GUARANTEED BOUNDS FOR THE SOLUTION OF THE WAVE EQUATION

Abstract

Numerical solution of the wave equation in the form of close lower and upper bounds provides a secure a posteriori error estimate that can be used for efficient accuracy control. The method considered in this paper uses some monotone properties of the differential operator in the wave equation to construct bounds for the solution in the form of trigonometric polynomials of x. Aspects of the numerical implementation, the accuracy of the computed bounds and some numerical examples are discussed.     

Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into such that the values on any given discrete sequence can be arbitrarily prescribed. Kutzschebauch supported by Schweizerische Nationalfonds grant 200021-107477/1.     

One-parameter toric deformations of cyclic quotient singularities

In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.     

Superconvergence of discontinuous Galerkin methods for hyperbolic systems

In this paper, the discontinuous Galerkin method for the positive and symmetric, linear hyperbolic systems is constructed and analyzed by using bilinear finite elements on a rectangular domain, and an O(h²)$O\left(h^2\right)$-order superconvergence error estimate is established under the conditions of almost uniform partition and the H³$H^3$-regularity for the exact solutions. The convergence analysis is based on some superclose estimates derived in this paper. Finally, as an application, the numerical treatment of Maxwell equation is discussed and computational results are presented.     

Two-grid methods for finite volume element approximations of nonlinear parabolic equations

Two-grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method. The methods are based on one coarse-grid space and one fine-grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine-grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H³|lnH|)$h=O\left(H^3|lnH|\right)$. As a result, solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation.     

Total Synthesis of (±)‐Eusiderin G and (±)‐Eusiderin M

(±)‐Eusiderin G and (±)‐Eusiderin M were first synthesized from pyrogallol, in which the Claisen Rearrangement was used to afford two important C₆‐C₃ units.     

Detecting unalloyed tin in LiSn2(PO4)3-based anodes with Mössbauer spectroscopy

The first discharge of the Li⁺ ion anode material LiSn₂(PO₄)₃ was investigated with Mössbauer spectroscopy and electrochemical techniques. Mössbauer spectroscopy provided insight into the structure of the tin atoms of the fully discharged anode materials. Spectra consist of overlapping peaks, which are assigned to noncrystalline β-Sn and Li–Sn alloy domains. An analysis of the relative intensities of the Mössbauer spectra shows the relative abundance of β-Sn increases at the expense of the Li–Sn alloy as the discharge rate increases. Cell polarization occurs at higher discharge rates, leading to inefficient electrode utilization and poor cycling performance. Sluggish Li⁺ ion diffusion through the amorphous Li₃PO₄ network that is formed early in the discharge process might be responsible for the poor electrochemical performance and the accumulation of unalloyed tin.     

Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

The deformation of a liquid capsule enclosed by a thin shell in a simple shear flow is studied numerically using an implicit immersed boundary method. We present a thin-shell model for computing the forces acting on the shell middle surface during the deformation within the framework of the Kirchhoff–Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. For hyperelastic materials, the plane-stress assumption is used to compute the hydrostatic pressure and the incompressibility condition yields the thickness strain component and the corresponding change in the thickness. The stresses developing over the cross-section of the shell are integrated over the thickness to yield the stress and moment resultants which are then used to compute the forces acting on the shell middle surface. The immersed boundary method is employed for calculating the hydrodynamics and fluid–structure interaction effects. The location of the thin shell is updated implicitly using the Newton–Krylov method. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical and oblate spheroidal capsules in the shear flow.     

Optimized POCl3-assisted synthesis of 2-amino-1,3,4-thiadiazole/1,3,4-oxadiazole derivatives as anti-influenza agents

Although recent decades have witnessed the synthesis of 1,3,4-thiadiazoles via phosphorus POCl₃-promoted cyclization reaction, simultaneous access to 2-amino-1,3,4-thiadiazole and 2-amino-1,3,4-oxadiazole analogs remains unexpected and elusive. Herein, a detailed regiocontrolled synthesis of 2-amino-1,3,4-thiadiazoles in good to high yields with good regioselectivities from readily available thiosemicarbazides using POCl₃ was disclosed. Meantime, to establish a comprehensive structure–activity relationship, 2-amino-1,3,4-oxadiazole derivatives as single regioisomers were prepared via EDCI·HCl-triggered cyclization of the thiosemicarbazide intermediates. The in vitro anti-influenza assays proved that the selected compounds with the pyrazine/pyridine ring exhibited certain inhibitory activities against influenza A virus strains A/HK/68 (H3N2) and A/PR/8/34 (H1N1) in MDCK cells. Among them, N-(adamantan-1-yl)-5-(5-(azepan-1-yl)pyrazin-2-yl)-1,3,4-thiadiazol-2-amine (4j) was the most active compound, and exhibited favorable activity with EC₅₀ values of 3.5 μM and 7.5 μM, respectively. In addition, the molecular docking results explained the reason why compound 4j had dual inhibitory activity and revealed the reasonable binding mode of this compound with the M2-S31N and M2-WT ion channels. This compound had the potential to be further developed as an anti-influenza drug.     

Brassinin inhibits proliferation and induces cell cycle arrest and apoptosis in nasopharyngeal cancer C666-1 cells

BackgroundNasopharyngeal cancer is a tumor that occurs in the mucous epithelium of the nasopharynx. Due to its rapid growth and early metastatic nature, the successful treatment of nasopharyngeal cancer is highly challenging.ObjectiveHere, we intended to assess the in vitro anticancer property of brassinin against the nasopharyngeal cancer C666-1 cells.MethodologyThe in vitro free radical scavenging property of the brassinin was assessed by various free radical scavenging activities such as FRAP, DPPH, chemiluminescence (CL), and ORAC assays. The cytotoxic level of the brassinin (1–50 µM) against the nasopharyngeal cancer C666-1 cells and normal Vero cells were assessed by the MTT cytotoxicity assay. The levels of TBARS, GSH, and the SOD activity was assessed using kits. The level of ROS generation, MMP, and apoptosis were investigated by the respective fluorescent staining techniques. The flow cytometry analysis was done to scrutinize the cell cycle arrest. The Bax/Bcl-2 level and caspase activities were examined using respective kits.ResultsThe brassinin treatment effectively scavenged the free radicals, which are assessed by the FRAP, DPPH, chemiluminescence (CL), and ORAC assays. The proliferation of brassinin treated C666-1 cells were decreased remarkably, while the same concentration of brassinin did not disturbed the Vero cell viability. The 30 µM of brassinin effectively increased the ROS production, depleted the MMP, and stimulated the apoptosis in the C666-1 cells. The brassinin increased the TBARS and depleted the GSH and SOD in the C666-1 cells. The flow cytometry analysis revealed that the brassinin administration improved the G0/G1 ratio and decreased the proportion of cells with ‘S’ and ‘G2/M’ phase. The Bax, caspase-3 and ?9 were elevated and Bcl-2 level was decreased in the brassinin administered C666-1 cells.ConclusionOur findings discovered that the brassinin has the capacity to prevent the proliferation and stimulate the apoptotic cell death C666‐1 cells via blocking cell cycle and increasing oxidative stress and apoptotic markers. Hence, it can be a talented therapeutic agent to treat the nasopharyngeal cancer in the future.