18,382 research outputs found

    Formula partitioning revisited

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    Dividing a Boolean formula into smaller independent sub-formulae can be a useful technique for accelerating the solution of Boolean problems, including SAT and #SAT. Nevertheless, and despite promising early results, formula partitioning is hardly used in state-of-the-art solvers. In this paper, we show that this is rooted in a lack of consistency of the usefulness of formula partitioning techniques. In particular, we evaluate two existing and a novel partitioning model, coupled with two existing and two novel partitioning algorithms, on a wide range of benchmark instances. Our results show that there is no one-size-fits-all solution: for different formula types, different partitioning models and algorithms are the most suitable. While these results might seem negative, they help to improve our understanding about formula partitioning; moreover, the findings also give guidance as to which method to use for what kinds of formulae

    Injective coloring of product graphs

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    The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph coloring and domination theory. We make use of these three points of view in this paper so as to investigate the injective coloring of some well-known graph products. We bound the injective chromatic number of direct and lexicographic product graphs from below and above. In particular, we completely determine this parameter for the direct product of two cycles. We also give a closed formula for the corona product of two graphs

    Coloring of two-step graphs: open packing partitioning of graphs

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    The two-step graphs are revisited by studying their chromatic numbers in this paper. We observe that the problem of coloring of two-step graphs is equivalent to the problem of vertex partitioning of graphs into open packing sets. With this remark in mind, it can be considered as the open version of the well-known 22-distance coloring problem as well as the dual version of total domatic problem. The minimum kk for which the two-step graph N(G)\mathcal{N}(G) of a graph GG admits a proper coloring assigning kk colors to the vertices is called the open packing partition number po(G)p_{o}(G) of GG, that is, p_{o}(G)=\chi\big{(}\mathcal{N}(G)\big{)}. We give some sharp lower and upper bounds on this parameter as well as its exact value when dealing with some families of graphs like trees. Relations between pop_{o} and some well-know graph parameters have been investigated in this paper. We study this vertex partitioning in the Cartesian, direct and lexicographic products of graphs. In particular, we give an exact formula in the case of lexicographic product of any two graphs. The NP-hardness of the problem of computing this parameter is derived from the mentioned formula. Graphs GG for which po(G)p_{o}(G) equals the clique number of N(G)\mathcal{N}(G) are also investigated

    Glyoxal uptake on ammonium sulphate seed aerosol: reaction products and reversibility of uptake under dark and irradiated conditions

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    Chamber studies of glyoxal uptake onto ammonium sulphate aerosol were performed under dark and irradiated conditions to gain further insight into processes controlling glyoxal uptake onto ambient aerosol. Organic fragments from glyoxal dimers and trimers were observed within the aerosol under dark and irradiated conditions. Glyoxal monomers and oligomers were the dominant organic compounds formed under the conditions of this study; glyoxal oligomer formation and overall organic growth were found to be reversible under dark conditions. Analysis of high-resolution time-of-flight aerosol mass spectra provides evidence for irreversible formation of carbon-nitrogen (C-N) compounds in the aerosol. We have identified 1H-imidazole-2-carboxaldehyde as one C-N product. To the authors' knowledge, this is the first time C-N compounds resulting from condensed phase reactions with ammonium sulphate seed have been detected in aerosol. Organosulphates were not detected under dark conditions. However, active photochemistry was found to occur within aerosol during irradiated experiments. Carboxylic acids and organic esters were identified within the aerosol. An organosulphate, which had been previously assigned as glyoxal sulphate in ambient samples and chamber studies of isoprene oxidation, was observed only in the irradiated experiments. Comparison with a laboratory synthesized standard and chemical considerations strongly suggest that this organosulphate is glycolic acid sulphate, an isomer of the previously proposed glyoxal sulphate. Our study shows that reversibility of glyoxal uptake should be taken into account in SOA models and also demonstrates the need for further investigation of C-N compound formation and photochemical processes, in particular organosulphate formation

    Partitioning of the molecular density matrix over atoms and bonds

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    A double-index atomic partitioning of the molecular first-order density matrix is proposed. Contributions diagonal in the atomic indices correspond to atomic density matrices, whereas off-diagonal contributions carry information about the bonds. The resulting matrices have good localization properties, in contrast to single-index atomic partitioning schemes of the molecular density matrix. It is shown that the electron density assigned to individual atoms, when derived from the density matrix partitioning, can be made con- sistent with well-known partitions of the electron density over AIM basins, either with sharp or with fuzzy boundaries. The method is applied to a test set of about 50 molecules, representative for various types of chemical binding. A close correlation is observed between the trace of the bond matrices and the SEDI (shared electron density index) bond index.Comment: 25 pages, 8 figures, preprin

    Numerical Evidence Invalidating Finite-Temperature Many-Body Perturbation Theory

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    Low-order perturbation corrections to the electronic grand potential, internal energy, chemical potential, and entropy of a gas of noninteracting, identical molecules at a nonzero temperature are determined numerically as the λ\lambda-derivatives of the respective quantity calculated exactly (by thermal full configuration interaction) with a perturbation-scaled Hamiltonian, H^0+λV^\hat{H}_0 + \lambda\hat{V}. The data thus obtained from the core definition of any perturbation theory serve as a benchmark against which analytical formulas can be validated. The first- and second-order corrections from finite-temperature many-body perturbation theory disagree with these benchmark data. This is because the theory neglects the variation of chemical potential with λ\lambda, thereby failing to converge at the exact, full-interaction (λ=1\lambda=1) limit, unless the exact chemical potential is known in advance. The renormalized finite-temperature perturbation theory [S. Hirata and X. He, J. Chem. Phys., 138, 204112 (2013)] is also found to be incorrect
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