A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and its Algorithmic Applications

A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously satisfy at least \pp m clauses, where \pp =(\sqrt{5}-1)/2. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula $F$ with $m$ clauses we can find, in polynomial time, a subformula $F'$ with $m'$ clauses such that we can simultaneously satisfy at least \pp m+(1-\pp)m'+(2-3\pp)n"/2 clauses (in $F$), where $n"$ is the number of variables in $F$ which are not in $F'$. We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds $m/2$ and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most $6k+3$ variables and $10k$ clauses. We improve this to $4k$ variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above

Eliciting a directed acyclic graph for a multivariate time series of vehicle counts in a traffic network

The problem of modelling multivariate time series of vehicle counts in traffic networks is considered. It is proposed to use a model called the linear multiregression dynamic model (LMDM). The LMDM is a multivariate Bayesian dynamic model which uses any conditional independence and causal structure across the time series to break down the complex multivariate model into simpler univariate dynamic linear models. The conditional independence and causal structure in the time series can be represented by a directed acyclic graph (DAG). The DAG not only gives a useful pictorial representation of the multivariate structure, but it is also used to build the LMDM. Therefore, eliciting a DAG which gives a realistic representation of the series is a crucial part of the modelling process. A DAG is elicited for the multivariate time series of hourly vehicle counts at the junction of three major roads in the UK. A flow diagram is introduced to give a pictorial representation of the possible vehicle routes through the network. It is shown how this flow diagram, together with a map of the network, can suggest a DAG for the time series suitable for use with an LMDM

Hospital food service: a comparative analysis of systems and introducing the ‘Steamplicity’ concept

Background Patient meals are an integral part of treatment hence the provision and consumption of a balanced diet, essential to aid recovery. A number of food service systems are used to provide meals and the Steamplicity concept has recently been introduced. This seeks, through the application of a static, extended choice menu, revised patient ordering procedures, new cooking processes and individual patient food heated/cooked at ward level, to address some of the current hospital food service concerns. The aim of this small-scale study, therefore, was to compare a cook-chill food service operation against Steamplicity. Specifically, the goals were to measure food intake and wastage at ward level; ‘stakeholders’ (i.e. patients, staff, etc.) satisfaction with both systems; and patients’ acceptability of the food provided. Method The study used both quantitative (self-completed patient questionnaires, n = 52) and qualitative methods (semi-structured interviews, n = 16) with appropriate stakeholders including medical and food service staff, patients and their visitors. Results Patients preferred the Steamplicity system overall and in particular in terms of food choice, ordering, delivery and food quality. Wastage was considerably less with the Steamplicity system, although care must be taken to ensure that poor operating procedures do not negate this advantage. When the total weight of food consumed in the ward at each meal is divided by the number of main courses served, at lunch, the mean intake with the cook-chill system was 202 g whilst that for the Steamplicity system was 282 g and for the evening meal, 226 g compared with 310 g. Conclusions The results of this small study suggest that Steamplicity is more acceptable to patients and encourages the consumption of larger portions. Further evaluation of the Steamplicity system is warranted. The purpose of this study was to directly compare selected aspects (food wastage at ward level; satisfaction with systems and food provided) of a traditional cook-chill food service operation against ‘Steamplicity’. Results indicate that patients preferred the ‘Steamplicty’ system in all areas: food choice, ordering, delivery, food quality and overall. Wastage was considerably less with the ‘Steamplicity’ system; although care must be taken to ensure that poor operating procedures do not negate this advantage. When the total weight of food consumed in the ward at each meal is divided by the number of main courses served, results show that at lunch, mean intake with the cook-chill system was 202g whilst that for the ‘Steamplicity’ system was 282g and for the evening meal, 226g compared with 310g

