Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.     

Multiple-solution problems in a statistics classroom: an example

The mathematics education literature shows that encouraging students to develop multiple solutions for given problems has a positive effect on students’ understanding and creativity. In this paper, we present an example of multiple-solution problems in statistics involving a set of non-traditional dice. In particular, we consider the exact probability mass distribution for the sum of face values. Four different ways of solving the problem are discussed. The solutions span various basic concepts in different mathematical disciplines (sample space in probability theory, the probability generating function in statistics, integer partition in basic combinatorics and individual risk model in actuarial science) and thus promotes upper undergraduate students’ awareness of knowledge connections between their courses. All solutions of the example are implemented using the R statistical software package.     

Synthesis,Structure, and Reactivity of a Tetranuclear Cerium(IV) Oxo Cluster Supported by the Kläui Tripodal Ligand [Co(η5‐C5H5){P(O)(OEt)2}3]−

A tetranuclear Ce^IV oxo cluster compound containing the Kläui tripodal ligand [Co(η⁵‐C₅H₅){P(O)(OEt)₂}₃]^? (L_OEt^?) has been synthesized and its reactions with H₂O₂, CO₂, NO, and Brønsted acids have been studied. The treatment of [Ce(L_OEt)(NO₃)₃] with Et₄NOH in acetonitrile afforded the tetranuclear Ce^IV oxo cluster [Ce₄(L_OEt)₄O₇H₂] ( 1 ) containing an adamantane‐like {Ce₄(μ₂‐O)₆} core with a μ₄‐oxo ligand at the center. The reaction of 1 with H₂O₂ resulted in the formation of the peroxo cluster [Ce₄(L_OEt)₄(μ₄‐O)(μ₂‐O₂)₄(μ₂‐OH)₂] ( 2 ). The treatment of 1 with CO₂ and NO led to isolation of [Ce(L_OEt)₂(CO₃)] and [Ce(L_OEt)(NO₃)₃], respectively. The protonation of 1 with HCl, ROH (R=2,4,6‐trichlorophenyl), and Ph₃SiOH yielded [Ce(L_OEt)Cl₃] ( 3 ), [Ce(L_OEt)(OR)₃] ( 4 ), and [Ce(L_OEt)(OSiPh₃)₃] ( 5 ), respectively. The chloride ligands in 3 are labile and can be abstracted by silver(I) salts. The treatment of 3 with AgOTs (OTs^?=tosylate) and Ag₂O afforded [Ce(L_OEt)(OTs)₃] ( 6 ) and 1 , respectively. The electrochemistry of the Ce‐L_OEt complexes has been studied by using cyclic voltammetry. The crystal structures of complexes 1 – 5 have been determined.     

Deciphering Photoluminescence Dynamics and Reactivity of the Luminescent Metal–Metal‐Bonded Excited State of a Binuclear Gold(I) Phosphine Complex Containing Open Coordination Sites

Luminescent metal complexes having open coordination sites hold promise in the design of sensory materials and photocatalysts. As a prototype example, [Au₂(dcpm)₂)]²⁺ (dcpm = bis(dicyclohexylphosphanyl) is known for its intriguing environmental sensitive photoluminescence. By integrating a range of complementary ultrafast time‐resolved spectroscopy to interrogate the excited state dynamics, this study uncovers that the events occurring in extremely rapid timescales and which are modulated strongly by environmental conditions play a pivotal role in the luminescence behavior and photochemical outcomes. Formed independent of the phase and solvent property within ～0.15 ps, the metal–metal bonded ³5dσ*6pσ state is highly reactive possessing strong propensity toward increasing coordination number at Au^I center, and with ～510 ps lifetime in dichloromethane is able to mediate light induced C–X bond cleavage.     

