Three-parameter Form Research Articles

Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{B_c}$$\\end{document} to A transition form factors and semileptonic decays in self-consistent covariant light-front approach

We present a comprehensive analysis of the semileptonic weak decays of Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c$$\\end{document} meson decaying to axial-vector (A) mesons for bottom-conserving and bottom-changing decay modes. We employ self-consistent covariant light-front quark model (CLF QM) that uses type-II correspondence to eliminate inconsistencies in the traditional type-I CLF QM. As a fresh attempt, we test the self-consistency in CLF QM through type-II correspondence for Bc→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow A$$\\end{document} meson transition form factors. We establish that in type-II correspondence the form factors for longitudinal and transverse polarization states are numerically equal and are free from zero-mode contributions, which confirms the self-consistency of type-II correspondence for Bc→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow A$$\\end{document} transition form factors. Furthermore, we ascertain that the problems of inconsistency and violation of covariance of CLF QM within the type-I correspondence are resolved in type-II correspondence for Bc→A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow A$$\\end{document} transitions. We thoroughly investigate the effects of self-consistency between type-I and type-II schemes using a comparative analysis. In this investigation, we employ a direct calculation approach to determine the form factors within the space-like region. Subsequently, we employ a vector meson dominance (VMD)-inspired three-parameter form to extrapolate these form factors into the physically accessible region. This enables us to study the q2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2$$\\end{document} dependence of the form factors in weak hadronic currents for the whole accessible kinematic range for both bottom-conserving as well as bottom-changing transitions. In addition, we extend our analysis to predict the branching ratios of the semileptonic weak decays of Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c$$\\end{document} meson involving axial-vector meson in the final state to quantify the effects of self-consistency in these decays that were not studied before. We evaluate the lepton mass effect on these branching ratios and various other important physical observables, such as forward-backward asymmetries, lepton-side convexity parameter, asymmetry parameter, and longitudinal polarization asymmetries and fractions. Finally, we obtain the lepton flavor universality ratios for various decays.

Analysis of the inflection points of height-diameter models

The inflection point is an important feature of sigmoidal height-diameter (H-D) models. It is often cited as one of the properties favoring sigmoidal model forms. However, there are very few studies analyzing the inflection points of H-D models. The goals of this study were to theoretically and empirically examine the behaviors of inflection points of six common H-D models with a regional dataset. The six models were the Wykoff (WYK), Schumacher (SCH), Curtis (CUR), Hossfeld IV (HOS), von Bertalanffy-Richards (VBR), and Gompertz (GPZ) models. The models were first fitted in their base forms with tree species as random effects and were then expanded to include functional traits and spatial distribution. The distributions of the estimated inflection points were similar between the two-parameter models WYK, SCH, and CUR, but were different between the three-parameter models HOS, VBR, and GPZ. GPZ produced some of the largest inflection points. HOS and VBR produced concave H-D curves without inflection points for 12.7% and 39.7% of the tree species. Evergreen species or decreasing shade tolerance resulted in larger inflection points. The trends in the estimated inflection points of HOS and VBR were entirely opposite across the landscape. Furthermore, HOS could produce concave H-D curves for portions of the landscape. Based on the studied behaviors, the choice between two-parameter models may not matter. We recommend comparing several three-parameter model forms for consistency in estimated inflection points before deciding on one. Believing sigmoidal models to have inflection points does not necessarily mean that they will produce fitted curves with one. Our study highlights the need to integrate analysis of inflection points into modeling H-D relationships.

The rigidity spectrum of the long-term cosmic ray variations during solar activity cycles 19–24

The monthly rigidity spectrum of cosmic ray (CR) variations in the 19-24 cycles of solar activity was obtained by the global survey method using the data of continuous ground and near-Earth monitoring of CRs, exempted from atmospheric and local effects.The changes of the spectrum, first obtained for such a long period, made it possible to reveal the features of large-scale effects in CR modulation, the presence of 22-year and 11-year CR variations in the spectrum, and confirm an abnormal spectrum change in the 70s. The paper assumes a rigidity spectrum of CRs, given in a three-parameter form. Analysis of the obtained long-term CR variations for particles with rigidity 10 GV shows that the amplitude of the 22-year wave in the CR intensity increases from cycle to cycle and reaches its maximum value at the minimum of 23/24th solar activity cycle. Softening of the spectrum at the cycle minima has been revealed for the negative polarity of the solar magnetic field (qA<0). The reasons for the abnormally high CR density at the minimum of the 24th cycle and the spectrum features in the 70s are discussed. The spectrum of long-term CR variations in the 19-24 solar activity cycles, determined from the experimental data, makes it possible to verify some conclusions of the theory of heliospheric CR modulation concerning the role of the magnetic drift of particles in cycles with the different polarity of the solar magnetic field. In particular, we propose the explanation for the observed R−2 spectrum of the variations in the minima of the negative solar activity cycles, related with the scattering of particles in the vicinity of the neutral current sheet.

