Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.     

On pairwise Lindelöf bitopological spaces

In this paper, we shall continue the study of bitopological separation axioms begun by Kelly and obtained some results. Furthermore, we introduce two concepts of pairwise Lindelöf bitopological spaces and the properties for them are established. We also show that a pairwise Lindelöf space is not hereditary property.     

Gender‐Related Beliefs and Mathematics Performance of Preservice Primary Teachers

The aim of this study was to examine whether there are gender differences in mathematics achievement and in beliefs about mathematics of preservice teachers over a period of four years. Data were collected from preservice teachers (156 males and 155 females) from the Ad?yaman University Faculty of Education in Turkey. The Mathematics as a Gendered Domain instrument was used to investigate preservice teachers' beliefs about the gender differences in mathematics. The results indicated that gender had no effect on mathematics performances of the preservice primary teachers. Findings of this research show that most of the male and female preservice primary teachers do not gender‐stereotype mathematics and believe that mathematics is gender neutral, although there are gender differences on some types of items of instrument.     

On the linear stability study of zonal incompressible flows on a sphere

The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P_n(μ) of degree n is stable to infinitesimal perturbations of every invariant set I_m with |m| ≥ n. For each zonal number m, I_m is here the span of all the spherical harmonics $Y^{m}_{k}(x)$, whose degree k is greater than or equal to m. It is also shown that such small-scale perturbations are stable not only exponentially, but also algebraically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 649–665, 1998     

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

On smoothers for multigrid of the second kind

We study smoothers for the multigrid method of the second kind arising from Fredholm integral equations. Our model problems use nonlocal governing operators that enforce local boundary conditions. For discretization, we utilize the Nyström method with the trapezoidal rule. We find the eigenvalues of matrices associated to periodic, antiperiodic, and Dirichlet problems in terms of the nonlocality parameter and mesh size. Knowing explicitly the spectrum of the matrices enables us to analyze the behavior of smoothers. Although spectral analyses exist for finding effective smoothers for 1D elliptic model problems, to the best of our knowledge, a guiding spectral analysis is not available for smoothers of a multigrid of the second kind. We fill this gap in the literature. The Picard iteration has been the default smoother for a multigrid of the second kind. Jacobi‐like methods have not been considered as viable options. We propose two strategies. The first one focuses on the most oscillatory mode and aims to damp it effectively. For this choice, we show that weighted‐Jacobi relaxation is equivalent to the Picard iteration. The second strategy focuses on the set of oscillatory modes and aims to damp them as quickly as possible, simultaneously. Although the Picard iteration is an effective smoother for model nonlocal problems under consideration, we show that it is possible to find better than ones using the second strategy. We also shed some light on internal mechanism of the Picard iteration and provide an example where the Picard iteration cannot be used as a smoother.     

On representations and <Emphasis Type="Italic">K</Emphasis>-theory of the braid groups

 Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50     

Immobilization of immunoglobulin G in a highly oriented manner on a protein-A terminated multilayer system

In this study, we have fabricated a multilayer system consisting of 3-glycidoxypropyldimethylmethoxysilane (GPDS), poly(dimethylsiloxane) bis 3-aminopropyl terminated (PDMS) and protein-A on a silicon wafer surface for oriented immobilization of immunoglobilin G (IgG). The multilayer system with a different component in each layer was characterized by ellipsometry, contact-angle goniometer, X-ray photoelectron spectroscopy (XPS) and atomic force microscopy (AFM) and fluorescence microscopy. The epoxy-terminated monolayer was formed by the chemisorption of GPDS molecules on the hydroxylated silicon surface. The PDMS film about 4.5 nm thick was produced on the GPDS-monolayer by the chemical reaction between the amine groups at the end of PDMS chain and the epoxy groups of GPDS molecules. By introducing the PDMS chains, the hydrophilic character of GPDS-monolayer decreased. Study of the time dependence of polymer grafting showed that the chemisorption of GPDS is fast, whereas at least 16 h is needed to generate the homogeneous PDMS layer. For immobilization of IgG molecules in a highly oriented manner, protein-A molecules were first chemically bound to an ultrathin (∼4.5 nm) PDMS reactive polymer layer and later used to capture IgG. It was shown that the existence of protein-A in the multilayer system has a strong influence on the binding properties of IgG not only in the efficiency of binding, but also in its specificity. In conclusion, the multilayer system with protein-A has the potential to be further developed into an efficient immunoassay protein chip.     

Fluorescent aminoarylcyclotetraphosphazenes

In the present work, octachlorocyclotetraphosphazatetraene (1), N₄P₄Cl₈, is reacted with aniline (2), 1-napthylamine (4) and 2-aminoanthracene (6) to give octakis(arylamino)cyclotetraphosphazenes (3, 5 and 7). These cyclotetraphosphazene compounds (3, 5 and 7) have been fully characterized by elemental analysis, mass (MS), FT-IR, ¹H and ³¹P NMR spectroscopies. The molecular and crystal structures of 5 have been characterized by X-ray crystallography. The structure of 5 is monoclinic with the space group P21/c. The octakis(1-napthylamino)-(5) and octakis(2-aminoanthracene)-(7) cyclotetraphosphazene compounds have been synthesised for the first time in this study. The fluorescence properties of 3, 5 and 7 have been investigated in tetrahydrofuran (THF) and have been shown to have highly fluorescence behavior. This work also presents the quenching of arylamino substituted cyclotetraphosphazene derivatives (3, 5 and 7) by p-benzoquinone (BQ) or hydroquinone (HQ).