New integral representations of nth order convex functions

AbstractIn this paper we give an integral representation of an n-convex function f in general case without additional assumptions on function f. We prove that any n-convex function can be represented as a sum of two (n+1)-times monotone functions and a polynomial of degree at most n. We obtain a decomposition of n-Wright-convex functions which generalizes and complements results of Maksa and Páles (2009) [13]. We define and study relative n-convexity of n-convex functions. We introduce a measure of n-convexity of f. We give a characterization of relative n-convexity in terms of this measure, as well as in terms of nth order distributional derivatives and Radon–Nikodym derivatives. We define, study and give a characterization of strong n-convexity of an n-convex function f in terms of its derivative f(n+1)(x) (which exists a.e.) without additional assumptions on differentiability of f. We prove that for any two n-convex functions f and g, such that f is n-convex with respect to g, the function g is the support for the function f in the sense introduced by Wąsowicz (2007) [29], up to polynomial of degree at most n

Resumption of studies on pollen deposition in the III Campus of the Jagiellonian University in Krakow

Abstract. Collected in 2016 samples from the surface and from the soil profiles, contained enough amounts of sporomorphs to have started studies on pollen deposition near the III Campus of the Jagiellonian University in Krakow. Preliminary results of the first year of research are presented (percentage pollen diagram and pollen accumulation rate) and compared with chosen results from previous years of research

On the classes of higher-order Jensen-convex functions and Wright-convex functions

The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number $n$ the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed

On the Raşa Inequality for Higher Order Convex Functions

We study the following (q−1)th convex ordering relation for qth convolution power of the difference of probability distributions μ and ν (ν−μ)∗q≥(q−1)cx0,q≥2, and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020)

Validation of the Polish version of WHO-5 as a screening instrument for depression in adults with diabetes

Aims The aim of this study was to validate and report the factorial analysis of the World Health Organization’s 5-item Well-being Index (WHO-5) among outpatients with type 2 diabetes. We investigated the psychometric properties of the WHO-5 and its suitability for identifying potential depressive symptoms in Polish adults with diabetes. Methods Participants were randomly chosen among Polish diabetes outpatients and invited to participate in the cross-sectional study (N = 216). Participants completed the Polish version of the WHO-5, Problem Areas in Diabetes Scale and Patient Health Questionnaire. Results Factor analyses identified the one-factor structure of the Polish version of the WHO-5. The internal consistency of the Polish version of the WHO-5 is satisfying. With regard to convergent validity, there were significant negative associations between the WHO-5 and PAID, the PHQ-9, HbA1c and the amount of medical complications. The AUC indicates that the WHO-5 is an effective measure for identifying depressive symptoms. The optimal cut off values of ≤12 yielded the best sensitivity/specificity trade-off for identifying depression among people with diabetes. Conclusions The Polish version of the WHO-5 is a reliable, valid outcome measure for outpatients with type 2 diabetes and can be a useful instrument for screening for depression in people with diabetes

Transformations of Religiosity during the SARS-CoV-2 Pandemic-On the Example of Catholic Religious Practices of Polish Students

This paper attempts to identify the changes in religiosity among Catholic practitioners in Poland that occurred during the COVID-19 pandemic, based on the changes in the realization of religious practices by students (N = 354). A questionnaire prepared by the authors was used for the study. We sought to answer the question of the extent to which the socio-demographic characteristics of the subjects, a history of COVID-19, and the death of a family member due to SARS-CoV-2 infection differentiated the subjects’ opinions about religiosity and their undertaking of religious practices. The paper also addresses the issue of media-facilitated religious practices. The study showed that the frequency of religious practices was influenced by the perceived religiosity of the family, the religiosity of the respondent, and the declaration concerning the belief in God. The same factors most often significantly differentiated respondents’ opinions on COVID-19. No relationship was confirmed between having COVID-19 and subjects’ beliefs about the pandemic or frequency of religious practice. Among the respondents, the pandemic did not intensify the practice of religion. In the face of danger, respondents did not turn to God; there was no revitalization of religion. The situation of limiting the physical experience communion of the Church was treated not as an acute undersupply but as an opportunity or a pretext to abandon the practice of religion