A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

Mycomedicine: A Unique Class of Natural Products with Potent Anti-tumour Bioactivities

Mycomedicine is a unique class of natural medicine that has been widely used in Asian countries for thousands of years. Modern mycomedicine consists of fruiting bodies, spores, or other tissues of medicinal fungi, as well as bioactive components extracted from them, including polysaccharides and, triterpenoids, etc. Since the discovery of the famous fungal extract, penicillin, by Alexander Fleming in the late 19th century, researchers have realised the significant antibiotic and other medicinal values of fungal extracts. As medicinal fungi and fungal metabolites can induce apoptosis or autophagy, enhance the immune response, and reduce metastatic potential, several types of mushrooms, such as Ganoderma lucidum and Grifola frondosa, have been extensively investigated, and anti-cancer drugs have been developed from their extracts. Although some studies have highlighted the anti-cancer properties of a single, specific mushroom, only limited reviews have summarised diverse medicinal fungi as mycomedicine. In this review, we not only list the structures and functions of pharmaceutically active components isolated from mycomedicine, but also summarise the mechanisms underlying the potent bioactivities of several representative mushrooms in the Kingdom Fungi against various types of tumour.     

Phase‐Junction Electrocatalysts towards Enhanced Hydrogen Evolution Reaction in Alkaline Media

To ensure sustainable hydrogen production by water electrolysis, robust, earth‐abundant, and high‐efficient electrocatalysts are required. Constructing a hybrid system could lead to further improvement in electrocatalytic activity. Interface engineering in composite catalysts is thus critical to determine the performance, and the phase‐junction interface should improve the catalytic activity. Here, we show that nickel diphosphide phase junction (c‐NiP₂/m‐NiP₂) is an effective electrocatalyst for hydrogen production in alkaline media. The overpotential (at 10 mA cm^?2) for NiP₂‐650 (c/m) in alkaline media could be significantly reduced by 26 % and 96 % compared with c‐NiP₂ and m‐NiP₂, respectively. The enhancement of catalytic activity should be attributed to the strong water dissociation ability and the rearrangement of electrons around the phase junction, which markedly improved the Volmer step and benefited the reduction process of adsorbed protons.     

Nonflammable nonaqueous electrolytes for lithium batteries

As the energy density of state-of-the-art lithium (Li)-ion batteries (LIBs) increases, the safety concern of LIBs using liquid electrolytes is drawing increasing attention. Flammability of electrolytes is a critical link of the overall safety performance of LIBs and Li metal batteries. For this reason, intensive efforts have been devoted to suppressing the flammability of liquid electrolytes. In this short review, the common approaches to reduce the flammability of the nonaqueous liquid electrolytes will be summarized. The advantages and limitations of these approaches will also be discussed.     

1H NMR Reassignment for Z/E-Benzomalvins B and Absolute Configuration of Benzomalvin C

Chemistry of Natural Compounds - Benzomalvins B, C, and E (1, 2, and 3) were simultaneously obtained from the marine fungus Aspergillus sp. isolated from the soft coral Sinularia sp., collected...     

A new 3D supramolecular Zn(II) compound: Crystal structure and important role in treatment of pulpitis

An innovative thermostable 2-D layered Zn(II) compound, chemical terms are written as [Zn(3-pysa)₂(H₂O)₂]_n (1, 3-Hpysa = 3-pyridinesulfonic acid), has been generated from the solvothermal reactions of 3-Hpysa and Zn(NO₃)₂·6H₂O. Its single crystal analysis was implemented via the single crystal X-ray diffraction analysis together with the powder X-ray diffraction, elemental analysis and thermogravimetric analysis. The crystal framework is of monoclinic P2₁/n space group and its crystal cell figures: a = 7.7499(5), b = 10.9923(6), c = 8.3430(5) Å, α = 90, β = 96.924(6), γ = 90°, V = 705.55(7) ų, Z = 4. The 2D layers were finally combined to a 3-D supramolecular conformation through intermolecular H-bonds existing between coordinated H₂O molecules and sulfonate oxygen atoms from adjacent layer. Its practical role in pulpitis treatment was estimated and the relevant mechanism was studied in the meantime.     

Characterization and properties of plywood bioadhesive derived from cottonseed protein and sawdust cellulose

Cellulose - The development of plant adhesive with good bonding strength, water resistance and thermal stability remains challenging to replace formaldehyde-based adhesive resins that usually...     

Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)^[1] and Xu et al. (1981).^[6]Nagano (1979)^[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ ₀ ² is negative in Nagano's paper.^[3]Moreover, even as early as γ ₀ ² → O⁺, the expansion becomes invalid because the 2nd order modifier γ₂ (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth. Theproject is supported by the National Natural Science Foundation of China.     

Three-dimensional measurements in non-axisymmetric flow experiment using a one-or two-component laser-Doppler anemometer

 A method to determine the velocity components in the third direction for steady-non-axisymmetric nozzle flow experiments has been developed in addition to the two components that could be directly measured by a one-or two-component laser-Doppler anemometer. With the aid of this method, rotation for either the test rig or the anemometer become unnecessary. Results obtained by this approach were used to compare with those directly measured in the downstream wake of a co-axial nozzle with a central five-lobe forced mixer where the flow is non-axisymmetric. Good agreement could be obtained with maximum discrepancy at about 5%. The present approach can be readily applied to many other non-axisymmetric nozzle flow situations with only some minor modifications. Received: 13 May 1996 / Accepted: 10 October 1996     

On 3-dimensional non-axisymmetrical deformation analyses for finite hollow circular cylinders

In this paper, 3-dimensional non-axisymmetrical deformation analyses for finite hollow circular cylinders have been carried out by Pickett's double series expansion method^[1]. Through expanding the displacement potentials as the sum of fourier series and Fourier-Bessel series, we could express the coefficients of one series by those of another under certain boundary conditions. Thus, a set of linear algebraic equations were derived. Solving these equations, we could obtain the solutions of the problems. Numerical examples have been given to show that the method presented here is workable for practical applications.