Discontinuous local semiflows for Kurzweil equations leading to LaSalle`s invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle`s invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle`s invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times. (C) 2011 Elsevier Inc. All rights reserved.

Sistemas dinâmicos e equações diferenciais ordinárias

Pós-graduação em Matemática Universitária - IGCE

Role of phosphatidylethanolamine in the function and assembly of phenylalanine-specific permease of Escherichia coli

Transmembrane segments of polytopic membrane proteins once inserted are generally considered stably oriented due to the large free energy barrier for topological reorientation of adjacent extra-membrane domains. However, proper topology and function of the polytopic membrane protein lactose permease (LacY) of Escherichia coli is dependent on the membrane phospholipid composition revealing topological dynamics of transmembrane domains (Bogdanov, M., Heacock, P. N., and Dowhan, W. (2002) EMBO J. 21, 2107–2116). The high affinity phenylalanine permease PheP shares many topological similarities with LacY. In this study, mutant E. coli cells lacking phosphatidylethanolamine (PE) as a membrane component were used to evaluate the role of PE in the function and assembly of PheP. Active transport of phenylalanine by cells lacking PE was severely inhibited (both Vmax and Km were altered), whereas the PheP protein level in membranes was unaffected. Cysteine residues were introduced into predicted periplasmic or cytoplasmic segments of cysteine-less PheP, and the topology of the protein was explored using a membrane-impermeable thiol-specific biotinylated probe. Based on the biotinylation patterns of PheP in whole cells, the N-terminus and adjoining transmembrane hairpin of PheP adopted an inverted topological orientation in PE-lacking cells. Introduction of PE following the assembly of PheP triggered a reorientation of the N-terminus and adjacent hairpin to their native orientation associated with regain of wild type transport function. These results coupled with the results for LacY support a specific role for membrane lipid composition in determining topological organization and function of membrane proteins. Several other secondary symporters are compromised for activity in PE-lacking cells suggesting that lipid-assisted topogenesis is a general property of such transporters. The reversible orientation of these secondary transport proteins in response to a change of phospholipid composition might be a result of inherent conformational flexibility necessary for transport function or during protein assembly. ^

Regular homomorphisms of minimal sets

The classification of minimal sets is a central theme in abstract topological dynamics. Recently this work has been strengthened and extended by consideration of homomorphisms. Background material is presented in Chapter I. Given a flow on a compact Hausdorff space, the action extends naturally to the space of closed subsets, taken with the Hausdorff topology. These hyperspaces are discussed and used to give a new characterization of almost periodic homomorphisms. Regular minimal sets may be described as minimal subsets of enveloping semigroups. Regular homomorphisms are defined in Chapter II by extending this notion to homomorphisms with minimal range. Several characterizations are obtained. In Chapter III, some additional results on homomorphisms are obtained by relativizing enveloping semigroup notions. In Veech's paper on point distal flows, hyperspaces are used to associate an almost one-to-one homomorphism with a given homomorphism of metric minimal sets. In Chapter IV, a non-metric generalization of this construction is studied in detail using the new notion of a highly proximal homomorphism. An abstract characterization is obtained, involving only the abstract properties of homomorphisms. A strengthened version of the Veech Structure Theorem for point distal flows is proved. In Chapter V, the work in the earlier chapters is applied to the study of homomorphisms for which the almost periodic elements of the associated hyperspace are all finite. In the metric case, this is equivalent to having at least one fiber finite. Strong results are obtained by first assuming regularity, and then assuming that the relative proximal relation is closed as well.

Topological phases, Majorana modes and quench dynamics in a spin ladder system

We explore the salient features of the `Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z(2) symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.

Topological blocking in quantum quench dynamics

We study the nonequilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one of their defining features: ground-state degeneracies and associated topological sectors. We present the notion of ``topological blocking,'' experienced by the dynamics due to a mismatch in degeneracies between two phases, and we argue that the dynamic evolution of the quench depends strongly on the topological sector being probed. We demonstrate this interplay between quench and topology in models stemming from two extensively studied systems, the transverse Ising chain and the Kitaev honeycomb model. Through nonlocal maps of each of these systems, we effectively study spinless fermionic p-wave paired topological superconductors. Confining the systems to ring and toroidal geometries, respectively, enables us to cleanly address degeneracies, subtle issues of fermion occupation and parity, and mismatches between topological sectors. We show that various features of the quench, which are related to Kibble-Zurek physics, are sensitive to the topological sector being probed, in particular, the overlap between the time-evolved initial ground state and an appropriate low-energy state of the final Hamiltonian. While most of our study is confined to translationally invariant systems, where momentum is a convenient quantum number, we briefly consider the effect of disorder and illustrate how this can influence the quench in a qualitatively different way depending on the topological sector considered.

The dynamics of a topological change in a system of soap films

Hutzler, S., Saadatfar, M., van der Net, A., Weaire, D. and Cox, S.J. (2007) The dynamics of a topological change in a system of soap films. Coll. Surf. A, 323:123-131. Sponsorship: This research was supported by the European Space Agency (contracts 14914/02/NL/SH and 14308/00/NL/SH), Science Foundation Ireland. (RFP 05/REP/PHY00/6), and the EU program COST P21 (The Physics of droplets). SJC acknowledges support from EPSRC (EP/D071127/1). MS is supported by the Irish Higher Education Authority (PRTLI-IITAC).

