Abstracts of Just's recent research papers

Recent research papers by Winfried Just

Preprints and research notes

W. Just; The steady state system problem is NP-hard even for monotone quadratic Boolean dynamical systems.

Abstract. Recently, Colon-Reyes, Laubenbacher and Pareigis developed a polynomial-time algorithm for deciding for a Boolean dynamical system in which each regulatory function is a monomial whether every limit cycle is a steady state. We show that the corresponding problem is NP-hard if the class of permissible regulatory functions contains the quadratic monotone functions x OR y and x AND y. We also show that the problem is NP-hard if the set of permissible regulatory functions includes all functions of the type xy and x + 1.

W. Just and B. Stigler; Efficiently Computing Groebner Bases of Ideals of Points.
arXiv:0711.3475v1

Abstract. We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the previously published one for the standard Buchberger-Moeller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Moeller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Groebner bases to the problem of network reconstruction in molecular biology.

W. Just and M. Korb; (In)consistency: Some low-dimensional examples.

Abstract. This note is a slightly revised version of an earlier note with the same title that was part of a larger research project of the Dynamical Systems Group at Ohio University. While the major findings of the project are described in the paper Two classes of ODE models with switch-like behavior by W. Just, M. Korb, B. Elbert, and T. R. Young, this note complements this paper as it contains a more extensive review of some basic low-dimensional examples of Boolean systems and their ODE counterparts and explores whether the ODE dynamics is consistent with the Boolean dynamics.

W. Just and S. Ahn; Lengths of attractors and transients in neuronal networks with random connectivities.
arXiv:1404.5536

Abstract. We study how the dynamics of a class of discrete dynamical system models for neuronal networks depends on the connectivity of the network. Specifically, we assume that the network is an Erdos-Renyi random graph and analytically derive scaling laws for the average lengths of the attractors and transients under certain restrictions on the intrinsic parameters of the neurons, that is, their refractory periods and firing thresholds. In contrast to earlier results that were reported in an earlier paper, here we focus on the connection probabilities near the phase transition where the most complex dynamics is expected to occur.

W. Just and Y. Xin; On the role of limsup in the definition of topological entropy via spanning or separation numbers. Part I: Basic examples.
arXiv:1707.09052

Abstract. The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution epsilon within T time units. It can then be formally dened as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the denition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature by constructing such counterexamples.

Y. Xin, D. Gerberry, and W. Just; Open-minded imitation can achieve near-optimal vaccination coverage.
arXiv:1808.08789

Abstract. Studies of voluntary vaccination decisions by rational individuals predict that the population will reach a Nash equilibrium with vaccination coverage below the societal optimum. Human decision-making involves mechanisms in addition to rational calculations of self-interest, such as imitation of successful others. Previous research had shown that imitation alone cannot achieve better results. Under realistic choices of the parameters it may lead to equilibrium vaccination coverage even below the Nash equilibrium. However, these findings rely on the widely accepted use of Fermi functions for modeling the probabilities of switching to another strategy. We consider here a more general functional form of the switching probabilities. It is consistent with functions that give best fits for empirical data in a widely cited psychological experiment and involves one additional parameter alpha. This parameter can be loosely interpreted as a degree of open-mindedness. We found both by means of simulations and analytically that sufficiently high values of alpha will drive the equilibrium vaccination coverage arbitrarily close to the societal optimum.

Recent articles, books, and book chapters

W. Just; COVID-19 Unmasked: The News, the Science, and Common Sense. World Scientific, March 2021.

Abstract. How can we keep up with the deluge of information about COVID-19 and tell which parts are most important and trustworthy? We read: "Scientists recommend", "Experts warn", "A new model predicts". How do scientific experts come up with their recommendations? What do their predictions really mean for us, for our friends, and our families? How can we make rational decisions? And how can we have sensible conversations about the pandemic when we disagree?
These are the questions that this book is trying to address. It is written in the form of dialogues. Alice, a student of epidemiology, explains the science to three of her fellow students who have a lot of questions for her. The students have the same concerns that we all share to varying degrees: What the pandemic is doing to our health, our economy, and our cherished freedoms. In their conversations, they discover how the science relates to these questions. The book focuses on epidemiology, the science of how infections spread and how the spread can be mitigated. The science of how many infections can be prevented by certain kinds of actions. This is what we need to understand if we want to act wisely, as individuals and as a society.
The author's goal is to help the reader think about the COVID-19 pandemic like an epidemiologist. About the various preventive measures, what they are trying to accomplish, what the obstacles are. About what is likely to be most effective in the long run at moderate economic and personal cost. About the likely consequences of personal decisions. About how to best protect oneself and others while allowing all of us to lead lives that are as close as possible to normal. While some chapters present slightly more advanced material than others, no scientific background is needed to follow the conversations. The technical concepts are explained in small steps and the occasional calculations in the book require only high-school mathematics.

W. Just and Y. Xin; Limsup is needed in the definitions of topological entropy via spanning or separation numbers. Dynamical Systems 35(3) (2020), 430-489.

Abstract. The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution epsilon within T time units. It can then be formally dened as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the denition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature by constructing such counterexamples.

Y. Xin, D. Gerberry, and W. Just; Open-minded imitation can achieve near-optimal vaccination coverage. Journal of Mathematical Biology 79(4) (2019), 1491-1514.

