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Following the pandemic scenario, researchers from all over the world, including Italians, have undertaken fervent research activity using the epidemiological data available on the sites of government and national and international research institutes. The objectives of our study were: (1) to analyze the load and trend of the COVID-19 pandemic in Italy, from the beginning to October 2021; (2) to analyze vaccination coverage by age groups and types of vaccines administered and check how the vaccination campaign has influenced the course of the disease and deaths; (3) to evaluate the Italian situation in the European context, comparing the incidence and mortality of Italy with respect to European countries; (4) finally, to evaluate how much vaccination coverage may have had an effect on mortality in the various European countries. Databases were structured to archive Italian and European COVID-19 data provided by Our World in Data, and data came from the Ministry of Health, to evaluate percentage of vaccines administered. The monthly trend of the cumulative incidence per 100,000 inhabitants in the period January 2020–October 2021 was evaluated. It is important to underline 3 peaks of incidence and mortality rates that occurred during the three waves of COVID-19: March–April–May 2020, October–November–December 2020, and March–April–May 2021. There is a slight increase in incidence in August 2021 and in mortality in September 2021. The three mortality rate peaks, related to the three waves of COVID-19, are always higher in Italy than in Europe, particularly in April 2020, December 2020, and March 2021. From May 2021 to October 2021, the mortality trend reversed, and it turned out to be higher in Europe than in Italy. Regarding vaccination, Italy currently has an important coverage, not only in the most fragile population, where it exceeds 90%, but also in the 12–19 age group, with percentages above 65%. The Pfizer/BioNTech vaccine was used widely in all age groups (first and second dose), with highest administration in 12–19 age groups and 80+, while the lowest was recorded in the 70–79 age group. In conclusion, these data confirm the importance of vaccination in the management of the COVID-19 pandemic.

This paper concerns the folklore statement that ``entropy is a lower bound for compression''. More precisely we derive from the entropy theorem a simple proof of a pointwise inequality firstly stated by Ornstein and Shields and which is the almost-sure version of an average inequality firstly stated by Khinchin in 1953. We further give an elementary proof of original Khinchin inequality that can be used as an exercise for Information Theory students and we conclude by giving historical and technical notes of such inequality., Compared to version 1, in version 2 we added a simpler proof than the one given by Shields of a more general theorem (Theorem 4, pg. 7) presented by Ornstein and Shields. Consequently we also modified the title of the paper. In version 3 we have reordered the sections of the paper, simplified the proof of Theorem 4 (now Theorem 3) and significantly reduced the proof of Theorem 3 (now Theorem 4)

We performed an updated meta-analysis to robustly quantify admission trends of patients with ST-segment elevation MI (STEMI) and non-ST-segment elevation MI (NSTEMI) during the first wave of the pandemic and to characterize on a large basis the risk profile and early prognosis. Studies having the same observation period for the comparison between SARS-CoV-2 outbreak in 2020 versus control period in 2019 were included. Primary endpoints were the relative variation of hospital admissions, the difference of in-hospital mortality for STEMI and NSTEMI. Secondary were: mortality according to countries, income levels and data quality; cardiogenic shock, mechanical complications, door-to-balloon time, time from symptom onset to first medical contact, left ventricular ejection fraction (LVEF) and troponin. In total, 61 observational studies with 125,346 patients were included. Compared with 2019, during the pandemic for STEMI were observed: a 24% reduction of hospitalizations with an impact on early survival (OR = 1.33 in-hospital mortality); the time from symptom onset to first medical contact was 91.31 min longer, whereas door-to-balloon time was increased (+5.44 min); after STEMI, the rate of cardiogenic shock was 33% higher; LVEF at discharge was decreased (−3.46); elevated high-sensitivity troponin levels (1.52) on admission. For NSTEMI, in the COVID-19 period, we observed a 31% reduction of hospitalizations and higher in-hospital deaths (OR = 1.34). The highest mortality rates among countries were: Italy OR = 3.71 (high income), Serbia OR = 2.15 (upper middle) and Pakistan OR = 1.69 (lower middle). Later hospital presentation was associated with larger infarctions, as well as with increased cardiogenic shock and in-hospital mortality.

It is well known from a theoretical point of view that LZ78 have an asymptotic convergence to the entropy faster than LZ77. A faster rate of convergence to the theoretical compression limit should lead to a better compression ratio. In effect, early LZ78-like and LZ77-like compressors behave accordingly to the theory. On the contrary, it seems that most of the recent commercial LZ77-like compressors perform better than the other ones. Probably this is due to a strategy of optimal parsing, which is used to factorize the text and can be applied to both LZ77 and LZ78 cases, as recent results suggest. To our best knowledge there are no theoretical results concerning the rate of convergence to the entropy of both LZ77-like and LZ78-like case when a strategy of optimal parsing is used. In this paper we investigate how an optimal parsing affect the rate of convergence to the entropy of LZ78-like compressors. We discuss some experimental results on LZ78-like compressors and we consider the ratio between the speed of convergence to the entropy of a compressor with optimal parsing and the speed of convergence to the entropy of a classical LZ78-like compressor. This ratio presents a kind of wave effect that become bigger and bigger as the entropy of the memoryless source decreases but it seems always to slowly converge to one. These results suggest that for non-zero entropy sources the optimal parsing does not improve the speed of convergence to the entropy in the case of LZ78-like compressors.