Inclusion-exclusion: Exact and approximate

It is often required to find the probability of the union of givenn eventsA ₁ ,...,A _n . The answer is provided, of course, by the inclusion-exclusion formula: Pr(A _i )= _i – _i<j Pr(A _i A _j )±.... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, #P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(A _i ), from the above data with an additive error of exp . In the present article we develop an expression that can be computed in polynomial time, that, given the sums _|S|=j Pr( _iS A _i ) forj=1,...k, approximates Pr(A _i ) with an additive error of exp . This error is optimal, up to the logarithmic factor implicit in the notation.The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v _l ^m C _i is an instance of the general problem that had been extensively studied [7]. HereA _i is the set of assignments that satisfyC _i , and Pr( _iS A _i )=a _S /2ⁿ where _iS C _i is satisfied bya _S assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa _S for, say, all cardinalities . Quite surprisingly, already the numbersa _S over |S|log(n+1)uniquely determine the number of satisfying assignments for F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0,1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Work supported in part by a grant of the Binational Israel-US Science Foundation.Work supported in part by a grant of the Binational Israel-US Science Foundation and by the Israel Science Foundation.     

Preparation of optically active allylic hydroperoxy alcohols and 1, 3-diols by enzyme-catalyzed kinetic resolution and photooxygenation of chiral homoallylic alcohols

All four possible enantiomers of the 3-hydroperoxy-4-penten-1-ols 2a, b and their corresponding 4-pentene-1,3-diols 4a,b have been prepared for the first time in high enantiomeric purity (up to 98% ee) and in preparative amounts according to two distinct ways: First the photooxygenation of the racemic homoallylic alcohols 1 gave the racemic hydroperoxy alcohols 2, which have subsequently been kinetically resolved by horseradish peroxidase (HRP); alternatively, first the lipase-catalyzed resolution afforded the optically active homoallylic alcohols 1 and subsequent photooxygenation led to the enantiomerically enriched hydroperoxy alcohols 2.     

Charge separation versus recombination in dye-sensitized nanocrystalline solar cells: the minimization of kinetic redundancy

In this paper we focus upon the electron injection dynamics in complete dye-sensitized nanocrystalline metal oxide solar cells (DSSCs). Electron injection dynamics are studied by transient absorption and emission studies of DSSCs and correlated with device photovoltaic performance and charge recombination dynamics. We find that the electron injection dynamics are dependent upon the composition of the redox electrolyte employed in the device. In a device with an electrolyte composition yielding optimum photovoltaic device efficiency, electron injection kinetics exhibit a half time of 150 ps. This half time is 20 times slower than that for control dye-sensitized films covered in inert organic liquids. This retardation is shown to result from the influence of the electrolyte upon the conduction band energetics of the TiO2 electrode. We conclude that optimum DSSC device performance is obtained when the charge separation kinetics are just fast enough to compete successfully with the dye excited-state decay. These conditions allow a high injection yield while minimizing interfacial charge recombination losses, thereby minimizing "kinetic redundancy" in the device. We show furthermore that the nonexponential nature of the injection dynamics can be simulated by a simple inhomogeneous disorder model and discuss the relevance of our findings to the optimization of both dye-sensitized and polymer based photovoltaic devices.     

Large volume sample application by a simplified “closed” on-column injector in capillary gas chromatography

Summary A new simplified version of a closed on-column injector is introduced. Because of its design isobaric injection conditions do not have to be followed and a wide range of injection temperatures above the boiling point of the sample solvent can be chosen for on-column injections in capillary gas chromatography. Also, when following certain basic injection rules, injections of large sample volumes (20 l or more) give accurate and reproducible results without further problems.Presented at the 17th International Symposium on Chromatography, September 25–30, 1988, Vienna, Austria.     

Continuous flow in open microfluidics using controlled evaporation

This paper presents a method for programming the flow rate of liquids inside open microfluidic networks (MFNs). A MFN comprises a number of independent flow paths, each of which starts with an open filling port, has a sealed microchannel in which assays can be performed, and an open capillary pump (CP). The MFN is placed over Peltier elements and its flow paths initially fill owing to capillary forces when liquids are added to the filling ports. A cooling Peltier element underneath the filling ports dynamically prevents evaporation in all filling ports using the ambient temperature and relative humidity as inputs. Another Peltier element underneath the CPs heats the pumps thereby inducing evaporation in the CPs and setting the flow rate in the microchannels. This method achieves flow rates in the microchannels ranging from approximately 1.2 nL s(-1) to approximately 30 pL s(-1), and is able to keep 90% of a 0.6 microL solution placed in an open filling port for 60 min. This simple and efficient method should be applicable to numerous assays or chemical reactions that require small and precise flow of liquids and reagents inside microfluidics.     

PGSE diffusion, 1H-19F HOESY and NMR studies on several [Rh(1,5-COD)(Biphemp)]X complexes: detecting positional anion effects

Three new salts of Rh(I), [Rh(1,5-COD)(Biphemp)]X, Biphemp = (6,6'-dimethylbiphenyl-2,2'-diyl)bis(diphenylphosphine), X = BF4-, 2a, PF6-, 2b, and CF3SO3-, 2c, were prepared and studied using PGSE diffusion, 1H-19F HOESY and inverse 103Rh (amongst other) NMR methods. Although the immediate coordination sphere of the cation does not sense the change in the anion, the PGSE diffusion and 1H-19F HOESY data suggest that, in addition to some ion pairing, each of the anions in 2 demonstrates selectivity in its approach towards the cation.     

Bestimmung des hei?wasserl?slichen Borgehalts in B?den mit der induktionsgekoppelten Plasma-Atom-Emissionsspektrometrie

Summary The purpose of this study is to find out whether the previously applied procedures for determining hot water soluble boron trace contents in soils using dianthrimide or azomethin-H can be substituted by ICP-AES. Research was made into chemical and spectral interferences which can arise in all three methods. The 249.678 nm boron emission line is not affected by the sample matrix indicated. The boron content from more than 60 extracts was determined using the three methods mentioned and the results were compared by means of regression calculations. It was found that ICP-AES and the dianthrimide procedure do not differ significantly, whereas the azomethin-H technique leads to somewhat higher values. The accuracy of the results achieved by means of ICP-AES was confirmed by a collaborative study. In order to determine the concentration of boron in soils in the range of 0–2 mg/kg the ICP-AES can be applied following the hot water extraction without further pretreatment of samples. This method is superior in particular when additional elements are to be considered in the soil.