Generic Constants Research Articles

Mixed volumes of networks with binomial steady-states

The steady-state degree of a chemical reaction network is the number of complex steady-states for generic rate constants and initial conditions. One way to bound the steady-state degree is through the mixed volume of an associated steady-state system. In this work, we show that for partitionable binomial chemical reaction systems, whose resulting steady-state systems are given by a set of binomials and a set of linear (not necessarily binomial) conservation equations, computing the mixed volume is equivalent to finding the volume of a single mixed cell that is the translate of a parallelotope. Additionally, we give a coloring condition on cycle networks to identify reaction systems with binomial steady-state ideals. We highlight both of these theorems using a class of networks referred to as species-overlapping networks and give a formula for the mixed volume of these networks.

Generic phosphate affinity constants of the CD-MUSIC-eSGC model to predict phosphate adsorption and dominant speciation on iron (hydr)oxides

Estimating the availability of phosphorus in soils and sediments is complicated by the diverse mineralogical properties of iron (hydr)oxides that control the environmental fate of phosphorus. Despite various surface complexation models have been developed, lack of generic phosphate affinity constants (logKPO4s) for iron (hydr)oxides hinders the prediction of phosphate adsorption to iron (hydr)oxides in nature. The aim of this work is to derive generic logKPO4s for the Charge Distribution-Multisite Complexation extended-Stern-Gouy-Chapman (CD-MUSIC-eSGC) model using a large phosphate adsorption database and previously derived generic protonation parameters. The optimized logKPO4s of goethite, hematite and ferrihydrite are located in a much narrower range than those in the RES3T database. Specifically, the logKPO4 ranges of FeOPO3, FeOPO2OH, FeOPO(OH)2, (FeO)2PO2, and (FeO)2POOH complexes were 17.40–18.00, 24.20–27.40, 27.90–29.80, 26.50–29.60, and 30.70–33.40, respectively. A simplified CD-MUSIC-eSGC model with species FeOPO2OH and (FeO)2PO2 and generic logKPO4 values 26.0 ± 0.9 and 27.9 ± 0.8, respectively, provides an accurate prediction of phosphate adsorption and dominant speciation to the iron (hydr)oxides at environmental pH and phosphate levels. For ferrihydrite at low pH and high phosphate levels the species FeOPO(OH)2 and (FeO)2POOH cannot be neglected. The simplified model expands the application boundaries of CD-MUSIC-eSGC model in predicting the phosphate adsorption on natural iron (hydr)oxides without laborious characterization.

An Equilibrated Error Estimator for the Multiscale Finite Element Method of a 2-D Eddy Current Problem

The multiscale finite element method (MSFEM) is a valuable tool to solve the eddy current problem in laminated materials consisting of many iron sheets, which would be prohibitively expensive to resolve in a finite element mesh. It allows using a coarse mesh that does not resolve each sheet and constructs the local fields using predefined micro-shape functions. This article presents for the first time an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> error estimator for the MSFEM, which considers the error with respect to the exact solution. It is based on flux equilibration and a modification of the theorem of Prager and Synge and provides an upper bound for the error that does not include generic constants. Numerical examples show a good performance in both linear and nonlinear cases.

Evaluation of the role of peripheral artery plaque geometry and composition on stent performance

Peripheral stent fracture is a major precursor to restenosis of femoral artery atherosclerosis that has been treated with stent implantation. In this work, we validate a workflow for performing in silico stenting on a patient specific peripheral artery with heterogeneous plaque structure. Six human cadaveric femoral arteries were imaged ex vivo using intravascular ultrasound virtual histology (IVUS-VH) to obtain baseline vessel geometry and plaque structure. The vessels were then stented and the imaging repeated to obtain the stented vessel lumen area. Finite element (FE) models were then constructed using the IVUS-VH images, where the material property constants for each finite element were calculated using the proportions of each plaque component in the element, as identified by the IVUS-VH images. A virtual stent was deployed in each FE model, and the model lumen area was calculated and compared to the experimental lumen area to validate the modeling approach. The model was then used to compare stent performance for heterogeneous and homogeneous artery models, to determine whether plaque geometry or composition had added effects on stent performance. We found that the simulated lumen areas were similar to the corresponding experimental values, despite using generic material constants. Additionally, the heterogeneous and homogeneous lumen areas were also similar, implying that plaque geometry is a stronger predictor of stent expansion performance than plaque composition. Comparing stent stress and strain for heterogeneous and homogeneous models, it was found that stress from these two models had a strong linear correlation, while the strain correlation was weaker but still present. This implies that stent performance may be predicted with a simple homogeneous material models accounting for overall geometry of the plaque, providing that stent fatigue is calculated using stress criteria.

