From probability to sequences and back

This is a survey covering sequential structures and their applications to the foundations of probability theory. Sequential convergence, convergence groups and the extension of sequentially continuous maps belong to general topology and Trieste for long has been a center of sequential topology. We begin with some personal reflections, con- tinue with topological problems motivated by the extension of probability measures, and close with some recent results related to the categorical foundations of probability theory

Fuzzification of crisp domains

summary:The present paper is devoted to the transition from crisp domains of probability to fuzzy domains of probability. First, we start with a simple transportation problem and present its solution. The solution has a probabilistic interpretation and it illustrates the transition from classical random variables to fuzzy random variables in the sense of Gudder and Bugajski. Second, we analyse the process of fuzzification of classical crisp domains of probability within the category $ID$ of $D$-posets of fuzzy sets and put into perspective our earlier results concerning categorical aspects of fuzzification. For example, we show that (within $ID$) all nontrivial probability measures have genuine fuzzy quality and we extend the corresponding fuzzification functor to an epireflector. Third, we extend the results to simplex-valued probability domains. In particular, we describe the transition from crisp simplex-valued domains to fuzzy simplex-valued domains via a “simplex” modification of the fuzzification functor. Both, the fuzzy probability and the simplex-valued fuzzy probability is in a sense minimal extension of the corresponding crisp probability theory which covers some quantum phenomenon

Hepcidin and ferritin levels as markers of immune cell activation during septic shock, severe COVID-19 and sterile inflammation

IntroductionMajor clinically relevant inflammatory events such as septic shock and severe COVID-19 trigger dynamic changes in the host immune system, presenting promising candidates for new biomarkers to improve precision diagnostics and patient stratification. Hepcidin, a master regulator of iron metabolism, has been intensively studied in many pathologies associated with immune system activation, however these data have never been compared to other clinical settings. Thus, we aimed to reveal the dynamics of iron regulation in various clinical settings and to determine the suitability of hepcidin and/or ferritin levels as biomarkers of inflammatory disease severity.CohortsTo investigate the overall predictive ability of hepcidin and ferritin, we enrolled the patients suffering with three different diagnoses – in detail 40 patients with COVID-19, 29 patients in septic shock and eight orthopedic patients who were compared to nine healthy donors and all cohorts to each other.ResultsWe showed that increased hepcidin levels reflect overall immune cell activation driven by intrinsic stimuli, without requiring direct involvement of infection vectors. Contrary to hepcidin, ferritin levels were more strongly boosted by pathogen-induced inflammation – in septic shock more than four-fold and in COVID-19 six-fold in comparison to sterile inflammation. We also defined the predictive capacity of hepcidin-to-ferritin ratio with AUC=0.79 and P = 0.03.DiscussionOur findings confirm that hepcidin is a potent marker of septic shock and other acute inflammation-associated pathologies and demonstrate the utility of the hepcidin-to-ferritin ratio as a predictor of mortality in septic shock, but not in COVID-19

Upgrading Probability via Fractions of Events

summary:The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for“ an upgrade: (i)~classical random events are black-and-white (Boolean); (ii)~classical random variables do not model quantum phenomena; (iii)~basic maps (probability measures and observables – dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the $\{0,1\}$-valued indicator functions of sets) into upgraded random events (represented by measurable $[0,1]$-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable