Two-parameter Formula Research Articles

Application of two-parameter scoliometer values for predicting scoliotic Cobb angle

BackgroundAdolescent idiopathic scoliosis, in which obvious curves are visible in radiographic images, is also seen in combination with lumps in the back. These lumps contribute to inclination, which can be measured by a scoliometer. To the authors’ knowledge, there are no previous formulas combining thoracic and lumbar scoliometer values simultaneously to predict thoracic and lumbar Cobb angles, respectively. This study aimed to create more accurate two-parameter mathematical formulas for predicting thoracic and lumbar Cobb angles.MethodsBetween Dec. 2012 and Jan. 2013, patients diagnosed with idiopathic scoliosis in an outpatient clinic were enrolled. The maximal trunk rotations at the thoracic and lumbar regions were recorded with a scoliometer. Right asymmetry hump was deemed positive (+), and left asymmetry hump was deemed negative (−). The Cobb angles were measured with a Picture Archiving and Communication System. Statistical analysis included Pearson’s correlation coefficient, multivariate regression and Bland–Atman analysis.ResultsOne-hundred and one patients were enrolled in our study. The average thoracic curve (TC) was 23.3 ± 1.8°, while the average lumbar curve (LC) was − 23.3 ± 1.4°. The thoracic inclination (TI) and lumbar inclination (LI) were 4.5 ± 0.7 and − 5.9 ± 0.6, respectively. The one-parameter formula for the thoracic curve was TC = 2.0 TI + 14.3 (r = 0.813); for the lumbar curve, it was LC = 0.9 LI − 16.9 (r = 0.409). By multivariate regression, the two-parameter formulas for the thoracic and lumbar curves were TC = 2.6 TI − 1.4 LI (r = 0.931) and LC = − 1.5 TI + 2.0 LI (r = 0.874), respectively. The two-parameter formulas were more accurate than the one-parameter formulas.ConclusionsBased on the results of these two-parameter formulas for thoracic and lumbar curves, the Cobb angles can be predicted more accurately by the readings of the scoliometer. Physicians and other healthcare practitioners can thus evaluate patients with scoliosis more precisely than before with a scoliometer.

Isospin dependence of the nucleon density distributions

The nucleon density distributions were usually described by the two-parameter Fermi shape formula. In this work, the isospin effects in the nucleon density distributions are considered, and a three-parameter formula is proposed and modified by simultaneously considering the experimental data of 885 root-mean-square (rms) charge radii and neutron skin thickness of 26 stable nuclei. The improved nucleon density formula is physically reasonable and achieves a good agreement for the charge radii and neutron skin thickness calculated with the new nucleon density formula comparing the available experimental data.

A Simple and Versatile Field Prediction Model for Indoor and Indoor-to-Outdoor Propagation

A simple field prediction model based on a combination of a two-parameter propagation formula and a multi-wall model is proposed for fast and yet accurate indoor and indoor-to-outdoor field prediction. The model’s approach is based on: 1) simplicity; 2) physical soundness; and 3) adaptability to the available environment-database format. The model is validated versus both ray tracing and measurements in different environments and it is shown to perform very well in all cases. Moreover, the model is very fast and can exploit the accuracy plus of deterministic prediction based on the 3-D indoor building map whenever it is available.

A simpler ingredient for a complex calculation

Using density functional theory to determine electron configurations requires detailed computations with multiple approximations. A two-parameter formula simplifies one part.

Two-term formula for ground band energy symmetry in low-lying levels of light Mg-Zr nuclei

In this paper, two parameter single-term energy formula [Formula: see text] is used to study the energy spin relationship within the ground bands of even–even Mg-Zr nuclei. The formula works better for the [Formula: see text]-soft nuclei as well as vibrational nuclei. We also compared it with other two-parameter formulas: Ejiri, [Formula: see text], [Formula: see text] and soft rotor formula (SRF). We also study the symmetry of the nuclei in the framework of interacting boson model (IBM-1). The IBM-1 was employed to determine the most appropriate Hamiltonian, the Hamiltonian of the IBM-1 and [Formula: see text](6) symmetry calculation, for the study of these isotopes. We have also calculated energy levels and B(E2) values for number of transitions in these [Formula: see text]Se and [Formula: see text]Kr isotopes and there is a good agreement between the presented results and the previous experimental data.

