Fast and Accurate Floating Point Summation with Application to Computational Geometry

We present several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator. Let f and F be the number of significant bits in the summands and the accumulator, respectively. Then assuming gradual underflow, no overflow, and round-to-nearest arithmetic, up to ?2 ^F?f /(1?2^?f )?+1 numbers can be accurately added by just summing the terms in decreasing order of exponents, yielding a sum correct to within about 1.5 units in the last place. In particular, if the sum is zero, it is computed exactly. We apply this result to the floating point formats in the IEEE floating point standard, and investigate its performance. Our results show that in the absence of massive cancellation (the most common case) the cost of guaranteed accuracy is about 30–40% more than the straightforward summation. If massive cancellation does occur, the cost of computing the accurate sum is about a factor of ten. Finally, we apply our algorithm in computing a robust geometric predicate (used in computational geometry), where our accurate summation algorithm improves the existing algorithm by a factor of two on a nearly coplanar set of points.     

Preparation and analytical properties of a chelating resin containing phenylalanine groups

A polystyrene-based macroreticular resin containing phenylalanine groups has been prepared and its analytical properties have been investigated and compared with Dowex A-1. The phenylalanine resin shows high selectivity for mercury(II) and copper(II) in the pH region 2-3. The sorption behaviour of copper has been examined in detail, with the intention of using the resin analytically. The important characteristics of the resin are fast equilibration, high selectivity and small volume change between its hydrogen form and metal forms. These enable it to be applied for the rapid concentration of trace amounts of copper in the presence of large amounts of diverse metals. It may be used for the determination of copper in sea-water and the separation of copper/cobalt and copper/nickel.     

Quasiperiodic hubbard chains

Low-energy properties of half-filled Fibonacci Hubbard models are studied by weak-coupling renormalization group and density matrix renormalization group methods. In the case of diagonal modulation, weak Coulomb repulsion is irrelevant and the system behaves as a free Fibonacci chain, while for strong Coulomb repulsion the charge sector becomes a Mott insulator and the spin sector behaves as a uniform Heisenberg antiferromagnetic chain. The off-diagonal modulation always drives the charge sector to a Mott insulator and the spin sector to a Fibonacci antiferromagnetic Heisenberg chain.     

Boson fock representations of stochastic processes

A classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented. Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 3–14, January, 2000.     

Rheological properties of poly(vinyl chloride)/plasticizer systems—relation between sol–gel transition and elongational viscosity

Poly(vinyl chloride) (PVC)/di-isononyl phthalate systems with PVC content of 45.5 (PVC8) and 70.4 wt% (PVC6) were prepared by a hot roller at 150 °C and press molded at 180 °C. The dynamic viscoelasticity and elongational viscosity of PVC8 and PVC6 were measured in the temperature range from 150 to 220 °C. We have found that the storage and loss shear moduli, G′ and G″, of PVC8 and PVC6 exhibited the power-law dependence on the angular frequency ω at 190 and 210 °C, respectively. Correspondingly, the tan δ values did not depend on ω. These temperatures indicate the critical gel temperature T _gel of each system. The critical relaxation exponent n obtained from these data was 0.75 irrespective of PVC content, which was in agreement with the n values reported previously for the low PVC concentration samples. These results suggest that the PVC gels of different plasticizer content have a similar fractal structure. Below T _gel, the gradual melting of the PVC crystallites takes place with elevating temperature, and above T _gel, a densely connected network throughout the whole system disappears. Correspondingly, the elongational viscosity behavior of PVC8 and PVC6 exhibited strong strain hardening below T _gel, although it did not show any strain hardening above T _gel. These changes in rheological behavior are attributed to the gradual melting of the PVC crystallites worked as the cross-linking domains in this physical gel, thereby inapplicability of the of time–temperature superposition for PVC/plasticizer systems.     

Dirichlet forms and white noise analysis

We use the white noise calculus as a framework for the introduction of Dirichlet forms in infinite dimensions. In particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability. If the forms are closable, we show that their closures are Markovian (in the sense of Fukushima).