Abstract. Studies of voluntary vaccination decisions by rational individuals predict that the population will reach a Nash equilibrium with vaccination coverage below the societal optimum. Human decision-making involves mechanisms in addition to rational calculations of self-interest, such as imitation of successful others. Previous research had shown that imitation alone cannot achieve better results. Under realistic choices of the parameters it may lead to equilibrium vaccination coverage even below the Nash equilibrium. However, these findings rely on the widely accepted use of Fermi functions for modeling the probabilities of switching to another strategy. We consider here a more general functional form of the switching probabilities. It is consistent with functions that give best fits for empirical data in a widely cited psychological experiment and involves one additional parameter alpha. This parameter can be loosely interpreted as a degree of open-mindedness. We found both by means of simulations and analytically that sufficiently high values of alpha will drive the equilibrium vaccination coverage arbitrarily close to the societal optimum.

B. Oduro, M. Grijalva, and W. Just; A model of insect control with imperfect treatment. Journal of Biological Dynamics 13(1) (2019).

Abstract. Insecticide spraying of housing units is an important control measure for vector-borne infections such as Chagas disease. As vectors may invade both from other infested houses and sylvatic areas and as the effectiveness of insecticide wears off over time, the dynamics of (re)infestations can be approximated by SIRS-type models with a reservoir, where housing units are treated as hosts, and insecticide spraying corresponds to removal of hosts. Here we investigate a model of this type that takes into account the possibility that some vectors may survive treatment, due to imperfect spraying by the operator or because they hide deep in the cracks or other places, and re-emerge in the same unit when the effect of the insecticide wears off.
In particular, we demonstrate that a previously described dual-rate effect can occur in this model. In the presence of this effect, an initially very high spraying rate may push the system into a region of the state space with low endemic levels of infestation that can be maintained in the long run at relatively moderate cost, while in the absence of an aggressive initial intervention the same average cost would only allow a much less significant reduction in long-term infestation levels. We determine conditions under which this effect occurs.

Y. Xin, C. K. Mudiyanselage, and W. Just; Development of epithelial tissues: How are cleavage planes chosen? PLoS ONE 13(11):e0205834 (2018). https://doi.org/10.1371/journal.pone.0205834

Abstract. Epithelia are sheets of tightly adherent cells that line both internal and external surfaces in metazoans. Mathematically, a cell in an epithelial tissue can be modeled as a k-sided polygon. Empirically studied distributions of the proportions of k-sided cells in epithelia show remarkable similarities in a wide range of evolutionarily distant organisms. A variety of mathematical models have been proposed to explain this phenomenon. Among the most parsimonious of such models are topological ones that take into account only the number of sides of a given cell and the neighborhood relation between cells. The model studied here is a refinement of a previously published such model (Patel et al., PLoS Comput. Biol., 2009). While using the same modeling framework as that paper, we introduce additional options for the choice of the endpoints of the cleavage plane in the simulated development of epithelial tissues. Some of these options appear to be more biologically realistic approximations of known mechanisms that influence the cleavage plane orientation. A comparative analysis of comprehensive simulations for relevant combinations of both previously studied and our newly designed options is reported here. We compared the outcomes with the empirical distributions of the same five organisms as in (Patel et al., 2009), Drosophila, Hydra, Xenopus, Cucumber, and Anagallis. We found that combinations of some of our new options consistently gave better fits with each of these data sets than combinations of previously studied options for choosing the cleavage plane.

W. Just; Approximating Network Dynamics: Some Open Problems.
Discrete and Continuous Dynamical Systems B (DCDS-B) (2018), 23(5): 1917-1930.

Abstract. Discrete-time finite-state dynamical systems on networks are often conceived as tractable approximations to more detailed ODE-based models of natural systems. Here we review research on a class of such discrete models N that approximate certain ODE models M of mathematical neuroscience. In particular, we outline several open problems on the dynamics of the models N themselves, as well as on structural features of ODE models M that allow for the construction of discrete approximations N whose predictions will be consistent with those of M.

B. Oduro, M. Grijalva, and W. Just; Models of disease vector control: When can aggressive initial intervention lower long-term cost?
Bulletin of Mathematical Biology, 80(4) (2018), 788-824.

Abstract. Insecticide spraying of housing units is an important control measure for vector-borne infections such as Chagas disease. As vectors may invade both from other infested houses and sylvatic areas and as the effectiveness of insecticide wears off over time, the dynamics of (re)infestations can be approximated by SIRS-type models with a reservoir, where housing units are treated as hosts, and insecticide spraying corresponds to removal of hosts. Here we investigate three ODE-based models of this type. We describe a dual-rate effect where an initially very high spraying rate can push the system into a region of the state space with low endemic levels of infestation that can be maintained in the long run at relatively moderate cost, while in the absence of an aggressive initial intervention the same average cost would only allow a much less significant reduction in long-term infestation levels. We determine some sufficient and some necessary conditions under which this effect occurs, and show that it is robust in models that incorporate some heterogeneity in the relevant properties of housing units.

W. Just and H. Callender Highlander; Vaccination strategies for small worlds.
In A. Wootton, V. Peterson, C. Lee, eds., A Primer for Undergraduate Research: From Groups and Tiles to Frames and Vaccines. Springer Verlag, 2018, 223-264.

Abstract. Infectious diseases are a serious threat to our health. Vaccination often can prevent their spread, but typically it is not feasible to vaccinate absolutely everybody. Sometimes it is necessary to carefully target the group of individuals to whom a limited supply of vaccine should be administered in order to achieve the largest amount of overall protection for the whole population. A method for choosing the group to be targeted for maximal effect is called a vaccination strategy. The development of optimal vaccination strategies leads to interesting mathematical problems and requires some knowledge of the contact network of the given population. Here we focus on contact networks that are so-called small-world networks. Through a sequence of background information and preliminary exercises, we take the readers to the point where they can explore some open research problems that are formulated in the last section.

