Scattering for the L2supercritical point NLS

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point"means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blowup profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ±∞ in the L2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation

Fast solitons on star graphs

We define the Schr\"odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so called $\delta$ and $\delta'$ boundary conditions. Moreover, in the same setting we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem we follow ideas borrowed from the seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.Comment: Sec. 2 revised; several misprints corrected; added references; 32 page

Kritisk tenkning i samfunnsfag. En kvalitativ studie om læreplanenes utvikling av begrepet kritisk tenkning og læreres forståelse av begrepet

Temaet for denne masteroppgaven har vært kritisk tenkning, med et søkelys på utviklingen av begrepet fra Kunnskapsløftet (2006) til Fagfornyelsen (2020), og hvordan begrepet forstås av lærere. Kritisk tenkning er særs dagsaktuelt ettersom elevene trenger ferdigheter for å imøtekomme et samfunn hvor falske nyheter preger mediebildet. Videre implementeres kritisk tenkning inn som en kjernekompetanse i fagfornyelsen som skal iverksettes august, 2020. Med bakgrunn i teori og intervju myntet på kritisk tenkning, er denne dokumentanalysen en gjennomgang av læreplanenes utvikling av tematikken, og har en kvalitativ tilnærming. Nærmere bestemt belyses spørsmålet om hvordan kritisk tenkning har utviklet seg i samfunnsfaglige læreplaner, og hvilken forståelse lærere har rundt dette begrepet. Gjennom dokumentanalyse av læreplanene har jeg sammenlignet definisjoner, sett på tilstedeværelse eller mangel på tilstedeværelse av begrepet, og videre sammenlignet læreplanene. For å tydeliggjøre fokuset for studien, ble John Goodlads (1979) fem læreplannivå redegjort for. Gjennom analysen kunne jeg se en klar utvikling og vektlegging av begrepet. I LK06 fremstår både definisjon og redegjørelse for hva kritisk tenkning innebærer i opplæringen som utydelig. Dette virket å ha en innvirkning på lærere, ettersom de utviste ulik forståelse av begrepet. I henhold til beskrivelsen av kritisk tenkning, uttrykte lærerne at kritisk tenkning er viktig, men at de ikke hadde et avklart forhold til begrepet. I LK20, er det derimot nedført en tydelig definisjon av begrepet og hvilket formål kritisk tenkning har i opplæringen. Videre i studiet så jeg på intensjonene bak bestemmelser i skolesektoren, og hvorfor kritisk tenkning er viktig. Kritisk tenkning er blant annet viktig for at elevene skal kvalifiseres til en fremtid hvor en må skille mellom sannferdig og falsk informasjon. Målstyringsdebatten ble også tatt med i studien, og jeg fant ut at målstyring i skolen kan true læreplanverkenes ideal om kritisk tenkende elver, og videre undergrave formålet med opplæringen

On the 2d Zakharov system with L^2 Schr\"odinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time.Comment: 30 pages, 2 figures. Minor revision. Title has been change

Last Bite: No Fish Oil No Footnotes

The exponential growth of off-shore mariculture that has occurred worldwide over the last 10 years has raised concern about the impact of the waste produced by this industry on the ecological integrity of the sea bottom. Investigations into this potential source of impact on the biochemistry of the sea floor have provided contrasting results, and no compelling explanations for these discrepancies have been provided to date. To quantify the impact of fish-farm activities on the biochemistry of sediments, we have investigated the quantity and biochemical composition of sediment organic matter in four different regions in the temperate-warm Mediterranean Sea: Akrotiri Bay (Cyprus), Sounion Bay (Greece), Pachino Bay (Italy), and the Gulf of Alicante (Spain). In these four study regions, the concentrations of phytopigments, proteins, carbohydrates, and lipids in the sediments were measured, comparing locations receiving wastes from fish farms to control locations in two different habitats: seagrass beds and soft nonvegetated substrates. Downward fluxes were also measured in all of the regions, up to 200 m from the fish farms, to assess the potential spatial extent of the impact. In all four regions, with the exception of seagrass sediments in Spain, the biochemistry of the sediments showed significant differences between the control and fish-farm locations. However, the variables explaining the differences observed varied among the regions and between habitats, suggesting idiosyncratic effects of fish-farm waste on the biochemistry of sediments. These are possibly related to differences in the local physicochemical variables that could explain a significant proportion of the differences seen between the control and fish-farm locations. Biodeposition derived from the fish farms decreased with increasing distance from the fish-farm cages, but with different patterns in the four regions. Our results indicate that quantitative and qualitative changes in the organic loads of the sediments that arise from intensive aquaculture are dependent upon the ecological context and are not predictable only on the basis of fish-farm attributes and hydrodynamic regimes. Therefore, the siting of fish farms should only be allowed after a case-by-case assessment of the ecological context of the region, especially in terms of the organic matter load and its biochemical composition

