An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

Chitin and Lignin: Old Polymers and New Bio-Tissue- Carriers

Worldwide consumers are nowadays much more focusing on their wealth and appearance, having increased their worry caused from the pollution, plastic wastes and the earth' disasters further increased for the COVID-19 pandemic. This trend has created an heightened demand for products which, formulated with natural and functional ingredients and carried by sustainable delivery systems, should be produced and packed with biodegradable compounds. The paper suggests to formulate innovative cosmetic and medical products based on the use of carriers made by biodegradable polysaccharide-tissues embedded by micro-nano particles of chitin nano fibril-nano lignin complexes, encapsulating different active ingredients. Thus, data on chitin, lignin and their complexes are reported and discussed, focusing the attention on their possible use to make innovative products, characterized for their effectiveness, safeness, and biodegradability

Genome-wide association study and scan for signatures of selection point to candidate genes for body temperature maintenance under the cold stress in Siberian cattle populations

Design of new highly productive livestock breeds, well-adapted to local climatic conditions is one of the aims of modern agriculture and breeding. The genetics underlying economically important traits in cattle are widely studied, whereas our knowledge of the genetic mechanisms of adaptation to local environments is still scarce. To address this issue for cold climates we used an integrated approach for detecting genomic intervals related to body temperature maintenance under acute cold stress. Our approach combined genome-wide association studies (GWAS) and scans for signatures of selection applied to a cattle population (Hereford and Kazakh Whiteheaded beef breeds) bred in Siberia. We utilized the GGP HD150K DNA chip containing 139,376 single nucleotide polymorphism markers

The origins and spread of domestic horses from the Western Eurasian steppes

This is the final version. Available on open access from Nature Research via the DOI in this recordData availability: All collapsed and paired-end sequence data for samples sequenced in this study are available in compressed fastq format through the European Nucleotide Archive under accession number PRJEB44430, together with rescaled and trimmed bam sequence alignments against both the nuclear and mitochondrial horse reference genomes. Previously published ancient data used in this study are available under accession numbers PRJEB7537, PRJEB10098, PRJEB10854, PRJEB22390 and PRJEB31613, and detailed in Supplementary Table 1. The genomes of ten modern horses, publicly available, were also accessed as indicated in their corresponding original publications57,61,85-87.NOTE: see the published version available via the DOI in this record for the full list of authorsDomestication of horses fundamentally transformed long-range mobility and warfare. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling at Botai, Central Asia around 3500 BC. Other longstanding candidate regions for horse domestication, such as Iberia and Anatolia, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 BC, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 BC driving the spread of Indo-European languages. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium BC Sintashta culture

Compatibilized Polyimide (R-BAPS)/BAPS-Modified Clay Nanocomposites With Improved Dispersion and Properties

There is a need for clav modifiers that will not thermally degrade at elevated temperatures commonly used in polymer processing operations such as extrusion and injection molding. In this context, natural montmorillonite clay (Na-MMT) was organically modified by varying concentrations of 4.4\u27-bis(4 \u27\u27-aininophenoxy)diphenylsulfone (BAPS) using different chemical dispersion methods to yield new chemically modified clays that are relatively thermally stable at elevated temperatures compared with current commercial modified clays. This paper shows that the Na-MMT chemical modification (BAPS-MMT) was confirmed by XRD that showed a shift of diffraction peak at 2 Theta = 7.3 degrees for Na-MMT towards lower 2 Theta = 5.8 degrees. Thermogravimetric analysis of the samples showed a weight loss of organically modified clay that started at a temperature of 350 degrees C. corresponding to the degradation temperature of the BAPS monomer. Rheological measurements in combination with XRD data showed clearly that the quality of dispersion of BAPS-MMT type particles in R-BAPS type polyimide and oligoimides strongly depends on the clay surface modification, the specific chemical modification method used, and on the polymer molecular weight. Note that the oligoimides were specifically used as model systems to confirm our expectation of improved chemical compatibility between the BAPS-MMT and the polyimide system. This study may stimulate a better understanding of the effects of rational chemical modification methods on the quality of clay dispersion in polyimide matrices, enhancing our ability to prepare useful polyimide/clay nanocomposites with improved properties for targeted high-temperature applications where current polymer nanocomposite systems are not useable. (C) 2007 Elsevier Ltd. All rights reserved

Novel Semicrystalline Thermoplastic R-BAPB Type Polyimide Matrix Reinforced By Graphite Nanoplatelets and Carbon Nanoparticles