W. Just, J. Saldaña, and Y. Xin; Oscillations in epidemic models with spread of awareness.
Journal of Mathematical Biology, 76(4) (2018), 1027-1057.
Preliminary version available at arXiv:1606.08788

Abstract. We study ODE models of epidemic spreading with a preventive behavioral response that is triggered by awareness of the infection. Previous studies of such models have mostly focused on the impact of the response on the initial growth of an outbreak and the existence and location of endemic equilibria. Here we study the question whether this type of response is sufficient to prevent future flare-ups from low endemic levels if awareness is assumed to decay over time. In the ODE context, such flare-ups would translate into sustained oscillations with significant amplitudes. Our results show that such oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible models with a single compartment of aware hosts, but can occur if we consider two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts.

W. Just and S. Ahn; Lengths of attractors and transients in neuronal networks with random connectivities.
SIAM Journal on Discrete Mathematics 30(2) (2016) 912�933.

Abstract. We study how the dynamics of a class of discrete dynamical system models for neuronal networks depends on the connectivity of the network. Specifically, we assume that the network is an Erdos-Renyi random graph and analytically derive scaling laws for the average lengths of the attractors and transients. In contrast to previously published results, here we focus on the connection probabilities near the phase transition where the most complex dynamics is expected to occur.

W. Just, H. Callender, M. D. LaMar, and N. Toporikova; Transmission of infectious diseases: Data, models, and simulations.
In Raina Robeva (ed.), Algebraic and Discrete Mathematical Methods for Modern Biology. Academic Press, 2015, 193-215.

Abstract. This book chapter introduces students to various aspects of mathematical modeling in epidemiology, including data collection, development of models, and deriving and interpreting predictions of models. The agent-based models of this chapter are explored by using a teaching tool IONTW that was developed by the authors using the NetLogo software.
The chapter discusses in detail the process of building mathematical models of disease transmission. Students will frequently be challenged to critically examine the simplifying assumptions that are involved in this process. It also provides an intuitive introduction to thinking about simulation models in terms of pseudocode, intended to empower students with no prior knowledge of computer programming to analyze simulation models at a semiformal level.
The chapter includes numerous conceptual and simulation exercises aimed at developing deeper understanding of the concepts.

W. Just, H. Callender, and M. D. LaMar; Disease transmission dynamics on networks: Network structure vs. disease dynamics.
In: Raina Robeva (ed.), Algebraic and Discrete Mathematical Methods for Modern Biology. Academic Press, 2015, 217-235.

Abstract. The first part of this book chapter covers compartment-based models that are constructed under the uniform mixing assumption. The role of the basic reproductive ratio in these models is illustrated both by theoretical results and simulations with IONTW, a teaching tool that was developed by the authors and uses the NetLogo software.
The second part introduces students to network-based models of disease transmission. In these models it is assumed that the infection can be transmitted only during a direct contact between two hosts who are adjacent in the underlying contact network. Models of this kind allow for incorporation of more realistic details than compartment-based models that assume uniform mixing.
Students are guided through sequences of conceptual and simulation exercises to discover how the network structure influences the predictions of the model, including predictions about the effectiveness of possible control measures.

W. Just and G. A. Enciso; Ordered Dynamics in Biased and Cooperative Boolean Networks.
Advances in Difference Equations 2013, 2013:313.

Abstract. This paper contributes to the theoretical analysis of the qualitative behavior of two types of Boolean networks: biased and cooperative ones. A Boolean network is biased if at least a specified fraction of its regulatory functions returns one Boolean value more often than the other and is cooperative if there are no negative interactions between the variables. We prove nontrivial upper bounds on the maximum length of periodic orbits in such networks under the assumption that the maximum number of inputs and outputs per node is a fixed constant r. For the case of n-dimensional networks with r = 2 in which only AND and OR are allowed, we find an upper bound of10^n/4, which is asymptotically optimal in view of previously published counterexamples. The theoretical results are supplemented by simulations of the generic behavior of cooperative networks which indicate that for large in-degrees, trajectories tend to converge rapidly towards a steady state or small orbit. The latter starkly contrasts with the behavior of random arbitrary Boolean networks.

W. Just, M. Korb, B. Elbert, and T. R. Young; Two classes of ODE models with switch-like behavior.
Physica D 264 (2013) 35-48.

Abstract. In cases where the same real-world system can be modeled both by an ODE system D and a Boolean system B it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that under suitable assumptions about B consistency between D and B will be implied by sufficient separation of time scales in one class of our models while it may fail in the other class. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.

W. Just and M. Malicki; Cooperative Boolean systems with generically long attractors II.
Advances in Difference Equations 2013, 2013:268.

Abstract. The purpose of this paper is to investigate to what extent cooperativity, that is, the absence of negative interactions, in Boolean networks with synchronous updating, limits the amount of chaos that is possible in such systems. We prove that a strong notions of sensitive dependence on initial conditions is precluded by cooperativity. Weaker notions of sensitive dependence are shown to be consistent with cooperativity, but if each regulatory functions is binary AND or binary OR, in N-dimensional networks they impose an upper bound of approximately sqrt(3)^N on the lengths of attractors that can be reached from an arbitrarily large fraction of initial conditions. The upper bound is shown to be sharp. These results indicate that transfer of analogous results for differential equations models crucially depends on the precise conceptualization of chaos in the Boolean context.

W. Just and M. Malicki; Cooperative Boolean systems with generically long attractors I.
Journal of Difference Equations and Applications 19(5) (2013) 772-795.

Abstract. We study the class of cooperative Boolean networks whose only regulatory functions are COPY, binary AND, and binary OR. We prove that for all sufficiently large N and c < 2 there exist Boolean networks in this class that have an attractor of length > c^N whose basin of attraction comprises an arbitrarily large fraction of the state space. The existence of such networks sharply contrasts with results on continuous dynamical systems that imply nongenericity of non-steady state attractors under the assumption of cooperativity.