Efficacy, Safety, and Timing of Anticoagulant Thromboprophylaxis for the Prevention of Venous Thromboembolism in Patients With Acute Spinal Cord Injury: A Systematic Review

Study Design: Systematic review. Objectives: The objective of this study was to answer 5 key questions: What is the comparative effectiveness and safety of (1a) anticoagulant thromboprophylaxis compared to no prophylaxis, placebo, or another anticoagulant strategy for preventing deep vein thrombosis (DVT) and pulmonary embolism (PE) after acute spinal cord injury (SCI)? (1b) Mechanical prophylaxis strategies alone or in combination with other strategies for preventing DVT and PE after acute SCI? (1c) Prophylactic inferior vena cava filter insertion alone or in combination with other strategies for preventing DVT and PE after acute SCI? (2) What is the optimal timing to initiate and/or discontinue anticoagulant, mechanical, and/or prophylactic inferior vena cava filter following acute SCI? (3) What is the cost-effectiveness of these treatment options? Methods: A systematic literature search was conducted to identify studies published through February 28, 2015. We sought randomized controlled trials evaluating efficacy and safety of antithrombotic strategies. Strength of evidence was evaluated using the Grading of Recommendations Assessment, Development and Evaluation (GRADE) system. Results: Nine studies satisfied inclusion criteria. We found a trend toward lower risk of DVT in patients treated with enoxaparin. There were no significant differences in rates of DVT, PE, bleeding, and mortality between patients treated with different types of low-molecular-weight heparin or between low-molecular-weight heparin and unfractionated heparin. Combined anticoagulant and mechanical prophylaxis initiated within 72 hours of SCI resulted in lower risk of DVT than treatment commenced after 72 hours of injury. Conclusion: Prophylactic treatments can be used to lower the risk of venous thromboembolic events in patients with acute SCI, without significant increase in risk of bleeding and mortality and should be initiated within 72 hours. © 2017, © The Author(s) 2017

Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ′\delta^\prime interaction

We determine and study the ground states of a focusing Schr\"odinger equation in dimension one with a power nonlinearity $|\psi|^{2\mu} \psi$ and a strong inhomogeneity represented by a singular point perturbation, the so-called (attractive) $\delta^\prime$ interaction, located at the origin. The time-dependent problem turns out to be globally well posed in the subcritical regime, and locally well posed in the supercritical and critical regime in the appropriate energy space. The set of the (nonlinear) ground states is completely determined. For any value of the nonlinearity power, it exhibits a symmetry breaking bifurcation structure as a function of the frequency (i.e., the nonlinear eigenvalue) $\omega$. More precisely, there exists a critical value \om^* of the nonlinear eigenvalue \om, such that: if \om_0 < \om < \om^*, then there is a single ground state and it is an odd function; if \om > \om^* then there exist two non-symmetric ground states. We prove that before bifurcation (i.e., for \om < \om^*) and for any subcritical power, every ground state is orbitally stable. After bifurcation (\om =\om^*+0), ground states are stable if $\mu$ does not exceed a value $\mu^\star$ that lies between 2 and 2.5, and become unstable for $\mu > \mu^*$. Finally, for $\mu > 2$ and \om \gg \om^*, all ground states are unstable. The branch of odd ground states for \om \om^*, obtaining a family of orbitally unstable stationary states. Existence of ground states is proved by variational techniques, and the stability properties of stationary states are investigated by means of the Grillakis-Shatah-Strauss framework, where some non standard techniques have to be used to establish the needed properties of linearization operators.Comment: 46 pages, 5 figure

