ALBERT

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 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 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 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 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.

Notes: Abstract The Balian-Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system $\{e^{2\pi imbt} \, g(t-na)\}_{m,n \in \mbox{\bf Z}}$ with $ab=1$ forms an orthonormal basis for $L^2({\bf R}),$ then $\left(\int_{-\infty}^\infty |t \, g(t)|^2 \, dt\right) \, \left(\int_{-\infty}^\infty |\gamma \, \hat g(\gamma)|^2 \, d\gamma\right) = +\infty.$ The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that $(g')^{\wedge}(\gamma) = 2 \pi i \gamma \, \hat g(\gamma)$ , the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form $\{e^{2\pi ib_mt\} \, g(t-a_n)}$ such that $\{(a_n,b_m)\}$ has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems $\{e^{2\pi imbt} \, g(t-na)\}$ that form exact frames, and a new proof of the BLT for exact frames that does not require differentiation and relies only on classical real variable methods from harmonic analysis.

Notes: Abstract In this paper we present a technique for proving bounds of the Boas-Kac-Lukosz type for unsharply restricted functions with nonnegative Fourier transforms. Hence we consider functions F(x) ≥ 0, the Fourier transform f(u) of which satisfies |f(u)| ≤ ε for all u in a subset of (-∞,-1] ⋃ [1,∞), and are interested in bounds on |f(u)| for |u| ≤ 1. This technique gives rise to several "epsilonized" versions of the Boas-Kac-Lukosz bound (which deals with the case f(u) = 0, |u| ≥ 1). For instance, we find that |f(u)| ≤ L(u) + O(ε2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. The technique also turns out to be sufficiently powerful to yield the best bound as ε ↓ 0 in various other cases with less severe restrictions on f.

Notes: Abstract The classical Rudin–Shapiro construction produces a sequence of polynomials with ±1 coefficients such that on the unit circle each such polynomial P satisfies the "flatness" property ||P||∞ ≤ √2||P||2. It is shown how to construct blocks of such flat polynomials so that the polynomials in each block form an orthogonal system. The construction depends on a fundamental generating matrix and a recursion rule. When the generating matrix is a multiple of a unitary matrix, the flatness, orthogonality, and other symmetries are obtained. Two different recursion rules are examined in detail and are shown to generate the same blocks of polynomials although with permuted orders. When the generating matrix is the Fourier matrix, closed-form formulas for the polynomial coefficients are obtained. The connection with the Hadamard matrix is also discussed.

Notes: Abstract This paper presents an expansion for radial tempered distributions on ${\bf R}^n$ in terms of smooth, radial analyzing and synthesizing functions with space-frequency localization properties similar to standard wavelets. Scales of quasi-norms are defined for the coefficients of the expansion that characterize, via Littlewood-Paley-Stein theory, when a radial distribution belongs to a Triebel-Lizorkin or Besov space. These spaces include, for example, the $L^p$ spaces, $1 〈 p 〈 \infty,$ Hardy spaces $H^p, 0 〈 p \leq 1,$ Sobolev spaces $L^p_k,$ and Lipschitz spaces $\Lambda_\alpha, \alpha 〉 0.$ We also present a smooth radial atomic decomposition and norm estimates for sums of smooth radial molecules. The radial wavelets, atoms, and molecules that we consider are localized near certain annuli, as opposed to cubes in the usual, nonradial setting. The radial wavelet expansion is multiscale, where the functions in the different scales are related by dilation. However, there is no translation structure within a given scale, unlike the situation with standard wavelet systems.

Notes: Abstract Are there enough mast cells in denervated skeletal muscle to account for autopharmacological mediation of the antigen potentials (APs) elicited by microtaps? Through rough qualitative estimations, some authors have suggested a positive answer to this question. However, in view of measurements performed in this investigation of both the density of mast cells and the diffusion coefficient of antigens, the probability of such mediated effects was found to be relatively low:P=0.016 for egg albumin andP=0.004 for ferritin. Therefore, most APs induced by microtaps should be attributed to the direct effect of antigen over the sensitized muscle fibers. Yet, both the density of mast cells found in this work and the known amount of histamine they are capable of releasing when challenged with antigen, support the hypothesis regarding the involvement of these cells when antigen is massively superfused so as to induce Schultz-Dale reactions in muscle strips. Under this circumstance, the direct and mediated mechanisms may coexist.