W. Just, S. Ahn, and D. Terman; Neuronal Networks: A Discrete Model. In Mathematical Concepts and Methods in Modern Biology, R. Robeva and T. Hodge, eds., Academic Press, 2013, 179-211.

Abstract. Simple discrete dynamical system models of neuronal dynamics can be constructed by assuming that at any given time step each neuron can either fire or be at rest, that after it has fired each neuron needs to be at rest for a specified refractory period and the firing of a neuron is induced by firing of a sufficient number of other neurons with synaptic connections to it. This book chapter explores the relation between the network connectivity and important features of the network dynamics such as the number and lengths of attractors, lengths of transients, and sizes of the basin of attraction. A variety of mathematical tools, ranging from combinatorics to probability theory, are used. Moreover, the chapter discusses some issues involved in choosing the appropriate model for a given biological system, including a result on the relation between our discrete models and more detailed differential equation models. Numerous exercises and eight undergraduate research projects are included in the chapter and its online supplement.

S. Ahn and W. Just; Digraphs vs. Dynamics in Discrete Models of Neuronal Networks.
Discrete and Continuous Dynamical Systems - Series B (DCDS-B) 17(5) (2012) 1365-1381.

Abstract. It has recently been shown that discrete-time finite-state models can reliably reproduce the ODE dynamics of certain neuronal networks. We study which dynamics are possible in these discrete models for certain types of network connectivities. In particular we are interested in the number of different attractors and bounds on the lengths of attractors and transients. We completely characterize these properties for cyclic connectivities and derive structural properties of the underlying digraph that are necessary for the existence of very long attractors.

G. A. Enciso and W. Just; Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems.
Journal of Difference Equations and Applications 18(2) (2012) 223-238.

Abstract. Discrete dynamical systems defined on the state space {0,1,...,p-1}ⁿ have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n = m+1, as in the Smale theorem for the continuous case, but we show that this is possible for n = m+2 as long as p is sufficiently large. We also prove that a natural discrete analogue of strong cooperativity implies nontrivial bounds on the lengths of periodic orbits and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strong cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strong cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.

W. Just and G. A. Enciso; Exponentially long orbits in Boolean networks with exclusively positive interactions.
Nonlinear Dynamics and Systems Theory 11(3) (2011) 275-284.

Abstract. The absence of negative feedback in Boolean networks tends to result in systems with relatively short orbits. We present a construction of N-dimensional Boolean networks that use only AND, OR, COPY gates and nevertheless have an exponentially large orbit (of size c^N for arbitrary c < 2). The construction is based on pseudorandom number generation algorithms. A previously obtained nontrivial upper bound on the orbit length under certain limitations on the number of outputs per node is shown to be optimal.

W. Just and A. Nevai; A Kolmogorov-type Competition Model with Finitely Supported Allocation Profiles and its Applications to Plant Competition for Sunlight.
Journal of Biological Dynamics 3(6) (2009) 599-619.

Abstract. A Kolmogorov-type competition model featuring species allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the Canopy Partitioning Model (R.R. Vance and A.L. Nevai, J. Theor. Biol. 245 (2007) 210--219), the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine an "interior" species and an "exterior" species. It is shown that when the allocation profile of the interior species is singular (i.e., a Dirac-delta function) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or multi-singular (i.e., a weighted sum of Dirac-delta functions) then multiple coexistence states may also occur. These results have important implications for big-leaf plants that compete for sunlight.

W. Just, S. Ahn, and D. Terman; Minimal attractors in digraph system models of neuronal networks.
Physica D 237 (2008) 3186-3196.

Abstract. We study a class of discrete dynamical systems models of neuronal networks. In these models, each neuron is represented by a finite number of states and there are rules for how a neuron transitions from one state to another. In particular, the rules determine when a neuron fires and how this affects the state of other neurons. In an earlier paper by Terman, Ahn, Wang, and Just, we demonstrate that a general class of excitatory-inhibitory networks can, in fact, be rigorously reduced to the discrete model. In the present paper, we analyze the discrete model. Taken together, these two papers allow us to characterize properties of the dynamics of the original neuronal systems. We demonstrate, for example, that these networks typically exhibit a large number of stable oscillatory patterns. Moreover, there are two phase transitions that depend on the probability that two cells are synaptically connected to each other: if the probability is very low, then the network will typically evolve to a frozen state in which cells remain at a stable resting state. However, if the probability of connections is above the first but below the second phase transition, then starting in a generic initial state, most but not all cells will fire at all times along the trajectory as soon as they reach the end of their refractory period. Above the second phase transition, this will be true for all cells in a typical initial state; thus most states will belong to a minimal attractor of oscillatory behavior (in a sense that is defined precisely in the paper). The exact position of the phase transitions depends on intrinsic properties of the cells including the lengths of the cells' refractory periods and the thresholds for firing.

W. Just and A. Nevai; A Kolmogorov-type Competition Model with Multiple Coexistence States and its Applications to Plant Competition for Sunlight.
Journal of Mathematical Analysis and its Applications 348(2) (2008), 620-636.

Abstract. It is demonstrated that a Kolmogorov-type competition model featuring species allocation and gain functions can possess multiple coexistence states. Two examples are constructed: one in which the two competing species possess rectangular allocation functions but distinct gain functions, and the other in which one species has a rectangular allocation function, the second species has a bi-rectangular allocation function, and the two species share a common gain function. In both examples, it is shown that the species nullclines may intersect multiple times within the interior of the first quadrant, thus creating both locally stable and unstable equilibrium points. These results have important applications in the study of plant competition for sunlight, in which the allocation functions describe the vertical placement of leaves for two competing species, and the gain functions represent rates of photosynthesis performed by leaves at different heights when shaded by overlying leaves belonging to either species.

