Monotone flows with dense periodic orbits

The main result is Theorem 1: A flow on a connected open set X ⊂ Rd is globally periodic provided (i) periodic points are dense in X, and (ii) at all positive times the flow preserves the partial order defined by a closed convex cone that has nonempty interior and contains no straight line. The proof uses the analog for homeomorphisms due to B. Lemmens et al. [27], a classical theorem of D. Montgomery [31, 32], and a sufficient condition for the nonstationary periodic points in a closed order interval to have rationally related periods (Theorem 2)

Zero sets of abelian Lie algebras of vector fields

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X ∈ A. Theorem If K is a locally maximal compact set of zeroes of X ∈ A and the Poincaré-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish

Primary singularities of vector fields on surfaces

Unless another thing is stated one works in the C∞ category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M. We say that Y tracks X if [Y, X] = fX for some continuous function f: M→ R. A subset K of the zero set Z(X) is an essential block for X if it is non-empty, compact, open in Z(X) and its Poincaré-Hopf index does not vanishes. One says that X is non-flat at p if its ∞-jet at p is non-trivial. A point p of Z(X) is called a primary singularity of X if any vector field defined about p and tracking X vanishes at p. This is our main result: consider an essential block K of a vector field X defined on a surface M. Assume that X is non-flat at every point of K. Then K contains a primary singularity of X. As a consequence, if M is a compact surface with non-zero characteristic and X is nowhere flat, then there exists a primary singularity of X

A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds

The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordinates, and not any Riemannian structure, are shown to play a natural role in lifting the Newton method to a manifold. The framework also gives new insight into the design of Newton methods in general.Comment: 36 page

Symmetries of degenerate center singularities of plane vector fields

Let $D$ be a closed unit $2$-disk on the plane centered at the origin $O$, and $F$ be a smooth vector field such that $O$ is a unique singular point of $F$ and all other orbits of $F$ are simple closed curves wrapping once around $O$. Thus topologically $O$ is a "center" singularity. Let also $\mathrm{Diff}(F)$ be the group of all diffeomorphisms of $D$ which preserve orientation and orbits of $F$. In arXiv:0907.0359 the author described the homotopy type of $\mathrm{Diff}(F)$ under assumption that the $1$-jet of $F$ at $O$ is non-degenerate. In this paper degenerate case is considered. Under additional "non-degeneracy assumptions" on $F$ the path components of $\mathrm{Diff}(F)$ with respect to distinct weak topologies are described.Comment: 21 page, 3 figure

Applications of dynamical systems to deterministic and stochastic economic models

Markets and related games are modeled by several types of deterministic and stochastic dynamical systems. Results are applied to B. Arthur's urn models of stochastic allocation processes, and to repeated games in which players use the long term strategy of fictitious play

Evolutionary Toggling of Vpx/Vpr Specificity Results in Divergent Recognition of the Restriction Factor SAMHD1

SAMHD1 is a host restriction factor that blocks the ability of lentiviruses such as HIV-1 to undergo reverse transcription in myeloid cells and resting T-cells. This restriction is alleviated by expression of the lentiviral accessory proteins Vpx and Vpr (Vpx/Vpr), which target SAMHD1 for proteasome-mediated degradation. However, the precise determinants within SAMHD1 for recognition by Vpx/Vpr remain unclear. Here we show that evolution of Vpx/Vpr in primate lentiviruses has caused the interface between SAMHD1 and Vpx/Vpr to alter during primate lentiviral evolution. Using multiple HIV-2 and SIV Vpx proteins, we show that Vpx from the HIV-2 and SIVmac lineage, but not Vpx from the SIVmnd2 and SIVrcm lineage, require the C-terminus of SAMHD1 for interaction, ubiquitylation, and degradation. On the other hand, the N-terminus of SAMHD1 governs interactions with Vpx from SIVmnd2 and SIVrcm, but has little effect on Vpx from HIV-2 and SIVmac. Furthermore, we show here that this difference in SAMHD1 recognition is evolutionarily dynamic, with the importance of the N- and C-terminus for interaction of SAMHD1 with Vpx and Vpr toggling during lentiviral evolution. We present a model to explain how the head-to-tail conformation of SAMHD1 proteins favors toggling of the interaction sites by Vpx/Vpr during this virus-host arms race. Such drastic functional divergence within a lentiviral protein highlights a novel plasticity in the evolutionary dynamics of viral antagonists for restriction factors during lentiviral adaptation to its hosts. © 2013 Fregoso et al

Conformational adaptation of Asian macaque TRIMCyp directs lineage specific antiviral activity

TRIMCyps are anti-retroviral proteins that have arisen independently in New World and Old World primates. All TRIMCyps comprise a CypA domain fused to the tripartite domains of TRIM5α but they have distinct lentiviral specificities, conferring HIV-1 restriction in New World owl monkeys and HIV-2 restriction in Old World rhesus macaques. Here we provide evidence that Asian macaque TRIMCyps have acquired changes that switch restriction specificity between different lentiviral lineages, resulting in species-specific alleles that target different viruses. Structural, thermodynamic and viral restriction analysis suggests that a single mutation in the Cyp domain, R69H, occurred early in macaque TRIMCyp evolution, expanding restriction specificity to the lentiviral lineages found in African green monkeys, sooty mangabeys and chimpanzees. Subsequent mutations have enhanced restriction to particular viruses but at the cost of broad specificity. We reveal how specificity is altered by a scaffold mutation, E143K, that modifies surface electrostatics and propagates conformational changes into the active site. Our results suggest that lentiviruses may have been important pathogens in Asian macaques despite the fact that there are no reported lentiviral infections in current macaque populations