（n-3）-Filiform李代数的极大环面

求出了（n-3）-filiform李代数的极大环面，并证明了（n-3）-filiform李代数是可完备化的．     

Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

Chaos in the softening duffing system under multi-frequency periodic forces

The chaotic dynamics of the softening-spring Duffing system with multi-frequency external periodic forces is studied. It is found that the mechanism for chaos is the transverse heteroclinic tori. The Poincare map, the stable and the unstable manifolds of the system under two incommensurate periodic forces were set up on a two-dimensional torus. Utilizing a global perturbation technique of Melnikov the criterion for the transverse interaction of the stable and the unstable manifolds was given. The system under more but finite incommensurate periodic forces was also studied. The Melnikov‘s global perturbation technique was therefore generalized to higher dimensional systems. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies will increase the area in parameter space where chaotic behavior can occur.     

A torus bifurcation theorem with symmetry

A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decouple into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D₃) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptotically stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory.     

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.     

Toric cycles in the complement to a complex curve in (C×)2

The amoeba of a complex curve in the 2‐dimensional complex torus is its image under the projection onto the real parts of the logarithmic coordinates. A toric cycle in the complement to a curve is a fiber of this projection over a point in the complement to the amoeba of the curve. We consider amoebas of complex algebraic curves defined by so‐called Harnack polynomials. We prove that toric cycles are homologically independent in the complement to a such curve.     

Magnetic Torus Microreactor as a Novel Device for Sample Treatment via Solid-Phase Microextraction Coupled to Graphite Furnace Atomic Absorption Spectroscopy: A Route for Arsenic Pre-Concentration

This work studied the feasibility of using a novel microreactor based on torus geometry to carry out a sample pretreatment before its analysis by graphite furnace atomic absorption. The miniaturized retention of total arsenic was performed on the surface of a magnetic sorbent material consisting of 6 mg of magnetite (Fe₃O₄) confined in a very small space inside (20.1 µL) a polyacrylate device filling an internal lumen (inside space). Using this geometric design, a simulation theoretical study demonstrated a notable improvement in the analyte adsorption process on the solid extractant surface. Compared to single-layer geometries, the torus microreactor geometry brought on flow turbulence within the liquid along the curvatures inside the device channels, improving the efficiency of analyte–extractant contact and therefore leading to a high preconcentration factor. According to this design, the magnetic solid phase was held internally as a surface bed with the use of an 8 mm-diameter cylindric neodymium magnet, allowing the pass of a fixed volume of an arsenic aqueous standard solution. A preconcentration factor of up to 60 was found to reduce the typical “characteristic mass” (as sensitivity parameter) determined by direct measurement from 53.66 pg to 0.88 pg, showing an essential improvement in the arsenic signal sensitivity by absorption atomic spectrometry. This methodology emulates a miniaturized micro-solid-phase extraction system for flow-through water pretreatment samples in chemical analysis before coupling to techniques that employ reduced sample volumes, such as graphite furnace atomic absorption spectroscopy.     

厚壁空心微球的球形度和壁厚均匀性的表征研究

为建立高效简洁的厚壁空心微球球形度和壁厚均匀性的表征和测量方法,以内径850 m、壁厚25 m的厚壁聚苯乙烯（PS）微球为测量对象,分析球形度和壁厚均匀性的表征方法,探讨测量过程中采样方式对表征微球球形度和壁厚均匀性的影响。研究表明：对于批量微球球形度和壁厚均匀性的表征,当抽样数目不小于30时,测量结果之间的差异很小;而对于单一微球的球形度和壁厚均匀性的表征,至少3个不同投影面的测量结果的平均值才趋于稳定。不同投影面导致球形度或壁厚均匀性较差的批次样品的测量结果有一定差异,多次投影测量可提高测量结果的可靠性。     

On Completely Integrable Systems with Local Torus Actions

This paper concerns with symplectic topology of compact completely integrable Hamiltonian systems with only stable nondegenerate elliptic singularities. We describe all systems whose universal coverings admit actian-angle coordinates and, in particular, prove that some finite cover of a base space is diffeomorphic to a product of a convex polytope and a solvmanifold. We also construct an obstruction, vanishing of which guarantees splitting of some finite cover of a phase space as a toric variety and a torus fibering over a solvmanifold.     

An Estimate of Smoothing and Composition with Applications to Conjugation Problems

An estimation of the difference between smoothing a composition of two functions and composing their smoothings is given. The smoothing operator considered here is defined by the convolution operator with a holomorphic kernel. It is also shown, by means of an example, how to apply the estimate given here to obtain finite differentiable versions of theorems on conjugation of maps in non-perturbative settings.