Neal Madras’s research while affiliated with New York University and other places

We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number c approaches $\sqrt{{2c}/{\pi}}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number n and show that the average 4-genus is sublinear in n. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number c and gives a specific sublinear upper bound of $9.75c/\log c$.

We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our “topology” refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption.

We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our "topology" refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption.

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size N that avoid the monotone decreasing pattern of fixed size m. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\rightarrow \infty$. For instance, the number of bounded affine permutations of size N that avoid 321 is asymptotically equal to $4^N (N/4\pi )^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $m\cdots 1$ looks like $m-1$ random lines of slope 1 whose y intercepts sum to 0.

We use analytical and numerical methods to investigate an adaptive vaccination strategy’s effects on the infectious disease dynamics in a demographically open population. The methodology and key assumptions are based on Breda et al. (2012). We show that the endemic force of infection in the demographically open population can be reduced significantly by two factors: the vaccine effectiveness and the vaccination rate. The impact of these factors can transform an endemic steady state into a disease free state.

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size n. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations. Comment: 35 pages

We study here the role of polyelectrolyte chain length, that is number of repeat units (mers), in the competitive adsorption of a simple model polyanion, poly(acrylic acid), onto 85 nm spherical silica particles capped with a model polycation, poly(allylamine hydrochloride). Performing fluorescence spectroscopy experiments, we measured chain-length dependence of dilute aqueous polyelectrolyte adsorption, at full surface coverage, onto an oppositely charged polyelectrolyte overtop spherical silica nanoparticles (10−3 g/L). Preferential adsorption was determined by comparing the characteristic fluorescence intensities of the two fluorophore-labeled and narrowly disperse polyacrylic acid samples (NMA-PAA450K and Dan-PAA2K) of 450K- and 2K- molecular weight (6250-and 28-mers), respectively. To compare and validate experimental results, a lattice model was developed for computing the probabilities of the different arrangements of two polymer chain lengths of polyacrylic acid on the surface of the silica nanosphere. We then determined which numbers of long and short adsorbed chains corresponded to the most configurations in our model. Both spectroscopic experiment results and the combinatorial model demonstrated that there is an entropic preference for complete adsorption of the longer 450K polyacrylic acid chain vs 2K. This study provides insights on entropy driven chain-length dependence of polyelectrolyte adsorption onto spherical nanoparticle surfaces for directing and optimizing their layer-by-layer self-assembly in organic films.

We use analytical methods to investigate a continuous vaccination strategy's effects on the infectious disease dynamics in a closed population and a demographically open population. The methodology and key assumptions are based on Breda et al. (J Biol Dyn 6(Sup2):103-117, 2012). We show that the cumulative force of infection for the closed population and the endemic force of infection in the demographically open population can be reduced significantly by combining two factors: the vaccine effectiveness and the vaccination rate. The impact of these factors on the force of infection can transform an endemic steady state into a disease-free state.

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size N that avoid the monotone decreasing pattern of fixed size m. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size N that avoid 321 is asymptotically equal to $4^N (N/4\pi)^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $m\cdots1$ looks like $m-1$ random lines of slope 1 whose y intercepts sum to 0.

We use analytical methods to investigate a continuous vaccination strategy effects on the infectious disease dynamics in a closed population and a demographically open population. The methodology and key assumptions are based on Breda et al (2012). We show that the cumulative force of infection for the closed population and the endemic force of infection in the demographically open population can be reduced significantly by combining two factors: the vaccine effectiveness and the vaccination rate. The impact of these factors on the force of infection can transform an endemic steady state into a disease-free state. Keywords: Force of infection, Cumulative force of infection, Scalar-renewal equation, Per capita death rate, Lambert function, adaptive vaccination strategy