Higher derivations of ore extensions

Let R be a prime ring and δ a derivation of R. Divided powers $ D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} $ D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that $ S^{\underline{\underline {def.}} } Q[x;\delta ] $ S^{\underline{\underline {def.}} } Q[x;\delta ] is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form

Discreteness and rationality of F-jumping numbers on singular varieties

We prove that the F-jumping numbers of the test ideal t(X; D, \mathfraka^t){\tau(X; \Delta, \mathfrak{a}^t)} are discrete and rational under the assumptions that X is a normal and F-finite scheme over a field of positive characteristic p, K _X  + Δ is \mathbb Q{\mathbb {Q}}-Cartier of index not divisible p, and either X is essentially of finite type over a field or the sheaf of ideals \mathfraka{\mathfrak{a}} is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.     

Prime radicals of constants of algebraic derivations

Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors ${\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}}$ . Set ? to be the maximum of n _i . Let ${R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}}$ be the subring of constants of δ on R. Denote the prime radical of a ring A by ${{\mathcal{P}}(A)}$ . It is shown among other things that $${\mathcal{P}}(R^{(\delta)})^{2^\ell-1}=0\quad\text{and}\quad{\mathcal{P}}(R^{(\delta)})=R^{(\delta)}\cap {\mathcal{P}}(C_R(b))$$ .     

Preparation, crystal structure, and thermal stability of the cadmium sulfide nanoclusters Cd6S44+ and Cd2Na2S4+in the sodalite cavities of zeolite A (LTA)

The crystal structure and thermal stability of two cadmium sulfide nanoclusters prepared in zeolite A (LTA) have been studied by XPS, TGA, and single-crystal and powder XRD. The crystal structures of Cd2.4Na3.2(Cd6S4)0.4(Cd2Na2S)0.6(H2O)> or =5.8[Si12Al12O48]-LTA (a = 12.2919(7) A, crystal 1 (hydrated)) and /Cd4Na2(Cd2O)(Na2O)/[Si12Al12O48]-LTA (a = 12.2617(4) A, crystal 2 (dehydrated)) were determined by single-crystal methods in the cubic space group Pm3m at 294(1) K. Crystal 1 was prepared by ion exchange of Na12-LTA in an aqueous stream 0.05 M in Cd2+, followed by washing in a stream of water, followed by reaction in an aqueous stream 0.05 M in Na2S. Crystal 2 was made by dehydrating crystal 1 at 623 K and 1 x 10(-6) Torr for 3 days. In crystal 1, Cd6S4(4+) nanoclusters were found in and extending out of about 40% of the sodalite cavities. Central to each Cd6S4(4+) cluster is a Cd4S4 unit (interpenetrating Cd2+ and S2- tetrahedra with near Td symmetry, Cd-S = 2.997(24) A, Cd-S-Cd = 113.8(12) degrees, and S-Cd-S = 58.1(24) degrees). Each of the two remaining Cd2+ ions bonds radially through a 6-ring of the zeolite framework to a sulfide ion of this Cd4S4 unit (Cd-S = 2.90(8) A). In each of the remaining 60% of the sodalite cavities of crystal 1, a planar Cd2Na2S4+ cluster was found (Cd-S/Na-S = 2.35(5)/2.56(14) A and Cd-S-Cd/Na-S-Na = 122(5)/92(7) degrees). Cd6S4(4+) and Cd2Na2S4+ are stable within the zeolite up to about 700 K in air. Upon vacuum dehydration at 623 K, all sulfur was lost (crystal 2). Instead as anions, only two oxide ions remain per sodalite unit. One bridges between two Cd2+ ions (Cd2O2+, Cd-O = 2.28(3) A) and the other between two Na+ ions (Na2O, Na-O = 2.21(10) A).     

A thermally stable Pt/Y-based metal-organic framework: Exploring the accessibility of the metal centers with spectroscopic methods using H2O, CH3OH, and CH3CN as probes

A metal-organic framework (MOF) based on Pt, Y, and 2,2'-bipyridine-5,5'-dicarboxylate (BPDC), stable up to 400 degrees C, has been synthesized and characterized. In this MOF, the Pt centers are coordinated to Cl and the N atoms of the BPDC unit, giving a local environment similar to that found in a series of Pt-organic complexes with catalytic activity toward C-H bond cleavage of alkanes. This new material is a heterogeneous counterpart to the corresponding metal-organic complex. The structure, determined by single-crystal XRD data, is the repetition of three covalently bonded layers. These layers form a block, which is stacking as an (a)(b)(c) sequence along the crystallographic b-axis. Each layer contains the Pt-organic unit, while Y atoms represent the connection between adjacent layers. No covalent connection is present between layer (a) of a block and layer (c) of an adjacent block. EXAFS (BM29 at the ESRF) analysis supports the XRD data. As this MOF crystallizes under hydrothermal conditions, water acts both as solvent and as a direct ligand of Y. Accessibility to the metal centers is demonstrated by reversible water desorption/readsorption, as determined by TPA/TPD, FTIR, UV-vis, EXAFS, and XANES. Importantly, the results show that the as-synthesized material will not suffer a permanent loss in porosity upon solvent removal. In addition to water, methanol, ethanol, and acetonitrile can also access the internal void of the dehydrated phase.     

