Energy Growth in Schrödinger's Equation with Markovian Forcing

Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ,t)=λV(θ)X(t), where V(θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^(s/4A)

The radial defocusing energy-supercritical cubic nonlinear wave equation in R^{1+5}

In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and scattering. The main tool that we use is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schr\"odinger equation.Comment: AMS Latex, 20 page

Absence of reflection as a function of the coupling constant

We consider solutions of the one-dimensional equation $-u'' +(Q+ \lambda V) u = 0$ where $Q: \mathbb{R} \to \mathbb{R}$ is locally integrable, $V : \mathbb{R} \to \mathbb{R}$ is integrable with supp$(V) \subset [0,1]$, and $\lambda \in \mathbb{R}$ is a coupling constant. Given a family of solutions $\{u_{\lambda} \}_{\lambda \in \mathbb{R}}$ which satisfy $u_{\lambda}(x) = u_0(x)$ for all $x<0$, we prove that the zeros of $b(\lambda) := W[u_0, u_{\lambda}]$, the Wronskian of $u_0$ and $u_{\lambda}$, form a discrete set unless $V \equiv 0$. Setting $Q(x) := -E$, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment $-u'' + \lambda V u = Eu$ gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then $V \equiv 0$.Comment: To appear in Journal of Mathematical Physic

Energy Growth in Schrödinger's Equation with Markovian Forcing

Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ,t)=λV(θ)X(t), where V(θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^(s/4A)

An unbiased genetic screen reveals the polygenic nature of the influenza virus anti-interferon response.

Influenza A viruses counteract the cellular innate immune response at several steps, including blocking RIG I-dependent activation of interferon (IFN) transcription, interferon (IFN)-dependent upregulation of IFN-stimulated genes (ISGs), and the activity of various ISG products; the multifunctional NS1 protein is responsible for most of these activities. To determine the importance of other viral genes in the interplay between the virus and the host IFN response, we characterized populations and selected mutants of wild-type viruses selected by passage through non-IFN-responsive cells. We reasoned that, by allowing replication to occur in the absence of the selection pressure exerted by IFN, the virus could mutate at positions that would normally be restricted and could thus find new optimal sequence solutions. Deep sequencing of selected virus populations and individual virus mutants indicated that nonsynonymous mutations occurred at many phylogenetically conserved positions in nearly all virus genes. Most individual mutants selected for further characterization induced IFN and ISGs and were unable to counteract the effects of exogenous IFN, yet only one contained a mutation in NS1. The relevance of these mutations for the virus phenotype was verified by reverse genetics. Of note, several virus mutants expressing intact NS1 proteins exhibited alterations in the M1/M2 proteins and accumulated large amounts of deleted genomic RNAs but nonetheless replicated to high titers. This suggests that the overproduction of IFN inducers by these viruses can override NS1-mediated IFN modulation. Altogether, the results suggest that influenza viruses replicating in IFN-competent cells have tuned their complete genomes to evade the cellular innate immune system and that serial replication in non-IFN-responsive cells allows the virus to relax from these constraints and find a new genome consensus within its sequence space. IMPORTANCE In natural virus infections, the production of interferons leads to an antiviral state in cells that effectively limits virus replication. The interferon response places considerable selection pressure on viruses, and they have evolved a variety of ways to evade it. Although the influenza virus NS1 protein is a powerful interferon antagonist, the contributions of other viral genes to interferon evasion have not been well characterized. Here, we examined the effects of alleviating the selection pressure exerted by interferon by serially passaging influenza viruses in cells unable to respond to interferon. Viruses that grew to high titers had mutations at many normally conserved positions in nearly all genes and were not restricted to the NS1 gene. Our results demonstrate that influenza viruses have fine-tuned their entire genomes to evade the interferon response, and by removing interferon-mediated constraints, viruses can mutate at genome positions normally restricted by the interferon response

Irreversible quantum graphs

Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a {\it single} harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a "bath" which consists of a {\it single} harmonic oscillator

Deep retrofit approaches: managing risks to minimise the energy performance gap

Energy use in buildings remains a significant part of overall energy demand. Deep renovation projects, delivered at scale, remain a challenging task to achieve a lower carbon building stock.The complexity of building renovation beyond standards and building specifications is related to inherent characteristics of buildings which require distinct project management techniques. While there are now more projects focusing on achieving operational performance, there is still very little research on the management of the renovation and retrofit process itself. Recognising that each project working on an existing building is unique in type, timing, energy goals and the roles/characteristics of people involved, the aim of this paper is to add to the current debate of how intervention approaches (one-off or over-time, whole-house, fabric-first room-by-room, measure-by-measure) are promoted by different policies, and with what impact. The paper discusses the complexity of a deep renovation project in terms of planning and management and the ways current policies can lead to unintended consequences in the short and long term, as well in lock-in effects that contribute to energy performance, and to the gap between designed and actual energy performance. Using a typology of risks, the issues associated with renovation processes and technologies were explored in a sample of cases studies from deep retrofits across the EU. The evidence from these shows that despite holistic planning for renovation, interventions tend to be carried out in phases. These contrasting time dimensions and the different retrofit approaches are discussed with risk profiles for each retrofit project, suggesting how risks emerge throughout a project. A series of risk mitigation strategies are suggested which, taken in combination to suit a specific project’s risk profile, may serve to reduce and potentially eliminate the building renovation energy performance gap

The dynamics of the 3D radial NLS with the combined terms

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\"{o}dinger equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u \tag{CNLS} in the energy space $H^1(\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, $\dot H^1$-subcritical perturbation $|u|^2u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.Comment: 46page

Asymptotic behavior of small solutions for the discrete nonlinear Schr\"odinger and Klein-Gordon equations

We show decay estimates for the propagator of the discrete Schr\"odinger and Klein-Gordon equations in the form \norm{U(t)f}{l^\infty}\leq C (1+|t|)^{-d/3}\norm{f}{l^1}. This implies a corresponding (restricted) set of Strichartz estimates. Applications of the latter include the existence of excitation thresholds for certain regimes of the parameters and the decay of small initial data for relevant $l^p$ norms. The analytical decay estimates are corroborated with numerical results.Comment: 13 pages, 4 figure