First-order Reductions Research Articles

Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

A new method to obtain either first- or second-order reductions for parametric polynomial ODEs

For a n th-order ordinary differential equation that is either polynomial on the dependent variable and its derivatives or polynomial in the derivatives of the dependent variable, a new method to obtain either first- or second-order reductions, respectively, is introduced. The corresponding reduced equations are generalized polynomials in the dependent variable or in its derivative. The method can be applied to equations that may have integer parameters in the exponents of the polynomial of the initial equation. A procedure to determine, in function of the parameters, if the given equation may admit such reductions and to restrict the possible variations of the exponents in the reduced equations is also provided. The introduced method strictly generalizes several methods that have recently appeared in the literature. As applications of the method, this study considers first-order reductions for a generalized Emden–Fowler equation and second-order reductions for the travelling-waves equation of a fifth-order Korteweg–de Vries equation and for a Painlevé–Chazy equation. As far as we know, some of the reported reductions are new. Several Maple programs to determine if a given polynomial equation admits any of the considered reductions and, if this is the case, to determine the reduced equations are also included.

Complexity and polymorphisms for digraph constraint problems under some basic constructions

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context, we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the “few subpowers algorithm” if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of “strict width” and solvability by few subpowers are unstable under first-order reductions. The analysis also yields a complete characterization of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.

Challenges in the application of spatial computable general equilibrium models for transport appraisal

The use of spatial computable general equilibrium (SCGE) models for assessing the economic impacts of transport projects is one of the key items on the research agenda for project appraisal in the Netherlands. These models are particularly suitable for analysing indirect effects of transport projects through linkages between the transport sector and the wider economy. Potentially, according to the literature, indirect effects that are additional to first-order direct cost reductions can turn out to be up to almost 80% in magnitude of the direct impacts. Given the relevance of these models for policy appraisal, experiences with this new modelling approach are important to report. After two years of development and application of SCGE models for transport appraisal, we found that the translation of theory behind the spatial equilibrium models into practical model specifications and empirical applications is a challenging task, and may lead to problems in project appraisal in terms of inaccuracies in the assessment of impacts. This paper discusses some key challenges we encountered with the specification of the Dutch SCGE model RAEM. This chapter is especially useful for researchers developing SCGE applications for use in transport appraisal and those who want to get a better understanding of differences between theoretical and computable SCGE modelling.

Extensions of MSO and the monadic counting hierarchy

In this paper, we study the expressive power of the extension of first-order logic by the unary second-order majority quantifier Most1. In 1 it was shown that the extension of FO by second-order majority quantifiers of all arities describes exactly the problems in the counting hierarchy. We consider first certain sublogics of FO(Most1) over unary vocabularies. We show that over unary vocabularies the logic MSO(R), where MSO is monadic second-order logic and R is the first-order Rescher quantifier, can be characterized by Presburger arithmetic, whereas the logic MSO(Rn)n∈Z+, where Rn is the nth vectorization of R, corresponds to the Δ0-fragment of arithmetic. Then we show that FO(Most1)⩾MSO(Rn)n∈Z+ and that, on unary vocabularies, FO(Most1) collapses to uniform-TC0. Using this collapse, we show that first-order logic with the binary second-order majority quantifier is strictly more expressive than FO(Most1) over the empty vocabulary. On the other hand, over strings, FO(Most1) is shown to capture the linear fragment of the counting hierarchy. Finally we show that, over non-unary vocabularies, FO(Most1) can express problems complete via first-order reductions for each level of the counting hierarchy.

Potential for ill-posedness in several second-order formulations of the Einstein equations

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false.

Simple expressions for Bragg reflection from axially excited chiral sculptured thin films

Reported here is the derivation of analytic expressions for the reflection coeffcients of a halfspace filled by a chiral sculptured thin film (STF) that is axially excited by a normally incident plane wave at the centrewavelength of the Bragg regime. Although these expressions are correct to the second order with respect to a small parameter that delineates the local anisotropy of the chiral STF, their first-order reductions are particularly simple and very adequate for practical purposes. A small difference between the two cross-polarized reflection coeffcients has also been found. The results are explained on the basis of a simple model which utilizes the standard Fresnel equations for a scalar dielectric halfspace, combined with coupled wave theory.

