S~1 × S~1-Borsuk-Ulam Theorem on Product Spaces

Ｓ~１×Ｓ~１－Ｂｏｒｓｕｋ－ＵｌａｍＴｈｅｏｒｅｍｏｎＰｒｏｄｕｃｔＳｐａｃｅｓＺｈｏｎｇＣｈｅｎｇｋｕｉ（钟承奎）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＬａｎｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｌａｎｚｈｏｕ，７３００００）Ａｂｓｔｒａｃｔ：...     

Existence of Positive Solutions for the Singular Equation (ϕ p (y′))′ + g(t,y, y′) = 0

Two theorems concerning the existence of positive solutions for the singular equation are presented. The results are obtained by using the nonlinear Leray-Schauder alternative and the lower-upper solution method.     

Rainbows and glories in the angular scattering of the state-to-state F + H2 reaction at E(trans)=0.04088 eV

State-of-the-art differential cross sections (DCSs) have been reported by Wang et al. [Proc. Nat. Acad. Sci. (U.S.), 2008, 105, 6227] for the state-to-state F + H(2)→ FH + H reaction using fully quantum-state-selected crossed molecular beams. We theoretically analyze the angular scattering of this reaction, in order to quantitatively understand the physical content of structure in the DCSs. Three transitions are studied, v(i)=0, j(i)=0, m(i)=0 → v(f)=3, j(f)=0, 1, 2, m(f)=0 at a translational energy of 0.04088 eV, where v, j, m are the vibrational, rotational and helicity quantum numbers respectively for the initial and final states. The input to our analyses consists of accurate quantum scattering (S) matrix elements computed for the Fu-Xu-Zhang potential energy surface, as used by Wang et al. in a computational simulation of their experimental DCSs. We prove that the pronounced peak at forward angles observed in the experimental and simulated DCSs for all three transitions is a glory. At larger angles, it is demonstrated that the 000 → 300 and 000 → 310 DCSs both possess a broad farside rainbow, which is accompanied by diffraction oscillations. We confirm the conjecture of Wang et al. that these diffraction oscillations arise from nearside-farside (NF) interference. We find that the reaction is N dominant for all three transitions. The theoretical techniques used to analyze the angular scattering include uniform semiclassical theories of glory and of rainbow scattering. We also make the first application of a semiclassical formula that is uniform for both glory + rainbow scattering. In addition, structure in the DCSs is analyzed using NF theory and local angular momentum theory, in both cases with three resummations of the partial wave series for the scattering amplitude. We make the first explicit application of the Thiele rational interpolation formula to extract the position and residue of the leading Regge pole from a set of S matrix elements, thereby making contact with complex angular momentum theories of DCSs, which interpret the angular scattering in terms of Regge resonances. Our calculations complement the exit-valley vibrationally-adiabatic analysis of Wang et al.     

The attractors for the nonhomogeneous nonautonomous Navier-Stokes equations

In this paper, we consider the attractors for the two-dimensional nonautonomous Navier-Stokes equations in nonsmooth bounded domain Ω with nonhomogeneous boundary condition u=φ on ∂Ω. Assuming , which is translation compact and φ∈L^∞(∂Ω), we establish the existence of the uniform attractor in L²(Ω) and .     

Attractors for the nonclassical diffusion equations with fading memory

We discuss long-time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical. The existence and regularity of global attractors in weak topological space and strong topological space are obtained, while the forcing term only belongs to H⁻¹(Ω) and L²(Ω) respectively. The results in this part are new and appear to be optimal corresponding to the forcing term.     

Boundary behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up

By the Karamata regular variation theory and constructing comparison function, we show the exact asymptotic behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up.     

Method validation,residue analysis and dietary risk assessment of trifloxystrobin and trifloxystrobin acid in milk,eggs and pork

Trifloxystrobin (TFS) is a widely used strobilurin fungicide and its residues accumulating in animal-derived food could result in potential harm to consumers. By optimization of extraction solvents and cleanup sorbents, a residue analysis method for TFS and its metabolite trifloxystrobin acid (TFSA) was established in milk, eggs and pork based on QuEChERS sample preparation and LC–MS/MS. The calibration curves exhibited good linearity with determination coefficients (R²) >0.9930 over the range of 0.5–250 ng/ml for both TFS and TFSA. The recoveries of the two analytes were 81–100% with RSD 3–10% and 76–96% with RSD 2–13%, respectively. The limit of quantification (LOQ) was 1 ng/g for both analytes. The milk, egg and pork samples, 30 each, were collected from the 30 main producing regions in China, and residues of TFS and TFSA were analyzed. The concentrations of both analytes were lower than the corresponding LOQs and maximum residue limits. Long-term dietary risk assessment showed that the hazard quotients were 0.001–0.003%, indicating an absence of unacceptable risks in milk, eggs and pork to the health of common consumers in China.     

Luminescent properties and applications of Tb3+ doped silicate glasses with industrial scales

Tb³⁺ doped X-ray conversion glassy screen with an industrial scale (50 mm × 50 mm × 12 mm) was successfully fabricated, and its luminescent properties and applications in CCD imaging system were investigated. Results showed that Tb³⁺ doped silicate glasses mainly emit weak blue (400–460 nm) and strong green (480–570 nm) fluorescence. With the increase of Tb³⁺ ion concentration, the intensity of green emission increases, but that of blue emission decreases. Gd³⁺ ions can sensitize the luminescence of Tb³⁺ ions among silicate glasses. With the increase of CeO₂ concentration, the luminescent intensity of Tb³⁺ doped silicate glasses at 550 nm quickly decreases. However, the irradiation resistance of Tb³⁺ doped silicate glasses can be effectively improved by CeO₂ addition. The imaging quality of the luminescent glass screen is more excellent than that of Gd₂O₂S polycrystalline screens.     

Global attractors for plate equations with critical exponent in locally uniform spaces

A plate equation with critical exponent in locally uniform spaces with a coefficient β(x) belonging to the locally uniform spaces is studied. This equation is shown to generate a dissipative semigroup in locally uniform spaces , which possesses global attractors in weighted spaces .     

Local angular momentum-local impact parameter analysis: derivation and properties of the fundamental identity, with applications to the F+H2, H+D2, and Cl+HCl chemical reactions

The technique of local angular momentum-local impact parameter (LAM-LIP) analysis has recently been shown to provide valuable dynamical information on the angular scattering of chemical reactions under semiclassical conditions. The LAM-LIP technique exploits a nearside-farside (NF) decomposition of the scattering amplitude, which is assumed to be a Legendre partial wave series. In this paper, we derive the "fundamental NF LAM identity," which relates the full LAM to the NF LAMs (there is a similar identity for the LIP case). Two derivations are presented. The first uses complex variable techniques, while the second exploits an analogy between the motion of the scattering amplitude in the Argand plane with changing angle and the classical mechanical motion of a particle in a plane with changing time. Alternative forms of the fundamental LAM-LIP identity are described, one of which gives rise to a CLAM-CLIP plot, where CLAM denotes (Cross section) x LAM and CLIP denotes (Cross section) x LIP. Applications of the NF LAM theory, together with CLAM plots, are reported for state-to-state transitions of the benchmark reactions F+H2-->FH+H, H+D2-->HD+D, and Cl+HCl-->ClH+Cl, using as input both numerical and parametrized scattering matrix elements. We use the fundamental LAM identity to explain the important empirical observation that a NF cross section analysis and a NF LAM analysis provide consistent (and complementary) information on the dynamics of chemical reactions.