The algorithm proceeds by changing the conditional expectation into a morphological filter while at the same time increasing the mean-square error by a minimal amount. It does so by switching observations between the 1-set and the 0-set of the conditional expectation. The switching algorithm is extremely efficient in many noise environments, and therefore provides a filter design that can be useful for online structuring-element updating. Owing to the relationship between stack and morphological filters, the algorithm is at once useful for finding optimal binary stack filters.

Notes: Abstract A basic control engineer's adage-the poles of a feedback compensator become zeros of the closed-loop system-admits difficulties of interpretation even in the most simple of cases; that of single-input, single-output. An earlier investigation has provided an analysis of this adage in a module-theoretic context for systems in state space form while avoiding restrictive assumptions on system minimality or squareness. The main result is expressed concisely in terms of an exact sequence of modules which include Ω-zero modules corresponding to the feedback system and the plant. Extended zero modules of Ω-type incorporate both finite invariant zero structure, and generic zero information which occurs when a system fails to be right-invertible. In the case of compensation in the feedback path, this main exact sequence reduces to a mathematically clear expression of the aforementioned adage: the Ω-zero module of the feedback system is precisely the direct sum of the Ω-zero module of the plant and the system pole module of the feedback compensator. This paper extends the previous work in order to avoid assumptions on causality in the plant. Implicit dynamical systems are employed, in lieu of systems in state space form. Once again, it is not assumed that the system is one-to-one or onto; and so the concepts of generic zeros and their modules are brought into the arena of implicit systems. The implicit system itself is assumed in this work to be regular; however, decoupling zeros are permitted. Moreover, input-decoupling zeros and system pole feedback relationships are considered.

Notes: Abstract When the problem is considered of obtaining a periodic description in state-space form of a linear process which can be modelled by linear difference equations with periodic coefficients, it is natural to ask whether it is possible to preliminarily derive a polynomial equivalent form of such equations, which in the periodic case plays a role similar to the Rosenbrock's polynomial matrix description of a linear time-invariant process. In this paper a polynomial time-invariant description of a linear periodic process is introduced. It is shown that such a polynomial description gives a simple characterization of the dimension of the space of the solutions corresponding to the null input function, i.e., of the order of the periodic model under consideration. In addition, it allows us to introduce a transfer matrix for the computation of the output responses corresponding to null initial conditions, and to deduce conditions for the periodic model to be causal. These results, as well as the possibility of defining strict system equivalence between two periodic models through their time-invariant polynomial descriptions, in a similar sense as in the time-invariant case, show the relevance of such a polynomial time-invariant description for the problem under consideration.

Notes: Abstract An actual sampling process can be modeled as a random process, which consists of the regular (uniform) deterministic sampling process plus an error in the sampling times which constitutes a zero-mean noise (the jitter). In this paper we discuss the problem of estimating the jitter process. By assuming that the jitter process is an i.i.d. one, with standard deviation that is small compared to the regular sampling time, we show that the variance of the jitter process can be estimated from thenth order spectrum of the sampled data,n=2, 3, i.e., the jitter variance can be extracted from the 2nd-order spectrum or the 3rd-order spectrum (the bispectrum) of the sampled data, provided the continuous signal spectrum is known. However when the signal skewness exceeds a certain level, the potential performance of the bispectrum-based estimation is better than that of the spectrum-based estimation. Moreover, the former can also provide jitter variance estimates when the continuous signal spectrum is unknown while the latter cannot. This suggests that the bispectrum of the sampled data is potentially better for estimating any parameter of the sampling jitter process, once the signal skewness is sufficiently large.

Notes: Abstract Functions belonging to various Paley-Wiener spaces have representations in sampling series. When a function does not belong to such a space, the sampling series may converge, not to the object function but to an "alias" of it, and an aliasing error is said to occur. Aliasing error bounds are derived for one- and two-channel sampling series analogous to the Whittaker-Kotel’nikov-Shannon series, and for the multi-band sampling series, and a "derivative" extension of it, due to Dodson, Beaty, et al. The Poisson summation formula is a basic tool throughout. Aliasing in the one-channel case is shown to arise from a transformation with similarities to a projection. Where possible, the sharpness of the error bounds is discussed.

