Stochastic Arithmetic in Multiprecision

Floating-point arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32-bit (respectively 64-bit) long. Extended precision formats can be up to 128-bit long. However some problems require a longer floating-point format, because of round-off errors. Such problems are usually solved in arbitrary precision, but round-off errors still occur and must be controlled. Interval arithmetic has been implemented in arbitrary precision, for instance in the MPFI library. Interval arithmetic provides guaranteed results, but it is not well suited for the validation of huge applications. The CADNA library estimates round-off error propagation using stochastic arithmetic. CADNA has enabled the numerical validation of real-life applications, but it can be used in single precision or in double precision only. In this paper, we present a library called SAM (Stochastic Arithmetic in Multiprecision). It is a multiprecision extension of the classic CADNA library. In SAM (as in CADNA), the arithmetic and relational operators are overloaded in order to be able to deal with stochastic numbers. As a consequence, the use of SAM in a scientific code needs only few modifications. This new library SAM makes it possible to dynamically control the numerical methods used and more particularly to determine the optimal number of iterations in an iterative process. We present some applications of SAM in the numerical validation of chaotic systems modeled by the logistic map.     

Structure and impact resistance of short carbon fiber reinforced polyamide 6 composites

Short carbon fiber (CF) reinforced polyamide 6 (PA6) composites were prepared by homogenization of the components in a twin-screw extruder and by injection molding. Fiber content was varied between 0 and 16 vol%, while specimens were injection molded at rates between 2.0 and 22.6 cm/s. Average fiber length and orientation were measured to characterize structure. Average fiber length decreased with increasing fiber content and processing rate. The observed structure is contradictory to those reported in the literature for short glass fiber reinforced composites. Fibers were oriented randomly relative to the mold fill direction in the skin layer, while they were oriented parallel to this direction in the middle of the specimen. The thickness of the skin decreased with increasing injection rate and decreasing fiber content. Although instrumented impact testing indicated brittle failure at all combinations of the variables, the strain energy release rate could not be determined by the usual technique using varying notch depths because of the different properties of the skin and the core. Also, the mechanism of failure seems to be different in the two layers. A minimum appears in the fracture toughness and impact resistance at low fiber contents, indicating that fibers might promote fracture initiation at such compositions. Fiber length changed in a narrow range in the studied composites; thus, properties are determined mainly by orientation. As a consequence, both increased fiber content and injection rate lead to an increase of stiffness and toughness.     

Preferential Siting of Aluminum Heteroatoms in the Zeolite Catalyst Al‐SSZ‐70

The adsorption and reaction properties of heterogeneous zeolite catalysts (e.g. for catalytic cracking of petroleum, partial oxidation of natural gas) depend strongly on the types and distributions of Al heteroatoms in the aluminosilicate frameworks. The origins of these properties have been challenging to discern, owing in part to the structural complexity of aluminosilicate zeolites. Herein, combined solid‐state NMR and synchrotron X‐ray powder diffraction analyses show the Al atoms locate preferentially in certain framework sites in the zeolite catalyst Al‐SSZ‐70. Through‐covalent‐bond 2D ²⁷Al{²⁹Si} J‐correlation NMR spectra allow distinct framework Al sites to be identified and their relative occupancies quantified. The analyses show that 94 % of the Al atoms are located at the surfaces of the large‐pore interlayer channels of Al‐SSZ‐70, while only 6 % are in the sub‐nm intralayer channels. The selective siting of Al atoms accounts for the reaction properties of catalysts derived from SSZ‐70.     

A study on Fe+ centers in NaCl crystals

An ESR study on Fe⁺ centers in NaCl at sites with various symmetries is presented. Such centers, with axial, octahedral or orthorhombic symmetry, can be observed below 32 K, in crystals freshly quenched at RT, before irradiating at 77 K. On warming to 180 K the conversion of Fe⁺ (orthorhombic) to Fe⁺ (axial) occurs. A second conversion of Fe⁺ (axial) to Fe⁺ (octahedral) takes place at 220 K. Both conversions are due to the tendency of the positive-ion vacancy to move away from the Fe⁺ ions.     

Compromise values in cooperative game theory

Summary The aim of this paper is to give a survey on several well-known compromise values in cooperative game theory and its applications. Special attention is paid to the τ-value for TU-games, the Raiffa-Kalai-Smorodinsky solution for bargaining problems, and the compromise value for NTU-games.     