Notes: Abstract We consider the phenomenon of self-focusing pattern formation of motile micro-organisms (“streamer”). The focusing mechanism is based on gyrotaxis, a physical phenomenon, and it results from the balance between viscous and gravitational moments on the organisms. Under particular circumstances such streamers are stabilized and sink, resulting in vertical transport of micro-organisms as well as horizontal heterogeneity in cell concentration. We develop a plume model for the streamer, which consists of the equations of continuity, momentum, and cell concentration. These equations are further simplified to three basic equations representing volume, momentum and concentration fluxes of the steady-state plume. Asymptotic analytical solutions for the plume are obtained. From numerical solutions we find the shape of the plume which takes the form of a vertical string with lengths ranging from 1 to 10 cm. The streamer formation can be related to algal blooms occurring in the sea and to algal cultures grown in the laboratory.

Notes: Abstract A mathematical model of the renal vascular and tubular systems was used to examine the possibility that synergistic interactions might occur between the tubuloglomerular feedback (TGF) and myogenic autoregulatory mechanisms in the kidney. To simulate the myogenic mechanism, the renal vasculature was modelled with a resistance network where the total preglomerular resistance varies with intravascular pressure. In addition, a steady-state model of glomerular filtration, proximal and Henle's loop reabsorption, and TGF-modulation of afferent arteriolar resistance was derived. The results show that, if TGF acts on the distal portion of the preglomerular vasculature, then any TGF-induced vasoconstriction should raise upstream intravascular pressure and, thereby, trigger a myogenic response in the more proximal vascular segments, a phenomenon referred to as an ascending myogenic (AMYO) response. The model further predicts that the magnitude of the AMYO response can be similar in magnitude to the TGF-induced increment in afferent resistance. Hence, the effects of TGF excitation on whole kidney hemodynamics may be much greater than pedicted from measurements in single nephrons. Moreover, a significant fraction of the intrinsic myogenic autoregulatory response to increased renal perfusion pressure may result from a synergistic interaction between the TGF and myogenic mechanisms.

Notes: Abstract To test whether a mathematical model combining dynamic models of the tubuloglomerular feedback (TGF) mechanism and the myogenic mechanism was sufficient to explain dynamic autoregulation of renal blood flow, we compared model simulations with experimental data. To assess the dynamic characteristics of renal autoregulation, a broad band perturbation of the arterial pressure was employed in both the simulations and the experiments. Renal blood flow and tubular pressure were used as response variables in the comparison. To better approximate the situationin vivo where as large number of individual nephrons act in parallel, each simulation was performed with 125 parallel versions of the model. The key parameters of the 125 versions of the model were chosen randomly within the physiological range. None of the constituent models, i.e., the TGF and the myogenic, could alone reproduce the experimental observations. However, in combination they reproduced most of the features of the various transfer functions calculated from the experimental data. The major discrepancy was the presence of a bimodal distribution of the admittance phase in the simulations. This is not consistent with most of the experimental data, which shows a unimodal curve for the admittance phase. The ability of the model to reproduce the experimental data supports the hypothesis that dynamic autoregulation of renal blood flow is due to the combined action of TGF and the myogenic response.

Notes: Abstract A model is presented of solute and water reabsorption along the proximal tubule of the rat kidney based on kinetic descriptions of the main membrane transport systems, in order to assess the extent to which these kinetics suffice to explain certain aspects of the global transport behaviour in this segment, especially with respect to bicarbonate reabsorption. The model includes in the apical membrane, an active proton pump, Na+/H+ antiport, Na-coupled transport of organic solutes, Cl−/formate exchange with formic acid recycling, and membrane conductances to protons and K+. In the baso-lateral membrane, besides the Na+/K+ pump, the model includes Na+-3HCO 3 − and electroneutral K+-Cl− cotransporters, and membrane conductances for K+, H+, and, optionally, for Cl−. Appropriate passive diffusional pathways were included in both cell membranes and in the paracellular pathway. Using mass balance and electoneutrality constraints, these transport systems were built into an epithelial model which was then integrated (by finite difference approximation) into a model of a longitudinal tubule. Simulated cellular solute concentrations and luminal concentration profiles were in good agreement with reported experimental observations. We show that, given the reported transport kinetics for the Na+/H+ antiporter, a hitherto unexplained observation concerning load-dependent bicarbonate reabsorption can be shown mainly to result from the nonlinear longitudinal concentration profile for bicarbonate and pH. We also discuss problems of transcellular Cl− transport in the light of recent reports of basolateral Cl− conductance and observations relevant to apical Cl−/formate (or other base) exchange.