Notes: Abstract This paper is devoted to a study of the Hausdorff-Young theorems from a historical perspective, beginning with the F. Riesz-Fischer theorem. Introduced by W. H. Young (1912), these theorems were considered and extended by F. Hausdorff (1923), F. Riesz (1923), E.C. Titchmarsh (1924), G. H. Hardy and J.E. Littlewood (1926), M. Riesz (1927), and O. Thorin (1939/48). Special emphasis is placed upon the development of the proofs of the two Hausdorff-Young inequalities and their impact upon Fourier analysis as a whole, in particular on the M. Riesz-Thorin convexity theoremand on the interpolation of operators. The golden thread connecting the various extensions and generalizations is the concept of logarithmic convexity, one that goes back to the work of J. Hadamard (1896), A. Liapounoff (1901), J.L.W.V. Jensen (1906), and O. Blumenthal (1907).

Notes: Abstract In this paper we give a further investigation of the method introduced by the author in [1, Frequency-domain bounds for nonnegative unsharply band-limited functions] for proving bounds for functions with nonnegative Fourier transforms. We also dealt with the question of how large the supremum KS of all numbers |f(u)| is with f the Fourier transform of a nonnegative integrable function F and f(0) = 1, |f(ku)| ≤ ε for k ∈ S. Here u 〉 0 and S ⊂ {2, 3, . . .}. This problem was related in [1] to finding the infimum MS of all numbers Mh = maxϑ [(1−h(ϑ))/(1− cos ϑ)] over all 2π-periodic even, smooth functions h whose Fourier cosine coefficients ak vanish for k ∉ S, and it was proved and announced for several cases that MS (1−KS ) = 1. In this paper we prove the results announced in [1]. To that end we generalize the method given in [1] to include Fourier transforms f of probability measures on R and a certain generalized function h, and we show that the numbers KS, MS are assumed as |f(u)|, Mh for certain allowed f,h. Moreover, we establish a fundamental relation between finding the numbers KS, MS and the numbers KT, MT where T = {2, 3, . . .}\S. In particular, we show that MT = 2KS (2KS − 1)−1,KT = 1/2 MS(MS − 1)−1 and that MT (1 − KT) = 1,KSKT = 1/2 , whenever MS (1 − KS) = 1.

Notes: Abstract Finite energy band-limited functions are reconstructed iteratively from nonuniform sample values of the functions and its derivatives. It is shown that the maximum gap allowed between the sampling points increases linearly with the number of derivatives considered. Moreover, a more precise result is presented for the first derivative case and another reconstruction of the functions using the frame algorithm is deduced.

Notes: Abstract In the early 1960s research into radar signal synthesis produced important formulas describing the action of the two-dimensional Fourier transform on auto- and crossambiguity surfaces. When coupled with the Poisson Summation formula, these results become applicable to the theory of Weyl-Heisenberg systems, in the form of lattice sum formulas that relate the energy of the discrete crossambiguity function of two signals f and g over a lattice with the inner product of the discrete autoambiguity functions of f and g over a "complementary" lattice. These lattice sum formulas provide a framework for a new proof of a result of N.J. Munch characterizing tight frames and for establishing an important relationship between l1-summability (condition A) of the discrete ambiguity function of g over a lattice and properties of the Weyl-Heisenberg system of g over the complementary lattice. This condition leads to formulas for upper frame bounds that appear simpler than those previously published and provide guidance in choosing lattice parameters that yield the most snug frame at a stipulated density of basis functions.

Notes: Abstract Let $a〉0, b〉0, ab〈1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$ $t\in{\Bbb R}$ , for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$ $B〈\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.

Notes: Abstract For any ε 〉 0, we construct an orthonormal Schauder basis of C(K) consisting of trigonometric polynomials Tn n = 1, 2, . . . , such that deg(Tn) ≤ (1/2)(1 + ε)n. This is best possible with regard to the degree. The construction uses wavelet techniques.

Notes: Abstract We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.

Notes: Abstract In the spirit of work of Kerman and Sawyer, a condition is given that is necessary and sufficient for the Fourier transform norm inequality $\Big(\int_{{\Bbb R}_d} \vert\hat{f}\vert^q d\mu\Big)^{1/q} \leq C\Big(\int_{{\Bbb R}_d} \vert f\vert^p v\Big)^{1/p}$ provided v is a radial weight for which v−1/p is convexly decreasing and μ is a suitable measure. We also establish alternative conditions for such inequalities by proving corresponding trace type inequalities and maximal function inequalities that underlie the Fourier transform estimates. Our conditions are relatively simple to compute. Among applications we give extensions of a Sobolev restriction theorem.