Phenomenological description of the Coulomb energies of medium-heavy and superheavy nuclei

A phenomenological description of Coulomb energies is presented for more than 400 nuclei in the range of mass numbers A = 5–244. The Coulomb displacement energies between neighboring isobaric nuclei ΔE C (A, Z) are approximated using a two-parameter formula. The influence of deformation is taken into account for heavy nuclei. It is shown that the mean difference is less than 100 keV, which is better than for microscopic calculations. Magnitudes of ΔE C (A, Z) are calculated for heavy and superheavy nuclei up to mass number A = 300.

Investigations on the nuclear charge radii

Experimental nuclear charge radii for 885 nuclei with N8 and Z8 have been systematically investigated. Results show that the formula for single parameter Z1/3 law is superior to that for the A1/3 law in describing nuclear charge radii. For two-parameter and three-parameter formulae, the Z1/3 law is as good as the A1/3 law. Considering the importance of shell effect and deformations for nuclear charge radii, we add a term including the Casten factor P into the conventional three-parameter formula and thus obtain very good results. The corresponding root-mean-square deviation falls to =0.0273 fm, i.e. reduced by about 50% when compared with the result obtained with the old three-parameter formula. Shell effect can be well reproduced for some elements by adding the Casten factor term. It is shown that the Casten factor plays a key role for nuclear charge radii. The odd-even staggering is a common phenomenon in many nuclear fields. This phenomenon can be observed with nuclear charge radii for most elements. For this reason, we add a term into the formula (10) in this paper. The root-mean-square deviation falls to =0.0266 fm. A five-parameter formula can well reproduce the variation of the nuclear charge radii for most elements. Calculated results are well consistent with the experimental data available. The differences between the experimental nuclear charge radii and the results calculated using the conventional three-parameter formula and the present five-parameter formula for the 885 selected nuclei are presented. A comparison of the formulae mentioned in this paper is given. The present five-parameter formula including the Casten factor P and the odd-even staggering is the best formula to fit available RC data and gives the smallest root-mean-square deviation . Our calculated results may be useful for future experiments.

Comparison of Reduced and Conventional Two-Phase Flash Calculations

Summary Phase-equilibrium calculations become computationally intensive in compositional simulation as the number of components and phases increases. Reduced methods were developed to address this problem, where the binary-interaction-parameter (BIP) matrix is approximated either by spectral decomposition (SD), as performed by Hendriks and van Bergen (1992), or with the two-parameter BIP formula of Li and Johns (2006). Several authors have recently stated that the SD method—and by reference all reduced methods—is not as fast as previously reported in the literature. In this paper we present the first study that compares all eight reduced and conventional methods published to date by use of optimized code and compilers. The results show that the SD method and its variants are not as fast as other reduced methods, and can be slower than the conventional approach when fewer than 10 components are used. These conclusions confirm the findings of recently published papers. The reason for the slow speed is the requirement that the code must allow for a variable number of eigenvalues. We show that the reduced method of Li and Johns (2006) and its variants, however, are faster because the number of reduced parameters is fixed to six, which is independent of the number of components. Speed up in flash calculations for their formula is achieved for all fluids studied when more than six components are used. For example, for 10-component fluids, a speed up of 2–3 in the computational time for Newton-Raphson (NR) iterations is obtained compared with the conventional method modeled after minimization of Gibbs energy. The reduced method modeled after the linearized approach of Nichita and Graciaa (2011), which uses the two-parameter BIP formula of Li and Johns (2006), is also demonstrated to have a significantly larger radius of convergence than other reduced and conventional methods for five fluids studied.