Notes: Abstract Substitution of measured permeabilities into mathematical models of the concentrating mechanism of the renal inner medulla yields less than the known urine osmolalities. To gain a better understanding of the mechanism we analyse a model in which a force of unspecified origin [expressed as fraction, ɛ, of entering descending thin limb (DTL) concentration] drives fluid from DTL to interstitial vascular space (CORE), thus concentrating the solution in DTL. When flow in the DTL reverses at the hairpin bend of the loop of Henle, the high solute permeability of ascending thin limb (ATL) permits solute to diffuse into the CORE thus permitting ɛ to be multiplied many-fold. Behavior of the model is described by two non-linear differential equations. In the limit for infinite salt permeability of ATL the two equations reduce to a single equation that is formally identical with that for the Hargitay and Kuhn multiplier, which assumes fluid transport directly from DTL to ATL (Z. Electrochem. Angew. Phys. Chem. 55, 539, 1951). Solutions of the equations describing the model with parameters taken from perfused thin limbs show that urine osmolalities of the order of 5000 mosm L−1 can be generated by forces of the order of 20 mosm L−1. It seems probable that mammals including desert rodents use some variant of this basic mechanism for inner medullary concentration.

Notes: Abstract Previous models of the renal concentrating mechanism employ ideal approximations of solution thermodynamics for membrane transport calculation. In three-dimensional models of the renal medulla, predicted urine concentrations reach levels where there idealized approximations begin to break down. In this paper we derive equations that govern membrane transport for non-dilute solutions and use these equations in a three-dimensional model of the concentrating mechanism. New numerical methods were employed that are more stable than those employed previously. Compared to ideal solution models, the urea non-ideality tends to increase predicted osmolarities, whereas NaCl non-ideality decreases predictions.

Notes: Abstract Chloride/formate exchange, in parallel with Na+/H+ exchange and nonionic diffusion of H2CO2, has been proposed as a mechanism of electroneutral transcellular Cl− reabsorption by the proximal tubule. However, the measured brush border H2CO2 permeability of the rat proximal tubule is at least an order of magnitude too low to support sufficient H2CO2 recycling. To investigate the possibility that an unstirred layer within the brush border might depress the measured H2CO2 permeability, we constructed a mathematical model of a villous membrane. Axial fluxes along villous and intervillous spaces were specified by Nernst-Planck diffusion equations. Model parameters were set to achieve agreement with ion and water fluxes measured in the rat proximal tubule. The equations were solved numerically to generate steady-state concentration profiles in the villous and intervillous spaces. An apparent brush border H2CO2 permeability was determined by perturbing luminal [H2CO2] and calculating the change in H2CO2 flux. Overall, the ratio of apparent brush border H2CO2 permeability to cell membrane H2CO2 permeability was greater than 90%. Contributing to the small decrease in apparent permeability are finite diffusion coefficients, folding of the membrane, and acidification of the luminal solution. An approximate analysis of this system shows the critical parameters of brush border formate transport to be the actual membrane H2CO2 permeability, and the diffusion coefficients of HCO 3 − and HCO 3 − . Nevertheless, decreasing the diffusion coefficients by one order of magnitude failed to depress apparent brush border H2CO2 permeability by more than an additional 25%. We conclude that although permeability is systematically underestimated across a villous membrane, unstirred layer effects in the brush border are still too small to resolve the discrepancy between the measured value of H2CO2 permeability and the value needed to allow recycling.

Notes: Abstract We show that an explicit method for solving hyperbolic partial differential equations can be applied to a model of a renal tubule to obtain both dynamic and steady-state solutions. Appropriate implementation of this method eliminates numerical instability arising from reversal of intratubular flow direction. To obtain second-order convergence in space and time, we employ the recently developed ENO (Essentially Non-Oscillatory) methodology. We present examples of computed flows and concentration profiles in representative model contexts. Finally, we indicate briefly how model tubules may be coupled to construct large-scale simulations of the renal counterflow system.