D. Terman, S. Ahn, X. Wang, and W. Just; Reducing neuronal networks to discrete dynamics.
Physica D 237(3) (2008) 324-338.

Abstract. We consider a general class of purely inhibitory and excitatory-inhibitory neuronal networks, with a general class of network architectures, and characterize the complex firing patterns that emerge. Our strategy for studying these networks is to first reduce them to a discrete model. In the discrete model, each neuron is represented as a finite number of states and there are rules for how a neuron transitions from one state to another. In this paper, we rigorously demonstrate that the continuous neuronal model can be reduced to the discrete model if the intrinsic and synaptic properties of the cells are chosen appropriately. In a companion paper, we analyze the discrete model.

W. Just; Data requirements of reverse-engineering algorithms.
Annals of the New York Academy of Sciences 1115 (2007) 142-153.

Abstract. Data sets used in reverse-engineering of biochemical networks usually contain relatively few, but high-dimensional, data points, which makes the problem in general vastly underdetermined. It is therefore important to estimate the probability that a given algorithm will return a model of acceptable quality when run on a data set of small size but high dimension. We propose a mathematical framework for investigating such questions. We then demonstrate that without assuming any prior biological knowledge, in general no theoretical distinction between the performance of different algorithms can be made. We also give an example of how expected algorithm performance can in principle be altered by utilizing certain features of the data collection protocol. We conclude the article with some examples of theorems that were proven within the proposed framework.

W. Just, M. R. Morris, and X. Sun; The evolution of aggressive losers.
Behavioural Processes 74 (2007) 342-350.

Abstract. We examine the question of when aggressive behavior of likely losers should be part of an evolutionarily stable strategy. We modified an earlier model by the authors that found situations where likely losers initiate aggressive interactions more often than likely winners. The modifications allowed us to examine the robustness of the previous study by including an unusually high number of possible strategies (n = 81) and to examine a wide range of parameter settings. First, we show that restricting attention to only a few most plausible strategies may change the overall results. Second, within the space where escalation is predicted, for a large percentage of the parameter settings (85%), an ESS exists that leads to a somewhat counterintuitive situation where escalation is more often initiated by the likely loser than by the likely winner of the contest. In contrast, an ESS that favors escalation by likely winners was found only for about 3% of parameter settings. Furthermore, we use simulations of evolution in a finite population to verify for certain parameter settings that the analytically predicted ESSs could in fact evolve.

W. Just and B. Stigler; Computing Groebner Bases of Ideals of Few Points in High Dimensions..
Communications in Computer Algebra 40(3) (2006) 65-76.

Abstract. A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Groebner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Groebner bases, to date none has been specifically designed to cope with large numbers of variables and few data points. In this paper, we present an algorithm for computing Groebner bases of zero-dimensional ideals that is optimized for the case when the number of points m is much smaller than the number of indeterminates n. The algorithm identifies those variables that are essential, that is, in the support of the standard monomials associated to a polynomial ideal, and computes the relations in the Groebner basis in terms of these variables. When n is much larger than m, the complexity is dominated by nm³. The algorithm has been implemented and tested in the computer algebra system Macaulay 2. We provide a comparison of its performance to the Buchberger-Moeller algorithm, as built into the system.

W. Just; Reverse engineering discrete dynamical systems from data sets with random input vectors. Journal of Computational Biology 13(8) (2006) 1435-1456.

Abstract. Recently a new algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is based on methods from computational algebra and finds most parsimonious models for a given data set. We derive mathematically rigorous estimates for the expected amount of data needed by this algorithm to find the correct model. In particular, we demonstrate that for one type of input parameter (graded term orders), the expected data requirements scale polynomially with the number n of chemicals in the network, while for another type of input parameters (randomly chosen lex orders) this number scales exponentially in n. We also show that for a modification of the algorithm, the expected data requirements scale as the logarithm of n.

W. Just and G. Della Vedova; Multiple Sequence Alignment as a Facility Location Problem. INFORMS Journal on Computing 16(4) (2004) 430-440.

Abstract. This is a substantially expanded version of a workshop paper by the same authors. A connection is made between certain multiple sequence alignment problems and facility location problems, and the existence of a PTAS (polynomial time approximation scheme) for these problems is shown. Moreover, it is shown that multiple sequence alignment with SP-score and fixed gap penalties is MAX SNP-hard.

W. Just, I. Shmulevich, and J. Konvalina; The number and probability of canalizing functions. Physica D 197(3-4) (2004) 211-221.

Abstract. Canalizing functions have important applications in physics and biology. For example, they represent a mechanism capable of stabilizing chaotic behavior in Boolean network models of discrete dynamical systems. When comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems, it is informative to know the number of canalizing functions with a given number of input variables. This is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data. To this end, we derive an exact formula for the number of canalizing Boolean functions of n variables. We also derive a formula for the probability that a random Boolean function is canalizing for any given bias p of taking the value 1. In addition, we consider the number and probability of Boolean functions that are canalizing for exactly k variables. Finally, we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables, which is needed for Monte Carlo simulations of Boolean networks.

W. Just and X. Sun; Is the Predicted ESS in the Sequential Assessment Game Evolvable? Proceedings of GECCO 2004 (Genetic and Evolutionary Computation Conference 2004), Kalyanmoy Deb et al. (eds.), Lecture Notes in Computer Science 3102, Springer Verlag: Berlin, 499-500.