For completeness, sublogarithmic space is no space

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions.

On fixed-point logic with counting

One of the fundamental results of descriptive complexity theory, due to Immerman [13] and Vardi [18], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way.The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [14] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [15, 9, 10]).The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to clarify the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions.

Antibacterial action of 2-bromo-2-nitropropane-1,3-diol (bronopol)

Patterns of growth inhibition of Escherichia coli in the presence of 2-bromo-2-nitropropane-1,3-diol (bronopol) indicate a period of biocide-induced bacteriostasis followed by growth at an inhibited rate. The length of the bacteriostatic period, but not the subsequent growth inhibition, was reduced by the addition of excess cysteine. Patterns of growth inhibition were unaffected by catalase or superoxide dismutase. The bactericidal concentrations (100 to 500 micrograms/ml) were considerably in excess of the MIC (13 micrograms/ml) and generally produced first-order reductions in viability. Bactericidal activity was considerably reduced by anoxic conditions and by the presence of catalase or superoxide dismutase. Results indicate that there are two distinct reactions between bronopol and thiols. Under aerobic conditions, bronopol catalytically oxidizes thiol-containing materials such as cysteine, with atmospheric oxygen as the final oxidant. By-products of this reaction are active oxygen species such as superoxide and peroxide, which are directly responsible for the bactericidal activity of the compound and for the reduced growth rate after the bacteriostatic period. The latter effect probably results from the oxidation of intracellular thiols such as glutathione and cysteine. Catalytic oxidation of thiols in the presence of excess thiol leads to the creation of an anoxic state. Under these conditions, the slower reaction with thiols, which consumes bronopol, predominates. Consumption of bronopol by its reaction with thiols, without the involvement of oxygen, leads to the eventual removal of bronopol from treated suspensions and the resumption of growth.

Change in state of spin labels bound to sarcoplasmic reticulum with change in enzymic state, as deduced from ascorbate-quenching studies.

The ATPase (EC 3.6.1.3) of sarcoplasmic reticulum vesicles was reacted to various extents with thiol-directed spin labels. By suspension of the preparation in appropriate solutions, the enzyme could be placed and held in certain intermediate states of the ATPase cycle, or it could be set into steady-state catalysis. Ascorbate added to the system destroyed the spin-label signals with undetectable distortion of the electron paramagnetic resonance spectrum. In general, in the presence of ascorbate, undestroyed signal as a function of time could be described as the sum of two first-order reductions going on in separate compartments with different ascorbate concentrations. In different enzymatic states the proportion of total signal in the two compartments was different, but the first-order velocity constants remained the same. If the labeled membrane was first attacked with Triton, then exposed to ascorbate, signal was destroyed according to a single first-order constant, equal to the faster of the two constants observed with intact membrane, and equal to the constant whereby ascorbate attacks free label in solution. The data were reconciled by a simple rotary model, envisioning that an enzymatic state corresponds to an average angular position of the ATPase and thereby determines the proportion of labeled thiols exposed to external and internal ascorbate concentrations.

Utilization of the peat bogs of Canada

The use of spatial computable general equilibrium (SCGE) models for assessing the economic impacts of transport projects is one of the key items on the research agenda for project appraisal in the Netherlands. These models are particularly suitable for analysing indirect effects of transport projects through linkages between the transport sector and the wider economy. Potentially, according to the literature, indirect effects that are additional to first-order direct cost reductions can turn out to be up to almost 80% in magnitude of the direct impacts. Given the relevance of these models for policy appraisal, experiences with this new modelling approach are important to report. After two years of development and application of SCGE models for transport appraisal, we found that the translation of theory behind the spatial equilibrium models into practical model specifications and empirical applications is a challenging task, and may lead to problems in project appraisal in terms of inaccuracies in the assessment of impacts. This paper discusses some key challenges we encountered with the specification of the Dutch SCGE model RAEM. This chapter is especially useful for researchers developing SCGE applications for use in transport appraisal and those who want to get a better understanding of differences between theoretical and computable SCGE modelling.