Original Linear Equation Research Articles

The Appendicular Osteology of Patagotitan Mayorum (Dinosauria, Sauropoda)

ABSTRACT Its huge size, excellent preservation, and completeness make Patagotitan mayorum a unique opportunity to explore the anatomy, paleobiological, and phylogenetic aspects linked to gigantism within Sauropoda. In this regard, we describe the appendicular skeleton of this titanosaurian species from the late Albian-aged Cerro Barcino Formation of Chubut Province, Argentina. The diagnosis of Patagotitan mayorum is revised, increasing the number of identified autapomorphies (i.e., lateral surface of the scapular blade with two divergent crests; anterior surface of proximal humerus with paired muscle scars; combined bulges on the deltopectoral area of the humerus; ischium with well-developed and sharp ridge projecting from the ischial tuberosity to the distal blade). Several diagnostic characters of this species correspond to osteological correlates associated to appendicular musculature (e.g., Mm. deltoideus scapularis, deltoideus clavicularis, and teres major; M. coracobrachialis; Mm. supracoracoideus/deltoideus clavicularis and latissimus dorsi; Mm. flexor tibialis 3 and adductor femoris 2), which we discuss in the context of sauropod evolution. In the light of a modification of the scaling equation previously proposed and adjusting the long bone circumference for the humeri of Patagotitan, a new body mass estimate of this species ranges between 42–71 tons, with a mean value of 57 tons. Although considerably less than the value obtained by the original linear equation, the corrected quadratic equation used here provides a mean body mass estimate that is more consistent with those derived from volumetric reconstructions of Patagotitan.

Automatic calibration and improvements on an instream chlorophyll a simulation in the HSPF model

Accurate prediction of chlorophyll a (Chl a) concentration in surface water bodies such as lakes or rivers is crucial for water quality management. This study improved the predictive simulation of instream Chl a with the Hydrological Simulation Program-FORTRAN (HSPF) by adding automatic calibration and modifying the growth-temperature formulation of phytoplankton in the original HSPF model. A total of 62 model parameters, selected from a series of sensitivity analyses, were automatically calibrated in a stepwise manner for different variables in the order of flow, sediment, water temperature, ammonia/nitrate couple, and phosphate/Chl a couple. With finer temporal resolution (5–8 days) data than those of majority of the existing HSPF studies, the automatic calibration procedure provided the model with performance ratings of ‘satisfactory’ or better for all the variables including nutrients and Chl a: The percent bias values ranged from -18% - 54% and -20% – 62% for nutrients and Chl a, respectively. The original linear equation on the growth-temperature relationship of phytoplankton in simulating instream Chl a was modified using a quadratic equation and an exponential equation. The exponential equation outperformed the original linear and quadratic equations, particularly in simulating the excess concentrations of Chl a observed during summer seasons. For the validation data set, the exponential equation predicted 78% of the eutrophic cases while the linear and quadratic equation only predicted 53% and 13% of the eutrophic cases, respectively. The modified HSPF model offers an improved prediction of instream Chl a. This approach will be useful for providing early warning of algal blooms, facilitating the implementation of effective management of stream water quality.

The prediction of in-flight hypoxaemia using non-linear equations

Respiratory disease may cause profound hypoxaemia during flight. Previously derived linear equations poorly predict the need for supplemental oxygen during air travel. The current gold standard assessment is the hypoxic challenge test (HCT). Recent guidelines recommend HCT is performed for those patients with SpO2<95% at sea level. The HCT protocol is a costly and time consuming investigation. Retrospective clinical and HCT data from 138 patients were applied to previous linear equations to assess predictive value. Novel non-linear predictive models (NLMs) were constructed from these data. The linear equations and the NLMs were then applied prospectively to 44 patients undergoing HCT. Overall, 39% of historic patients had a positive HCT (PaO2N2<50mmHg). Existing linear equations varied in sensitivity (52-87%) and specificity (40-74%) at predicting positive HCT results. Seven novel NLMs (NLM1 to NLM7) were developed from the historic dataset. All NLMs predicted PaO2N2 more accurately than the original linear equations when tested prospectively. The best fit was observed using NLM2 which uses PaO2RA and PaCO2RA as input terms. The NLMs are applicable to a broad range of conditions. The novel NLMs represent a low cost option for the prediction of significant hypoxia during flight and perform better than SpO2 in identifying those patients who require more formal assessment with HCT.

Historical aspects of capillarity and capillary condensation

The basic equation of capillarity is generally associated with the names of Thomas Young and Pierre Simon Laplace. Careful perusal of the early literature has revealed some unexpected historical aspects. For example, Young’s 1805 essay on the ‘cohesion of fluids’ was purely descriptive, but it did provide a basis for the ‘classical’ treatment of capillarity and wetting. In contrast, Laplace’s 1806 work involved a more ‘modern’ energetic approach to capillary attraction, but avoided any direct reference to surface tension. The related capillary condensation equation is now universally known as the Kelvin equation. However, this exponential relationship was not the form of the original equation proposed in 1871 by William Thomson (later Lord Kelvin). Zsigmondy (1911) [11] applied Thomson’s original linear equation in his seminal study of silica gels. Although the Kelvin equation is still widely used for mesopore size analysis, its limitations began to be recognised in the 1930s.