Notes: Abstract Recently algorithms for parametric alignment (Watermanet al., 1992,Natl Acad. Sci. USA 89, 6090–6093; Gusfieldet al., 1992,Proceedings of the Third Annual ACM-SIAM Discrete Algorithms) find optimal scores for all penalty parameters, both for global and local sequence alignment. This paper reviews those techniques. Then in the main part of this paper dynamic programming methods are used to compute ensemble alignment, finding all alignment scores for all parameters. Both global and local ensemble alignments are studied, and parametric alignment is used to compute near optimal ensemble alignments.

Notes: Abstract A class of minimal models is constructed that can exhibit several salient phenomena associated with T-cell inoculations that prevent and cure autoimmune disease. The models consist of differential equations for the magnitude of two populations, the effectorsE (which cause the disease), and an interacting regulator populationR. In these models, normality, vaccination and disease are identified with stable steady-states of the differential equations. Thereby accommodated by the models are a variety of findings such as the induction of vaccination or disease, depending on the size of the effector inoculant. Features such as spontaneous acquisition of disease and spontaneous cure require that the models be expanded to permit slow variation of their coefficients and hence slow shifts in the number of steady-states. Other extensions of the basic models permit them to be relevant to vaccination by killed cells or by antigen, or to the interaction of a larger number of cell types. The discussion includes an indication of how the highly simplified approach taken here can serve as a first step in a modeling program that takes increasing cognizance of relevant aspects of known immunological physiology. Even at its present stage, the theory leads to several suggestions for experiments.

Notes: Abstract A mathematical model for steady flow through a discontinuity in the tight junction of an endothelial intercellular cleft is presented. Subject to plausible assumptions the problem of calculating the flow in the cleft, in either the presence or the absence of a fibre matrix, reduces to the solution of Laplace's equation in a two-dimensional domain. For an idealized geometry representing a discontinuity between two semi-infinite tight junction regions, a general analytic solution is found by means of conformal mappings. The model geometry, unlike those assumed in previous studies, allows the tight junction regions to be out of alignment with each other, and even to overlap, modelling flow through a tortuous, rather than a direct, pathway. Useful asymptotic approximations for the flow rate are derived when the discontinuity is either very small or very large. For small discontinuities, the predicted flow rate is much greater than a naïve estimate based on uniform parallel flow through the discontinuity. For the special case where the tight junction regions are aligned with each other, comparison of our results with those of an approximate treatment due to Tsayet al. [Chem. Engng Commun. 82, 67–102 (1989)] shows generally very close agreement.

Notes: Abstract A general description is given for the meristem of elongating organs such as for stem internode, monocotyledonous leaf and roots. The meristem is viewed as a cellular organization rather than traditionally as a region with certain growth dynamics; cell lineages are inspected and analyzed by cell packet features. Three features of the meristem are described by the displacement of the distal (bottom) transverse wall of a cell: (a) the duration of time the cell and its derivatives spend in the meristem, (b) the duration of cell cycle and (c) the number of cells in a clone derived from a meristematic cell. Cell packet analyses of data from the internode of the Purple Heart (Setcreasea) and the top epidermis of the leaf of the Wandering Jew (Zebrina) indicate meristem lengths of 33 and 50 cells, respectively, while growth and cell division rates in the meristem for both species follow Gaussian distributions.

Notes: Abstract I seek to explain phenomena observed in simulations of populations of gap junction-coupled bursting cells by studying the dynamics of identical pairs. I use a simplified model for pancreatic β-cells and decompose the system into fast (spike-generating) and slow subsystems to show how bifurcations of the fast subsystem affect bursting behavior. When coupling is weak, the spikes are not in phase but rather are anti-phase, asymmetric or quasi-periodic. These solutions all support bursting with smaller amplitude spikes than the in-phase case, leading to increased burst period. A key geometrical feature underlying this is that the in-phase periodic solution branch terminates in a homoclinic orbit. The same mechanism also provides a model for bursting as an emergent property of populations; cells which are not intrinsic bursters can burst when coupled. This phenomenon is enhanced when symmetry is broken by making the cells differ in a parameter.

Notes: Abstract Twining plants exhibit a striking oscillation of their stems in their quest for a support. The oscillations, called circumnutation, have periods generally of 1–5 hr, and virtually all species have a preferred direction of twining. I seek to explain these chiral asymmetries in plant behavior by hypothesizing a chiral asymmetry in plant anatomy. Such asymmetries already exist, for example, in phyllotaxis. I explore wave phenomena on asymmetric but isotropic rings, and seek systems which will only support (stable) waves in one direction around the ring, and not in the other. Simulations indicate that (1) oscillatory reaction-diffusion systems do not support unidirectional waves on rings; (2) excitable reaction-diffusion systems do support unidirectional waves on rings; and (3) unidirectional phase-locking (discrete unidirectional waves) occurs in rings of coupled oscillators. Thus, chiral asymmetries of circumnutating plants cannot be explained by continuum oscillator phenomena, but can be explained by general discrete oscillators, or excitable phenomena on the continuum.

