Replication models have been developed to describe the replication dynamics of a variety of viruses to better understand the kinetics and key contributing factors affecting infectivity and spread. However, accurate representations of the dynamics of virus replication observed in vitro and in vivo are often limited due to the failure of these models to account for both environmental influences, such as temperature, and the variety of possible mechanisms employed by viruses to spread. Several major families of viruses including paramyxoviruses, pneumoviruses and coronaviruses, induce and use the formation of syncytia, large multinucleated cell masses formed through fusion of cells, to aid in spread to neighbouring susceptible cells. In this study, we evaluate and compare both the dynamics and roles of temperature and syncytia formation on the replication of two different fusogenic viruses in vitro : human respiratory syncytial virus (hRSV) and a murine coronavirus, mouse hepatitis virus (MHV). Thermal stability, replication kinetics and both the rates and dynamics of syncytia formation were evaluated for hRSV and MHV. These data were then incorporated into a novel and improved replication model for each of the two viruses, which provides new insights into the contributions of both temperature and syncytia formation in the replication of fusogenic viruses.

In this paper we study properties of endomorphisms of P k \mathbb {P}^k using a symmetric product construction ( P 1 ) k / S k ≅ P k (\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k . Symmetric products have been used to produce examples of endomorphisms of P k \mathbb {P}^k with certain characteristics, k ≥ 2 k\geq 2 . In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study k k -deep post-critically finite maps and characterize families of Lattès maps.

Respiratory syncytial virus (RSV) is a leading viral cause of pediatric respiratory infections and early infant mortality. Despite extensive development efforts currently underway, there remain no vaccines available for the prevention of RSV. RSV is an enveloped, negative-strand RNA virus that utilizes two different proteins (G and F) to mediate attachment and entry into host cells. These G and F proteins are the primary determinants of viral strain-specific differences and elicit protective neutralizing antibodies during natural infection in humans. Earlier studies have demonstrated that these proteins play an additional role in regulating the stability of RSV particles in response to temperature and pH. However, it remains unclear how much variability exists in the stability of RSV strains and what contribution changes in temperature and pH make to the clearance of virus during an active infection. In this study, we evaluated the impacts of changes in temperature and pH on the inactivation of four different chimeric recombinant RSV strains that differ exclusively in G and F protein expression. Using these data, we developed predictive mathematical models to examine the specific contributions and variations in susceptibility that exist between viral strains. Our data provide strain-specific clearance rates and temperature-pH landscapes that shed light on the optimal contributions of temperature and pH to viral clearance. These provide new insight into how much variation exists in the clearance of a major respiratory pathogen and may offer new guidance on optimization of viral strains for development of live-attenuated vaccine preparations.

For maps of one complex variable, f, given as the sum of a degree n power map and a degree d polynomial, we provide necessary and sufficient conditions that the geometric limit as n approaches infinity of the set of points that remain bounded under iteration by f is the closed unit disk or the unit circle. We also provide a general description, for many cases, of the limiting set.

We explore student performance on True-False assessments with statements in the conditional form “If P then Q” in order to better understand how students process conditional logic and to see whether logical misconceptions impede students’ ability to demonstrate mathematical knowledge. We administered an online assessment to a population of Calculus II students. We find that students make logical errors on True-False items of a certain logical form, and that these errors are unrelated to their calculus knowledge.

We place Schwartz's work on the real dynamics of the projective heat map H into the complex perspective by computing its first dynamical degree and gleaning some corollaries about the dynamics of H.

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

AbstractLet $f: X~\dashrightarrow ~X$ be a dominant meromorphic self-map, where $X$ is a compact, connected complex manifold of dimension $n\gt 1$. Suppose that there is an embedded copy of ${ \mathbb{P} }^{1} $ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose that $f$ restricted to this line is given by $z\mapsto {z}^{b} $, with resulting invariant circle $S$. We prove that if $a\geq b$, then the local stable manifold ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition $a\geq b$ cannot be relaxed without adding additional hypotheses by presenting two examples with $a\lt b$ for which ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is not real analytic in the neighborhood of any point.

We show that the geometric limit as n → ∞ of the Julia sets J(Pn,c) for the maps Pn,c(z) = zn + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle.

We show that the geometric limit as n →∞ of the filled Julia sets K(P_n,c) for the maps P_n,c(z) = z^n + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle, and this is used to show that for certain parameters, the geometric limit of the Julia sets J(P_n,c) is the unit circle.

We present a simple rational map of the complex projective plane whose first and second dynamical degrees coincide, but which does not have any invariant foliation.