Paulik revisited: Statistical framework and estimation performance of multistage recruitment functions

Multiple processes act at different stages and different intensities within the timeline between spawning and the age designated as “recruitment”. However, common practice is to model only a single step between spawning stock and recruits. Reasons for this practice include lack of data on the intermediate stages, lack of understanding of the mechanisms and functional form governing intermediate stages, and lack of computational resources to model a multistage process in the appropriate statistical framework. We develop a state-space framework and, using a simulation study, we explore the estimation of multistage stock-recruit functions. We evaluated four different functions (Ricker, Beverton-Holt, Shepherd, and Generalized), and examined the effects on estimation of several factors, including the form of density dependence, the magnitude of measurement error associated with each stage, type of prior on measurement error, and the magnitude of process error between stages. Three-parameter stock-recruit functions (Shepherd, Generalized) correctly identified the form of density dependence in each stage, although the Shepherd model exhibited problems with convergence. Model misspecification resulted in bias, especially in parameters specifying measurement and process error; an informative prior on measurement error improved precision and bias. The Deviance Information Criterion selected the Ricker model too often, even when the true model was Beverton-Holt. Sequential density-dependent stages, even multiple overcompensatory stages, lead to an overall function that appears fairly flat, suggesting that a function capable of producing asymptotic dynamics is a practical default. The common practice of bypassing stages between the first (spawning stock) and the last (recruits) worked reasonably well, except when fitting a Ricker model, most likely because the true function was nearly flat over most of the range of the first stage. An application to data on North Sea herring illustrates that a multi-stage stock-recruit model can generate a stock-recruit function that is intermediate between Ricker and Beverton-Holt models, and that does not match existing three-parameter forms.

Objectively combining AR5 instrumental period and paleoclimate climate sensitivity evidence

Combining instrumental period evidence regarding equilibrium climate sensitivity with largely independent paleoclimate proxy evidence should enable a more constrained sensitivity estimate to be obtained. Previous, subjective Bayesian approaches involved selection of a prior probability distribution reflecting the investigators’ beliefs about climate sensitivity. Here a recently developed approach employing two different statistical methods—objective Bayesian and frequentist likelihood-ratio—is used to combine instrumental period and paleoclimate evidence based on data presented and assessments made in the IPCC Fifth Assessment Report. Probabilistic estimates from each source of evidence are represented by posterior probability density functions (PDFs) of physically-appropriate form that can be uniquely factored into a likelihood function and a noninformative prior distribution. The three-parameter form is shown accurately to fit a wide range of estimated climate sensitivity PDFs. The likelihood functions relating to the probabilistic estimates from the two sources are multiplicatively combined and a prior is derived that is noninformative for inference from the combined evidence. A posterior PDF that incorporates the evidence from both sources is produced using a single-step approach, which avoids the order-dependency that would arise if Bayesian updating were used. Results are compared with an alternative approach using the frequentist signed root likelihood ratio method. Results from these two methods are effectively identical, and provide a 5–95% range for climate sensitivity of 1.1–4.05 K (median 1.87 K).

Properties and prediction accuracy of a sigmoid function of time-determinate growth

The properties and short-term prediction accuracy of mathematical model of sigmoid time-determinate growth, denoted as “KM-function”, are presented. Comparative mathematical analysis of the function revealed that it is a model of asymmetrical sigmoid growth, which starts at zero size of an organism and terminates when it reaches its final size. The function assumes a finite length of the growth period and includes a parameter interpretable as the expected lifespan of the organism. Moreover, the possibility for growth curve inflexion at any age is possible, so the function can be used for modelling of S-shaped growth trajectories with various degree of asymmetry. These good theoretical predispositions for realistic growth predictions were empirically evaluated. The KM-function used in three and four-parameter forms was compared with three classical (Richards, Korf and Weibull) growth functions employing two parameterisation methods - nonlinear least squares (NLS) and Bayesian method. The evaluation was conducted on the basis of the tree diameter series obtained from stem analyses. The main empirical findings are: (i) if the minimisation of the prediction bias is required, the KM-function in three-parameter form in connection with Bayes parameterisation can be recommended; (ii) if the minimisation of root square error (RMSE) is required, the best short-term prediction results for a particular dataset were obtained with four-parameter Weibull function employing NLS parameterisation; (iii) moreover, three-parameter functions parameterised by Bayesian methods show a considerably smaller RMSE by 15-25% as well as smaller biases by 40-60% than four-parameter functions employing NLS. Overall, all analyses confirmed relative usefulness of the KM-function in comparison with classical growth functions, especially in connection with Bayesian parameterisation.

