Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form where 0 ∈ Ω is a smooth bounded domain in , , and λ > 0 is a real parameter. Combining with the variational method and Nehari manifold method, two positive solutions of the system are obtained.     

A Series Solution of the Unsteady Von Kármán Swirling Viscous Flows

A new analytic technique is applied to solve the unsteady viscous flow due to an infinite rotating disk, governed by a set of two fully coupled nonlinear partial differential equations deduced directly from the exact Navier-Stokes equations. The system of coupled nonlinear partial differential equations is replaced by a sequence of uncoupled systems of linear ordinary differential equations. Different from all other previous analytic results, our series solution is accurate and valid for all time in the whole spatial region. Accurate expressions for skin friction coefficients are given, which are valid for all time. Such kind of series solutions have not been reported, to the best of our knowledge.      

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Let M ⁿ be an n-dimensional compact C ^∞-differentiable manifold, n ≥ 2, and let S be a C ¹-differential system on M ⁿ . The system induces a one-parameter C ¹ transformation group φ _t (−∞ < t < ∞) over M ⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M ⁿ . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M ⁿ of the differential system S. With respect to a given C ^∞ Riemannian metric of M ⁿ , we consider the bundle of all (n−2) spheres Q _x ⁿ⁻², x∈M, where Q _x ⁿ⁻² for each x consists of all unit tangent vectors of M ⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ _t ^# (−∞<t<∞) of . For an l-frame α = (u ₁, u ₂,⋯, u _l ) of M ⁿ at a point x in M, 1 ≥ l ≥ n−1, each u _i being in , we shall denote the volume of the parallelotope in the tangent space of M ⁿ at x with edges u ₁, u ₂,⋯, u _l by υ(α), and let . This is a continuous real function of t. Let

体上二次矩阵方程的解

本文得到了体上二次矩阵方程XAY= B有解的充要条件.并给出了当r(A)= r(B)时矩阵方程XAY= B的通解.     

中立双曲型时滞偏微分方程解振动的充要条件

本文讨论一类线性中立双曲型时滞微分方程解的振动性质，获得了其一切解振动的充要条件．     

A study of direct- and pulse-current chromium electroplating on rotating cylinder electrode (RCE)

Direct- and pulse-current (DC and PC) chromium electroplating on Cr-Mo steel were performed in a sulfate-catalyzed chromic acid solution at 50 °C using a rotating cylinder electrode (RCE). The electroplating cathodic current densities were at 30, 40, 50 and 60 A dm⁻², respectively. The relationship between electroplating current efficiency and the rotating speed of the RCE was studied. The cross-sectional microstructure of Cr-deposit was examined by transmission electron microscope (TEM). Results showed that DC-plating exhibited higher current efficiency than the PC-plating under the same conditions of electroplating current density and the rotating speed. We found the critical rotating speed of RCE used in the chromium electroplating, above this rotating speed the chromium deposition is prohibited. At the same plating current density, the critical rotating speed for DC-plating was higher than that for PC-plating. The higher plating current density is, the larger difference in critical rotating speeds appears between DC- and PC-electroplating. Equiaxed grains, in a nanoscale size with lower dislocation density, nucleate on the cathodic surface in both DC- and PC-electroplating. Adjacent to the equiaxed grains, textured grains were found in other portion of chromium deposit. Fine columnar grains were observed in the DC-electroplated deposit. On the other hand, very long slender grains with high degree of preferred orientation were detected in PC-electroplated deposit.     

Crystal structure of (1,10-phenanthroline-N,N')(1,4,7,10-tetraazacyclododecane-N,N',N",N"')nickel(II) diperchlorate.

The crystal structure of the title compound, [Ni(C8H20N4)(C12H8N2)](ClO4)2, has been determined by X-ray diffraction. The Ni(II) ion is six coordinated with four nitrogen atoms of the tetradentate macrocyclic ligand and two nitrogen atoms of the bidentate ligand in a distorted octahedron geometry. The folded tetradentate macrocyclic ligand adopts a configuration having four five-membered chelate rings in distorted eclipsed conformations. The four hydrogen atoms of the amine groups of the macrocyclic ligand are on the same side towards the bidentate ligand.     

Synthesis, characterization and magnetic properties of novel μ-isophthalato oxovanadium(IV) binuclear complexes

Four novel oxovanadium(IV) binuclear complexes have been synthesized, namely [(VO)₂(IPHTA) (L)₂SO₄ (L denotes 2,2′-bipyridine (bpy); 1,10-phenanthroline (phen); 4,4′-dimethyl-2,2′-bipyridine (Me₂bpy) and 5-nitro-1,10-phenanthroline (NO₂-phen)), where IPHTA is the isophthalate dianon. Based on elemental analyses, molar conductivity measurements, IR and electronic spectra studies, it is proposed that these complexes have IPHTA-bridged structures and consist of two vanadium(IV) atoms in a square-pyramidal environment. The complexes [(VO)₂(IPHTA)(Me₂bpy)₂]SO₄ (1) and [(VO)₂(IPHTA)(bpy)₂]SO₄ (2) were characterized by variable temperature magnetic susceptibility (4–300 K) and the data could be well fitted by the least-squares method to a susceptibility equation derived from the spin Hamiltonian operator, . The exchange integral, J, was found to be −26.8 cm⁻¹ for (1) and −31.0 cm⁻¹ for (2). These results are commensurate with antifferomagnetic interactions between two oxovanadium(IV) ions within each molecule. The influence of different terminal ligands on magnetic interactions between the metals of this kind of complexes is also discussed.     

Microstructure and photoluminescence of a-SiO x :H

Two strong photoluminescence (PL) bands in the spectral range of 550–900 nm have been observed at room temperature from a series of a-SiO _x :H films fabricated by plasma-enhanced chemical vapor deposition (PECVD) technique. One is composed of a main band in the red-light region and a shoulder; the other is located at about 850 nm, only found after 1170°C annealing in N₂ atmosphere. In conjunction with infrared (IR) and micro-Raman spectra, it is thought that the two PL bands are associated with a-Si clusters in the SiO _x network and nanocrystalline silicon in SiO₂, respectively. Project supported by the National Natural Science Foundation of China (Grant No. 69576025).