A family of Nikishin systems with periodic recurrence coefficients

Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval \Delta_k\subset\er with $\Delta_k\cap\Delta_{k+1}=\emptyset$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $\Delta_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^{\infty}$, the so-called \emph{Chebyshev-Nikishin polynomials}. We show that the polynomials $P_{n}$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $\Delta_{k}$. In this way we generalize a result of the third author and Rocha \cite{LopRoc} for the case $p=2$. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for the second kind functions of the Nikishin system for $\{P_{n}\}_{n=0}^{\infty}$.Comment: 30 pages, minor change

Evolution of the Greater Caucasus Basement and Formation of the Main Caucasus Thrust, Georgia

Along the northern margin of the Arabia‐Eurasia collision zone in the western Greater Caucasus, the Main Caucasus Thrust (MCT) juxtaposes Paleozoic crystalline basement to the north against Mesozoic metasedimentary and volcaniclastic rocks to the south. The MCT is commonly assumed to be the trace of an active plate‐boundary scale structure that accommodates Arabia‐Eurasia convergence, but field data supporting this interpretation are equivocal. Here we investigate the deformation history of the rocks juxtaposed across the MCT in Georgia using field observations, microstructural analysis, U‐Pb and 40Ar/39Ar geochronology, and 40Ar/39Ar and (U‐Th)/He thermochronology. Zircon U‐Pb analyses show that Greater Caucasus crystalline rocks formed in the Early Paleozoic on the margin of Gondwana. Low‐pressure/temperature amphibolite‐facies metamorphism of these metasedimentary rocks and associated plutonism likely took place during Carboniferous accretion onto the Laurussian margin, as indicated by igneous and metamorphic zircon U‐Pb ages of ~330–310 Ma. 40Ar/39Ar ages of ~190–135 Ma from muscovite in a greenschist‐facies shear zone indicate that the MCT likely developed during Mesozoic inversion and/or rifting of the Caucasus Basin. A Mesozoic 40Ar/39Ar biotite age with release spectra indicating partial resetting and Cenozoic (<40 Ma) apatite and zircon (U‐Th)/He ages imply at least ~5–8 km of Greater Caucasus basement exhumation since ~10 Ma in response to Arabia‐Eurasia collision. Cenozoic reactivation of the MCT may have accommodated a fraction of this exhumation. However, Cenozoic zircon (U‐Th)/He ages in both the hanging wall and footwall of the MCT require partitioning a substantial component of this deformation onto structures to the south.Plain Language SummaryCollisions between continents cause deformation of the Earth’s crust and the uplift of large mountain ranges like the Himalayas. Large faults often form to accommodate this deformation and may help bring rocks once buried at great depths up to the surface of the Earth. The Greater Caucasus Mountains form the northernmost part of a zone of deformation due to the ongoing collision between the Arabian and Eurasian continents. The Main Caucasus Thrust (MCT) is a fault juxtaposing old igneous and metamorphic (crystalline) rocks against younger rocks that has often been assumed to be a major means of accommodating Arabia‐Eurasia collision. This study examines the history of rocks along the MCT with a combination of field work, study of microscopic deformation in rocks, and dating of rock formation and cooling. The crystalline rocks were added to the margins of present‐day Eurasia about 330–310 million years ago, and the MCT first formed about 190–135 million years ago. The MCT is likely at most one of many structures accommodating present‐day Arabia‐Eurasia collision.Key PointsAmphibolite‐facies metamorphism and plutonism in the Greater Caucasus basement took place ~330–310 MaThe Main Caucasus Thrust formed as a greenschist‐facies shear zone during Caucasus Basin inversion and/or rifting (~190–135 Ma)The Main Caucasus Thrust may have helped facilitate a portion of at least 5–8 km of basement exhumation during Arabia‐Eurasia collisionPeer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/1/tect21292-sup-0002-2019TC005828-ts01.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/2/tect21292-sup-0006-2019TC005828-ts05.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/3/tect21292_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/4/tect21292-sup-0003-2019TC005828-ts02.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/5/tect21292-sup-0005-2019TC005828-ts04.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/6/tect21292.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154626/7/tect21292-sup-0004-2019TC005828-ts03.pd

Ladder operators and differential equations for multiple orthogonal polynomials

In this paper, we obtain the ladder operators and associated compatibility conditions for the type I and the type II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann-Hilbert problems and the Christoffel-Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.Comment: 28 page

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

Влияние нокдауна кавеолина-1 на белковый состав экстраклеточных везикул, секретируемых клетками немелкоклеточного рака легких

Background. Recent data show evidence that lipid rafts (LR) proteins could be involved in the formation of exosomes and the sorting of proteins that make up the exosomal cargo. Such data are available for flotillins, structural and functional components of flatted rafts. The presence of the main component of caveolar rafts, caveolin-1 (Cav-1), has been shown in exosomes produced by some cancer cells; however, its possible participation in the regulation of the protein composition of exosomes has not been studied previously.Materials and methods. Knockdown of Cav-1 by transduction of a lentiviral vector expressing precursors of short hairpin ribonucleic acid to Cav-1; isolation (by ultracentrifugation) and analysis (transmission electron microscopy, nanoparticle tracking analysis) of extracellular vesicles (EVs) from non-small cell lung cancer cells (NSCLC) H1299; analysis of proteins in cells and in EVs by immunoblotting.Results. Analysis of the effect of Cav-1 expression on the composition of EV proteins associated with exosome biogenesis revealed a decrease in the level of Alix and TSG101, an increase in the level of LR proteins and the absence of changes in the level of tetraspanin CD9. Conclusion. The obtained data demonstrate a Cav-1-dependent changes in the composition of EVs, indicating a change in the ratio of vesicles formed by the various molecular mechanisms.Введение. Данные исследований последних лет свидетельствуют о том, что белки, входящие в состав липидных рафтов, могут быть задействованы в формировании экзосом и отборе белков, входящих в состав экзосомального карго. Такие данные получены для флотиллинов, структурно-функциональных компонентов плоских рафтов. Для кавеолина-1 (Cav-1), основного компонента кавеолярных рафтов, показано присутствие в экзосомах некоторых опухолевых клеток, однако его возможное участие в регуляции белкового состава экзосом ранее не исследовалось.Материалы и методы. Нокдаун Cav-1 проводили методом трансдукции лентивирусного вектора, экспрессирующего предшественников малых шпилечных рибонуклеиновых кислот к Cav-1. Экстраклеточные везикулы (ЭКВ) выделяли из клеток линии Н1299 немелкоклеточного рака легких методом дифференциального ультрацентрифугирования. Препараты ЭКВ верифицировали с помощью трансмиссионной электронной микроскопии (анализ размера и морфологии) и методом анализа траекторий движения наночастиц (среднеразмерное распределение и концентрация). Для анализа экзосомальных маркеров и Cav-1 в клетках и ЭКВ применяли метод иммуноблоттинга.Результаты. Анализ влияния экспрессии Cav-1 на состав белков ЭКВ, ассоциированных с биогенезом экзосом, выявил снижение уровня Alix и TSG101, повышение уровня белков липидных рафтов и отсутствие изменений уровня тетраспанина CD9.Заключение. Полученные данные демонстрируют Cav-1-зависимое изменение состава ЭКВ, свидетельствующее об изменении соотношения везикул, образованных с помощью различных молекулярных механизмов.