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.     

Janus Emulsion Drops: Equilibrium Calculations

Experimental results indicated the contact angles in the drops of Janus emulsions formed in a one-step mixing process to be invariant within a significant range the oil volume ratios, similar to the results from microfluidics emulsification. Since this result points to a connection between the kinetically formed emulsions and the local equilibrium topology of emulsion drops, the effect of interfacial tensions on the morphology of Janus emulsions was estimated from the equilibrium interfacial tensions at the line of contact. Realistic values of the tensions revealed the limited range of these to obtain Janus drops and also offered correlation between the equilibrium entities and the curvature of the interface between the two oils.     

Destabilization of a dual emulsion to form a Janus emulsion

A vegetable oil (VO) was added to an emulsion of silicone oil in water (SO/W) with mixing limited to once turning the test tube upside down. Initially, the VO was dispersed into virtually centimeter-sized drops and the emulsion contained effectively no Janus drops, while after 1 h of agitation at a low level to prevent creaming, drops of 50–100-μm size of the two oils were observed: in addition to an insignificant number of Janus drops. The topology of the latter showed them to emanate from flocculated individual drops of the two oils, but with no discernible effect by the interfacial tension equilibrium on the drop topology. Continued gentle mixing gave increasing fraction of Janus drops of increased size with a topology gradually approaching the one expected from the interfacial equilibrium at the contact line. The spontaneous formation of Janus drops indicated a reduction of the interfacial free energy in the process and the interfacial energy difference between separate and Janus drops was calculated for an appropriate range of interfacial tensions and for all oil fractions. The calculations enabled a distinction of the decrease due to interfacial area changes from the reduction of interfacial tensions per se, with the latter only a minor fraction. Figure

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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.