Cavity ring down absorption spectroscopy of the BΣ-XΣ transition of YO

The absorption spectrum of the (2, 0) and (2, 1) bands of the B²Σ⁺-X²Σ⁺ transition of YO between 442 and 478 nm were recorded using laser vaporization/reaction with free-jet expansion and cavity ring down laser absorption spectroscopy. Local rotational perturbations have been found for both spin components of the v = 2 level of the B²Σ⁺ state. The observed perturbations could be ascribed to a degenerate perturbing state interacting with the B²Σ⁺ state. Least-squares fit of the observed upper state term values yielded molecular constants for the v = 2 level of the B²Σ⁺ state and the perturbing ²Π state. Earlier ab initio calculations [J. Chem. Phys. 89 (1988) 2160] indicated that the C²Π state is nearby, it is plausible that the C²Π state is the perturbing state.     

Drying methods for XPS analysis of PureVision™, Focus Night&Day™ and conventional hydrogel contact lenses

The surface composition of hydrogel contact lenses that contain silicon-based monomers, PureVision™ (balafilcon A) and Focus^® Night&Day™ (lotrafilcon A), were investigated by X-ray photoelectron spectroscopy (XPS). Conventional and daily disposable hydrogel lenses based on hydroxyethyl methacrylate (HEMA) were also studied, with the commonly prescribed 1-day Acuvue^® lens (etafilcon A) used as a control. All the lenses were pre-washed and dehydrated by three different methods, including drying in air, drying in nitrogen or freezing with subsequent freeze-drying, before the XPS analysis. The lenses dried in air had more impurities on the surface, and the lenses that were freeze-dried lost transparency, suggesting that drying lenses in nitrogen is the preferred preparation method for XPS analysis. Surface compositions for all lens materials were obtained and this data can be used as a control/base-value for future analysis of the interactions of soft contact lens materials with chemicals such as drugs or tear components.     

Some permanence properties of C-unique groups

We will study some permanence properties of C^*-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C^*-unique group. Moreover, any amenable discrete group is a retract of a discrete C^*-unique group.     

Synthesis and Characterization of Oligo(2,7-biphenylenylene-(E)-vinylene)s: Conjugated Oligomers Containing Antiaromatic 12-π Electron Units

The syntheses and the electronic properties of oligo(2,7-biphenylenylene-(E)-vinylene)s are reported. According to our spectral analysis, a limit of 2.5 ev for the peak band-gap of poly(2,7-biphenylenylene-(E)-vinylene)s is estimated. In addition, we discovered that oligo(2,7-biphenylenylene-(E)-vinylene)s are amphoteric and can be either oxidized or reduced electrochemically.     

Symplectic analysis for periodical electro-magnetic waveguides

Symplectic analysis is introduced into electro-magnetic waveguide theory, by using Hamiltonian system theory in which the transverse electric and magnetic field vectors are the dual vectors. The method can accommodate arbitrary anisotropic material and includes the interface conditions between adjacent segments of the waveguide. An electro-magnetic stiffness matrix is introduced which relates to the two ends of each segment of the waveguide. Both the pass- and stop-band stiffness matrices for plane waveguides with constant cross-section are given analytically and also a transformation matrix is given to permit abrupt changes of cross-section to occur. The variational principle is applied to obtain the segment combination algorithm needed to generate the electro-magnetic stiffness matrix related to the two ends of the fundamental periodical segment. Then the Wittrick-Williams algorithm is used to extract the eigenvalues. Thereafter, an energy band analysis is performed for a periodical waveguide, e.g., a grating, by using the symplectic eigensolutions.     

Crystallographic report: A two‐dimensional layer structure containing calcium: [Ca3(1,3,5‐benzenetricarboxylate)2(H2O)12]n

In the crystal structure, an eight‐coordinated calcium center is connected to adjacent seven‐coordinated calcium ions by a pair of µ‐aqua bridging ligands, leading to the formation of a chain, which is linked via benzenetricarboxylate ligands to neighboring chains to furnish a two‐dimensional layer structure. Layers are connected into a three‐dimensional structure facilitated by hydrogen bonding interactions. Copyright © 2004 John Wiley & Sons, Ltd.