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.     

Validation of a high performance liquid chromatography analysis for the determination of noradrenaline and adrenaline in human urine with an on-line sample purification

A high performance liquid chromatography (HPLC) method with fluorescence detection including an on-line purification was established for determination of catecholamines in human urine. The method was evaluated using samples of pooled urine spiked with catecholamines and validated for measurements of catecholamines in urine of healthy individuals in a field study. The laboratory method evaluation study showed that the recovery of the method was 0.82 (confidence interval (CI): 0.79-0.86) and 0.92 (CI: 0.89-0.95) for noradrenaline and adrenaline, respectively. Thus, correction factors of 0.82(-1) and 0.92(-1) were applied to correct the measurement results for this systematic effect. Furthermore, an uncertainty budget was generated for the analytical method using the BIPM-approach recommended by the International Organization for Standardization. The relative uncertainty of the method was estimated to be 10-12%, which was consistent with the observed relative variability found in the method evaluation. The method was evaluated in accordance with EURACHEM Guidance Document No 1 concerning accreditation for chemical laboratories with respect to accuracy and precision. The field study showed that the standard deviation of the method was sufficiently low to distinguish between persons working with two different processes of garbage collection, i.e. collection using four wheeled containers versus collection using bins.     

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.     

Wetting transitions of ionic solutions

Cahn's phenomenological theory of wetting of a solid substrate by a saturated vapor is generalized to the case where the substrate is charged and the wetting film contains counterions, with or without added salt. The electrostatic contribution to the grand potential associated with these ions is calculated within a nonlinear Poisson-Boltzmann theory. In the salt-free case, when the wetting film includes only counterions released by the substrate, the wetting transition is always first order, regardless of its nature in a neutral system. When salt is present, other wetting scenarios may arise, depending on the salt concentration and substrate surface charge. Over a restricted range of salt concentrations, a wetting scenario similar to that of prewetting, is predicted to occur along the liquid-vapor coexistence line. This scenario includes a discontinuous wetting transition between microscopic and mesoscopic film thicknesses, followed by a continuous divergence of the film thickness at higher temperatures.