Unravelling the controls on the molybdenum isotopic composition of rivers

Formation and crystallisation of the Lunar Magma Ocean (LMO) was one of the most incisive events during the early evolution of the Moon. Lunar Magma Ocean solidification concluded with the coeval formation of K-, REE- and P-rich components (KREEP) and an ilmenite-bearing cumulate (IBC) layer. Gravitational overturn of the lunar mantle generated eruptions of basaltic rocks with variable Ti contents, of which their δ49Ti variations may now reflect variable mixtures of ambient lunar mantle and the IBC. To better understand the processes generating the spectrum of lunar low-Ti and high-Ti basalts and the role of Ti-rich phases such as ilmenite, we determined the mass dependent Ti isotope composition of four KREEP-rich samples, 12 low-Ti, and eight high-Ti mare basalts by using a 47Ti-49Ti double spike. Our data reveal significant variations in δ49Ti for KREEP-rich samples (+0.117 to +0.296 ‰) and intra-group variations in the mare basalts (-0.030 to +0.055 ‰ for low-Ti and +0.009 to +0.115 ‰ for high-Ti basalts). We modelled the δ49Ti of KREEP using previously published HFSE data as well as the δ49Ti evolution during fractional crystallisation of the LMO. Both approaches yield δ49TiKREEP similar to measured values and are in excellent agreement with previous studies. The involvement of ilmenite in the petrogenesis of the lunar mare basalts is further evaluated by combining our results with element ratios of HFSE, U and Th, revealing that partial melting in an overturned lunar mantle and fractional crystallisation of ilmenite must be the main processes accounting for mass dependent Ti isotope variations in lunar basalts. Based on our results we can also exclude formation of high-Ti basalts by simple assimilation of ilmenite by ascending melts from the depleted lunar mantle. Rather, our data are in accord with melting of these basalts from a hybrid mantle source formed in the aftermath of gravitational lunar mantle overturn, which is in good agreement with previous Fe isotope data

Unravelling the controls on the molybdenum isotope ratios of river waters

The molybdenum (Mo) isotope ratios (δ98/95Mo) of river waters control the δ98/95Mo values of seawater and impact on the use of Mo isotope ratios as a proxy of past redox conditions. The δ98/95Mo values of river waters vary by more than 2 ‰, yet the relative roles of lithology versus fractionation during weathering remain contested. Here, we combine measurements from river waters (δ98/95Modiss), river bed materials (δ98/95MoBM) and soils from locations with contrasting lithology. The δ98/95Mo values of river bed materials (δ98/95MoBM), set by rock type, vary by ~1 ‰ between rivers in New Zealand, the Mackenzie Basin, and Iceland. However, the difference between dissolved and solid phase Mo isotopes (Δ98/95Modiss-BM) varies from +0.3 ‰ to +1.0 ‰. We estimate Mo removal from solution using the mobile trace element rhenium and find that it correlates with Δ98/95Modiss-BM across the sample set. The adsorption of Mo to Fe-Mn-(oxyhydr)oxides can explain the observed fractionation. Together, the amount of Mo released through dissolution and taken up by (oxyhydr)oxide formation on land may cause changes in the δ98/95Mo values of rivers, driving long term changes in the Mo isotope ratios of seawater

Confinement and scaling in deep inelastic scattering

We show that parton confinement in the final state generates large $1/Q^2$ corrections to Bjorken scaling, thus leaving less room for the logarithmic corrections. In particular, the $x$-scaling violations at large $x$ are entirely described in terms of power corrections. For treatment of these non-perturbative effects, we derive a new expansion in powers of $1/Q^2$ for the structure function that is free of infra-red singularities and which reduces corrections to the leading term. The leading term represents scattering from an off-mass-shell parton, which keeps the same virtual mass in the final state. It is found that this quasi-free term is a function of a new variable $\bar x$, which coincides with the Bjorken variable $x$ for $Q^2\to\infty$. The two variables are very different, however, at finite $Q^2$. In particular, the variable $\bar x$ depends on the invariant mass of the spectator particles. Analysis of the data at large $x$ shows excellent scaling in the variable $\bar x$, and determines the value of the diquark mass to be close to zero. $\bar x$-scaling allows us to extract the structure function near the elastic threshold. It is found to behave as $F_2\sim (1-x)^{3.7}$. Predictions for the structure functions based on $\bar x$-scaling are made.Comment: Discussion of target mass corrections is added. Accepted for publication in Phys. Rev.

