Minimising the number of gap-zeros in binary matrices

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.     

Upper (lower) bounds of the eigenvalues,spread and the open problems for the real symmetric interval matrices

This work is concerned with exploring the upper bounds and lower bounds of the eigenvalues of real symmetric matrices of order n whose entries are in a given interval. It gives the maximum and minimum of the eigenvalues and the upper bounds of spread of real symmetric interval matrices in all cases. It also gives the answers of the open problems for the maximum and minimum of the eigenvalues of real symmetric interval matrices. Copyright © 2012 John Wiley & Sons, Ltd.     

RANK t ℋ-PRIMES IN QUANTUM MATRICES

ABSTRACT

Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (2002 Goodearl , K. R. , Lenagan , T. H. ( 2002 ). Prime ideals invariant under winding automorphisms in quantum matrices . Internat. J. Math 13 : 497 – 532 . [CROSSREF]  [Google Scholar]) have shown that the number of ?-primes in R = O _q (? _n (𝕂)) that contain all (t + 1) × (t + 1) quantum minors but not all t × t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!) ² S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (1997 Kaneko , M. ( 1997 ). Poly-Bernoulli numbers . J. Théorie Nombres Bordeaux 9 : 221 – 228 . [Google Scholar]) and Arakawa and Kaneko (1999 Arakawa , T. , Kaneko , M. ( 1999 ). On poly-Bernoulli numbers . Comment Math. Univ. St. Paul 48 ( 2 ): 159 – 167 . [Google Scholar]).     

Error Bounds for Low-Rank Approximations of the First Exponential Integral Kernel

A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented.     

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.     

Generating Birth and Death Processes

Associated with an ordered sequence of an even number 2N of positive real numbers is a birth and death process (BDP) on {0, 1, 2,…, N} having these real numbers as its birth and death rates. We generate another birth and death process from this BDP on {0, 1, 2,…, 2N}. This can be further iterated. We illustrate with an example from tan(kz). In BDP, the decay parameter, viz., the largest non-zero eigenvalue is important in the study of convergence to stationarity. In this article, the smallest eigenvalue is found to be useful.     

Field-theoretical Renormalization-Group approach to critical dynamics of crosslinked polymer blends

We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, ζ. First, by averaging over ζ, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q ⁻¹, a time length scale R(t) ∼ t ^1/z , and the mesh size ξ ^*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of ξ ^*, the dynamics is stopped. The final time, t ^*, then scales as t ^* ∼ ξ ^{* z}, with the dynamic exponent z = 6−η. Here, η is the usual Ising critical exponent. Since the final size of microdomains ξ ^* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.     

单分散键合型含铕聚苯乙烯微球的制备与荧光传感性能

制备了一种可定性定量检测水溶液中三价铁离子的含铕聚苯乙烯微球, 分别用固体核磁碳谱(¹³C CP/MAS NMR)、 傅里叶变换红外光谱(FTIR)、 X射线光电子能谱(XPS)、 扫描电子显微镜(SEM)、 透射电子显微镜(TEM)、 元素分析、 粒度分析和ζ电位分析等对其化学组成和结构形貌进行表征. 当铕配合物单体用量低于2.5%时, 可以得到稳定的单分散键合型含铕聚苯乙烯微球. 用紫外光激发时, 该含铕聚苯乙烯微球发射铕离子的特征红光. Fe³⁺能猝灭该微球的荧光, 酸根离子和其它金属离子对其干扰较少; 猝灭效率与Fe³⁺浓度在0~300 μmol/L浓度范围内均呈线性关系; 随着铕配合物单体用量的增加, 微球的荧光增强, 其在检测Fe³⁺的荧光时, 猝灭常数(K_SV)增加, 检测限(LOD)下降. 调节铕配合物单体的用量, 可获得热性能优异、 红光发射强度高且稳定性好的单分散聚苯乙烯荧光微球, 对Fe³⁺荧光检测显示出较高的选择性, 在生物检测和环境保护等领域具有较高的应用价值.