New reduced parameters for flash calculations based on two-parameter BIP formula

Phase equilibrium calculations constitute a significant percentage of computational time in compositional simulation, especially as the number of components and phases increase. Reduced methods address this problem by carrying out phase equilibrium calculations using a reduced number of independent parameters. These methods have shown to speed up flash calculations, decrease simulation times, and also improve convergence behavior. In this paper, we present new reduced parameters using the two-parameter binary interaction parameter formula originally proposed by Li and Johns (2006). The new reduced parameters are applied to solve two-phase flash calculations for five different fluid descriptions. The results show a significant reduction in the number of iterations required to achieve convergence compared to the Li and Johns original approach. The improved method is also shown to converge more often than other flash calculation methods for the cases studied. We further compare computational times with the new reduced approach to conventional flash calculations based on the minimization of Gibbs energy using an optimized compiler.

Micromagnetic Dynamics of Single-Domain Grain in Thin-Film Magnetic Recording Media

This paper reports a micromagnetic study of single-domain-grain dynamics in a thin-film magnetic recording medium which is used in the hard disk drive industry today. The behavior of the magnetization-vector of an individual grain is modeled by a modified Landau-Lifshitz (MLL) equation with the damping torque originating from the coupling between the magnetization and stresses of the intrinsic and extrinsic magnetic anisotropy. An analytic two-parameter formula governing the magnetization-vector reversal as a function of precession frequency is derived from the equation of magnetization energy loss. This equation depends on material-dependent parameters which are deduced from ferromagnetic resonance measurements. The results show a characteristic feature of magnetization reversal, a transient quasi-stationary state. This state is generally a property of multilayered thin-film recording media and can cause shortening of the relaxation time to the picosecond regime.

Anharmonic vibrational frequency calculations for solvated molecules in the B3LYP Kohn–Sham basis set limit

The solvent dependence of harmonic and anharmonic vibrational wavenumbers of water, formaldehyde and formamide was studied using the B3LYP method. The results obtained with the hierarchy of Jensen's polarization-consistent basis sets were fitted with two-parameter formula toward the B3LYP Kohn–Sham complete basis set (CBS) limit. Anharmonic corrections have been obtained by a second order perturbation treatment (VPT2) and vibrational configuration interaction (VCI) method. The solvent environment was treated according to the self-consistent reaction field polarizable continuum model (SCRF PCM) approach.

Statistical nuclear properties (level densities, spin distributions)

A general overview is given on the phenomenological methods used to describe the level densities in nuclei. Two well-known two-parameter formulas of level densities, the Back-Shifted Fermi Gas (BSFG) model and the Constant Temperature (CT) model, were used. A common ingredient of both is the spin distribution function, which contains in Ericsons's parametrization the spin-cutoff parameter σ. A realistic description of the parameters of both spin distribution function and the two level density models has been obtained by fitting the experimental data of 310 nuclei between 18F and 251Cf, consisting of the complete level schemes at low excitation energies and the s-wave neutron resonance spacings at the neutron binding energy. We determine a simple formula for the spin-cutoff parameter as a function of mass number and excitation energy. Also, an even-odd spin staggering in the spin distribution of the even-even nuclei was observed, and described with a simple formula. Using this newly defined spin distribution function, an empirical set of parameters of the BSFG and CT models was determined by fitting both the low-energy levels and the neutron resonance spacings. For these parameters, simple formulas were proposed that involve only quantities available from the mass tables, and allow reasonable estimations of the level density parameters for nuclei far from stability. Both the BSFG and CT models describe equally well the level densities at energies up to at least the neutron binding energy. Finally, we discuss recent experimental evidence that the CT model is the more correct description of the nuclei in the low-excitation energy (pairing) regime.

