Generalized and weighted Strichartz estimates

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schr\"odinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied Analysis. 33 pages. 2 more references adde

On the supercritical KDV equation with time-oscillating nonlinearity

For the initial value problem (IVP) associated to the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, \begin{equation*} u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, \end{equation*} numerical evidence [Bona J.L., Dougalis V.A., Karakashian O.A., McKinney W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 351, 107–164 (1995) ] shows that, there are initial data $\phi\in H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [Abdullaev F.K., Caputo J.G., Kraenkel R.A., Malomed B.A.: Controlling collapse in Bose–Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 012605 (2003) and Konotop V.V., Pacciani P.: Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005) ], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0, \end{equation*} where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large.M. P. was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and through the project PTDC/MAT/109844/2009, and M. S. was partially supported by FAPESP Brazil

A copula model for marked point processes

The final publication (Diao, Liqun, Richard J. Cook, and Ker-Ai Lee. (2013) A copula model for marked point processes. Lifetime Data Analysis, 19(4): 463-489) is available at Springer via http://dx.doi.org/10.1007/s10985-013-9259-3Many chronic diseases feature recurring clinically important events. In addition, however, there often exists a random variable which is realized upon the occurrence of each event reflecting the severity of the event, a cost associated with it, or possibly a short term response indicating the effect of a therapeutic intervention. We describe a novel model for a marked point process which incorporates a dependence between continuous marks and the event process through the use of a copula function. The copula formulation ensures that event times can be modeled by any intensity function for point processes, and any multivariate model can be specified for the continuous marks. The relative efficiency of joint versus separate analyses of the event times and the marks is examined through simulation under random censoring. An application to data from a recent trial in transfusion medicine is given for illustration.Natural Sciences and Engineering Research Council of Canada (RGPIN 155849); Canadian Institutes for Health Research (FRN 13887); Canada Research Chair (Tier 1) – CIHR funded (950-226626

Combining genetic resources and elite material populations to improve the accuracy of genomic prediction in apple

Genomic selection is an attractive strategy for apple breeding that could reduce the length of breeding cycles. A possible limitation to the practical implementation of this approach lies in the creation of a training set large and diverse enough to ensure accurate predictions. In this study, we investigated the potential of combining two available populations, i.e., genetic resources and elite material, in order to obtain a large training set with a high genetic diversity. We compared the predictive ability of genomic predictions within-population, across-population or when combining both populations, and tested a model accounting for population-specific marker effects in this last case. The obtained predictive abilities were moderate to high according to the studied trait and small increases in predictive ability could be obtained for some traits when the two populations were combined into a unique training set. We also investigated the potential of such a training set to predict hybrids resulting from crosses between the two populations, with a focus on the method to design the training set and the best proportion of each population to optimize predictions. The measured predictive abilities were very similar for all the proportions, except for the extreme cases where only one of the two populations was used in the training set, in which case predictive abilities could be lower than when using both populations. Using an optimization algorithm to choose the genotypes in the training set also led to higher predictive abilities than when the genotypes were chosen at random. Our results provide guidelines to initiate breeding programs that use genomic selection when the implementation of the training set is a limitation

Recent progress in understanding and projecting regional and global mean sea-level change