Notes: Abstract A mathematical model has been considered in which the known equation of McKendrick and Von Foerster for cell age distribution is combined with that for substrate concentration. The dependence of cell division rate on cell age has been taken as a step function. The interrelation between culture parameters describing the substrate consumption and cell division has been found. The shape of cell age distribution as well as the values of substrate and cell concentrations in steady and transient states have been investigated. Stationary regimes at the initial culture state synchronized by ages have been found to be established as damped oscillations and age waves. Under definite conditions the transition from one steady growth regime to another includes sharp single-time age synchronization of the culture.

Notes: Abstract The simplex methods of nonlinear forecasting are used to study the data sets of hepatitis A and AIDS in various regions of the United States. The results are compared with those obtained from the traditional ARIMA methods. In many regions, the simplex methods developed from nonlinear dynamical system theory give smaller errors for the data of hepatitis A. A combination of the simplex methods and the traditional ARIMA methods can produce better results for the AIDS data sets.

Notes: Abstract We consider optimal strategies for harvesting a population that is composed of two local populations. The local populations are connected by the dispersal of juveniles, e.g. larvae, and together form a metapopulation. We model the metapopulation dynamics using coupled difference equations. Dynamic programming is used to determine policies for exploitation that are economically optimal. The metapopulation harvesting theory is applied to a hypothetical fishery and optimal strategies are compared to harvesting strategies that assume the metapopulation is composed either of single unconnected populations or of one well-mixed population. Local populations that have high per capita larval production should be more conservatively harvested than would be predicted using conventional theory. Recognizing the metapopulation structure of a stock and using the appropriate theory can significantly improve economic gains.

Notes: Abstract A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.

Notes: Abstract Most theoretical analyses of tracer kinetics in capillaries contain an implicit assumption that the tissues to which they are connected have homogeneous material properties. The microscopic description of the exchange of tracer molecules and tissues is then modeled in terms of first-order kinetics. We consider a class of more general models allowing us to assess the robustness of simplifying assumptions made above. It is shown that when amorphous properties are important the kinetics of the system may differ considerably from those predicted by standard theories.

Notes: Abstract An extension of an earlier model of the p170 glycoprotein pump is presented. In an earlier work (Michelson and Slate,Bull. math. Biol. 54, 1023–1038, 1992), the pump was modeled using an energy-dependent model of facilitated diffusion. In this paper we add an inhibitor to the model. New equations are derived which represent either competitive or non-competitive inhibition in the pumping action of the glycoprotein. Numerical simulations were run which provide a response surface (initial loading concentration of inhibitor and its ability to compete with an ideal anti-cancer drug vs a summary measure of cytoplasmic exposure) for each scenario. The importance of the exposure profile, how it is related to ultimate tumor cell survival, and the binding requirements for developing multidrug resistance inhibitors are discussed.

Notes: Abstract In this note, we consider orthogonal wavelets with the oversampling property. We prove that if an orthogonal scaling function with exponential decay has the oversampling property, then it has the sampling property (i.e., it takes values 1 at 0 and 0 at other integers); therefore, an orthogonal scaling function with compact support has the oversampling property if and only if it is the Haar function.

Notes: Abstract A simple parametrization is given for the set of positive measures with finite support on the circle group T that are solutions of the truncated trigonometric moment problem: $\hat{\mu}(k)=s_k, |k|\le N,$ where the parameters are, up to nonzero multiplicative constants, the polynomials whose roots all have a modulus less than one. This result is then used to characterize, on a certain natural Hilbert space of polynomials associated with the problem, all finite "weighted" tight frames of evaluation polynomials. Finally, a new and simple way of parametrizing the whole set of positive Borel measures on T, solutions of the given moment problem is deduced, via a limiting argument.

