Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

On Haar measurable additive maps on B(N)

Let B(N) be the power set of the set of natural numbers endowed with the usual structure of a compact abelian group and letμ be the normed Haar measure on it. The paper studies the properties ofμ-measurable additive maps of B(N) into Hausdorff abelian groups.     

De novo structural prediction of transition metal complexes: application to technetium

De novo structural prediction of transition metal complexes is investigated. Technetium complexes are chosen given their importance in medical imaging and nuclear waste remediation and for the chemical diversity they display. A new conformational searching algorithm (LIGB) for transition metals is described that allows one to search for different conformational and geometric isomers within a single simulation. In the preponderance of cases, both conformational searching techniques (LIGB and high-temperature molecular dynamics/simulated annealing) provide comparable results, while LIGB is superior for macrocyclic complexes. A genetic algorithm-optimized PM3(tm) parametrization for Tc is compared with the standard implementation and found to yield a significant improvement in predictive ability for the most prevalent Tc structural motifs. The utility of a coupled molecular mechanics-semiempirical quantum mechanics protocol is demonstrated for very rapid, efficient, and effective de novo prediction of transition metal complex geometries.     

Low-coordinate chromium siloxides: the "box" [Cr(mu-Cl)(mu-OSitBu3)]4, distorted trigonal [(tBu3SiO)3Cr][Na(benzene)] and [(tBu3SiO)3Cr][Na(dibenzo-18-c-6)], and trigonal (tBu3SiO)3Cr

Treatment of CrCl(2)(THF)(2) with NaOSi(t)Bu(3) afforded the tetrameric "box" [Cr(mu-Cl)(mu-OSi(t)Bu(3))](4) (1, X-ray). THF cleaved 1 to provide trans-(silox)ClCr(THF)(2) (2), whereas degradation of 1 with 4-picoline caused disproportionation and the generation of trans-Cl(2)Cr(4-pic)(2) and trans-(silox)(2)Cr(4-pic)(x) (n = 2, 3; 3, 3-4-pic). Chromous centers in 1 were antiferromagnetically coupled, and density functional calculations on the high-spin (multiplicity = 17) model [Cr(mu-Cl)(mu-OH)](4) (1') revealed that its singly occupied 3d orbitals spanned an energy range of approximately 2 eV. The addition of 8 equiv of Na(silox) to 1 yielded [((t)Bu(3)SiO)Cr(mu-OSi(t)Bu(3))(2)]Na.C(6)H(6) (4, Y shaped, angle OCrO(Na) = 91.28(7) degrees), and treatment of 4 with dibenzo-18-crown-6 produced [(silox)(3)Cr][Na(dibenzo-18-crown-6)] (5, angle OCrO = approximately 120 degrees, (120 + alpha) degrees, (120 - alpha) degrees). Calculations of [((t)Bu(3)SiO)Cr(mu-OSi(t)Bu(3))(2)]Na (4') and Cr(silox)(3)(-) (5') provided reasonable matches with the experimental geometries (X-ray). The trigonal chromic derivative (silox)(3)Cr (6) was synthesized from CrCl(3)(THF)(3) for structural and calculational comparisons to the chromous derivatives.     

Stability studies of transition-metal linkage isomers using quantum mechanical methods. Groups 11 and 12 transition metals

Several hypotheses to elucidate the linkage isomer preference of the thiocyanate (SCN(-)) ion have been offered. For complexes with small coordination numbers (i.e., 1 and 2) and groups 11 (Cu-triad) and 12 (Zn-triad) metals, different levels of theory and a variety of basis sets have been employed to study linkage isomerism. Similar results are obtained for all density functionals tested, pure and hybrid. Overall, good agreement, vis-à-vis experimentally identified linkage isomers, is achieved for ab initio techniques, whereas semiempirical quantum mechanical methods show a bias toward S-ligated isomers. Despite the seeming ease for the a priori prediction of the most stable thiocyanate isomers using acid/base principles, this research highlights the sensitivity of quantitative calculations of transition-metal linkage isomerism to the choice of basis set and electron correlation, particularly with post-Hartree-Fock treatments.     

Synthesis and Characterization of Ge(II), Sn(II), and Pb(II) Monoamides with -NH2 Ligands

The synthesis and characterization of the first divalent germanium, tin, and lead monoamide derivatives of the parent amide group -NH(2) are presented. They have the general formula (ArMNH(2))(2) (M = Ge, Ar = Ar'(C(6)H(3)-2,6-Pr(i)(2)) or Ar* (C(6)H(3)-2,6(C(6)H(2)-2,4,6-Pr(i)(3))); M = Sn, Ar = Ar*; M = Pb, Ar = Ar*). For germanium and tin, they were obtained by reacting the corresponding terphenyl halides of the group 14 elements with liquid ammonia in diethyl ether. The lead amide derivative (Ar*PbNH(2))(2) was synthesized by reaction of LiNH(2) with Ar*PbBr in diethyl ether. The compounds were characterized by IR and multinuclear NMR spectroscopies and by X-ray crystallography in the case of the (Ar'GeNH(2))(2) or (Ar*SnNH(2))(2) derivatives. They possess dimeric structures with two -NH(2) groups bridging the germanium and tin centers. For lead, the reaction with ammonia led to isolation of a stable ammine complex of formula Ar*PbBr(NH(3)) which was characterized by IR and NMR spectroscopies and by X-ray crystallography. It is the first structural characterization of a divalent lead ammine complex.     

Effect of conformational parameters on physical properties of polymers containing pendant phenoxyphthalonitrile substituents

Heteroaromatic polymers are considered to be high performance organic materials due to their unique and highly attractive properties, including outstanding thermal and mechanical resistance, that arise from their aromatic structure and strong interactions between macromolecular chains. Modification or designing new molecular architectures with tailored physico-chemical characteristics allows expanding the applications of these materials in various advanced technologies. Herein, a series of polymers containing bulky phenoxyphthalonitrile pendant units was synthesized and their physical properties were studied and correlated with their conformational parameters, as well as free and van der Waals volumes. For comparison, the related polymers without lateral moieties were also investigated to highlight the effect of bulky substituent on the polymer rigidity. Thus, it is shown that conformational rigidity determines the packing of macromolecules in solid state, and, therefore, the free volume, glass transition, and decomposition temperatures. The values found experimentally for T _g correlate well with those obtained using the conformational rigidity parameters. The dependence of T _g of these polymers on Kuhn segment is described by linear equations, with very good factors of convergence. The correlations established by Monte Carlo method allow obtaining the T _g values for related polymers where the experimental measurement of this parameter is difficult.     

Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic–plastic formulation

This paper proposes an extension of the SHB8PS solid–shell finite element to large strain anisotropic elasto-plasticity, with application to several non-linear benchmark tests including sheet metal forming simulations. This hexahedral linear element has an arbitrary number of integration points distributed along a single line, defining the “thickness” direction; and to control the hourglass modes inherent to this reduced integration, a physical stabilization technique is used. In addition, the assumed strain method is adopted for the elimination of locking. The implementation of the element in Abaqus/Standard via the UEL user subroutine has been assessed through a variety of benchmark problems involving geometric non-linearities, anisotropic plasticity, large deformation and contact. Initially designed for the efficient simulation of elastic–plastic thin structures, the SHB8PS exhibits interesting potentialities for sheet metal forming applications—both in terms of efficiency and accuracy. The element shows good performance on the selected tests, including springback and earing predictions for Numisheet benchmark problems.