Considerable progress has been made in understanding the present and future regional and global sea level in the 2 years since the publication of the Fifth Assessment Report (AR5) of the Intergovernmental Panel on Climate Change. Here, we evaluate how the new results affect the AR5’s assessment of (i) historical sea level rise, including attribution of that rise and implications for the sea level budget, (ii) projections of the components and of total global mean sea level (GMSL), and (iii) projections of regional variability and emergence of the anthropogenic signal. In each of these cases, new work largely provides additional evidence in support of the AR5 assessment, providing greater confidence in those findings. Recent analyses confirm the twentieth century sea level rise, with some analyses showing a slightly smaller rate before 1990 and some a slightly larger value than reported in the AR5. There is now more evidence of an acceleration in the rate of rise. Ongoing ocean heat uptake and associated thermal expansion have continued since 2000, and are consistent with ocean thermal expansion reported in the AR5. A significant amount of heat is being stored deeper in the water column, with a larger rate of heat uptake since 2000 compared to the previous decades and with the largest storage in the Southern Ocean. The first formal detection studies for ocean thermal expansion and glacier mass loss since the AR5 have confirmed the AR5 finding of a significant anthropogenic contribution to sea level rise over the last 50 years. New projections of glacier loss from two regions suggest smaller contributions to GMSL rise from these regions than in studies assessed by the AR5; additional regional studies are required to further assess whether there are broader implications of these results. Mass loss from the Greenland Ice Sheet, primarily as a result of increased surface melting, and from the Antarctic Ice Sheet, primarily as a result of increased ice discharge, has accelerated. The largest estimates of acceleration in mass loss from the two ice sheets for 2003–2013 equal or exceed the acceleration of GMSL rise calculated from the satellite altimeter sea level record over the longer period of 1993–2014. However, when increased mass gain in land water storage and parts of East Antarctica, and decreased mass loss from glaciers in Alaska and some other regions are taken into account, the net acceleration in the ocean mass gain is consistent with the satellite altimeter record. New studies suggest that a marine ice sheet instability (MISI) may have been initiated in parts of the West Antarctic Ice Sheet (WAIS), but that it will affect only a limited number of ice streams in the twenty-first century. New projections of mass loss from the Greenland and Antarctic Ice Sheets by 2100, including a contribution from parts of WAIS undergoing unstable retreat, suggest a contribution that falls largely within the likely range (i.e., two thirds probability) of the AR5. These new results increase confidence in the AR5 likely range, indicating that there is a greater probability that sea level rise by 2100 will lie in this range with a corresponding decrease in the likelihood of an additional contribution of several tens of centimeters above the likely range. In view of the comparatively limited state of knowledge and understanding of rapid ice sheet dynamics, we continue to think that it is not yet possible to make reliable quantitative estimates of future GMSL rise outside the likely range. Projections of twenty-first century GMSL rise published since the AR5 depend on results from expert elicitation, but we have low confidence in conclusions based on these approaches. New work on regional projections and emergence of the anthropogenic signal suggests that the two commonly predicted features of future regional sea level change (the increasing tilt across the Antarctic Circumpolar Current and the dipole in the North Atlantic) are related to regional changes in wind stress and surface heat flux. Moreover, it is expected that sea level change in response to anthropogenic forcing, particularly in regions of relatively low unforced variability such as the low-latitude Atlantic, will be detectable over most of the ocean by 2040. The east-west contrast of sea level trends in the Pacific observed since the early 1990s cannot be satisfactorily accounted for by climate models, nor yet definitively attributed either to unforced variability or forced climate change

A century of sea level measurements at Newlyn, SW England

The Newlyn Tidal Observatory is the most important sea level station in the UK. It commenced operations in 1915 as part of the Second Geodetic Levelling of England and Wales, and the mean sea level determined from the tide gauge during the first six years (May 1915-April 1921) defined Ordnance Datum Newlyn (ODN) which became the national height datum for the whole of Great Britain. The 100 years of sea level data now available have contributed significantly to many studies in oceanography, geology and climate change. This paper marks the centenary of this important station by reviewing the sea level (and, more recently, detailed land level) measurements and Newlyn’s contributions to UK cartography, geodesy and sea-level science in general. Recommendations are made on how sea and land level measurements at Newlyn might be enhanced in the future

Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Conversely, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disk composed of radial trajectories connecting a saddle equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and synchronized equilibria. We prove in particular non-linear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero.Comment: revised version, 28 pages, 4 figure

Continental mass change from GRACE over 2002-2011 and its impact on sea level

Present-day continental mass variation as observed by space gravimetry reveals secular mass decline and accumulation. Whereas the former contributes to sea-level rise, the latter results in sea-level fall. As such, consideration of mass accumulation (rather than focussing solely on mass loss) is important for reliable overall estimates of sea-level change. Using data from the Gravity Recovery And Climate Experiment satellite mission, we quantify mass-change trends in 19 continental areas that exhibit a dominant signal. The integrated mass change within these regions is representative of the variation over the whole land areas. During the integer 9-year period of May 2002 to April 2011, GIA-adjusted mass gain and mass loss in these areas contributed, on average, to −(0.7 ± 0.4) mm/year of sea-level fall and + (1.8 ± 0.2) mm/year of sea-level rise; the net effect was + (1.1 ± 0.6) mm/year. Ice melting over Greenland, Iceland, Svalbard, the Canadian Arctic archipelago, Antarctica, Alaska and Patagonia was responsible for + (1.4±0.2) mm/year of the total balance. Hence, land-water mass accumulation compensated about 20 % of the impact of ice-melt water influx to the oceans. In order to assess the impact of geocentre motion, we converted geocentre coordinates derived from satellite laser ranging (SLR) to degree-one geopotential coefficients. We found geocentre motion to introduce small biases to mass-change and sea-level change estimates; its overall effect is + (0.1 ± 0.1) mm/year. This value, however, should be taken with care owing to questionable reliability of secular trends in SLR-derived geocentre coordinates