Hydroxycarbonyl anion (m/z 45), a diagnostic marker for α‐hydroxy carboxylic acids

Collision‐induced dissociation mass spectra of anions derived from α‐hydroxy carboxylic acids (AHAs) show a diagnostic peak at m/z 45. Product ion spectra recorded from this m/z 45 ion confirm that it represents the hydroxycarbonyl anion ( ), and not the formate anion ( ) as sometimes described in the literature. For example, the formate anion is not only defiant to further fragmentation but is also unreactive toward CO₂. In contrast, the hydroxycarbonyl anion easily fragments to produce a peak at m/z 17 for the hydroxyl anion, and also readily reacts with CO₂ to produce a peak at m/z 61 for the bicarbonate anion. The hydrogen atom in the hydroxycarbonyl anion and that in the formate anion are not mobile within the skeletal framework of the ions, since the two ions did not manifest any interconversion under the conditions and time scales of our mass spectrometric experiments. The other significant product ion peak in the spectra of deprotonated AHAs represents a 46‐Da loss. MS/MS data from appropriately deuteriated compounds confirmed that one hydrogen atom from the C‐2 position, and the other from the hydroxy group are specifically removed for this loss of elements of formic acid. Moreover, the two oxygen atoms eliminated for the HCOOH loss originate exclusively from the carboxylate group. Copyright © 2008 John Wiley & Sons, Ltd.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

A new method of numerical evaluation of singular integrals occurring in two-dimensional BIEM

A new quadrature rule for integrands having logarithmical singularities has been developed. This rule proved to be efficient especially in the context of the BIEM.     

Legendre Gauss Spectral Collocation for the Helmholtz Equation on a Rectangle

A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation –u+(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satisfying Dirichlet boundary conditions. A matrix decomposition algorithm is developed for the solution of the collocation problem in the case the coefficient (x,y) is a constant. This algorithm is then used in conjunction with the preconditioned conjugate gradient method for the solution of the spectral collocation problem with the variable coefficient (x,y).     

Collision‐induced dissociation mass spectra of glucosinolate anions

Collision‐induced dissociation (CID) mass spectra of differently substituted glucosinolates were investigated under negative‐ion mode. Data obtained from several glucosinolates and their isotopologues (³⁴S and ²H) revealed that many peaks observed are independent of the nature of the substituent group. For example, all investigated glucosinolate anions fragment to produce a product ion observed at m/z 195 for the thioglucose anion, which further dissociates via an ion/neutral complex to give two peaks at m/z 75 and 119. The other product ions observed at m/z 80, 96 and 97 are characteristic for the sulfate moiety. The peaks at m/z 259 and 275 have been attributed previously to glucose 1‐sulfate anion and 1‐thioglucose 2‐sulfate anion, respectively. However, based on our tandem mass spectrometric experiments, we propose that the peak at m/z 275 represents the glucose 1‐thiosulfate anion. In addition to the common peaks, the spectrum of phenyl glucosinolate (β‐D ‐Glucopyranose, 1‐thio‐, 1‐[N‐(sulfooxy)benzenecarboximidate] shows a substituent‐group‐specific peak at m/z 152 for C₆H₅‐C(?NOH)S^?, the CID spectrum of which was indistinguishable from that of the anion of synthetic benzothiohydroxamic acid. Similarly, the m/z 201 peak in the spectrum of phenyl glucosinolate was attributed to C₆H₅‐C(?S)OSO₂^?. Copyright © 2009 John Wiley & Sons, Ltd.     

A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

Negative refraction

Light bends the wrong way in materials where both ε and μ are negative as was pointed out in 1968, but the absence of natural materials with this property led to neglect of the subject until 1999 when it was shown how to make artificial materials, metamaterials, with negative μ. The rapid advance of the subject since that date, both in theory and experiment, is reflected in the exponential growth of publications now at the 200 per year level and still growing. This interest is explained by the sudden availability of a qualitatively different class of electromagnetic materials combined with the quite startling properties which these materials appear to have; all of which provokes debate as each new facet of their behaviour is revealed. Experiment has been vital to resolution of controversy and has chiefly been in the microwave region of the spectrum though there is potential in the optical region currently being explored by several groups.     

Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.     

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C¹ smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.