Celebrating Cercignani's conjecture for the Boltzmann equation

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani, powerful mind and great scientist, one of the founders of the modern theory of the Boltzmann equation. 24 pages. V2: correction of some typos and one ref. adde

Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coeffcients depending on the probability ux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study ....Preprin

Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coeffcients depending on the probability ux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study ...

Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coeffcients depending on the probability ux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study ...

Metrics of Gender Differences in Mortality Risk after Diabetic Foot Disease

Background: The aim of this study was to clarify any gender differences in the mortality risk of people with DFD since patients with diabetic foot disease (DFD) are at a high risk of mortality and, at the same time, are more likely to be men. Methods: From regional administrative sources, the survival probability was retrospectively evaluated by the Kaplan-Meier method and using the Cox proportional-hazards model comparing people with DFD to those without DFD across the years 2011-2018 in Tuscany, Italy. Gender difference in mortality was evaluated by the ratio of hazard ratios (RHR) of men to women after initial DFD hospitalizations (n = 11,529) or in a cohort with prior history of DFD hospitalizations (n = 11,246). Results: In both cohorts, the survival probability after DFD was lower among women. Compared to those without DFD, after initial DFD hospitalizations, the mortality risk was significantly (18%) higher for men compared to women. This excess risk was particularly high after major amputations but also after ulcers, infections, gangrene, or Charcot, with a lower reduction after revascularization procedures among men. In the cohort that included people with a history of prior DFD hospitalizations, except for the risk of minor amputations being higher for men, there was no gender difference in mortality risk. Conclusions: In people with DFD, the overall survival probability was lower among women. Compared to those without DFD after a first DFD hospitalization, men were at higher risk of mortality. This excess risk disappeared in groups with a history of previous DFD hospitalizations containing a greater percentage of women who were older and probably had a longer duration of diabetes and thus becoming, over time, progressively frailer than men

Increasing prevalence of multiple sclerosis in Tuscany, Italy

Background and rationale: An increase of prevalence and incidence of multiple sclerosis (MS) has been reported in several countries, especially taking into account a long-term evaluation. This increasing trend often reflects improved case identification and ascertainment due to the refinement of diagnostic criteria. The aim of this study was to update the prevalence rate of MS in Tuscany (central Italy) as of 2017, and to assess if there has been an increasing trend of prevalence in this Region considering a short period of analysis, from 2014 to 2017. Methods: To capture prevalent cases, a case-finding algorithm based on administrative data, previously created and validated, was used. As data sources, we considered hospital discharge records, drug-dispensing records, disease-specific exemptions from copayment to health care, home and residential long-term care, and inhabitant registry. Results: As of January 1, 2017, 7809 cases were identified, of which 69.4% were females and 30.6% were males. Considering temporal variation, an increasing trend was observed, with standardized rates rising from 189.2 in 2014 to 208.7 per 100,000 in 2017. Conclusions: Results confirm that prevalence increases every year, probably mainly due to the difference between incidence and mortality, resulting in an increasing trend. Moreover, administrative data may accurately identify MS patients in a routinary way and monitor this cohort along disease care pathways