Combinatorial-topological framework for the analysis of global dynamics

We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conley's topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767672]

The dynamics of cellular automata on 2-manifolds is affected by topology

In this paper, we demonstrate that the distribution of Wolfram classes within a cellular automata rule space in the triangular tessellation is not consistent across different topological general. Using a statistical mechanics approach, cellular automata dynamical classes were approximated for cellular automata defined on genus-0, genus-1 and genus-2 2-manifolds. A distribution-free equality test for empirical distributions was applied to identify cases in which Wolfram classes were distributed differently across topologies. This result implies that global structure and local dynamics contribute to the long term evolution of cellular automata.

Fractal and complex network analyses of protein molecular dynamics

Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h(q)h(q) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h(2)h(2) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H≈0.5H≈0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h(2)h(2) of MF-DFA on the time series, exponent λλ of the exponential degree distribution and fractal dimension dBdB of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between 〈h(2)〉〈h(2)〉 (from MF-DFA on time series) and 〈dB〉〈dB〉 of the converted HVGs for different energy, pressure and volume.

Structure, function and intracellular dynamics of alphavirus replication complexes

Intracellular membrane alterations are hallmarks of positive-sense RNA (+RNA) virus replication. Strong evidence indicates that within these exotic compartments, viral replicase proteins engage in RNA genome replication and transcription. To date, fundamental questions such as the origin of altered membranes, mechanisms of membrane deformation and topological distribution and function of viral components, are still waiting for comprehensive answers. This study addressed some of the above mentioned questions for the membrane alterations induced during Semliki Forest virus (SFV) infection of mammalian cells. With the aid of electron and fluorescence microscopy coupled with radioactive labelling and immuno-cytochemistry techniques, our group and others showed that few hours after infection the four non structural proteins (nsP1-4) and newly synthesized RNAs of SFV colocalized in close proximity of small membrane invaginations, designated as spherules . These 50-70 nm structures were mainly detected in the perinuclear area, at the limiting membrane of modified endosomes and lysosomes, named CPV-I (cytopathic vacuoles type I). More rarely, spherules were also found at the plasma membrane (PM). In the first part of this study I present the first three-dimensional reconstruction of the CPV-I and the spherules, obtained by electron tomography after chemical or cryo-fixation. Different approaches for imaging these macromolecular assemblies to obtain better structure preservation and higher resolution are presented as unpublished data. This study provides insights into spherule organization and distribution of viral components. The results of this and other experiments presented in this thesis will challenge currently accepted models for virus replication complex formation and function. In a revisitation of our previous models, the second part of this work provides the first complete description of the biogenesis of the CPV-I. The results demonstrate that these virus-induced vacuoles, where hundreds of spherules accumulate at late stages during infection, represent the final phase of a journey initiated at the PM, which apparently serves as a platform for spherule formation. From the PM spherules were internalized by an endocytic event that required the activity of the class I PI3K, caveolin-1, cellular cholesterol and functional actin-myosin network. The resulting neutral endocytic carrier vesicle delivered the spherules to the membrane of pre-existing acidic endosomes via multiple fusion events. Microtubule based transport supported the vectorial transfer of these intermediates to the pericentriolar area where further fusions generated the CPV-I. A signal for spherule internalization was identified in one of the replicase proteins, nsP3. Infections of cells with viruses harbouring a deletion in a highly phosphorylated region of nsP3 did not result in the formation of CPV-Is. Instead, thousands of spherules remained at the PM throughout the infection cycle. Finally, the role of the replicase protein nsP2 during viral RNA replication and transcription was investigated. Three enzymatic activities, protease, NTPase and RNA-triphosphatase were studied with the aid of temperature sensitive mutants in vitro and, when possible, in vivo. The results highlighted the interplay of the different nsP2 functions during different steps of RNA replication and sub-genomic promoter regulation, and suggest that the protein could have different activities when participating in the replication complex or as a free enzyme.

Sharp raman anomalies and broken adiabaticity at a pressure induced transition from band to topological insulator in Sb2Se3

The nontrivial electronic topology of a topological insulator is thus far known to display signatures in a robust metallic state at the surface. Here, we establish vibrational anomalies in Raman spectra of the bulk that signify changes in electronic topology: an E-g(2) phonon softens unusually and its linewidth exhibits an asymmetric peak at the pressure induced electronic topological transition (ETT) in Sb2Se3 crystal. Our first-principles calculations confirm the electronic transition from band to topological insulating state with reversal of parity of electronic bands passing through a metallic state at the ETT, but do not capture the phonon anomalies which involve breakdown of adiabatic approximation due to strongly coupled dynamics of phonons and electrons. Treating this within a four-band model of topological insulators, we elucidate how nonadiabatic renormalization of phonons constitutes readily measurable bulk signatures of an ETT, which will facilitate efforts to develop topological insulators by modifying a band insulator. DOI: 10.1103/PhysRevLett.110.107401

Quench dynamics and parity blocking in Majorana wires

We theoretically explore quench dynamics in a finite-sized topological fermionic p-wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick's theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities-the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term `parity blocking'. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.

Streamline topology and dilute particle dynamics in a Karman vortex street flow

Three types of streamline topology in a Karman vortex street flow are shown under the variation of spatial parameters. For the motion of dilute particles in the Karman vortex street flow, there exist a route of bifurcation to a chaotic orbit and more attractors in a bifurcation diagram for the proportion of particle density to fluid density. Along with the increase of spatial parameters in the flow field, the bifurcation process is suspended, as well as more and more attractors emerge. In the motion of dilute particles, a drag term and gravity term dominate and result in the bifurcation phenomenon.