729-1 Modulation of Cardiac Hypertrophy by Blockade of the Renin Angiotensin System: Effects on LVH Regression, Gene Expression, and Survival

The aim of the present study was to investigate the contribution of the renin angiotensin system to maintenance of pressure overload left ventricular hypertrophy (LVH). Rats with fixed ascending aortic stenosis were treated with either vehicle (VEH, n=36), hydralazine (HYD, 20mg/kg/d, n=35). ramipril (RAM, 10mg/kg/d, n=35), or losartan (LOS, 40mg/kg/d, n=16) during weeks 6–12 after banding. We have previously demonstrated that compared to sham (n=36), VEH and HYD rats were characterized by a 1.8–1.9-fold increase of left ventricular to body weight ratios (LV/BW). whereas those aortic stenosis rats treated with RAM or LOS displayed a blunted increase of LV/BW (14-fold; p&lt;0.05, each vs. HYD). We now extend these observations demonstrating that myocyte cross sectional widths were increased by 150% in VEH and HYD rats (p&lt;0.001, vs. sham), whereas ramipril and losartan treatment resulted in myocyte widths that were only mildly elevated (53% and 28%, respectively). Furthermore, VEH and HYD displayed a 14–15 fold increase of LV atrial natriuretic peptide (ANP) mRNA as well as a 44% decrease of sarcoplasmatic reticulum (SR) Ca2+-ATPase (p&lt;0.001, vs. sham). In contrast, alterations of ANP and SR-Ca2+-ATPase mRNA levels were significantly blunted by both RAM and LOS. The attenuation of LVH by RAM or LOS was not explained by blood pressure reduction that was similar in the HYD group. Finally, RAM and LOS decreased mortality (6 out of 51 animals; 11%)as compared to 20% in HYD and 31% in VEH groups (p&lt;0.05).In summary, blockade of the renin angiotensin system may promote regression of pressure overload LVH on the macro-, and microscopical, as well as the molecular level by mechanisms that are, in part, independent of hemodynamic drug effects. LVH regression may improve survival despite persisting pressure overload

Effects of hypoxia on renin secretion and renal renin gene expression

Effects of hypoxia on renin secretion and renal renin gene expression. Plasma renin activity (PRA) and renal renin mRNA levels were measured in male rats exposed to hypoxia (8% O2) or to carbon monoxide (CO; 0.1%) for six hours. PRA increased fourfold and 3.3-fold, and renin mRNA levels increased to 220% and 200% of control, respectively. In primary cultures of renal juxtaglomerular (JG) cells, hypoxia (lowering medium O2 from 20% to 3% or 1%) for 6 or 20hours did not affect renin secretion or gene expression. Renal denervation did not prevent stimulation of the renin system by hypoxia. Because norepinephrine increased 1.7-fold and 3.2-fold and plasma epinephrine increased 3.9-fold and 7.8-fold during hypoxia and CO inhalation, respectively, circulating catecholamines might mediate the stimulatory effects of hypoxia on renin secretion and renin gene expression. Stimulation of β-adrenergic receptors by continuous infusion of 160 μg/kg/hr isoproterenol increased PRA 17-fold and 20-fold after three and six hours, respectively, and renin mRNA by 130% after six hours. In rats with a stimulated renin system (low-sodium diet), isoproterenol did not stimulate PRA or renal renin mRNA further. In summary, both arterial and venous hypoxia can stimulate renin secretion and renin gene expression powerfully in vivo but not in vitro. These effects seem not to be mediated by renal nerves or by a direct effect on JG cells but might be mediated by circulating catecholamines