In this article, our main purpose is to establish a new extension of Mittag-Leffler function by using the known extended beta function $\mathbf{B}_{\omega}(a,b;p)$ introduced in [1]. It led to a novel extension of the applicability of Mittag-Leffler function that introduced them as distributions defined for a specific set of functions. We also, investigate some of its important properties, namely recursion relation, Mellin transform and differential formulas.

In this study, the complex interactions between the well-known Mittag-Leffler functions with two, three, and four parameters are explored. We first study the integral forms of the Mittag-Leffler function $\mathsf{E}_{\alpha, \beta}^{\rho,\kappa}(z)$ with the goal of clarifying its mathematical behaviours and features. Moreover, we create relations between these functions and generalised hypergeometric functions and Fox-Wright functions, expanding the range of their use and comprehension. We explore a number of unique situations by in-depth examination, providing insight into the subtle characteristics of these functions. This investigation advances our theoretical knowledge and opens up possibilities for future applications in a number of scientific fields. As such, this study adds to the current conversation in mathematical analysis and is an important tool for both practitioners and researchers.

In the paper, the authors generalize the notion "k-Mittag-Leffler function'', establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function.

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function.

The aim of this article is to establish a new class of unified integrals associated with the generalized Lommel-Wright functions, which are expressed in terms of Wright Hypergeometric function.Some integrals involving trigonometric,generalized Bessel function and Struve functions are also indicated as special cases of our main results.further, with the help of our main results and their special cases, we obtain two reduction formulas for the Wright hypergeometric function.

This paper is devoted to the study of multi-dimensional integral transform with Fox H-function in the kernel in weighted spaces integrable functions in the domain R+n with positive coordinates. Mapping properties such as the boundedness, the range, the representations of the considered transformation are established.

The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor . To date, the captured flow friction factor can be extracted from the logarithmic form analytically only in the term of the Lambert -function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert -function also known as the Wright -function. The Wright -function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term of the Lambert -function to the series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert -function is identical to the original expression in term of accuracy, a further evaluation of the Lambert -function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contains only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with the relative error of no more than 0.0096%. The presented approximations are in the form suitable for everyday engineering use, they are both accurate and computationally efficient.

The main aim of this paper is to establish two generalized integral formulas involving the extended Mittag–Leffler function based on the well known Lavoie and Trottier integral formula and the obtain results are express in term of extended Wright-type function. Also, we establish certain special cases of our main result.

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized fractional differential and integral operators is demonstrated and discussed.

The red fox, found on all continents except Antarctica, occupies diverse habitats. In Croatia it is an indigenous wild species with a population density of 0.7 animals per square kilometer. While tumors in wild animals from the Canidae family are scarce, the true prevalence and diversity of tumors are likely underestimated due to limited research. So far, a limited number of tumors were observed among the red fox population, either in their natural habitat or in captivity. As part of the National rabies control program, we examined 1,890 red fox carcasses over a four-year period. Our focus was on identifying abnormalities on the skin and internal organs that suggest potential neoplastic proliferation. Five red foxes, three males and two females, were found to have growths resembling potential tumors. Their age distribution spanned from 2 to 7 years. Microscopic investigation revealed two collagenous hamartomas, two Meibomian gland adenomas and one intra-abdominal teratoma within a cryptorchid testis. This retrospective study aims to provide a comprehensive description of tumor and tumor-like lesions observed in free-range red foxes from Croatia, marking the first research of its kind in Croatia.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

The aim of this study was to examine whether H2O2, generated in such beverages as tea and coffee, is formed also in infusions of medicinal herbs and propose the possible mechanism of its formation. Infusions of sixteen herbs were studied and the presence of H2O2 was found in all cases. The highest concentrations of H2O2, exceeding 50 µM, were found in the infusions of leaves of Betula and Polygonum, and of the inflorescence of Tilia. These low amounts of H2O2 should not have a harmful effect and may even exert beneficial action, perhaps contributing to the health effects of the herbs. Generation of semiquinone and superoxide radicals was detected in herbal infusions, indicating that oxidation of phenolic compounds is the main source of H2O2 in these infusions. Herbal infusions were cytotoxic to human ovary cancer SKOV-3 and PEO1 cells; this cytotoxicity was compromised by catalase added to the growth medium, demonstrating the contribution of hydrogen peroxide to the cytotoxic action of herbal extracts.