The evolution of Zipf’s law indicative of city development

Zipf’s law of city-size distributions can be expressed by three types of mathematical models: one-parameter form, two-parameter form, and three-parameter form. The one-parameter and one of the two-parameter models are familiar to urban scientists. However, the three-parameter model and another type of two-parameter model have not attracted attention. This paper is devoted to exploring the conditions and scopes of application of these Zipf models. By mathematical reasoning and empirical analysis, new discoveries are made as follows. First, if the size distribution of cities in a geographical region cannot be described with the one- or two-parameter model, maybe it can be characterized by the three-parameter model with a scaling factor and a scale-translational factor. Second, all these Zipf models can be unified by hierarchical scaling laws based on cascade structure. Third, the patterns of city-size distributions seem to evolve from three-parameter mode to two-parameter mode, and then to one-parameter mode. Four-year census data of Chinese cities are employed to verify the three-parameter Zipf’s law and the corresponding hierarchical structure of rank-size distributions. This study is revealing for people to understand the scientific laws of social systems and the property of urban development.

Empirical characterisation and mathematical modelling of settlement in composting batch reactors

The settlement of organic matter during composting was measured at different levels during the active biodegradation phase in forced-aerated static reactors loaded with different mixtures of organic wastes. The temperature evolution and the concentration of oxygen and carbon dioxide were also recorded in the exhaust gases. Two two-parameter equations and their generalised three-parameter form were fitted to the experimental data and their capability to predict settlement as a function of time was discussed. The settlement field inside the reactors was successfully described with a linear profile. At any given time and vertical position, mass settlement was proportional to the initial height after reactor loading. Furthermore, a relationship linking settlement to biological activity was also proposed. Under all tested experimental conditions, settlement showed a clear correlation with the cumulative oxygen consumption that was modelled with a three parameter equation.

Generalized nonhyperbolic moveout approximation

Reflection moveout approximations are commonly used for velocity analysis, stacking, and time migration. A novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced. In comparison with the classic hyperbolic approximation, which uses only two parameters (zero-offset time and moveout velocity), this form involves five parameters that can be determined, in a known medium, from zero-offset computations and from tracing one nonzero-offset ray. It is called a generalized approximation because it reduces to some known three-parameter forms with a particular choice of coefficients. By testing the accuracy of the proposed approximation with analytical and numerical examples, the new approximation is shown to bring an improvement in accuracy of several orders of magnitude compared to known analytical approximations, which makes it as good as exact for many practical purposes.

A parameterization of cosmic ray shower profiles based on shower width

Cosmic ray (CR) air showers, detected via the air fluorescence technique, are reconstructed in part using functions that parameterize the longitudinal profile of each shower. The profile parameterization yields the position of shower maximum, Xmax in gm cm−2, which is sensitive to the incident CR particle type: e.g. p, C/N/O, Fe or γ. The integral of the profile is directly related to the shower energy. The Pierre Auger fluorescence reconstruction uses the Gaisser–Hillas (GH) four-parameter form [5]. The HiRes group has used both the Gaisser–Hillas form and a three-parameter Gaussian in age form [7]. Historically analytic shower theory suggested yet other forms; the best known is a three-parameter form popularized by Greisen [6]. Our work now uses the shower full width half-maximum, fwhm, and shower asymmetry parameter, f, to unify the parameterization of all three profile functions. Furthermore, shower profiles expressed in terms of the new parameters: fwhm, f, have correlations greatly reduced over e.g. Gaisser–Hillas parameters X0, λ. This allows shower profile reconstructions to add constraints (if needed) on the mostly uncorrelated parameters fwhm, f.

