几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

Oriented hypergraphs: Balanceability

An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camion's algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.     

k圈图的最大Laplace分离度

设$ G $是一个$ n $阶$ k $圈图, $ k $圈图为边数等于顶点数加$ k-1 $的简单连通图。$ \mu_{1}(G) $、$ \mu_{2}(G) $分别记为图$ G $的Laplace矩阵的最大特征值和次大特征值, 图$ G $的Laplace分离度定义为$ S_{L}(G)=\mu_{1}(G)-\mu_{2}(G) $。本文研究了给定阶数的$ k $圈图的最大Laplace分离度, 并刻画了相应的极图, 其结果推广了已有当$ k=1, 2, 3 $时的结论。     

分块算子矩阵的加权EP

本文在加权广义Schur补的基础上, 引入并研究了Hilbert空间上分块算子矩阵的加权Moore-Penrose逆和加权EP. 进一步, 给出了加权EP算子在算子方程中的一个应用.     

Fastiterative methods for least squares estimations

Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions() of the process is known, we prove that ifs() is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.Research supported in part by HKRGC grant no. 221600070, ONR contract no. N00014-90-J-1695 and DOE grant no. DE-FG03-87ER25037.     

The covariance matrix of the Potts model: A random cluster analysis

We consider the covariance matrix,G ^mm =q ²<(_x,m);(_y,m)>, of thed-dimensionalq-states Potts model, rewriting it in the random cluster representation of Fortuin and Kasteleyn. In any of theq ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometric quantities. We also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers. For dimensiond=2, we show that this correlation length and the correlation length of the two-point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two. For systems with first-order transitions, this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self-dual point _o. For systems with second order transitions, this relation implies the equality of the correlation length exponents from above and below threshold, as well as an amplitude ratio of two. In the course of proving the above results, we establish several properties of independent interest, including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions, and left continuity of the two-dimensional free boundary condition percolation probability at _o. We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events. The first is similar to the FKG inequality, but holds for events which are neither increasing nor decreasing; the second is similar to the van den Berg-Kesten inequality in standard percolation. Both inequalities hold for an arbitrary FKG measure.     

Method of polarization reflection matrix measurement in the submillimeter wave range

Homodyne method of measurement of polarization reflection matrix, providing the possibility of simultaneous measurement of all four complex coefficients of polarization reflection matrix in submillimeter quasi-optical (QO) circuits is presented. Technical realizability of the method for QO waveguides of the class of "hollow dielectric wavequide" is shown.     

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

We study methods for solving the constrained and weighted least squares problem min _x by the preconditioned conjugate gradient (PCG) method. HereW = diag (₁, , _m ) with _{1 m} 0, andA ^T = [T ₁ ^T , ,T _k ^T ] with Toeplitz blocksT _l R ^{n × n} ,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A ^T = 0, whereM =W ^–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E.     

Whitehead test modules

A (right -) module is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module , implies is projective. Dually, i-test modules are defined. For example, is a p-test abelian group iff each Whitehead group is free. Our first main result says that if is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring , there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring.

A non-semisimple ring is said to be fully saturated (-saturated) provided that all non-projective (-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, . The four parameters involved here are skew-fields and , and natural numbers . For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of -modules.

     

Dimer asymmetry in superoxide dismutase studied by molecular dynamics simulation

Summary Molecular dynamics (MD) simulations of 100 ps have been carried out to study the active-site behaviour of the Cu,Zn superoxide dismutase dimer (SOD) in water. The active site of each subunit was monitored during the whole simulation by calculating the distances between functional residues and the catalytic copper. The results indicate that charge orientation is maintained at each active site but the solvent accessibility varies. Analysis of the MD simulation, carried out by using the atomic displacement covariance matrix, has shown a different intra-subunit correlation pattern for the two monomers and the presence of inter-subunit correlations. The MD simulation presented here indicates an asymmetry in the two active sites and different dynamic behaviour of the two SOD subunits.