Notes: Abstract We present two-sided singular value estimates for a class of convolution-product operators related to time-frequency localization.

Notes: Abstract New characteristics of feedback neural networks are studied. We discuss in detail the question of updating of neurons given incomplete information about the state of the neural network. We show how the mechanism of self-indexing for such updating provides better results than assigning ‘don't know’ values to the missing parts of the state vector. Issues related to the choice of the neural model for a feedback network are also considered. Properties of a new complex valued neuron model that generalizes McCulloch-Pitts neurons are examined.

Notes: Abstract This paper deals with chained eigenstructure assignment for strongly controllable singular systems of the form Êx (t)=Â x(t)+B u(t) with state feedback control of the formu(t)=Kx(t)+w(t). The development of our method depends crucially on the properties of standard form singular systems. The closed-loop system will satisfy the following requirements: regularity, impulse-free response and rankÊ arbitrary eigenvalues assignment. This parametric characterization conveniently organizes the nonunique gain matrixK to modify the dynamic response of the systems. The result can be used for discrete-time descriptor systems, in which a zero-value eigenvalue may well be a desired closed-loop eigenvalue. One illustrative example is included.

Notes: Abstract An efficient bi-state stochastic gradient is proposed for spontaneous constrained time delay estimation. The quantized stochastic gradient is an approximation of the polarity of the instantaneous delay estimation error. It is adjusted in such a way that it has a much higher probability to move in the correct direction at each iteration so as to enable a speed-up in the delay estimate to converge to global minimum in steady state. The performance of the delay estimator is evaluated statistically and an analytical solution for its convergence behavior is established. It is demonstrated that the proposed algorithm has at least a two-fold improvement in convergence speed when compared with the conventional approach, and this is verified by extensive simulation results.

Notes: Abstract In this work, we extend the coding theory approach to error control in redundant residue number systems (RRNS). The concept of erasure correction capability in RRNS is introduced. We derive the relationship between the minimum distance and the error detection and error/erasure correction capability. New computationally efficient algorithms are derived for simultaneously correcting single errors and multiple erasures and detecting multiple errors. These algorithms reduce the computational complexity of the previously known algorithms by at least an order of magnitude. Another attractive feature of the algorithms is that all the arithmetic operations are modulo operations. Consequently, the need to process large valued integers is avoided.

Notes: Abstract This paper presents a general expression relating the complex-normalized scattering matrix of ann-port network to that of its augmentedn-port network normalizing to then 1 −Ω resistances, where the Darlington equivalent network may be either reciprocal or nonreciprocal.

Notes: Abstract A linear-quadratic (LQ) control problem subject to a standard continuous-time system is called regular if the input weighting matrix is invertible, and singular if this is not the case. Consequently, optimal inputs for regular LQ problems are ordinary functions (state feedbacks), whereas optical controls for singular problems are in general distributions, e.g., impulses. We will show that regularity and singularity in LQ problems subject to ageneral (implicit) system depends not so much on the input weighting matrix, as on the property that the integrand of the cost criterion is a function only if inputs and state trajectories are, as is the case for LQ problems, subject to astandard system. In particular, we will provide a simple criterion for distinguishing between regularity and singularity in LQ problems subject to a general system. Our criterion is expressed in the system coefficients only and reduces to the classical one if the underlying system is standard.

Notes: Abstract Completions of linear time varying singular systems of the formE(t)x′(t)+F(t)x′(t)=f(t) are explicitly computed using recent results on rational matrix functions. The algorithm and the theory behind it are carefully described. Computational issues are discussed.

Notes: Abstract In this paper a spectral method using orthogonal periodic basis functions for the analysis of linear time invariant descriptor systems is discussed, and the case of the trigonometric Fourier functions is investigated in detail. The method is shown to be convergent, in the distributional sense. However, for any finite number of basis functions, the periodicity induced by the chosen basis can give rise to spurious impulsive components in the computed system response, even in the case of correct initial conditions.