Abstract. The Sequential Assessment Game is one of the most important game-theoretic models of animal contests. It intends to model contests in which animals gain a progressively more accurate estimate of relative fighting ability by means of repeated bouts of fighting. The model predicts an evolutionarily stable strategy (ESS) that would correspond to an increasing sequence of thresholds for quitting the game. Tests of these predictions by means of simulated evolution reveal that the selection pressure on thresholds for higher-numbered bouts in this game is most likely too low to allow for the ESS to evolve in nature.

X. Sun and W. Just; Evolution of Strategies in Modified Sequential Assessment Games. Proceedings of CEC 2004 (2004 Congress on Evolutionary Computation), Garrison W. Greenwood (ed.), IEEE Press, 388-394.

Abstract. The Sequential Assessment Game is one of the most important game-theoretic models of animal contests. It intends to model contests in which animals gain a progressively more accurate estimate of relative fighting ability by means of repeated bouts of fighting. The model predicts an evolutionarily stable strategy (ESS) that would correspond to an increasing sequence of thresholds for quitting the game. We report on simulated evolution of strategies in modified versions of the game and compare our results with theoretical predictions for the original model. Outcomes of these simulations corroborate some, but not all theoretical predictions for the Sequential Assessment Game. In particular, our results suggest that theoretical analyses of the Sequential Assessment Game with information asymmetry need to take into account factors that have hitherto been ignored in the literature.

W. Just and F. Zhu; Individual-Based Simulations of the War of Attrition. Behavioural Processes 66(1) (2004) 53-62.

Abstract. The War of Attrition model of John Maynard Smith predicts a single, mixed evolutionarily stable strategy (ESS) for animal contests which are settled by conventional displays with no assessment of the opponent's fighting ability. We test the predictions of the model by simulating the evolution of strategies in a finite population of animals under various assumptions on how possible strategies are coded and mutated. While our simulations for the most part confirm the predictions of the model, we also discovered some significant deviations from the theoretically predicted ESS. Specifically, we found that if inheritance of strategies is somewhat imprecise, then a population can evolve that achieves on average a higher payoff than a population at the theoretically predicted ESS. Moreover, if the ESS is realized as a polymorphism of fixed persistence times, then for small populations, sufficiently stringent statistical tests will reject the hypothesis that these times are distributed as theoretically predicted.

W. Just and M. R. Morris; The Napoleon Complex: Why Smaller Males Pick Fights. Evolutionary Ecology 17(5) (2003) 509-522.

Abstract. Does it ever pay for smaller animals to be more aggressive than a larger opponent? One empirical study of swordtail fishes showed that while smaller males were more likely to lose, 78% of fights were initiated by the smaller contestants. Asymmetry in payoffs between opponents or a suboptimal strategy based on incorrect assessment of the probability of winning have both been proposed as possible explanations for the aggressive behavior of smaller males. It does not appear, however, that either hypothesis is a likely explanation of the aggressive behaviour in the swordtails. We present a model that examines the aggressive behavior of probable losers. This model suggests that in some cases, even without a payoff asymmetry and allowing only for a small error in perception, likely losers are expected to attack first. It is shown that if the value of the resource exceeds the cost of losing a fight, the cost of displaying is sufficiently small, and assessment of resource holding power is slightly less than perfect, the evolutionarily stable strategy (ESS) prompts those contestants who perceive themselves as the likely losers to initiate fights, while it prompts those contestants who perceive themselves as the likely winners to wait for the adversary to attack or retreat.

Y. Xiao and W. Just; A Possible Mechanism of Repressing Cheating Mutants in Myxobacteria. E. Cant�-Paz et al. (Eds.): Genetic and Evolutionary Computation Conference - GECCO 2003. LNCS 2723, Springer Verlag: Berlin, 154-155.

Abstract. The formation of fruiting bodies by myxobacteria colonies involves altruistic suicide by many individual bacteria and is thus vulnerable to exploitation by cheating mutants. We report results of simulations that show how in a structured environment with patchy distribution of cheating mutants the wild type might persist.

W. Just and X. Sun; Simulating the Evolution of Contest Escalation. Alwyn Barry (Ed.): Workshop Program for GECCO 2003 (Genetic and Evolutionary Computation Conference), 75-77.

Abstract. Empirical studies of animal contests have shown that escalation to costly fights is usually initiated by their likely winners; in some species however, it is the likely losers who tend to initiate escalation. Game-theoretic models developed by the first author and M. R. Morris show a possible explanation for the latter phenomenon. Here we test one of these models with simulated evolution.

W. Just and F. Zhu; Effects of Genetic Architecture on Simultaneous Evolution of Multiple Traits. Alwyn Barry (Ed.): Workshop Program for GECCO 2003 (Genetic and Evolutionary Computation Conference), 22-26.

Abstract.The phenomena of pleiotropy, the influence of individual genes on multiple traits of an organism, and epistasis, the interaction of multiple genes in influencing a single trait, are ubiquitous characteristics of organization of genomes of living organisms. In evolutionary computation, a high degree of epistasis and pleiotropy is often considered an indication of hardness of the problem. The experiments reported here indicate that even for a very simple separable fitness function, pleiotropy and epistasis can enhance the effectiveness of evolution as an optimizing procedure when compared to a straightforward problem representation.

W. Just; Computational complexity of multiple sequence alignment with SP-score. Journal of Computational Biology Vol 8, Number 6 (2001) 615-623.

Abstract. It is shown that the multiple alignment problem with SP-score is NP-hard for each scoring matrix in a broad class M that includes most scoring matrices actually used in biological applications. The problem remains NP-hard even if sequences can only be shifted relative to each other and no internal gaps are allowed. It is also shown that there is a scoring matrix M₀ such that the multiple alignment problem for M₀ is MAX-SNP-hard in the number of sequences, regardless of whether or not internal gaps are allowed.

