Generalizations of Weighted Trapezoidal Inequality for Monotonic Mappings and Its Applications

In this paper, we establish some generalizations of weighted trapezoid inequality for monotonic mappings, and give several applications for r − moment, the expectation of a continuous random variable and the Beta mapping

Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.Comment: 34 pages, 24 figure

Stationary wave patterns generated by an impurity moving with supersonic velocity through a Bose-Einstein condensate

Formation of stationary 3D wave patterns generated by a small point-like impurity moving through a Bose-Einstein condensate with supersonic velocity is studied. Asymptotic formulae for a stationary far-field density distribution are obtained. Comparison with three-dimensional numerical simulations demonstrates that these formulae are accurate enough already at distances from the obstacle equal to a few wavelengths.Comment: 7 pages, 3 figure

Bending-wave Instability of a Vortex Ring in a Trapped Bose-Einstein Condensate

Based on a velocity formula derived by matched asymptotic expansion, we investigate the dynamics of a circular vortex ring in an axisymmetric Bose-Einstein condensate in the Thomas-Fermi limit. The trajectory for an axisymmetrically placed and oriented vortex ring is entirely determined, revealing that the vortex ring generally precesses in condensate. The linear instability due to bending waves is investigated both numerically and analytically. General stability boundaries for various perturbed wavenumbers are computed. In particular, the excitation spectrum and the absolutely stable region for the static ring are analytically determined.Comment: 4 pages, 4 figure

Modeling of Polymer Clay Nanocomposite for a Multiscale Approach

The mechanical property enhancement of polymer reinforced with nano-thin clay platelets (of high aspect ratio) is associated with a high polymer-filler interfacial area per unit volume. The ideal case of fully separated (exfoliated) platelets is generally difficult to achieve in practice: a typical nanocomposite also contains multilayer stacks of intercalated platelets. Here we use numerical modelling to investigate how the platelet properties affect the overall mechanical properties. The configuration of platelets is modelled using a statistical interpretation of the Representative Volume Element (RVE) approach, in which an ensemble of "sample" heterogeneous material is generated (with periodic boundary conditions). A simple Monte Carlo algorithm is used to place non-intersecting platelets in the RVE according to a specified set of statistical distributions. The effective stiffness of the platelet-matrix system is determined by measuring the stress (using standard Finite Element analysis) produced as a result of applying a small deformation to the boundaries, and averaging over the entire statistical ensemble. In this work we determine the way in which the platelet properties (curvature, filling fraction, stiffness, aspect ratio) and the number of layers in the stack affect the overall stiffness enhancement of the nanocomposite. Thus, we bridge the gap between behaviour on the macroscopic scale with that on the scale of the nano-reinforcement, forming part of a multi-scale modelling framework.Comment: 39 pages, 19 figure

Simulations on High-z Long Gamma-Ray Burst Rate

Since the launch of Swift satellite, the detections of high-z (z>4) long gamma-ray bursts (LGRBs) have been rapidly growing, even approaching the very early Universe (the record holder currently is z=8.3). The observed high-z LGRB rate shows significant excess over that estimated from the star formation history. We investigate what may be responsible for this high productivity of GRBs at high-z through Monte Carlo simulations, with effective Swif/BAT trigger and redshift detection probabilities based on current Swift/BAT sample and CGRO/BATSE LGRB sample. We compare our simulations to the Swift observations via log N-log P, peak luminosity (L) and redshift distributions. In the case that LGRB rate is purely proportional to the star formation rate (SFR), our simulations poorly reproduce the LGRB rate at z>4, although the simulated log N-log P distribution is in good agreement with the observed one. Assuming that the excess of high-z GRB rate is due to the cosmic metallicity evolution or unknown LGRB rate increase parameterized as (1+z)^delta, we find that although the two scenarios alone can improve the consistency between our simulations and observations, incorporation of them gives much better consistency. We get 0.2<epsilon<0.6 and delta<0.6, where epsilon is the metallicity threshold for the production of LGRBs. The best consistency is obtained with a parameter set (epsilon, delta)=(~0.4, ~0.4), and BAT might trigger a few LGRBs at z~14. With increasing detections of GRBs at z>4 (~15% of GRBs in current Swift LGRB sample based on our simulations), a window for very early Universe is opening by Swift and up-coming SVOM missions.Comment: 9 pages, including 8 figures and 1 table, one more figure added. Accepted for publication in MNRA