Covariant light-front approach forBctransition form factors

In the covariant light-front quark model, we investigate the form factors of $B_c$ decays into $D, D^*, D_s, D_s^*, \eta_c, J/\psi, B, B^*, B_s, B_s^*$ mesons. The form factors in the spacelike region $q^2<0$ are directly evaluated. To extrapolate the form factors to the full kinematic region, we fit the form factors by adopting a three-parameter form from the spacelike region. $b\to u,d,s$ transition form factors at maximally recoiling point ($q^2=0$) are smaller than $b\to c$ and $c\to d,s$ transition form factors, while the $b\to d,s,c$ form factors at zero recoiling point are close to each other. In the fitting procedure, we find the parameters for the form factors $A_2(B_c\to B^*)$ and $A_2(B_c\to B^*_s)$ strongly depend on the decay constants of $B^*$ and $B_s^*$ mesons. Fortunately, the semileptonic and nonleptonic $B_c$ decays are not sensitive to these form factors. With the form factors, we also investigate the branching fractions, polarizations of the semileptonic $B_c\to Ml\nu$ decays. Semileptonic $B_c\to (\eta_c,J/\psi)l\nu$ and $B_c\to (B_s,B_s^*)l\nu$ decays have much larger branching fractions than $B_c\to (D,D^*,B,B^*)l\nu$. In the three kinds of $B_c\to Vl\nu$ decays, contributions from the longitudinal polarized vector is comparable with those from the transversely polarized vector. These predictions will be tested at the ongoing and forthcoming hadron colliders.

Direct determination of the Brooks–Corey hydraulic functions by fitting an extended power function to the outflow method data

Summary A method is developed for the direct determination of the parameters characterizing the Brooks and Corey closed-form hydraulic functions (water retention curve and hydraulic conductivity function) using data obtained only from one-step outflow experiments. Simple algebraic equations are derived to calculate directly the Brooks and Corey model parameters. The only parameters involved in these equations are empirical parameters obtained by a simple curve-fitting procedure applied to the experimental outflow data for a simple three-parameter power form equation. In addition, the method requires the value of the water content obtained at the pressure head of −150 m. The suggested method is verified numerically by simulating a one-step outflow experiment of a hypothetical soil. The proposed method was also verified from one-step outflow experimental data. The predicted soil water retention curve is compared favorably with measured water retention data. The coefficient of determination R2 is greater than 0.87 in all cases examined.

Logistic Equations Effectively Model Mammalian Cell Batch and Fed-Batch Kinetics by Logically Constraining the Fit

A four-parameter logistic equation was used to fit batch and fed-batch time profiles of viable cell density in order to estimate net growth rates from the inoculation through the cell death phase. Reduced three-parameter forms were used for nutrient uptake and metabolite/product formation rate calculations. These logistic equations constrained the fits to expected general concentration trends, either increasing followed by decreasing (four-parameter) or monotonic (three-parameter). The applicability of this approach was first verified for Chinese hamster ovary (CHO) cells cultivated in 15-L batch bioreactors. Cell density, metabolite, and nutrient concentrations were monitored over time and used to estimate the logistic parameters by nonlinear least squares. The logistic models fit the experimental data well, supporting the validity of this approach. Further evidence to this effect was obtained by applying the technique to three previously published batch studies for baby hamster kidney (BHK) and hybridoma cells in bioreactors ranging from 100 mL to 300 L. In 27 of the 30 batch data sets examined, the logistic models provided a statistically superior description of the experimental data than polynomial fitting. Two fed-batch experiments with hybridoma and CHO cells in benchtop bioreactors were also examined, and the logistic fits provided good representations of the experimental data in all 25 data sets. From a computational standpoint, this approach was simpler than classical approaches involving Monod-type kinetics. Since the logistic equations were analytically differentiable, specific rates could be readily estimated. Overall, the advantages of the logistic modeling approach should make it an attractive option for effectively estimating specific rates from batch and fed-batch cultures.

Assessing the Value of the Threshold Parameter in the Weibull Distribution Using Bayes Paradigm

The Weibull distribution represents a wide variety of situations. Usually, the distribution is considered as a two-parameter family with a scale, and a shape parameter. If, however, the given data reflect additional information in the form of a minimum guarantee, a positive value away from zero, it is better to go for a three-parameter model with the additional parameter known as the threshold. The threshold parameter is often very important, but increases the complexity of the model. Arbitrarily going for the three-parameter form is not advisable unless it is really required by the data. This article attempts to make a simulation-based Bayesian study for checking if the threshold parameter can be taken to be zero or positive in situations representing the two models. We study the compatibility of the models for the given data set. We conduct the posterior simulation in each case using Gibbs sampling.