W. Just; $\clubsuit$-like principles under CH. Fundamenta Mathematicae 170 (2001) 247-256.

Abstract. Devlin's result that the $\clubsuit$ principle in the presence of CH implies the $\diamondsuit$ principle is generalized. Some relatives of the Juhasz club principle are introduced and studied in the presence of CH.

J. Brendle, W. Just, and C. Laflamme; On the refinement and countable refinement numbers. Questions and Answers in General Topology 18 (2000) 123-128.

Abstract. This short note presents the authors' contributions to Vojtas' unsolved problem of whether the refinement and countable refinement numbers are the same. We present evidence that the construction of a model where these numbers are different would require novel techniques.

Z. T. Balogh, S. W. Davis, W. Just, S. Shelah, and P. J. Szeptycki; Strongly almost disjoint sets and weakly uniform bases. Transactions of the AMS 352 number 11 (2000) 4971-4987.

Abstract. A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of $\square_\lambda$ for singular $\lambda$ is proved. CECA is used to show that certain "almost point-$<\tau$" families can be refined to point-$<\tau$ families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of "every first countable $T_1$-space with a weakly uniform base has a point-countable base."

W. Just and Gianluca Della Vedova; Multiple Sequence Alignment as a Facility Location Problem. Proceedings of the Prague Stringology Club Workshop 2000, Collaborative Report DC-2000-03, M. Balik and M. Simanek, eds., Department of Computer Science and Engineering, Czech Technical University, 60-70.

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Abstract. A connection is made between certain multiple sequence alignment problems and facility location problems, and the existence of a PTAS (polynomial time approximation scheme) for these problems is shown. Moreover, it is shown that multiple sequence alignment with SP-score and fixed gap penalties is MAX SNP-hard.

W. Just, M. Wu, and J. Holt; How to Evolve a Napoleon Complex. Proceedings of the 2000 Congress on Evolutionary Computation, IEEE Press 2000, 851-856.

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Abstract. In some species of animals, fights for scarce resources tend to be initiated by the smaller contestant, who is also more likely to eventually lose the fight. An evolutionary algorithm is used to study under which conditions such a behavior would evolve. The simulations partially confirm predictions of an earlier mathematical model, but also show an unexpectedly complex evolutionary dynamic of the observed behavior patterns.

W. Just, M. V. Matveev, and P. J. Szeptycki; Some results on property (a). Topology and its Applications 100 (2000) 67-83.

Abstract. A space is absolutely countably compact if for each open cover $U$ of $X$ and each dense subset $D$ of $X$ there exists a finite $F \subseteq D$ such that $St(F,U) = X$. Property (a) is defined by replacing "finite" in the above definition by "closed discrete." In an earlier paper, Matveev noticed surprising similarities in the behaviour of property (a) and normality. In this paper we further scrutinize the analogies between these two concepts. We also investigate a weakening of property (a) that is obtained by dropping the word "closed" from its definition.

A. V. Arhangel'skii, W. Just, E. A. Reznichenko, and P. J. Szeptycki; Sharp bases and weakly uniform bases versus point-countable bases. Topology and its Applications 100 (2000) 39-46.

Abstract. A base $B$ for a topological space $X$ is said to be sharp if for every $x \in X$ and every sequence $(U_n)_{n \in \omega}$ of pairwise distinct elements of $B$ with $x \in U_n$ for all $n$ the set $\{ \bigcap_{i < n}: n \in \omega\}$ forms a local base at $x$. Sharp bases of $T_0$-spaces are weakly uniform, i.e., each two-element subset of $X$ is contained in only finitely many elements of $B$. We investigate which spaces with sharp bases or weakly uniform bases have point-countable bases. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform base, but without a point-countable base. They asked whether such an example can be constructed in ZFC. We partially answer this question by showing that under CH every first countable space with a weakly uniform base and at most $\aleph_\omega$ isolated points has a point-countable base.

W. Just, S. Shelah, and S. Thomas; The Automorphism Tower Problem Revisited. Advances in Mathematics 148(2) (1999) 243-265.

Abstract. It is well known that the automorphism towers of infinite centreless groups of cardinality $\kappa$ terminate in less than $(2^\kappa)^+$ steps. But an easy counting argument shows that $(2^\kappa)^+$ is not the best possible bound. However, in this paper, we shall show that it is impossible to find an explicit better bound using ZFC.

P. DeLaney and W. Just; Two remarks on weaker connected topologies. Comment. Math. Univ. Carolinae 40,2 (1999) 327-329.

Abstract. It is shown that no generalized Luzin space condenses onto the unit interval and that the discrete sum of $\aleph_1$ copies of the Cantor set consistently does not condense onto a connected compact Hausdorff space. This answers two questions from a paper by Tkachenko, Tkachuk, Uspenskii, and Wilson.

S. Garcia-Ferreira and W. Just; Some remarks on the $\gamma_p$-property. Questions and Answers in General Topology 17 (1999) 1-8.

Abstract. A natural generalization of the $\gamma$-property is the $\gamma_p$-property, where $p$ is an arbitrary free ultrafilter on $\omega$. We say that a space $X$ has the $\gamma_p$-property if for every open $\omega$-cover ${\cal O}$ of $X$ there is a sequence $(O_n)_{n < \omega}$ in ${\cal O}$ such that for every $x \in X$ the set $\{ n < \omega :\, x \in O_n \}$ is in $p$. It is well known that every subset of the reals $\R$ with the $\gamma$-property has strong measure zero. However, we know that there is a free ultrafilter $p$ on $\omega$ for which $\R$ has the $\gamma_p$-property, and if $q$ is a selective ultrafilter on $\omega$, then every subset of the reals $\R$ with the $\gamma_q$-property has strong measure zero. Let $p$ be an ultrafilter on $\omega$ and let us consider $p$ as a subspace of the Cantor space $2^\omega$. Then, it is possible to find a free ultrafilter $q$ on $\omega$ for which the space $p$ has the $\gamma_q$-property. We prove that the space $p$ never has the $\gamma_p$-property; if $p$ has the $\gamma_q$-property, then $\R$ has the $\gamma_q$-property; and if $p$ is a Q-point and $q$ is a P-point of $\omega^*$, then $p$ does not have the $\gamma_q$-property.