Smoothness of Invariant Manifolds for Nonautonomous Equations

For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C1+ α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v ′ = A ( t ) v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v ′ = A ( t ) v + f ( t , v ) , which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v ′ = A ( t ) v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.

Computation of the Adiabatic Eigenmodes of Stars Using the Riccati Method

Abstract In stellar oscillation problems, it is often necessary to compute a large number of stellar eigenmodes with wildly varying radial orders and angular degrees. Conventionally the relaxation method is used for this purpose. This method is, however, fraught with several numerical issues, which make it cumbersome and tedious to write the corresponding computer programs. For this reason we adopt the Riccati method as an alternative, in which the original system of linear equations is transformed into a nonlinear matrix Riccati equation. The dependent variable of the converted equation does not retain the dependent variables of the original linear equations themselves, but keeps the relations among them. This method is particularly useful when we compute eigenmodes, where the amplitudes of the eigenfunctions vary by many orders of magnitude, since the ratios of the amplitudes generally show much less variation than the amplitudes themselves. The Riccati method was successfully applied to the problems of nonadiabatic stellar oscillations in previous studies. Our aim is to do the same for the adiabatic case.

Linear Ill-Posed Problems and Dynamical Systems

A new approach to solving linear ill-posed problems is proposed. The approach consists of solving a Cauchy problem for a linear operator equation and proving that this problem has a global solution whose limit at infinity solves the original linear equation.

On the stabilization inherent in the EMS algorithm

In an attempt to overcome the EM (estimate, maximize) algorithm's lack of inherent stabilization for the solution of underdetermined linear equations, a multiplicative smoothing step was added as an essential component in the iteration, thus creating the EMS (estimate, maximize, smooth) algorithm. This paper gives an analysis of the stabilization produced through the introduction of this smoothing step. The nonlinear EMS equation are given a data smoothing only interpretation in the sense that their fixed points are solutions of a nonlinear system which takes the form of the original linear equations with only the right-hand side replaced by a nonlinear function of the fixed points. It is found that the stabilization inherent in this reformulation depends crucially on the structure of the smoothing introduced into EMS. Two types of nonuniqueness for EMS (namely the persistence of spurious solutions, associated with the underdeterminedness of the original linear system, and the artifactual presence of extraneous EM/EMS solutions, induced by the smoothing's structure) are shown to be eliminated by the use of irreducible smoothing, and thus the fate of the inherent stabilization lies with the structure of the smoothing. An exact nonnegative solution of the underlying linear system (in the case that one exists) can be obtained from the EMS algorithm if this solution is an eigenvector of the introduced smoothing matrix corresponding to the eigenvalue one. Hence, to choose the smoothing appropriately requires a priori knowledge of the desired solution.

Analysis of the kinetic hairpin transfer model for parvoviral DNA replication

All linear DNA molecules face special problems in replicating their 5' ends, as DNA polymerases add nucleotides only to pre-existing strands with free 3'-OH groups. Parvoviruses, a group of small animal viruses with a linear single-stranded DNA genome, cope with this problem by having palindromic terminal sequences that can fold back on themselves to form hairpin structures essential in priming DNA replication. The 3' terminal sequence that initiates replication becomes reversed in orientation during the process, and if the palindrome is imperfect, two different, reverse-complementary terminal sequences are generated. The relative abundances of the terminal sequence orientations at each end of the DNA molecules can be measured and give information about the replication process. From such clues, we developed a "kinetic hairpin transfer model" based on differential rates of hairpin formation and inversion processes depending on the conformations of the 3' termini. Numerical studies showed that this simple idea can account for the diverse pattern of DNA distributions observed in the family Parvoviridae. In this paper, we simplify the model to a set of coupled linear first-order ordinary differential equations in order to delineate its essential properties by Perron-Frobenius theory. Secondly, we examine our assumption of linear kinetics by modeling enzyme catalysis of the component steps of the hairpin transfer process. We show that the rate-determining step of the process is the binding of initiation complex to the self-priming hairpin structures. Furthermore, we find that if the replication machinery is saturated by DNA substrate late in an infection, the differential equations become non-linear but the steady-state DNA distribution is still given by the solution of our original linear equations.

Weighted Contact Packing: Improved Formula for Grain Packing of Quartz Arenites

Contact types between grains - floating, point, long, concavo-convex and suture - help define the fabric of the detrital framework m a sandstone and its pore geometry. These contact types are combined in a non­ linear equation based on powers of two, which is more discriminatory of fabrics in quartz arenites than the original linear equation proposed by Fuchtbauer in 1967.