On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity

We propose and analyze two locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity. One is a high-order scheme, and the other is a low-order scheme. For time discretization, we use the backward Euler scheme. The proposed methods are well-posed, and optimal error estimates of all the unknowns are obtained for fully discrete solutions. The generic constants in the estimates are uniformly bounded as the Lamé coefficient λ tends to infinity, and as the constrained specific storage coefficient is arbitrarily small. Therefore the methods are free of both Poisson locking and pressure oscillations. Numerical results illustrate the good performance of the methods and confirm our theoretical predictions.

Two-loop rational terms in Yang-Mills theories

Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to nf fermions with arbitrary masses.

A low-order locking-free virtual element for linear elasticity problems

We propose and analyze a low-order virtual element for linear elasticity problems. It can be seen as an evolution of the Bernardi–Raugel element to general polygonal meshes. Like the Bernardi–Raugel element, the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge. The proposed method is well-posed, and optimal error estimates are obtained in the L2 and H1 norms. Moreover, the generic constants in these estimates are shown to be uniform with respect to the Lamé coefficient λ. Numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions.

A two-energies principle for the biharmonic equation and ana posteriorierror estimator for an interior penalty discontinuous Galerkin approximation

We consider ana posteriorierror estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator.

Three First-Order Finite Volume Element Methods for Stokes Equations under Minimal Regularity Assumptions

Three first-order finite volume element methods, namely, conforming, nonconforming, and discontinuous Galerkin schemes for Stokes equations, are analyzed and compared using the medius analysis. The latter analysis is based on a combination of arguments from a priori and a posteriori error analyses under no extra regularity assumptions on the weak solution. Best-approximation results for the energy norms hold in all cases and allow for comparison results up to generic equivalence constants and higher-order data oscillation of the applied volume forces with little modification for different pressure approximations. A priori and a posteriori analyses of discontinuous Galerkin finite volume element methods with SIPG, IIPG, NIPG are included for completeness.

Uniform estimate for characteristics-mixed finite element method for two-dimensional advection-dominated transport problems

In this paper, we prove uniform optimal-order error estimates for characteristics-mixed finite element methods for two-dimensional convection-dominated diffusion equations. The generic constants in the error estimates do not explicitly depend on the scaling diffusion parameter ε, but depend linearly on certain Sobolev norms of the true solution. Combining the estimates with the stability estimates of the true solution, we prove that these constants depend only on the initial and the right-hand side data. Numerical experiments are presented to confirm our theoretical findings.

BPS boojums in ${\cal N}=2$ supersymmetric gauge theories I

We study 1/4 Bogomol'nyi-Prasad-Sommerfield (BPS) composite solitons of vortex strings, domain walls and boojums in ${\cal N}=2$ supersymmetric Abelian gauge theories in four dimensions. We obtain solutions to the 1/4 BPS equations with the finite gauge coupling constant. To obtain numerical solutions for generic coupling constants, we construct globally correct approximate functions which allow us to easily find fixed points of gradient flow equations. We analytically/numerically confirm that the negative mass of a single boojum appearingat the endpoint of the vortex string on the logarithmically bent domain wall is equal to the half-mass of the 't Hooft-Polyakov monopole. We examine various configurations and clarify how the shape of the boojum depends on the coupling constants and moduli parameters. We also find analytic solutions to the 1/4 BPS equations for specific values of the coupling constants.

Liquid crystal defects in the Landau–de Gennes theory in two dimensions — Beyond the one-constant approximation

We consider the two-dimensional (2D) Landau–de Gennes energy with several elastic constants, subject to general [Formula: see text]-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case [Formula: see text]. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parametres. We also numerically study the stability properties of the critical points of the Landau–de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system.

Eliminating the pollution effect in Helmholtz problems by local subscale correction

We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number $\kappa$ in bounded domains in $\mathbb{R}^d$. The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale corrections in the spirit of numerical homogenization. The precomputation of the corrections involves the solution of coercive cell problems on localized subdomains of size $\ell H$; $H$ being the mesh size and $\ell$ being the oversampling parameter. If the mesh size and the oversampling parameter are such that $H\kappa$ and $\log(\kappa)/\ell$ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to $H$; pollution effects are eliminated in this regime.

Evaluation of different approaches to describe the sorption and desorption of phosphorus in soils on experimental data

Phosphorus is an essential element to enhance the needed increase in crop production in the forthcoming century. On the other hand environmental losses of phosphorus cause eutrophication of surface waters. Both problems call for reliable models to predict the behaviour of phosphorus in agricultural soils. In this study the performances of five different sorption approaches were evaluated. The ultimate aim was to identify the most suitable concept for large scale predictions of P dynamics in soils, in terms of a high comparability between observations and predictions with a minimum amount of input data. The model results were compared with unique data from long term (10–15years) experimental field studies of grassland including situations with P mining, equilibrium P fertilization and P surpluses and a pot experiment with P mining. The model performance was evaluated while using site specific constants and generic constants for adsorption and desorption. Three rate limited models (DPPS, INITIATOR and ANIMO) showed good performance when site specific constants were used but the performance of the equilibrium model (NEWS-Dynamic) was reasonably comparable. Model performance was better for experiments with a P surplus than for P mining and was also better for sandy soils as compared to clay and peat soils. However, long term desorption rates had to be calibrated for each application rate. The performance of all models declined when generic data were used. We conclude that none of the included models properly describe what happens when the soil changes its P status, considering that parameterization needs to be treatment-specific to obtain reliable predictions. Considering this flaw, models of intermediate complexity, including both equilibrium and rate limited sorption, and a limited data demand, like DPPS and INITIATOR, seem most suited for regional model application.