DFT calculations of structures, 13C NMR chemical shifts, and Raman RBM mode of simple models of small‐diameter zigzag (4,0) carboxylated single‐walled carbon nanotubes

Linearly conjugated benzene rings (acenes), belt-shaped molecules (cyclic acenes), and models of single-walled carbon nanotubes (SWCNTs) with one carboxylic group at the open end were fully optimized at the B3LYP/6-31G* level of theory. These models were selected to obtain some insight into the nuclear isotropic changes resulting from systematically increasing the basic building units of open-tip-monocarboxylated SWCNTs. In addition, the position of radial breathing mode (RBM), empirically correlated with the SWCNT diameter, was directly related with the radius of model cyclic acene rings. A regular convergence of selected structural, NMR, and Raman parameters with the molecular system size increase was observed, and a simple two-parameter mathematical formula enabled their estimation in infinity. The predicted (13) C NMR chemical shifts of carbon atoms close to the substituted rim of carboxylated models of zigzag (4,0) SWCNTs differed significantly from the pristine nanotubes.

DFT calculation of structures and NMR chemical shifts of simple models of small diameter zigzag single wall carbon nanotubes (SWCNTs)

Linearly conjugated benzene rings (acenes), belt-shape molecules (cyclic acenes) and model single wall carbon nanotubes (SWCNTs) were fully optimized at the unrestricted level of density functional theory (UB3LYP/6-31G*). The models of SWCNTs were selected to get some insight into the potential changes of NMR chemical shift upon systematic increase of the molecular size. The theoretical NMR chemical shifts were calculated at the B3LYP/pcS-2 level of theory using benzene as reference. In addition, the change of radial breathing mode (RBM), empirically correlated with SWCNT diameter, was directly related with the radius of cyclic acenes. Both geometrical and NMR parameters were extrapolated to infinity upon increase in the studied systems size using a simple two-parameter mathematical formula. Very good agreement between calculated and available experimental CC bond lengths of acenes was observed (RMS of 0.0173 Å). The saturation of changes in CC bond lengths and (1)H and (13)C NMR parameters for linear and cyclic acenes, starting from 7-8 conjugated benzene rings, was observed. The (13)C NMR parameters of individual carbon atoms from the middle of ultra-thin (4,0) SWCNT formed from 10 conjugated cyclic acenes differ by about 130 ppm from the corresponding open end carbon nuclei.

Integrating holographic vector dominance to hidden local symmetry for the nucleon form factor

We derive a two-parameter formula for the electromagnetic form factors of the nucleon described as an instanton by "integrating out" all KK modes other than the lowest mesons from the infinite-tower of vector mesons in holographic QCD while preserving hidden local symmetry for the resultant vector fields. With only two parameters, the proton Sachs form factors can be fit surprisingly well to the available experimental data for momentum transfers $Q^2\lsim 0.5$ GeV$^2$ with $\chi^2$/dof $\lsim 2$. We interpret this agreement as indicating the importance of an infinite tower in the soliton structure of the nucleon. The prediction of the Sakai-Sugimoto holographic dual model is checked against the fit values to assess its accuracy in describing the proton structure. We find that the structure of the "core" of roughly 1/3 in the proton size indicated in experiments and commonly associated with an intrinsic quark-gluon structure in QCD is "hidden" in the infinite tower in the holographic model.

Empirical parametrization of the two-photon-exchange effect contributions to the electron-proton elastic scattering cross section