89Y magic-angle spinning NMR of Y2Ti2-xSnxO7 pyrochlores

The yttrium local environment in the series of pyrochlores Y2Ti2-xSnxO7 was studied using 89Y NMR. Oxides with the pyrochlore structure exhibit a range of interesting physical and chemical properties, resulting in many technological applications, including the encapsulation of lanthanide- and actinide-bearing radioactive waste. The use of the nonradioactive Y3+ cation provides a sensitive probe for any changes in the local structure and ordering with solid solution composition, through 89Y (I = 1/2) NMR. We confirm that a single pyrochlore phase is formed over the entire compositional range, with Y found only on the eight-coordinated A site. A significant (approximately 15 ppm) chemical shift is observed for each Sn substituted into the Y second neighbor coordination environment. The spectral signal intensities of the possible combinations of Sn/Ti neighbors match those predicted statistically assuming a random distribution of Sn4+/Ti4+ on the six-coordinated pyrochlore B site.     

Laser-assisted synthesis of Au-Ag alloy nanoparticles in solution

By using laser-induced heating, we prepared Au-Ag nanoalloys via three different procedures: (i) mixture of Au nanoparticles and Ag(+) ions irradiated by a 532 nm laser, (ii) mixture of Au and Ag nanoparticles irradiated by a 532 nm laser, and (iii) mixture of Au and Ag nanoparticles irradiated by a 355 nm laser. Procedure i is advantageous for the production of spherical alloy nanoparticles; in procedures ii and iii, nanoalloys with a sintered structure have been obtained. The morphology of the obtained nanoalloys depends not only on the laser wavelength but also on the concentration of nanoparticles in the initial mixture. When the total concentration of Ag and Au nanoparticles in the mixture is increased, large-scale interlinked networks have been observed upon laser irradiation. It is expected that this selective heating strategy can be extended to prepare other bi- or multi-metallic nanoalloys.     

Switch of electronic reactivity in fullerene C60: activation of three trans-4 positions via temporary saturation of the cis-1 positions

[Structure: see text] The regioselective functionalization of C60 with a trans-4,trans-4,trans-4 trisaddition pattern is not feasible directly. We have found an indirect approach taking advantage of the modified electronic reactivity of cis-1 bisadducts. The cis-1 addition pattern electronically activates three trans-4 C=C bonds on the opposite hemisphere of C60, allowing further highly regioselective additions at these positions. Thermal removal of the cis-1 blocking unit results in a trans-4,trans-4,trans-4 trisadduct with C3v symmetry.     

Limitations of quantum simulation examined by simulating a pairing Hamiltonian using nuclear magnetic resonance

Quantum simulation uses a well-known quantum system to predict the behavior of another quantum system. Certain limitations in this technique arise, however, when applied to specific problems, as we demonstrate with a theoretical and experimental study of an algorithm proposed by Wu, Byrd, and Lidar [Phys. Rev. Lett. 89, 057904 (2002).10.1103/PhysRevLett.89.057904] to find the low-lying spectrum of a pairing Hamiltonian. While the number of elementary quantum gates required scales polynomially with the size of the system, it increases inversely to the desired error bound E. Making such simulations robust to decoherence using fault tolerance requires an additional factor of approximately 1/E gates. These constraints, along with the effects of control errors, are illustrated using a three qubit NMR system.     

Highly Efficient Implementation of ANS/ EAS ‘Solid-Shell’ Finite Elements for Explicit Time Integration

In an explicit time integration algorithm, the most time-consuming part of an analysis is the computation of the internal nodal force vector. An implementation concept for element routines for volumetric shell ­ so-called Solid-Shell - elements based on the application of the symbolic programming tool ACEGEN, a plug-in for the computer algebra software MATHEMATICA is presented. Symbolic implementation means that vector and matrix operations and differentiations can be performed symbolically in order to generate program code, which supports the implementation, reduces programming effort and helps to avoid programming errors almost completely. The program code is generated in FORTRAN and simultaneously optimized automatically, which leads to very efficient routines compared to manually implemented code. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)