A Brief Story about the Operators of the Generalized Fractional Calculus

2000 Mathematics Subject Classification: 26A33, 33C60, 44A20In this survey we present a brief history and the basic ideas of the generalized fractional calculus (GFC). The notion “generalized operator of fractional integration” appeared in the papers of the jubilarian Prof. S.L. Kalla in the years 1969-1979 when he suggested the general form of these operators and studied examples of them whose kernels were special functions as the Gauss and generalized hypergeometric functions, including arbitrary G- and H-functions. His ideas provoked the author to choose a more peculiar case of such kernels and to develop a theory of the corresponding GFC that featured many applications. All known fractional integrals and derivatives and other generalized integration and differential operators in various areas of analysis happened to fall in the scheme of this GFC

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, two operators introduced by Saigo, one involving the Gauss hypergeometric function, and the other - the Appell (or Horn) F3-function, are rather popular. Here we view on these Saigo’s operators as cases of generalized fractional integration operators, and show that the techniques of the generalized fractional calculus and special functions are helpful to obtain explicit sufficient conditions that guarantee mappings as: S → S and K → S, that is, preserving the univalency of functions.* Partially supported by National Science Fund (Bulg. Ministry of Educ. and Sci.) under Project MM 1305

Degree Of Autonomy In The Provision Of Obstetric Care When Monitoring Normal Pregnancy

Introduction: Midwives are health professionals who provide care for women during pregnancy, childbirth, in the postpartum period and in gynecological diseases. In many developed countries around the world, the obstetric profession is practiced autonomously, which is the basis of greater satisfaction and higher prestige among society.Purpose: To analyze obstetric care in Bulgaria and in different countries around the world and to carry out an analysis of the conditions for autonomous exercise of the profession.Material and methods: A survey of literary sources and normative documents concerning the practice of the obstetric profession in Bulgaria and around the world for the period January - February 2020 was conducted. Used methods are – analysis of literature sources, documentary and comparative analysis.Results and discussion: A survey of obstetric practices was conducted in Bulgaria and abroad. Due to the ever-increasing health needs of society and an increase in the prices of medical services, in the context of a shortage of resources (material and human) and the search for an acceptable way to satisfy them, different health systems have adopted and strengthened different roles of healthcare professionals – midwives.Conclusion: The modern medical education of midwives in Bulgaria is in line with the European Staff Mobility Directive and provides the necessary basic knowledge and skills for practicing the profession. In Bulgaria, low levels of autonomy of the profession are established and there are no conditions for practicing it independently. The imperfection of regulations and the lack of conditions for obstetric autonomy are one of the reasons for low satisfaction, unsatisfactory status of the profession in society and a significant share of midwives wishing to emigrate abroad

The development of perceptual priors

Bayesian inference has come to be regarded as the best, statistically optimal, way to deal with the sensory uncertainty inherent in our natural environment. One way to cope with such uncertainty is to incorporate our pre-existing knowledge about the world. However, we know very little about the circumstances in which human observers integrate sensory information with prior knowledge in a way that is close to optimal. We understand even less about how the developing brain adapts to the environmental statistics, learns to use them efficiently, and what factors may underlie the development of this critical perceptual skill. We addressed these questions though a series of psychophysical experiments, in which adults and 6- to 11-year-old children estimated the location of unseen targets based on a noisy sensory cue and a prior distribution that can be learned over the course of the experiment. In Chapter 2, we showed that adult observers weighted sensory and prior information by their reliabilities but were far from optimal and struggled to generalise to untrained reliabilities in complex situations. The findings of Chapter 3 showed that 6- to 8-year-olds also weighted priors in proportion to their reliability, but they were slow to tune their behaviour to the statistics over time and remained furthest from optimal. Six- to -eight-year-olds’ performance reached adult-like levels when the priors were explicitly shown. Conversely, when the decision rule was made more complex, 6- to 8-year-olds’ abilities to distinguish between the priors broke down and adults’ performance became more child-like. These findings prompted us to investigate whether individual differences, specifically in working memory, may predict performance in adults. The distance from optimal was not predicted by working memory capacity, beyond general cognitive abilities. Together, these studies offer fresh insights into the capacity and limitations both adults and 6-11-year-old children have in learning and efficiently using novel environmental statistics

The Immersive Power of Augmented Reality

Augmented reality is one of the technologies that have received great attention and interest in recent years. In a world where the boundaries between the real and the virtual are blurring, this immersive technology enriches and complements the reality with digital content and allows people to gain a complete and real sense of the objects around them. Currently, the applications of augmented reality go beyond the domains of games and entertainment and are aimed at taking the full advantage of the technology in areas such as medicine, architecture, business, tourism, education and more. The current paper presents the essence of technology and types of augmented reality systems. The basic approaches for creating augmented reality applications are discussed. Specific examples of the application of the technology in the field of education are given-an augmented book and augmented reality educational projects, whose purpose is to make learning an interesting, immersive, engaging and motivating process

Explicit solutions to hyper-Bessel integral equations of second kind

AbstractIn earlier papers, the authors have used the transmutation method to find solutions to Volterra integral or differ-integral equations of second kind, involving Erdélyi-Kober fractional integration operators (see [1,2]), as well as to dual integral equations and some Bessel-type differential equations (see [3,4]). Here we consider the so-called hyper-Bessel integral equations whose kernel-function is a rather general special function (a Meijer's G-function). Such an equation can be written also in a form involving a product of arbitrary number of Erdélyi-Kober integrals. By means of a Poisson-type transmutation, we reduce its solution to the well-known solution of a simpler Volterra equation involving Riemann-Liouville integration only. In the general case, the solution is found as a series of integrals of G-functions, easily reducible to series of G-functions. For particular nonhomogeneous (right-hand side) parts, this solution reduces to some known special functions. The main techniques are based on the generalized fractional calculus

A Poster about the Recent History of Fractional Calculus

MSC 2010: 26A33, 05C72, 33E12, 34A08, 34K37, 35R11, 60G22In the last decades fractional calculus became an area of intense re-search and development. The accompanying poster illustrates the major contributions during the period 1966-2010

General covariant Horava-Lifshitz gravity without projectability condition and its applications to cosmology

We consider an extended theory of Horava-Lifshitz gravity with the detailed balance condition softly breaking, but without the projectability condition. With the former, the number of independent coupling constants is significantly reduced. With the latter and by extending the original foliation-preserving diffeomorphism symmetry ${{Diff}}(M, {\cal{F}})$ to include a local U(1) symmetry, the spin-0 gravitons are eliminated. Thus, all the problems related to them disappear, including the instability, strong coupling, and different speeds in the gravitational sector. When the theory couples to a scalar field, we find that the scalar field is not only stable in both the ultraviolet (UV) and infrared (IR), but also free of the strong coupling problem, because of the presence of high-order spatial derivative terms of the scalar field. Furthermore, applying the theory to cosmology, we find that due to the additional U(1) symmetry, the Friedmann-Robertson-Walker (FRW) universe is necessarily flat. We also investigate the scalar, vector, and tensor perturbations of the flat FRW universe, and derive the general linearized field equations for each kind of the perturbations.Comment: 19 pages, comments are welcome!!