The most recent electron-proton elastic scattering data were re-analyzed using an empirical parametrization of the two-photon-exchange (TPE) effect contributions to ${\ensuremath{\sigma}}_{R}$. The TPE effect contribution $F({Q}^{2},\ensuremath{\varepsilon})$ was double Taylor series expanded as a polynomial of order $n$ keeping only terms linear in $\ensuremath{\varepsilon}$ to account for the experimentally observed and verified linearity of the Rosenbluth plots. We fix the ratio $R={G}_{\mathit{Ep}}/{G}_{\mathit{Mp}}$ to be that obtained from a fit to the recoil-polarization data and parametrize ${\ensuremath{\sigma}}_{R}$ first by a three-parameter formula (fit I) and then by a two-parameter formula (fit III). In contrast to previous analyses, the fit parameter ${G}_{\mathit{Mp}}^{2}$ as obtained from these fits is either smaller or equal to the values obtained from our conventional Rosenbluth fit (fit II) but never larger. The ratio $g({Q}^{2})/{G}_{\mathit{Mp}}^{2}$ which represents the ratio of the TPE and one-photon-exchange (OPE) effect contributions to the intercept of ${\ensuremath{\sigma}}_{R}$ is large and it ranges 3%$\ensuremath{-}$88%. The ratio ${R}_{1\ensuremath{\gamma}\ensuremath{\bigotimes}2\ensuremath{\gamma}}=\ensuremath{\tau}f({Q}^{2})/{G}_{\mathit{Ep}}^{2}$ which represents the ratio of the TPE and OPE effect contributions to the slope of ${\ensuremath{\sigma}}_{R}$ is also large, reaching a value of 12.0$\ensuremath{-}$14.4 at ${Q}^{2}=$ 5.25 (GeV/c)${}^{2}$. The ratio ${R}_{1\ensuremath{\gamma}\ensuremath{\bigotimes}2\ensuremath{\gamma}}$ as obtained from fits I and III is consistent, within error, with those obtained from previous analyses. Our formulas seem to explain the linearity of ${\ensuremath{\sigma}}_{R}$. Moreover, our analysis shows that the extracted ${G}_{\mathit{Ep}}^{2}$ and ${G}_{\mathit{Mp}}^{2}$ using the conventional Rosenbluth separation method can in fact be broken into the usual OPE and TPE contributions. Therefore, ${\ensuremath{\sigma}}_{R}$ can in fact be derived under weaker conditions than those imposed by the Born approximation. Our results show that the TPE amplitudes, $g({Q}^{2})/{G}_{\mathit{Mp}}^{2}$ and $f({Q}^{2})/{G}_{\mathit{Mp}}^{2}$, are sizable and grow with ${Q}^{2}$ value up to ${Q}^{2}~$ 6 (GeV/c)${}^{2}$ in agreement with previous studies. A revision of and comparison to previous analyses are also presented.

Estimation of formamide harmonic and anharmonic modes in the Kohn-Sham limit using the polarization consistent basis sets

Formamide harmonic and anharmonic frequencies of fundamental vibrations in the gas phase and in several solvents were successfully estimated in the B3LYP Kohn-Sham complete basis set limit (KS CBS). CBS results were obtained by extrapolating a power function (two-parameter formula) to the results calculated with polarization-consistent basis sets. Anharmonic corrections using the second order perturbation treatment (PT2) and hybrid B3LYP functional combined with polarization consistent pc-n (n = 0, 1, 2, 3, 4) and several Pople’s basis sets were analyzed for all fundamental formamide vibrational modes in the gas phase and solution. Solvent effects were modeled within a PCM method. The anharmonic frequency of diagnostic amide vibration C = O in the gas phase and the CCl4 solution calculated with the VPT2 method was significantly closer to experimental data than the corresponding harmonic frequency. Both harmonic and anharmonic frequencies of C = O stretching mode decreased linearly with solvent polarity, expressed by relative environment permittivity (ε) ratio (ε − 1)/(2ε + 1). However, an unphysical behavior of solvent dependence of some low frequency anharmonic amide modes of formamide (e.g., CN stretch, NH2 scissoring, and NH2 in plane bend) was observed, probably due to the presence of severe anharmonicity and Fermi resonance. FigureFormamide harmonic and anharmonic frequencies of fundamental vibrations in the gas phase and in several solvents were successfully estimated in the B3LYP Kohn-Sham complete basis set limit (KS CBS). CBS results were obtained by extrapolating a power function (two-parameter formula) to the results calculated with polarization-consistent basis sets. Anharmonic corrections using the second order perturbation treatment (PT2) and hybrid B3LYP functional combined with polarization consistent pc-n (n = 0, 1, 2, 3, 4) and several Pople’s basis sets were analysed for all fundamental formamide vibrational modes in the gas phase and solution. Electronic supplementary materialThe online version of this article (doi:10.1007/s00894-010-0944-9) contains supplementary material, which is available to authorized users.