Variational Analysis of the Phenyl + O2 and Phenoxy + O Reactions

Variational transition state analysis was performed on the barrierless phenyl + O2 and phenoxy + O association reactions. In addition, we also calculated rate constants for the related vinyl radical (C2H3) + O2 and vinoxy radical (C2H3O) + O reactions and provided rate constant estimates for analogous reactions in substituted aromatic systems. Potential energy scans along the dissociating C-OO and CO-O bonds (with consideration of C-OO internal rotation) were obtained at the O3LYP/6-31G(d) density functional theory level. The CO-O and C-OO bond scission reactions were observed to be barrierless, in both phenyl and vinyl systems. Potential energy wells were scaled by G3B3 reaction enthalpies to obtain accurate activation enthalpies. Frequency calculations were performed for all reactants and products and at points along the potential energy surfaces, allowing us to evaluate thermochemical properties as a function of temperature according to the principles of statistical mechanics and the rigid rotor harmonic oscillator (RRHO) approximation. The low-frequency vibrational modes corresponding to R-OO internal rotation were omitted from the RRHO analysis and replaced with a hindered internal rotor analysis using O3LYP/6-31G(d) rotor potentials. Rate constants were calculated as a function of temperature (300-2000 K) and position from activation entropies and enthalpies, according to canonical transition state theory; these rate constants were minimized with respect to position to obtain variational rate constants as a function of temperature. For the phenyl + O2 reaction, we identified the transition state to be located at a C-OO bond length of between 2.56 and 2.16 A (300-2000 K), while for the phenoxy + O reaction, the transition state was located at a CO-O bond length of 2.00-1.90 A. Variational rate constants were fit to a three-parameter form of the Arrhenius equation, and for the phenyl + O2 association reaction, we found k(T) = 1.860 x 1013T-0.217 exp(0.358/T) (with k in cm3 mol-1 s-1 and T in K); this rate equation provides good agreement with low-temperature experimental measurements of the phenyl + O2 rate constant. Preliminary results were presented for a correlation between activation energy (or reaction enthalpy) and pre-exponential factor for heterolytic O-O bond scission reactions.

Propeller Blade Design Using Partial Differential Equations with Three Vector Valued Shape Parameter

Surface Partial Differential Equations (USP) is defined as a PPS solution has many advantages in surface blending, free-form surfaces and so on. In this paper, we will examine the design of the blade with three shape parameters. We will examine the effects of parameter changes on the shape of the blade is raised. A solution to the PPS degree will be carried out analytically four to take into account the performance of hardware. Keywords: Partial Differential Equations; blade design; three-parameter form.

Molecular dynamics simulations of monoclinic calcium apatites: A universal equation of state

Molecular dynamics simulations of monoclinic ( P2 1/ b) hydroxy- and chlorapatite were undertaken in the range 498 K < T < 1298 K, and for pressures up to 7.5 GPa. The all-atom Born–Huggins–Mayer force field, that had been previously used to successfully describe the room temperature isotherms of both compounds, was also used in this work. The isothermal sets of p– V data generated by simulation were each fitted to the three-parameter form of the isothermal Parsafar and Mason equation of state (EoS) with an accuracy better than 0.07%. Taking the temperature dependence of the coefficients into account, it was found that the MD data are satisfactorily reproduced by the universal EoS. The isothermal compressibility coefficient dependence with pressure can be described by a linear relation.

The log-normal distribution for hydraulic conductivity of compacted clays: Two or three parameters?

Risk and uncertainty assessments for waste containment systems employing clay barriers often include spatial variability in the hydraulic conductivity as part of the analysis. The two-parameter log-normal distribution is often used to describe the spatial variability, but for compacted clays the three-parameter form can be more appropriate. A statistical hypothesis test was developed that can be used to determine if the two- or three-parameter form of the log-normal distribution is more appropriate. The test is based on a likelihood ratio, comparing likelihood functions for the two- and three-parameter forms at their maximums. Likelihood functions are used assuming that a data set can be segregated into a set of distinct groups of hydraulic conductivities. A step-by-step calculation procedure is described and the test is applied to data collected from 45 sites.

TETRAHEDRAL FAMILY SYMMETRY AND THE NEUTRINO MIXING MATRIX

In a new application of the discrete non-Abelian symmetry A4 using the canonical seesaw mechanism, a three-parameter form of the neutrino mass matrix is derived. It predicts the following mixing angles for neutrino oscillations: θ13=0, sin 2θ23=1/2, and sin 2θ12 close, but not exactly equal to 1/3, in one natural symmetry limit.

Review of the Thermodynamic Properties of Mg(NO3)2(aq) and Their Representation with the Standard and Extended Ion‐Interaction (Pitzer) Models at 298.15 K

AbstractFor Abstract see ChemInform Abstract in Full Text.