Notes: Abstract Gabor time-frequency lattices are sets of functions of the form $g_{m \alpha , n \beta} (t) =e^{-2 \pi i \alpha m t}g(t-n \beta)$ generated from a given function $g(t)$ by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals that, like speech or music, seem over short intervals to have well-defined frequencies that, however, change with time. It was recently observed that the behavior of a lattice $(m \alpha , n \beta )$ can be connected to that of a dual lattice $(m/ \beta , n /\alpha ).$ Here we establish this interesting relationship and study its properties. We then clarify the results by applying the theory of von Neumann algebras. One outcome is a simple proof that for $g_{m \alpha , n \beta}$ to span $L^2,$ the lattice $(m \alpha , n \beta )$ must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having improved convergence and localization properties.

Notes: Abstract A method will be presented for the approximation of a desired two-dimensional frequency response by the frequency response of a two-dimensional finite-impulse-response digital filter. It is possible to match the functional values and an arbitrary number of derivatives of both responses for zero frequency, thus making the error flat up to a desired degree. Remaining degrees of freedom are used for anL 2-approximation. Closed form design formulae will be given.

Notes: Abstract This paper presents an efficient procedure for the design of interpolated FIR (IFIR) filters with linear phase. The algorithm uses the uniform B-spline function as an interpolator and solves the optimal Chebyshev approximation problem on the optimal subinterval. The technique can be used for the design of general lowpass, highpass and bandpass filters. While the number of multiplications of the IFIR filter is dependent on the bandwidth and the center frequency of the desired filter, it provides the minimum number of multiplications achievable and nearly always provides a substantial reduction when compared to Parks-McClellan designs.

Notes: Conclusion We have seen that the stationary solutions of the regularized equations are asymptotically stable if the stationary solution of the original quasilinear index-2 tractable DAE is so. In other words, the given regularizations supply a solution sharing the stability properties of the solution of the unperturbed equation.

Notes: Abstract A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays. This paper discusses progress on the implementation of this method, resolves some of the issues involved, and lists some remaining problems. Computational experience on two prescribed path control problems is presented showing that the approach should prove practical for many applications.

Notes: Abstract Some issues in the stability of differential delay systems in the linear and the nonlinear case are investigated. In particular, sufficient robustness conditions are derived under which a system remains stable, independent of the length of the delay(s). Applications in the design of delayed feedback systems are given. Two approaches are presented, one based on Lyapunov theory, the other on a transformation to Jordan form. In the former, sufficient conditions are obtained in the form of certain Riccati-type equations.

Notes: Abstract Image coding can be implemented through DPCM, transform, hybrid, or segmentation coding techniques. Some transform coding techniques, such as cosine and Hadamard, have been exhaustively analyzed and evaluated, while others, such as Legendre, have not. This paper introduces the use of Legendre transform in image coding. The transform matrix for different block sizes is calculated, the fast algorithm is derived, and the performance is evaluated through both mean square error and subjective quality. The results obtained have indicated that the system performance is comparable with that of optimum KLT and cosine transforms; moreover, it is simpler in implementation.

Notes: Abstract A method will be presented for the approximation of a desired frequency response by the frequency response of a FIR filter. It is possible to match the functional values and an arbitrary number of derivatives of both responses for zero frequency, thus making the error flat up to a desired degree. Remaining degrees of freedom are used for a weightedL 2-approximation. Closed form design formulae will be given.

Notes: Abstract The Stewart platform manipulator is a fully parallel kinematic linkage system that has great advantages over typical serial link robots. However, one major problem in controlling this kind of manipulator is that, due to its closed kinematic chain and parallel-link constrained structure, it is impossible to derive an explicit kinematic solution. This paper proposes a simplified algorithm to numerically solve the forward kinematics of a six-links Stewart platform. Taking advantage of some fundamental geometric operations, the proposed algorithm involves only 3 nonlinear simultaneous equations. The explicit expressions are derived for some special configurations which can directly give the geometric limitations to motion in terms of the geometric dimensions of the platforms and the legs.

Notes: Abstract We consider the numerical solution of an unconfined detonation problem modelled by differential/algebraic equations in a boundary value format. The problem has index one throughout the interval of integration, except at a free boundary point, where it has index two. Moreover, a nonlinear constraint is active at the free boundary point. We consider a simple mathematical model, as well as a more realistic and functionally complex model. In both cases, the resulting boundary value problem is solved directly, using a collocation method. This approach compares favorably with the solution techniques proposed previously in the literature and enables us to examine some interesting properties of the solution to the detonation problem.