Transition probabilities in146Eu

Transition probabilities in the odd-odd nucleus¹⁴⁶Eu are measured. A recent determination of the lifetime of the 115.6 keV level has indicated a strong retardation of the 115.6 keVM1 transition. In contrast, the present results areT _1/2(230.3)≦165 ps,T _1/2(115.6)≦160 ps. TheE2 cross-over transitions of 269.5 and 230.3 keV were not found in the Ge(Li) gamma spectrum. All results are in agreement with the single particle model.     

Effect of salt gradients on the separation of dilute mixtures of proteins by ion-exchange in simulated moving beds

Salt gradients can improve the efficiency during fractionation of proteins by ion-exchange in simulated moving beds (SMBs). The gradients are formed using feed and desorbent solutions of different salt concentrations. The thus introduced regions of high and low affinity may reduce eluent consumption and resin inventory compared to isocratic SMB systems. This paper describes a procedure for the selection of the flow-rate ratios that enables successful fractionation of a dilute binary mixture of proteins in a salt gradient. The procedure is based on the so-called "triangle theory" and can be used both for upward gradients (where salt is predominantly transported by the liquid) and downward gradients (where salt is predominantly transported by the sorbent). The procedure is verified by experiments.     

On the broadening of the annihilation line

The broadening of annihilation line was studied with a Ge detector. The annihilation line was deconvoluted taking into account the instrumental resolution of 1.08 keV. The P_f in several metals are found. The contribution of core electrons to annihilation is deduced.     

Ln12‐Containing 60‐Tungstogermanates: Synthesis,Structure, Luminescence,and Magnetic Studies

A new class of hexameric Ln₁₂‐containing 60‐tungstogermanates, [Na(H₂O)₆?Eu₁₂(OH)₁₂(H₂O)₁₈Ge₂(GeW₁₀O₃₈)₆]^39? ( Eu₁₂ ), [Na(H₂O)₆?Gd₁₂(OH)₆(H₂O)₂₄Ge(GeW₁₀O₃₈)₆]^37? ( Gd₁₂ ), and [(H₂O)₆?Dy₁₂(H₂O)₂₄(GeW₁₀O₃₈)₆]^36? ( Dy₁₂ ), comprising six di‐Ln‐embedded {β(4,11)‐GeW₁₀} subunits was prepared by reaction of [α‐GeW₉O₃₄]^10? with Ln^III ions in weakly acidic (pH 5) aqueous medium. Depending on the size of the Ln^III ion, the assemblies feature selective capture of two (for Eu₁₂ ), one (for Gd₁₂ ), or zero (for Dy₁₂ ) extra Ge^IV ions. The selective encapsulation of a cationic sodium hexaaqua complex [Na(H₂O)₆]⁺ was observed for Eu₁₂ and Gd₁₂ , whereas Dy₁₂ incorporates a neutral, distorted‐octahedral (H₂O)₆ cluster. The three compounds were characterized by single‐crystal XRD, ESI‐MS, photoluminescence, and magnetic studies. Dy₁₂ was shown to be a single‐molecule magnet.     

Flexibilized styrene–N‐substituted maleimide copolymers. II. Multiblock copolymers prepared from PTHF‐based iniferters

This article describes the synthesis and molecular characterization of thermal polymeric iniferters, based on hydroxy‐terminated poly(tetrahydrofuran) (PTHF), bearing thiuram disulfide groups along the chain. Thermal polymerization after the addition of styrene (S) and N‐methylmaleimide (MI) to these PTHF‐based polymeric iniferters yielded segmented PTHF (SMI‐block‐PTHF)_n block copolymers that proved to have a single T_g. The multiblock copolymers were molecularly characterized by elemental analysis, IR, and NMR. The thermal stability, as checked by thermogravimetric analysis, proved to be good up to about 350 °C. A size exclusion chromatography/differential viscosity (DV) analysis showed that the molecular weights of the synthesized single‐phase multiblock copolymers were sufficiently high (several times the estimated molecular weight between two adjacent entanglements) to determine the entanglement density from the rubbery plateau modulus, for which the method developed by S. Wu (J Polym Sci Part B: Polym Phys 1989, 27, 723–741) was applied. The entanglement density of flexibilized SMI proved to be about 20–25% higher than that of the nonflexibilized SMI. This increase is disappointing, and more work, based on the described concept, is required to achieve the desired enhancement of the toughness. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 3558–3568, 2000