Perturbative QCD and factorization of coherent pion photoproduction on the deuteron

We analyze the predictions of perturbative QCD for pion photoproduction on the deuteron, gamma D -> pi^0 D, at large momentum transfer using the reduced amplitude formalism. The cluster decomposition of the deuteron wave function at small binding only allows the nuclear coherent process to proceed if each nucleon absorbs an equal fraction of the overall momentum transfer. Furthermore, each nucleon must scatter while remaining close to its mass shell. Thus the nuclear photoproduction amplitude, M_{gamma D -> pi^0 D}(u,t), factorizes as a product of three factors: (1) the nucleon photoproduction amplitude, M_{gamma N_1 -> pi^0 N_1}(u/4,t/4), at half of the overall momentum transfer, (2) a nucleon form factor, F_{N_2}(t/4), at half the overall momentum transfer, and (3) the reduced deuteron form factor, f_d(t), which according to perturbative QCD, has the same monopole falloff as a meson form factor. A comparison with the recent JLAB data for gamma D -> pi^0 D of Meekins et al. [Phys. Rev. C 60, 052201 (1999)] and the available gamma p -> pi^0 p data shows good agreement between the perturbative QCD prediction and experiment over a large range of momentum transfers and center of mass angles. The reduced amplitude prediction is consistent with the constituent counting rule, p^11_T M_{gamma D -> pi^0 D} -> F(theta_cm), at large momentum transfer. This is found to be consistent with measurements for photon lab energies E_gamma > 3 GeV at theta_cm=90 degrees and \elab > 10 GeV at 136 degrees.Comment: RevTeX 3.1, 17 pages, 6 figures; v2: incorporates minor changes as version accepted by Phys Rev

Curvature correction to the mobility of fluid membrane inclusions

For the first time, using rigorous low-Reynolds-number hydrodynamic theory on curved surfaces via a Stokeslet-type approach, we provide a general and concise expression for the leading-order curvature correction to the canonical, planar, Saffman-Delbrück value of the diffusion constant for a small inclusion embedded in an arbitrarily (albeit weakly) curved fluid membrane. In order to demonstrate the efficacy and utility of this wholly general result, we apply our theory to the specific case of calculating the diffusion coefficient of a locally curvature inducing membrane inclusion. By including both the effects of inclusion and membrane elasticity, as well as their respective thermal shape fluctuations, excellent agreement is found with recently published experimental data on the surface tension dependent mobility of membrane bound inclusions

Neutron structure function and inclusive DIS from H-3 and He-3 at large Bjorken-x

A detailed study of inclusive deep inelastic scattering (DIS) from mirror A = 3 nuclei at large values of the Bjorken variable x is presented. The main purpose is to estimate the theoretical uncertainties on the extraction of the neutron DIS structure function from such nuclear measurements. On one hand, within models in which no modification of the bound nucleon structure functions is taken into account, we have investigated the possible uncertainties arising from: i) charge symmetry breaking terms in the nucleon-nucleon interaction, ii) finite Q**2 effects neglected in the Bjorken limit, iii) the role of different prescriptions for the nucleon Spectral Function normalization providing baryon number conservation, and iv) the differences between the virtual nucleon and light cone formalisms. Although these effects have been not yet considered in existing analyses, our conclusion is that all these effects cancel at the level of ~ 1% for x < 0.75 in overall agreement with previous findings. On the other hand we have considered several models in which the modification of the bound nucleon structure functions is accounted for to describe the EMC effect in DIS scattering from nuclei. It turns out that within these models the cancellation of nuclear effects is expected to occur only at a level of ~ 3%, leading to an accuracy of ~ 12 % in the extraction of the neutron to proton structure function ratio at x ~ 0.7 -0.8$. Another consequence of considering a broad range of models of the EMC effect is that the previously suggested iteration procedure does not improve the accuracy of the extraction of the neutron to proton structure function ratio.Comment: revised version to appear in Phys. Rev. C; main modifications in Section 4; no change in the conclusion