Two forms of the integral representations of the Mittag-Leffler function

The integral representation of the two-parameter Mittag-Leffler function $E_{\rho,\mu}(z)$ is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function $E_{\rho,\mu}(z)$ has two forms: the representation ``A'' and ``B''. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed

The calculation of the probability density and distribution function of a strictly stable law in the vicinity of zero

The problem of calculating the probability density and distribution function of a strictly stable law is considered at $x\to0$. The expansions of these values into power series were obtained to solve this problem. It was shown that in the case $\alpha<1$ the obtained series were asymptotic at $x\to0$, in the case $\alpha>1$ they were convergent and in the case $\alpha=1$ in the domain $|x|<1$ these series converged to an asymmetric Cauchy distribution. It has been shown that at $x\to0$ the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly stable laws

Conserved developmental processes and the formation of evolutionary novelties: examples from butterfly wings

The origin and diversification of evolutionary novelties—lineage-specific traits of new adaptive value—is one of the key issues in evolutionary developmental biology. However, comparative analysis of the genetic and developmental bases of such traits can be difficult when they have no obvious homologue in model organisms. The finding that the evolution of morphological novelties often involves the recruitment of pre-existing genes and/or gene networks offers the potential to overcome this challenge. Knowledge about shared developmental processes obtained from extensive studies in model organisms can then be used to understand the origin and diversification of lineage-specific structures. Here, we illustrate this approach in relation to eyespots on the wings of Bicyclus anynana butterflies. A number of spontaneous mutations isolated in the laboratory affect eyespots, lepidopteran-specific features, and also processes that are shared by most insects. We discuss how eyespot mutants with disturbed embryonic development may help elucidate the genetic pathways involved in eyespot formation, and how venation mutants with altered eyespot patterns might shed light on mechanisms of eyespot development

Involvement of the conserved Hox gene Antennapedia in the development and evolution of a novel trait

<p>Abstract</p> <p>Background</p> <p>Hox proteins specify segment identity during embryogenesis and have typical associated expression patterns. Changes in embryonic expression and activity of <it>Hox </it>genes were crucial in the evolution of animal body plans, but their role in the post-embryonic development of lineage-specific traits remains largely unexplored. Here, we focus on the insect <it>Hox </it>genes <it>Ultrabithorax </it>(<it>Ubx</it>) and <it>Antennapedia </it>(<it>Antp</it>), and implicate the latter in the formation and diversification of novel, butterfly-specific wing patterns.</p> <p>Results</p> <p>First, we describe a conserved pattern of <it>Ubx </it>expression and a novel pattern of <it>Antp </it>expression in wing discs of <it>Bicyclus anynana </it>butterflies. The discrete, reiterated domains of Antp contrast with the typical expression of Hox genes in single continuous regions in arthropod embryos. Second, we show that this pattern is associated with the establishment of the organizing centres of eyespots. <it>Antp </it>upregulation is the earliest event in organizer development described to date, and in contrast to all genes implicated in eyespot formation, is exclusive to those centres. Third, our comparative analysis of gene expression across nymphalids reveals unexpected differences in organizer determination.</p> <p>Conclusions</p> <p>We show that the Antp's recruitment for the formation of novel traits in butterfly wing discs involved the evolution of new expression domains, and is restricted to a particular lineage. This study contributes novel insights into the evolution of <it>Antp </it>expression, as well as into the genetic mechanisms underlying morphological diversification. Our results also underscore how a wider representation of morphological and phylogenetic diversity is essential in evolutionary developmental biology.</p

Historical Memory And Cultural Immunity Of Society As Resources Of The Transformation Of The Culture Of Education

This paper problematizes the current state of historical memory and cultural immunity as a cross-development trend in the field of education, noting that sociocultural thought has focused on understanding of historical memory and cultural immunity as parallel processes, and the transformation of the culture of education is described in relation to a new information reality, which gives rise to the dehistorization of the field of education. The author of the paper believes that the adoption of social and resource methodological schemes brings hope to get the answers for the two important aspects: first, cultural space of the educational society in the analysis of the effects of the wearable information technologies, and, in this regard, the degree of influence of historical memory and cultural immunity, second, the aspect that can be described as “specific” in the educational culture of Russian society, in which the “language” of globalization is translated into the culture-historical context of national education