A guaranteed equilibrated error estimator for the $\mathbf{A}-\varphi$ and $\mathbf{T}-\Omega$ magnetodynamic harmonic formulations of the Maxwell system

In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodynamic formulation of the Maxwell's system. This system is recast in two classical potential formulations, which are solved by a Finite Element method. The equilibrated estimator is built starting from these dual problems and is consequently available to estimate the error of both numerical resolutions. The estimator reliability and efficiency without generic constants are established. Then, numerical tests are performed, allowing to illustrate the obtained theoretical results.

Uniform estimates for characteristics-mixed finite method for transient advection-dominated diffusion problems in two-dimensional space

We prove uniform error estimates for the characteristics-mixed finite element method for two-dimensional transient convection-dominated diffusion equations, in the sense that the generic constants in the error estimates depend linearly on certain Sobolev norms of the true solution but not on the scaling diffusion parameter ε. Numerical experiments are presented to confirm our theoretical findings.

Entanglement of Dirac bi-spinor states driven by Poincaré classes of [formula omitted] coupling potentials

A generalized description of entanglement and quantum correlation properties constraining internal degrees of freedom of Dirac(-like) structures driven by arbitrary Poincaré classes of external field potentials is proposed. The role of (pseudo)scalar, (pseudo)vector and tensor interactions in producing/destroying intrinsic quantum correlations for SU(2)⊗SU(2) bi-spinor structures is discussed in terms of generic coupling constants. By using a suitable ansatz to obtain the Dirac Hamiltonian eigenspinor structure of time-independent solutions of the associated Liouville equation, the quantum entanglement, via concurrence, and quantum correlations, via geometric discord, are computed for several combinations of well-defined Poincaré classes of Dirac potentials. Besides its inherent formal structure, our results set up a framework which can be enlarged as to include localization effects and to map quantum correlation effects into Dirac-like systems which describe low-energy excitations of graphene and trapped ions.

Allometry and partitioning of individual tree biomass and carbon of Abies nephrolepis Maxim in northeast China

Abies nephrolepis Maxim is an important commercial conifer species in northeastern China. In this study, we compared the partitioning and variations of biomass and carbon concentration for four tree components (i.e. stem, root, branch, and foliage). The results indicated that foliage had the largest carbon concentration, which was significantly different from other tree components. Above-ground biomass or carbon accounted for about 78% of the total tree biomass or carbon, whereas below-ground biomass or carbon corresponded to about 22%. The root-to-shoot ratio averaged .28 for biomass and .27 for carbon. We applied likelihood analysis to investigate the error structure of allometric relationship W = a·Db and found that the multiplicative error structure was favored. Thus, the additive systems of biomass and carbon equations in a log-transformed scale were constructed using nonlinear seemly unrelated regression. The model fitting results showed that all biomass and carbon equations fitted the data well. Further, five approaches for calculating carbon stock of individual trees were evaluated and compared. The carbon allometric equations and the estimated biomass multiplied by weighted mean carbon concentration were more advantageous, whereas using the generic carbon concentration constants produced significant biases in estimating the carbon stock of individual trees.

Large deviations and exact asymptotics for constrained exponential random graphs

We present a technique for approximating generic normalization constants subject to constraints. The method is then applied to derive the exact asymptotics for the conditional normalization constant of constrained exponential random graphs.

A posteriori error estimators for convection–diffusion eigenvalue problems

A posteriori error estimators for convection–diffusion eigenvalue model problems are discussed in Heuveline and Rannacher (2001) [17] in the context of the dual-weighted residual method (DWR). This paper directly addresses the variational formulation rather than the non-linear ansatz of Becker and Rannacher for some convection–diffusion model problem and presents a posteriori error estimators for the eigenvalue error based on averaging techniques. Two different postprocessing techniques attached to the DWR paradigm plus two new dual-weighted a posteriori error estimators are also presented. The first new estimator utilises an auxiliary Raviart–Thomas mixed finite element method and the second exploits an averaging technique in combination with ideas of DWR. The six a posteriori error estimators are compared in three numerical examples and illustrate reliability and efficiency and the dependence of generic constants on the size of the eigenvalue or the convection coefficient.