S. Garcia-Ferreira and W. Just; Two examples of relatively pseudocompact spaces. Questions and Answers in General Topology 17 (1999) 35-45.

Abstract. It is shown that every discrete space of uncountable cardinality is potentially pseudocompact. This answers a question posed by A. V. Arhangelskii and H. M. M. Genedi. A subset $Y$ of a space $X$ is C-compact in $X$ if $f[Y]$ is a compact subset of $\bbB R$ for every real-valued continuous function $f$ on $X$. An example is given of a C-compact subset of a first countable Tychonoff space $X$ that is not strongly pseudocompact in $X$. An investigation of potential pseudocompactness in the class of first countable separable Tychonoff spaces leads to the introduction of two new cardinal invariants of the continuum.

W. Just and J. Tartir; A $\kappa$-normal, not densely normal Tychonoff space. Proceedings of the AMS 127 Number 3 (1999) 901-905.

Abstract. A space $X$ is $\kappa$-normal if every two disjoint canonical closed subsets of $X$ can be separated by disjoint open sets. A space $X$ is densely normal if there exists a dense subspace $Y$ of $X$ such that every two disjoint closed subsets $E,F$ of $X$ such that $E \cap Y$ is dense in $E$ and $F \cap Y$ is dense in $F$ can be separated by disjoint open subsets of $X$. We give an example of a $\kappa$-normal Tychonoff space that is not densely normal. This answers a question of Arhangel'skii.

W. Just and P. Vojtas; On matrix rapid filters. Fundamenta Mathematicae 154 (1997) 177-182.

Abstract. Galois-Tuckey equivalence between matrix summability and absolute convergence of series is shown and an alternative characterization of rapid ultrafilters on $\omega$ is derived.

W. Just and A. Tanner; Splitting $\omega$-covers. Comment. Math. Univ. Carolinae 38,2 (1997) 375-378.

Abstract. The authors give a ZFC example of a separable metrizable space $X$ such that every $\omega$-cover of $X$ can be split into two disjoint $\omega$-covers, but not every $\lambda$-cover can be split into two disjoint open covers. This answers a question of Just, Miller, Scheepers, and Szeptycki.

W. Just, M. Scheepers, J. Steprans, and P. J. Szeptycki; $G_\delta$-sets in topological spaces and games. Fundamenta Mathematicae 153 (1997) 41-58.

Abstract. Players ONE and TWO play the following game: in the n-th inning ONE chooses a set $O_n$ from a prescribed family $F$ of sbsets of a space $X$; TWO responds by choosing an open subset $T_n$ of $X$. The players must obey the rule that $O_n \subseteq O_{n+1} \subseteq T_{n+1} \subseteq T_n$ for each $n$. TWO wins if the intersection of TWO's sets is equal to the union of ONE's sets. If ONE has no winning strategy, then each element of $F$ is a $G_\delta$-set. To what extent is the converse true? We show that:

(A) For $F$ the family of countable subsets of $X$:

There are subsets of the real line for which neither player has a winning strategy in this game.
The statement "If $X$ is a set of real numbers, then ONE does not have a winning strategy if, and only if, every countable subset of $X$ is a $G_\delta$-set" is independent of ZFC.
There are spaces whose countable subsets are $G_\delta$-sets, and yet ONE has a winning strategy in this game.
For a hereditarily Lindelöf space $X$, TWO has a winning strategy if, and only if, $X$ is countable.

(B) For $F$ the collection of $F_\sigma$-subsets of a subset $X$ of the real line the determinacy of this game is independent of ZFC.

A. V. Arhangel'skii, W. Just, and G. Plebanek; Sequential continuity on dyadic compacta and topological groups. Comment. Math. Univ. Carolinae 37,4 (1996) 775-790.

Abstract. We study conditions under which sequentially continuous functions on topological spaces and sequentially continuous homomorphisms of topological groups are continuous.

W. Just, A. W. Miller, M. Scheepers, and P. J. Szeptycki; The combinatorics of open covers II. Topology and its Applications 73 (1996) 241-266.

Abstract. We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are equivalent to saying all finite powers have one of the classical properties above (Rothberger property in one case and Menger property in the other). We consider for each property the smallest cardinality of a metric space which fails to have the property. In each case this cardinal turns out to be another well-known cardinal invariant of the continuum. We also disprove (in ZFC) a long-standing conjecture of Hurewicz. Finally, we answer several questions from Part I concerning partition properties of covers.

W. Just; The sizes of relatively compact $T_1$-spaces. Comment. Math. Univ. Carolinae 37,2 (1996) 379-380.

Abstract. The relativization of Gryzlov's theorem about the size of compact $T_1$-spaces with countable pseudocharacter is false.

W. Just, O. Sipacheva, and P. J. Szeptycki; Nonnormal spaces $C_p(X)$ with countable extent. Proceedings of the AMS vol. 124(4) (1996) 1227-1235.

Abstract. Examples of spaces $X$ are constructed for which $C_p(X)$ is not normal but all closed discrete subsets are countable. A monolithic example is constructed in ZFC and a separable first countable example is constructed using $\diamondsuit$.