Notes: Abstract In this paper we will study topological properties of the class of proper and improperp×m transfer functions of a fixed McMillan degreen. A natural generalization of this class is all autoregressive systems of degreen under external system equivalence. The subset of irreducible systems has in a natural way the structure of a manifold and we show how to extend this topology to the set of all autoregressive systems of degree at mostn. We will describe the subset of systems with fixed Kronecker indicesv=(v1,...,v p ) as an orbit space, which will enable us to calculate the topological dimension for each collection of indicesv. Finally, we will describe the topological closure of those sets in the space of all autoregressive systems.

Notes: Abstract It has been shown in [B.M.90] that non-square implicit differential equations allow for the description of variable structure systems (variable order, variable sign, variable parameters). We combine here the possible control strategy developed in [L.91] for rectangular systems (insuring a unique output behavior for the system compensated with a proportional or proportional and derivative state feedback) with the detector and the observer introduced in [B.M.90] in order to obtain a closed-loop system where the initial structure variation disappears on the output. We also give necessary and sufficient conditions for the free assignment of the associated output dynamics.

Notes: Abstract An exposition of joint cumulants and cumulant spectra is presented. A distinction is emphasized in this paper between the cumulant spectrum of a time series and its stationary version, here called apolyspectrum. The variance and covariance of the sample bispectrum is then derived using a relationship between cumulant spectra of the finite Fourier transform for the 2nd and 4th cumulant function, and the bispectrum and trispectrum of the time series.

Notes: Abstract The detection of a general class of transient (i.e., finite energy) signals in additive stationary interference using the spectral correlation function (second order cumulant spectrum) is presented. Observable features in the two-dimensional spectral correlation function due to properties of signals in the assumed class of transients are exploited to derive a detection statistic. The performance of the proposed detection statistic relative to a conventional power spectral detector is presented.

Notes: Abstract Vibroacoustic signals of rotating machinery are composed of sums of modulated periodicities, broadband random components, and occasionally a set of transient responses. These signals are not ergodic as the modulated periodicities are partially coherent. Progressive wear of the rotating machine causes the nonlinear structure of the received signal to intensify, and nonlinearity results in transfer of energy between harmonics of the signal's periodic components. Statistics developed from bispectrum and second-order cumulant spectrum estimates of the measured signal are combined with power spectrum amplitudes as feature inputs for standard multivariate classifiers. The higher-order statistics measure, respectively, the extent of nonlinearity and intermodulation of the received signal. Classification results of simulated and actual incipient wear data collected from a controlled experiment drilling circuit boards illustrate the potential of this novel statistical signal processing approach.

Notes: Abstract A geometric interpretation of the Lewis Structure Algorithm (LSA) is given in terms of precise projection maps directly defined from the (E, A, B, C) maps of the system. An extended version of LSA is offered which, in addition to this geometric information, also provides in a direct way, and within the same (E, A, B, C) class of models, a left inverse (if any) of the system. An example is given.

Notes: Abstract We consider the problem of control of linear, time-invariant, multivariable descriptor (implicit) systems. In particular we examine the effectiveness of an algorithm (which is a generalization of previous work in state space systems) for the design of an output feedback control giving pole placement in such systems. Conditions are presented which ensure that the algorithm produces the required control. We also address the important issue of uniqueness of solutions.

Notes: Abstract Detecting active sonar returns in multipath media is a central underwater signal processing problem. This paper studies a new approach to active sonar detection using bispectral analysis of sonar data. Its sensitivity to non-stationarities is used to develop a threshold detector that can be applied to broad classes of signals and noise. Precise statistical descriptions of the underwater medium and noise are not required. Theoretical analyses predicting its performance as a function of signal-to-noise ratio and time-bandwidth product are presented. Computer simulation experiments verify the results and show that its performance compares favorably to that of conventional detectors.

Notes: Abstract The noise suppression capability of higher-order moments and spectra has made them attractive when the goal is to extract or reconstruct a signal that is contaminated by multivariate Gaussian noise or certain types of non-Gaussian noise. Two new detectors, one centralized and one distributed, which are based on the third-order moment of the data are proposed. The asymptotic performance of the centralized detector and the asymptotic distribution of the components of the distributed detector are analyzed. Further, the performance of these detectors is simulated and compared to that of the matched filter for three different types of interference: Gaussian noise, Gaussian noise corrupted by a sinusoid with random phase, and Arctic under-ice noise.