Extrapolation of water and formaldehyde harmonic and anharmonic frequencies to the B3LYP/CBS limit using polarization consistent basis sets

The harmonic and anharmonic frequencies of fundamental vibrations in formaldehyde and water were successfully estimated using the B3LYP Kohn-Sham limit. The results obtained with polarization- and correlation-consistent basis sets were fitted with a two-parameter formula. Anharmonic corrections were obtained by a second order perturbation treatment (PT2). We compared the performance of the PT2 scheme on the two title molecules using SCF, MP2 and DFT (BLYP, B3LYP, PBE and B3PW91 functionals) methods combined with polarization consistent pc-n (n = 0, 1, 2, 3, 4) basis sets, Dunning’s basis sets (aug)-cc-pVXZ where X = D, T, Q, 5, 6 and Pople’s basis sets up to 6-311++G(3df,2pd). The influence of SCF convergence level and density grid size on the root mean square of harmonic and anharmonic frequency deviations from experimental values was tested. The wavenumber of formaldehyde CH2 anharmonic asymmetric stretching mode is very sensitive to grid size for large basis sets; this effect is not observed for harmonic modes. BLYP-calculated anharmonic frequencies consistently underestimate observed wavenumbers. On the basis of formaldehyde anharmonic frequencies, we show that increasing the Pople basis set size does not always lead to improved agreement between anharmonic frequencies and experimental values. FigureSensitivity of water B3LYP calculated harmonic and anharmonic v s(OH) frequencies on selected Pople and polarization consistent basis sets size. The results for pc-n basis sets were fitted with two parameter formula and the CBS(2,3,4) estimated Electronic supplementary materialThe online version of this article (doi:10.1007/s00894-010-0913-3) contains supplementary material, which is available to authorized users.

Formulation of boundary conditions for vorticity in viscous incompressible flow problems

For the stream function-vorticity formulation of the Navier-Stokes equations, vorticity boundary conditions are required on the body surface and the far-field boundary. A two-parameter approximating formula is derived that relates the velocity and vorticity on the outer boundary of the computational domain. The formula is used to construct an algorithm for correcting the conventional far-field boundary conditions. Specifically, a soft boundary condition is set for the vorticity and a uniform flux is specified for the transversal velocity. A third-order accurate three-parameter formula for the vorticity on the wall is derived. The use of the formula does not degrade the convergence of the iterative process of finding the vorticity as compared with a previously derived and tested two-parameter formula.

Convergence of Nuclear Magnetic Shieldings in the Kohn-Sham Limit for Several Small Molecules.

Convergence patterns and limiting values of isotropic nuclear magnetic shieldings were studied for several small molecules (N2, CO, CO2, NH3, CH4, C2H2, C2H4, C2H6, and C6H6) in the Kohn-Sham limit. Individual results of calculations using dedicated families of Jensen's basis sets (pcS-n and pcJ-n) were fitted toward the complete basis set limit (CBS) using a simple two-parameter formula. Several density functionals were used; calculated vibrational corrections (ZPV) applied; and, for comparison purposes, similar calculations performed using RHF, MP2, SOPPA, SOPPA(CCSD), and CCSD(T) methods and additionally, the aug-cc-pVTZ-J basis set. Finally, the CBS estimated results were critically compared with earlier reported literature data and experimental results. Among 42 studied DFT methods, the KTn and "pure" functionals produced the most accurate heavy atom isotropic nuclear shieldings.