A deep review of the existing literature on the culpeo ecology is carried out, using scientific articles, book chapters and web resources. For information published before 1988, the synthesis made by a previous report was used. For subsequent information, bibliographic searches were carried out through the main servers, considering all of the generic names used to define the species so far. From this update, new general patterns on ecology, behavior and conservation concerns about culpeos are described. Gaps in current knowledge have been identified and new lines of research are proposed.Most of the studies focused on diet, conflicts with the species in livestock areas, and on the use of space and habitat. We found an incomplete, poor justification for all of the proposed subspecies and their supposed geographical distribution, as well as a scarcity of studies on genetic issues, population dynamics and conservation concerns. It is remarkable that vast regions in South America holding culpeos lacked basic information on the species.Diet studies describe a marked trend towards resources selection at the local level, which supports the view of the culpeo as a facultative trophic specialist. In addition, it has been confirmed that in the high Andes the culpeo is also a top predator that may regulate carnivorous communities, as well as that in arid environments culpeos can act as important seed dispersers. The assessment of the conservation status of the species differs among regions, although there is no sufficient information to reach clear conclusions in most cases. Even so, in Ecuador and Colombia the species has been listed as ‘Vulnerable’. Direct persecution and habitat alteration are considered to be the most important threats that the species is facing in many countries, although other risk factors such as climate change could also have serious consequences for the canid at the global scale.

The dramatic outbreak of the coronavirus disease 2019 (COVID-19) pandemics and its ongoing progression boosted the scientific community's interest in epidemic modelling and forecasting. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive explicit solution. The present work demonstrates that this is a non-trivial task. Notably, it is proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega function. The present manuscript reports new analytical results and numerical routines suitable for parametric estimation of the SIR model. The manuscript introduces iterative algorithms approximating the incidence variable, which allows for estimation of the model parameters from the numbers of observed cases. The numerical approach is exemplified with data from the European Centre for Disease Prevention and Control (ECDC) for several European countries in the period Jan 2020 -- Jun 2020.

In the last decade, an upsurge of human leishmaniasis has been reported in the Emilia-Romagna region, Northeast Italy. Epidemiologic data raised doubts about the role of dogs as the main reservoirs for Leishmania infantum. In the present study, a total of 1,077 wild animals were screened for L. infantum DNA in earlobe and spleen samples from 2019 to 2022. The lymph nodes were tested only in the 23 animals already positive in the earlobe and/or spleen. A total of 71 (6.6%) animals resulted positive in at least one of the sampled tissues, including 3/18 (16.7%) wolves, 6/39 (15.4%) European hares, 38/309 (12.3%) roe deer, 1/11 (9.1%) red deer, 8 (4.9%) wild boars, 13/319 (4.1%) red foxes, 1/54 (1.9%) porcupine, and 1/59 (1.7%) European badger. Most of the infected animals (62/71) tested positive only in the earlobe tissue, just 4 animals (2 roe deer and 2 wild boars) tested positive only in the spleen, and 5 animals (3 roe deer and 2 red foxes) resulted positive for both tissues. L. infantum DNA was detected in the lymph nodes of 6/23 ani-mals. L. infantum detection occurred in all seasons associated with low real-time PCR Ct values. Further research is needed in order to clarify the role of wildlife in the re-emerging focus of leishmaniasis in Northeast Italy.

Flying-fox populations are increasingly threatened by heat events, starvation events and other stressors due to habitat clearing and human/flying-fox conflict.These factors are unlikely to resolve, meaning that a well-coordinated and timely approach to flying-fox disasters is imperative for the mitigation of further flying-fox population impacts.

Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright ω-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in “wrightOmega” feature of Matlab R2016a, the herein introduced related approximations of the Wright ω-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers’ use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.

This research addresses the strategic issue of why key decision-makers fail to foresee potential extreme ‘black swan’ events. Following a review of the literature, a conceptual framework is developed that identifies two types of organisational blindness that are reflected in Tetlock’s hedgehog cognitive thinking style, being the oversimplification of uncertainty (e.g., inductive biases) and an unquestioned, top-down, reference narrative. This framework is tested using a case study approach and qualitative analysis of secondary data sources available from the Royal Commission of Inquiry and other published reports following the 2010 methane explosion at the Pike River Coal Ltd’s mine (Pike) in New Zealand, that killed 29 miners and caused the loss of all funds invested. The results indicate that the combined effect of both blindness’s meant that Pike’s collective intelligence was limited, and for the three key decision-makers at the Pike River mine, some type of extreme ‘black swan’ event was apparently inevitable. This research provides theoretical and practical contributions to the analysis of business and public policy decision-making under uncertainty.

In this paper, we study nonlocal problems for a fractional diffusion equation and a degenerate hyperbolic equation with singular coefficients in the lower terms. The uniqueness of the solution to the problem is proved by the method of energy integrals. The existence of a solution is equivalently reduced to the question of the solvability of Volterra integral equations of the second kind and a fractional differential equation. For a particular solution of the proposed problem, its visualization is carried out for various values of the order of the fractional derivative. It is shown that the order of the derivative affects the intensity of the diffusion process (subdiffusion), as well as the shape of the wave front.