Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form $L_1(x)+g(x)L_2(y)$, where $L_1(x)$ is one-dimensional and $g(x)$ may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our results do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.Comment: Accepted for publication at the Journal of Mathematical Analysis and Applications. 23 page

Development of strategies to improve modern wheat cultivars by adopting genetic diversity from wild relatives

Bread wheat is a hexaploid plant that contains three subgenomes derived from a hybridization event between tetraploid Triticum dicoccoides and diploid Aegilops taushii. These wild ancestors of wheat offer more genetic diversity, but they lack the domesticated traits critical for cultivating wheat and adapted to modern agricultural practices. In this study, we are optimizing the process of wild relative diversity introgression while reducing the negative impact of non-adaptive alleles on wheat performance. In a population created by crossing a wild relative and bread wheat, we remove a non-domesticated allele of Btr1, affecting traits important for mechanical harvesting, by genetic engineering or by using molecular markers. The resulting population of wheat lines has the domesticated allele at the Btr1 locus as well as novel genetic diversity from a wild relative for further evaluation

Issue of Waqf in the context of establishment of Muslim institutions in the post-Soviet Tatarstan

© the author(s). The importance of the issue under study is specified by the increasing significance of the religious factor in the social and political, religious and cultural life of global community and Russia in the 21st century. The aim of the article is to find out and analyze the tendencies of the development of Waqf system among Tatars at the end of the 20th and at the beginning of the 21st centuries, to determine the possibilities of religious institutions' financing by means of engaging and adapting such traditional instruments as Waqf to the modern conditions. The main method of the research is historical and comparative method, at the example of successful existence of Waqf property in the Russian Empire until 1917. Previous experience is compared with the modern state in the realities of the current Russian legislation. The article highlights the impossibility of the unimpaired operation of Waqf institution in the framework of available legal base and in the conditions of secular state. Alongside with this there offered ways of legitimization of separate principles of Waqf institution in the framework of current Russian legislation. The article presents the results indicating historical experience of the functioning of Waqf among Tatar in the past, rather successful existence of Waqf in the sovereign Tatarstan in the 1990s in the framework of local legislation, as well as failed attempts of the local authorities to solve the issue of Waqf property in the federal and all-Russia level. The article materials might be useful in the further study of institutional history of Russian Islam, as well as in modern social discussions about the role of religion in the secular space, and also in forming scientific and methodological, regulatory and legal framework to improve government policy about religion