A new semicrystalline polyimide, based on 1. 3-bis-(3,3\u27,4,4\u27-dicarboxyphenoxy) benzene (R) and 4,4\u27-bis(4-aminophenoxy)biphenyl (BAPB), was modified by exfoliated graphite nanoplatelets of different shape, sizes and structure. This R-BAPB type polyimide, with a molecular weight Mw = 30000 g/mol has a low melting temperature of 320 degrees C and a melt zero shear viscosity of about 1000 Pa s at 340 degrees C, making it possible to mix it with particulates using classical melt-blending technology. By using calorimetric and rheological measurements, this study reports a dramatic crystal nucleating effect of the polyimide caused by the special nanographite platelets. In addition, the study shows that classical melt blending of R-BAPB type PI with exfoliated graphite nanoplatelets can be used to prepare useful polyimide nanocomposites with the added benefit of possibly accelerating the crystallization process of the polyimide matrix. Further, it was observed that the time of crystallization at 300 degrees C for the nanocomposites filled with 5 wt% nanographite platelets was about 10 5 minutes, a value corresponding to one-half that of the unfilled R-BAPB type PI matrix

A Comparative Study on the Mechanical and Barrier Characteristics of Polyimide Nanocomposite Films Filled with Nanoparticles of Planar and Tubular Morphology

Polyimide (PI) films based on poly(pyromellitic dianhydride-co-4,4\u27-oxydianiline) (PI-PM) were filled with different nanoparticles, such as organically modified montmorillonite (MMT), vapor-grown carbon nanofibers (VGCF), and silicate nanotubes (SNT) of different concentration.. Rheological measurements and structural investigations showed a relatively good dispersion of the nanoparticles in the PI matrix to an extent that depended on the type and morphology of the nanoparticles used. The mechanical (tensile modulus, strength, and deformation at break) and the barrier (oxygen permeability) properties of PI-PM nanocomposite films were investigated. The polyimide nanocomposites filled with SNT and tubular VGCF nanoparticles showed an increased tensile modulus with increasing volume concentration of the nanoparticles without a catastrophic decrease in the elongation at break. In addition, the MMT particles, chemically modified with 4,4\u27-bis-(4 \u27\u27-aminophenoxy) diphenylsulfone, significantly improved the barrier properties of the PI-PM films, which exceeded those of the nanocomposites filled with VGCF or SNT. The relative poor oxygen barrier and mechanical properties of the PI-PM/VGCF nanocomposite films are ascribed to the relative weak adhesion between the VGCF and the polyimide matrix, which was confirmed by scanning electron microscopy of the fracture surface of these films

Gas Barrier Behavior of Polyimide Films Filled with Synthetic Chrysotile Nanotubes

Chrysotile nanotubes (ChNTs) were synthesized under hydrothermal conditions. These synthetic nanotubes crystallographically and morphologically mimic the nanofibrils of natural white asbestos but they are considerably shorter. ChNTs containing polyimide nanocomposites were prepared by a solution mixing/casting method. Oxygen and water vapor barrier of the nanocomposite films were tested and related to the amount, dispersion, and orientation of the nanotubes. The dispersion and orientation of the nanotubes were examined by transmission electron microscopy (TEM). The nanotubes were nanodispersed and oriented in the plane of the film in the nanocomposites with up to 4.5% (vol/vol) of ChNTs leading to a gradual increase of the gas barrier. The lowest gas permeability was 60% smaller than that for the pristine polyimide film. However, with the onset of nanotube micro aggregation at larger ChNTs loadings the nanotube dispersion and orientation were compromised and oxygen barrier was reduced. The efficacy of nanotubes to enhance polymer gas barrier was discussed and compared with that by nanoplatelets. (c) 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013, 51, 1184-119

Rheological and Mechanical Properties of Thermoplastic Crystallizable Polyimide-Based Nanocomposites Filled with Carbon Nanotubes: Computer Simulations and Experiments

Recently, a strong structural ordering of thermoplastic semi-crystalline polyimides near single-walled carbon nanotubes (SWCNTs) was found that can enhance their mechanical properties. In this study, a comparative analysis of the results of microsecond-scale all-atom computer simulations and experimental measurements of thermoplastic semi-crystalline polyimide R-BAPB synthesized on the basis of dianhydride R (1,3-bis-(3′,4-dicarboxyphenoxy) benzene) and diamine BAPB (4,4′-bis-(4″-aminophenoxy) biphenyl) near the SWCNTs on the rheological properties of nanocomposites was performed. We observe the viscosity increase in the SWCNT-filled R-BAPB in the melt state both in computer simulations and experiments. For the first time, it is proven by computer simulation that this viscosity change is related to the structural ordering of the R-BAPB in the vicinity of SWCNT but not to the formation of interchain linkage. Additionally, strong anisotropy of the rheological properties of the R-BAPB near the SWCNT surface was detected due to the polyimide chain orientation. The increase in the viscosity of the polymer in the viscous-flow state and an increase in the values of the mechanical characteristics (Young’s modulus and yield peak) of the SWCNT-R-BAPB nanocomposites in the glassy state are stronger in the directions along the ordering of polymer chains close to the carbon nanofiller surface. Thus, the new experimental data obtained on the R-BAPB-based nanocomposites filled with SWCNT, being extensively compared with simulation results, confirm the idea of the influence of macromolecular ordering near the carbon nanotube on the mechanical characteristics of the composite material