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. Cantú-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 $X$:

  1. There are subsets of the real line for which neither player has a winning strategy in this game.
  2. 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.
  3. There are spaces whose countable subsets are $G_\delta$-sets, and yet ONE has a winning strategy in this game.
  4. 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 $\diamondsuit$.