Modified Boussinesq System with Variable Coefficients： Classical Lie Approach and Exact Solutions

The Lie-group formalism is applied to investigate the symmetries of the modified Boussinesq system with variable coefficients. We derived the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The reduced systems of ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.     

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.     

On a motivic interpretation of primitive,variable and fixed cohomology

If M is a smooth projective variety whose motive is Kimura finite‐dimensional and for which the standard Lefschetz Conjecture B holds, then the motive of M splits off a primitive motive whose cohomology is the primitive cohomology. Under the same hypotheses on M, let X be a smooth complete intersection of ample divisors within M. Then the motive of X is the sum of a variable and a fixed motive inducing the corresponding splitting in cohomology. I also give variants with group actions.     

On the complexity of the Leibniz hierarchy

We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.     

On Generalized Algebraic Cone Metric Spaces and Fixed Point Theorems

In this paper, the author first introduces the concept of generalized algebraic cone metric spaces and some elementary results concerning generalized algebraic cone metric spaces. Next, by using these results, some new fixed point theorems on generalized (complete) algebraic cone metric spaces are proved and an example is given. As a consequence, the main results generalize the corresponding results in complete algebraic cone metric spaces and generalized complete metric spaces.     

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex. This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)     

由n次幂等矩阵确定的交换幺半群

设R是含幺结合环,n≥2为自然数.对所有的k≥1,本文给出了n次幂等矩阵集Pk^n（R）={P｜P^n=P∈Mk（R）}上的一种等价关系,证明了P^n（R）=∪k=1^∞Pk^n（R）中的等价类在给定的加法运算下构成一个交换幺半群.     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17     

基于代数滑移模型的光催化反应器内催化剂颗粒分布特性研究

针对光催化制氢反应器内液固两相流的特点,运用代数滑移模型,对反应器内的催化剂颗粒-水两相流动进行了数值模拟。采用分块的结构化网格,并运用多重网格方法进行求解。通过与前人实验中压力梯度结果的比较,对模型进行了验证,并最终得到了反应器内催化剂颗粒的典型分布及相应的类型图。