Inexact rational Krylov method for evolution equations

BIT Numerical Mathematics - Linear and nonlinear evolution equations have been formulated to address problems in various fields of science and technology. Recently, methods using an exponential...     

Dehydro[12]- and -[18]annulenes fused with tetrafluorobenzene: synthesis,electronic properties,packing structures,and reactivity in the solid state

Dehydro[12]- and -[18]annulenes 3 and 4 fused with tetrafluorobenzene were newly synthesized by the copper-mediated oxidative coupling of 1,2-diethynyltetrafluorobenzene. The UV-vis spectra of 3 and 4 showed the maximum absorption to be almost identical to that of the corresponding unsubstituted benzodehydro[12]- and -[18]annulenes 1 and 2, respectively, while the reduction waves in cyclic voltammetry observed at potentials of -1.48 and -1.56 V vs Fc/Fc(+) for 3 and 4 were less negative than those for 1 and 2. In agreement with these results, theoretical calculations (B3LYP/6-31G(d)) indicated that the HOMO-LUMO gap is similar for 1 and 3 and for 2 and 4 but that the LUMO levels of 3 and 4 are apparently lowered by the electronegative fluorine substituents. The X-ray crystallography of single crystals grown from 3 (crystal A), 3.C(6)H(6) (crystal B), and a mixture of 1 and 3 (crystal C) demonstrated that the molecules of 3 are stacked in a slanted manner in crystals A and B, while those of 1 and 3 form sandwichlike 1:2 complexes (3.1.3) that are stacked in a columnar arrangement in crystal C. Despite the suitable packing for topochemical polymerization, crystals A-C were quite stable against photochemical reaction. In contrast, differential scanning calorimetry showed that the thermal polymerization occurred explosively at 120-135 degrees C.     

X-ray energy dependence of the properties of the focused beams produced by polycapillary X-ray lens

We investigated X-ray energy distribution in an X-ray microbeam produced by a polycapillary X-ray lens in combination with a sealed-type X-ray tube. This polycapillary X-ray lens has an output focal distance (OFD) of approximately 15 mm. The size of the X-ray microbeam and its OFD were estimated by using a wire scanning method. In our case, the sizes of the X-ray microbeams at the output focal distance were 49 microm for Mo L(alpha), 36 microm for W L(alpha), and 28 microm for Mo K(alpha). The spot sizes depend on the energy of the X-ray fluorescence. The reason for the energy dependence is that X-ray capillary optics is based on the principle of propagation through glass capillaries by means of X-ray total external reflection. The evaluated OFD values of Mo L(alpha) and Mo K(alpha) were slightly changed in 17 microm. However, a deviation of 100 microm from the OFD caused only a 3% increase of the focal spot size. Therefore, we concluded that the OFD showed no significant dependence on X-ray energy.     

Improvement of spatial resolution of µ‐XRF by using a thin metal filter

An improvement of spatial resolution of µ‐XRF by using a thin metal filter was investigated. The size of the x‐ray beam focused by the polycapillary x‐ray lens depended on the energy of the characteristic x‐rays. Original spot sizes at the focal point were 48 µm for CrKα, 41 µm for NiKα, and 28 µm for MoKα, respectively. To make the x‐ray beam size small, Ti? Cu thin foil was placed between the output of the lens and the focal point as a metal filter to reduce the continuous x‐rays. Finally, the x‐ray microbeam size was improved to 30 µm by applying a filter. Clear 2D mapping images of Cr, Fe, and Ni in 300‐mesh stainless steel could be obtained by applying this filter. Copyright © 2008 John Wiley & Sons, Ltd.     

A Note on the GMRES Method for Linear Discrete Ill-Posed Problems

In this paper, we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES. It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems. In this paper we have focused on the residual norm and have come up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively. Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.     

A new scheme of computing the approximate inverse preconditioner for the reduced linear systems

In this paper, we propose a new implementation of the Newton scheme of an approximate preconditioner for the reduced linear system. In the original Newton scheme, the trouble is that the computation cost of the matrix–matrix product is always so expensive. On the other hand, the proposed implementation computes the preconditioner implicitly and reduces the cost of constructing the preconditioner by using the matrix–vector product form. We also show that the proposed implementation is less expensive than computing the preconditioner in explicit form.