Chinese Children’s School Experiences Represented in Picture Books

In a graduate level children’s literature class, Eun Hye, the second author, was facilitating the discussion of the importance of multicultural children’s literature from an assigned reading of Rudine Sims Bishop\u27s “Mirrors, Windows, and Sliding Glass Doors” (1990). As Lin, the first author, was engaged in the discussion by sharing different childhood experiences, one student, who was adopted from China by Americans as a toddler, said that she found herself rarely represented in picture books while growing up in the U.S. We both felt bad because we grew up in China and Korea respectively, surrounded by many children’s books representing children like ourselves. However, when we each came to the United States in our 20s to attend higher education, we quickly learned that there is a limited number of books portraying Chinese and Korean children living in the United States. At different time periods, we both developed intellectual interests toward critically examining how these two groups of children (Chinese and Korean) have been portrayed in multicultural children’s books

Petrogenesis of the Early Eocene adakitic rocks in the Napuri area, southern Lhasa: Partial melting of thickened lower crust during slab break-off and implications for crustal thickening in southern Tibet

Cenozoic adakitic rocks in the Lhasa block (southern Tibet) have been widely used to trace the lateral extent of crustal thickening. However, their petrogenesis remains controversial. Here, we report geochronological and geochemical data for the Napuri intrusive rocks in the core area of the Quxu batholith, southern Lhasa. Zircon U–Pb dating suggests that they were generated at approximately 48 Ma. The studied samples show significant geochemical variations, manifested by the coexistence of three types of igneous rocks. Groups I and II rocks exhibit variable and high SiO2 (66.4–73.9 wt.%), high Al2O3 (14.0–17.4 wt.%), K2O (3.9–5.3 wt.%), Sr (273–718 ppm) and Sr/Y (18.3 to 81.3) values, and low Y (3.6 to 16 ppm), heavy rare earth element (REE) (e.g., Yb = 0.48 to 1.8 ppm), MgO (0.4–1.0 wt.%), Cr (2.9–7.4 ppm) and Ni (1.6–4.5 ppm) contents, which are similar to those of thickened lower crust-derived adakitic rocks. The Group I rocks show higher Sr/Y (77.5–81.3) ratios and lower total REE (55.5–63.2 ppm) contents with clearly positive Eu and Sr anomalies, whereas the Group II rocks have relatively lower Sr/Y (18.3–65.7) ratios and higher total REE (115–375 ppm) contents with negligible or slightly negative Eu and Sr anomalies. Group III rocks have the highest SiO2 (74.5–76.0 wt.%), Y (17.0–23.7 ppm) and Yb (2.91–3.30 ppm) contents, and the lowest Al2O3 (12.5–13.2 wt.%), Sr (81.3–141 ppm) and Sr/Y (4.8–5.9) values with distinctly negative Eu and Sr anomalies. Compared with the Jurassic–Cretaceous granitoids in southern Lhasa, the relative enrichment in Sr–Nd–Hf isotopic compositions ((87Sr/86Sr)i = 0.7049–0.7055, εNd(t) = − 0.3 ± 0.7 and εHf(t)zircon = + 3.6 ± 11.4) for the Napuri intrusive rocks indicates that they likely contained Indian continental components. The Group I and Group II rocks most probably originated from thickened mafic lower crust (amphibolite eclogites or garnet amphibolites) with garnet + rutile ± plagioclase as residual minerals in the source at > 1.5 GPa, corresponding to depths of > 50 km, and Group III rocks were probably generated by fractional crystallization of plagioclase from the adakitic magmas. Taking into account the narrow linear nature of the Eocene magmatic belt and reported synchronous asthenosphere-derived basaltic rocks in southern Lhasa, we suggest that upwelling asthenosphere triggered by the break-off of subducted Neo-Tethyan slab probably provided the required thermal conditions for lower crustal melting. The identification of Indian continental components in the Napuri intrusive rocks probably indicates that the Asia–India collision had taken place prior to their emplacement. The dramatic changes in the (La/Yb)N ratios and εNd(t) and εHf(t) values of magmatic rocks in the Gangdese area at ca. 51–46 Ma indicate that the Cenozoic crustal thickening associated with the indentation of the Indian continent began in the Early Eocene (ca. 51–46 Ma) at the latest