Non-uniqueness for the nonlocal Liouville equation in R\mathbb{R} and applications

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-\Delta)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) = 1+\varepsilon \kappa(x)$ with $\varepsilon \in (0,1)$ small and $\kappa \in C^{1,\alpha}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $\alpha > 0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $\gamma\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma\neq1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena

Mediterranean Forecasting System: forecast and analysis assessment through skill scores

Abstract. This paper describes the first evaluation of the quality of the forecast and analyses produced at the basin scale by the Mediterranean ocean Forecasting System (MFS) (http://gnoo.bo.ingv.it/mfs). The system produces short-term ocean forecasts for the following ten days. Analyses are produced weekly using a daily assimilation cycle. The analyses are compared with independent data from buoys, where available, and with the assimilated data before the data are inserted. In this work we have considered 53 ten days forecasts produced from 16 August 2005 to 15 August 2006. The forecast skill is evaluated by means of root mean square error (rmse) differences, bias and anomaly correlations at different depths for temperature and salinity, computing differences between forecast and analysis, analysis and persistence and forecast and persistence. The Skill Score (SS) is defined as the ratio of the rmse of the difference between analysis and forecast and the rmse of the difference between analysis and persistence. The SS shows that at 5 and 30 m the forecast is always better than the persistence, but at 300 m it can be worse than persistence for the first days of the forecast. This result may be related to flow adjustments introduced by the data assimilation scheme. The monthly variability of SS shows that when the system variability is high, the values of SS are higher, therefore the forecast has higher skill than persistence. We give evidence that the error growth in the surface layers is controlled by the atmospheric forcing inaccuracies, while at depth the forecast error can be interpreted as due to the data insertion procedure. The data, both in situ and satellite, are not homogeneously distributed in the basin; therefore, the quality of the analyses may be different in different areas of the basin

Mediterranean Forecasting System: forecast and analysis assessment through skill scores

This paper describes the first evaluation of the quality of the forecast and analyses produced at the basin scale by the Mediterranean ocean Forecasting System (MFS) (http://gnoo.bo.ingv.it/mfs). The system produces short-term ocean forecasts for the following ten days. Analyses are produced weekly using a daily assimilation cycle. The analyses are compared with independent data from buoys, where available, and with the assimilated data before the data are inserted. In this work we have considered 53 ten days forecasts produced from 16 August 2005 to 15 August 2006. The forecast skill is evaluated by means of root mean square error (rmse) differences, bias and anomaly correlations at different depths for temperature and salinity, computing differences between forecast and analysis, analysis and persistence and forecast and persistence. The Skill Score (SS) is defined as the ratio of the rmse of the difference between analysis and forecast and the rmse of the difference between analysis and persistence. The SS shows that at 5 and 30m the forecast is always better than the persistence, but at 300m it can be worse than persistence for the first days of the forecast. This result may be related to flow adjustments introduced by the data assimilation scheme. The monthly variability of SS shows that when the system variability is high, the values of SS are higher, therefore the forecast has higher skill than persistence. We give evidence that the error growth in the surface layers is controlled by the atmospheric forcing inaccuracies, while at depth the forecast error can be interpreted as due to the data insertion procedure. The data, both in situ and satellite, are not homogeneously distributed in the basin; therefore, the quality of the analyses may be different in different areas of the basin

The Ljapunov-Schmidt reduction for some critical problems

This is a survey about the application of the Ljapunov-Schmidt reduction for some critical problems

Is There a Relationship between Laser Therapy and Root Canal Cracks Formation? A Systematic Review

Introduction: Crack formation has become an important issue for endodontists, as it can be decisive for the long-term prognosis of the endodontically treated tooth. Since the applicability of laser in endodontics has become frequent, this systematic review aimed to evaluate the association between laser therapy and the formation of cracks in the dentinal structure of the root canal. Materials and Methods: A search was performed in PubMed, Scopus, Web of Science, and Virtual Health Library, as well as in the gray literature, on September 24, 2021. Studies that evaluated the formation of cracks in human root dentin due to different types of lasers were included. The risk of bias was assessed following the modified version of the Consolidated Standards of Reporting Trials (CONSORT) checklist tool. A meta-analysis was performed to evaluate (i) the total number of crack incidences; (ii) complete crack formation; (iii) incomplete crack formation; (iv) intra-dentinal crack formation between ultrasonic tips and laser use. The mean difference was calculated with a 95% confidence interval in a fixed-effect model, the heterogeneity was tested using the I2 index with level of significance of 5%. Results: Of the 22 studies included in this review, 15 have shown that lasers can form cracks in root dentin, including those that performed baseline assessment of samples. The meta-analysis confirmed no difference in crack formation between ultrasonic tips and laser devices. Conclusions: Laser therapy has been gaining prominence in endodontics and that irradiation can form and propagate cracks in the dentinal structure of the root canal assessed by in vitro studies. This is a critical concern for endodontists as it affects the strength and longevity of the tooth. Future research is encouraged to seek the standardization of good methodological practices and achieve establishing parameters to minimize harmful effects of laser on dentin

Responsibility Analysis by Abstract Interpretation

Given a behavior of interest in the program, statically determining the corresponding responsible entity is a task of critical importance, especially in program security. Classical static analysis techniques (e.g. dependency analysis, taint analysis, slicing, etc.) assist programmers in narrowing down the scope of responsibility, but none of them can explicitly identify the responsible entity. Meanwhile, the causality analysis is generally not pertinent for analyzing programs, and the structural equations model (SEM) of actual causality misses some information inherent in programs, making its analysis on programs imprecise. In this paper, a novel definition of responsibility based on the abstraction of event trace semantics is proposed, which can be applied in program security and other scientific fields. Briefly speaking, an entity ER is responsible for behavior B, if and only if ER is free to choose its input value, and such a choice is the first one that ensures the occurrence of B in the forthcoming execution. Compared to current analysis methods, the responsibility analysis is more precise. In addition, our definition of responsibility takes into account the cognizance of the observer, which, to the best of our knowledge, is a new innovative idea in program analysis.Comment: This is the extended version (33 pages) of a paper to be appeared in the Static Analysis Symposium (SAS) 201