Notes: Abstract Recent research has shown that multilayer feedforward networks with sigmoidal activation functions are universal approximators, and that this holds for more general activations as well. The mathematical underpinning for these results has been various: Kolmogorov's resolution of Hilbert's thirteenth problem; the Stone-Weierstrass theorem; approximation of Fourier and Radon integral representations; and convergence of probability measures. This paper • Rigorously establishes the robustness of feedforward network realizations. • Uses a theorem of Wiener and ideas of translation invariant subspaces to provide conditions for universal approximations toL 1 andL 2 functions by networks, for quite general activation functions. The second result extends and simplifies some of the recent results of Stinchcombe and White, at least for the special cases ofL 1 andL 2 functions.

Notes: Abstract We investigate the controller design problem for linear systems in which the state and the controls are subject to static linear constraints. We give necessary and sufficient conditions for the existence, and present a complete parametrization of all stabilizing controllers. This parametrization allows us to transform the constrained control problem into a standard problem which can be solved using usualH 2 orH ∞ optimization methods. The approach is illustrated by a simple numerical example showing the various steps of the proposed algorithm.

Notes: Abstract A continuous stationary signal possessing non-Gaussian higher order statistics cannot be correctly modelled by any discrete process based on passing independently and identically distributed noise through a linear filter. In particular, it is shown that at third order there exists no discrete skewed linear model with a discrete bispectrum that is the same as that obtained from the Nyquist samples of any continuous stationary process. The nature of the problem is elucidated and an alternative method for modelling the third order statistics of continuous stationary processes is proposed.

Notes: Abstract We present an algorithm for allocating individual ants to tasks that relies solely on task change being caused by the unavailability of work. We prove that such an algorithm will allocate the correct number of individuals to each job. Furthermore, we can demonstrate that if such an algorithm is used then an age structure emerges over the ants performing the various tasks. This matches closely with the weak temporal structure over tasks that is observed in Sendova-Franks and Franks (1993. Division of labour in ants nests within highly variable environments. (A study of temporal polyethism: experimental).Bull. math. Biol. 55, 75–96).

Notes: Abstract Biological regulatory systems can be described in terms of non-linear differential equations or in logical terms (using an “infinitely non-linear” approximation). Until recently, only part of the steady states of a system could be identified on logical grounds. The reason was that steady states frequently have one or more variable located on a threshold (see below); those steady states were not detected because so far no logical status was assigned to threshold values. This is why we introduced logical scales with values 0,1θ, 12θ, 2, ..., in which1θ,2θ, ... are the logical values assigned to the successive thresholds of the scale. We thus have, in addition to the regular logical states,singular states in which one or more variables is located on a threshold. This permits identifyingall the steady states on logical grounds. It was noticed that each feedback loop (or reunion of disjointed loops) can be characterized by a logical state located at the thresholds at which the variables of the loop operate. This led to the concept ofloop-characteristic state, which, as we will see, enormously simplifies the analysis.The core of this paper is a formal demonstration that among the singular states of a system, only loop-characteristic states can be steady. Reciprocally, given a loop-characteristic state, there are parameter values for which this state is steady; in this case, the loop is effective (i.e. it generates multistationarity if it is a positive loop, homeostasis if it is a negative loop). This not only results in the above-mentioned radical simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.

Notes: Abstract In this short note, we establish a simple, yet precise, necessary and sufficient condition for the “right coprime factorization” of a nonlinear feedback control system. As a consequence, we also obtain similar conditions for the “stable right coprime factorizations ” of the nonlinear feedback control system.

Notes: Abstract The pseudorandom sequence of arrays (PRSA) and a method to generate it was reported earlier by the authors. This paper presents another method to generate a PRSA. The mathematical recursion describing the PRSA and some of its properties are discussed.

Notes: Abstract This is a survey of recent work involving concepts of self-similarity that relate to harmonic analysis. Perhaps the main theme is the question: how does the fractal or self-similar nature of an object express itself on the Fourier transform side? A wide range of related topics are discussed, including self-similar measures and distributions, fractal Plancherel theorems, Lp dimensions and densities of measures, multiperiodic functions and their asymptotic behavior, convolution equations with self-similar measures, self-similar tilings, and the development of self-similar analysis on stratified nilpotent Lie groups.