Parameter Β2 Research Articles

The explicit β1/β2-Bathe time integration method

In this paper we introduce the explicit β1/β2-Bathe method for solving dynamic problems, in particular wave propagations. Like for the implicit β1/β2-Bathe method, the proposed explicit scheme uses two sub-steps per time step and can be used directly as a first-order and a second-order method with the capability to suppress high spurious frequency response. In both sub-steps, standard Taylor series are employed resulting in an explicit solution scheme. The novelty is that we calculate the final displacements and velocities in each sub-step by applying correction terms using the generalized trapezoidal rule with control parameters β1 and β2. This approach makes the method a quite simple scheme. We consider the stability, accuracy and numerical dispersion and give recommendations on the parameter values β1 and β2 to be used in practice. We give the solutions of four problems, three of which are wave propagation problems, and compare the results with those obtained using other methods. While more experience in the use of the procedure is needed to understand its full solution capabilities, we can already conclude that the proposed method is effective in some wave propagation analyses.

Measurement and In-Depth Analysis of Higher Harmonic Generation in Aluminum Alloys with Consideration of Source Nonlinearity.

Harmonic generation measurement is recognized as a promising tool for inspecting material state or micro-damage and is an ongoing research topic. Second harmonic generation is most frequently employed and provides the quadratic nonlinearity parameter (β) that is calculated by the measurement of fundamental and second harmonic amplitudes. The cubic nonlinearity parameter (β2), which dominates the third harmonic amplitude and is obtained by third harmonic generation, is often used as a more sensitive parameter in many applications. This paper presents a detailed procedure for determining the correct β2 of ductile polycrystalline metal samples such as aluminum alloys when there exists source nonlinearity. The procedure includes receiver calibration, diffraction, and attenuation correction and, more importantly, source nonlinearity correction for third harmonic amplitudes. The effect of these corrections on the measurement of β2 is presented for aluminum specimens of various thicknesses at various input power levels. By correcting the source nonlinearity of the third harmonic and further verifying the approximate relationship between the cubic nonlinearity parameter and the square of the quadratic nonlinearity parameter (β∗β), β2≈β∗β, the cubic nonlinearity parameters could be accurately determined even with thinner samples and lower input voltages.

In the Quest of a Parametric Relation Between Ice Sheet Model Inferred Weertman's Sliding‐Law Parameter and Airborne Radar‐Derived Basal Reflectivity Underneath Thwaites Glacier, Antarctica

AbstractNumerical ice sheet models use sliding laws to connect basal shear stress and ice velocity to simulate ice sliding. A sliding‐law parameter β2 is used to control Weertman's sliding law in numerical ice sheet models. Basal reflectivity derived from ice‐penetrating radar also provides information about frozen or thawed conditions underneath glaciers. To assess whether basal reflectivity can be used to constrain β2, we carry out statistical experiments between two recently published datasets: β2 inferred from three numerical ice sheet models (ISSM, Úa and STREAMICE) and airborne radar‐derived relative basal reflectivity from the AGASEA‐BBAS mission over Thwaites Glacier (TG). Our results show no robust correlation between the β2–relative reflectivity pair. Pearson's correlation coefficient, a test of linearity, ranges from −0.26 to −0.38. Spearman's correlation coefficient, which does not require a linear assumption, is also modest (∼−0.35). We conclude that β2 and relative basal reflectivity underneath TG do not infer similar basal conditions.

Investigation of giant dipole resonance in Mo isotopes within TDHF theory

The isovector giant dipole resonance (IVGDR) in the chain of even-even Mo isotopes is investigated within the time-dependent Hartree-Fock (TDHF) using the Skyrme force Sly6. The GDR calculated in Mo92−108 are presented, and compared with the available experimental data. An overall agreement between them is obtained. Moreover, the dipole strength in Mo102−108 is predicted. Shape phase transition from spherical to oblate as well as shape coexistence (A ∼ 100) in Mo isotopes are also investigated in this work. In addition, the correlation between the deformation splitting ΔE and the quadrupole deformation parameter β2 is studied. The results confirm that ΔE is proportional to the deformation of nucleus. We also discuss the dependence of GDR strength on some nuclear properties of Skyrme forces, such as the asymmetry energy as. We find that as=32 MeV, corresponding to SLy6, is quite consistent with experimental data.

Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study

Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if Rv is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter β2 that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when ψ=1 for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (R0≤1) and endemic cases (R0>1). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number R0≈1.2591. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.

On Powell-Eyring hybridity nanofluidic flow based Carboxy-Methyl-Cellulose (CMC) with solar thermal radiation: A quadratic regression estimation

Examining the thorough double-hybrid nanofluid mode Yamada-Ota and Xue provide a detailed description of the Powell-Eyring fluid model (PEFM) with the grouping of the base streaming Carboxy-Methyl-Cellulose (CMC) and doubly changed Carbon nanotubes CNTs (Single-walled carbon nanotubes (SWCNT), Multi-walled carbon nanotubes (MWCNT)). The coordinate systems are considered normal vectors, which are all regarded to be components of the spinning cone system in which this fluid is thought to movement. The PEFM move and thermal characteristics are governed by the sliding boundary conditions, the porous medium, and buoyancy. Similarity transformations are employed to derive a set of ordinary differential equations (ODEs) indicated by the accurate initial requirement. These ODEs are after that numerically solved using a particular technique called the Galerkin finite element method (GFEM). In addition to the thermal dispersion, this investigation consumes likewise shaped graphical consequences for the velocity in different 2D-directions. Along with the dimensionless Nusselt number, the outcomes of the skin friction constants are tabulated under the influence of pertinent regulating factors. According to the findings, radiation increases the current field while the Darcy-Forchheimer limitation increases the movement of the 2D-axis. The frictional coefficient is additionally increased together with the x- and y-components by the buoyancy ratio parameter. Additionally, by controlling elements approximating radioactivity, absorbency, and sliding boundary conditions, the dimensionless Nusselt number is reduced (velocity and temperature). Furthermore, regarding the frictional factor Cfx, after testing with 100 combinations of Powell-Eyring-II parameter (β2) and speed slippage constrain (γ1) in between the 20% and 110%, it was noted that those constrains tends to resist frictional factor.

Relative acoustic nonlinearity parameter variation with local defect resonance frequencies and its different harmonics

In recent times, nonlinear ultrasonic inspection has become a very reliable technique to characterize micro-scale cracks and small-scale delaminations. In this paper, an analytical framework is developed for the orthotropic composite plate to calculate the variation of relative acoustic nonlinearity parameter, a quantitative measurement of nonlinearity at the delamination region. The analytical investigation is justified with numerical and experimental research using a carbon fiber-reinforced polymer (CFRP) plate with three delaminations. Two incident waves are used for excitation, out of which the first wave frequency is varied as local defect resonance (LDR) frequency ( fd), subharmonic LDR frequency (2 fd), second-order superharmonic LDR frequency ( fd/2), and third-order superharmonic LDR frequency ( fd/3) while the second incident wave is taken as a single periodic frequency. Different delamination sizes, positions, and depths were used to understand the second-order relative acoustic nonlinearity parameter ( β2) with other LDR frequencies as well as its harmonics. It has been observed from analytical, numerical, and experimental results that the relative acoustic nonlinearity parameter is maximum at LDR frequencies. It increases in case of surface level delaminations.

Deuteron scattering and (d,t) reaction on 11B at an energy of 14.5 MeV

At an energy of 14.5 MeV, the elastic and inelastic scattering of deuterons and the (d, t) reaction on 11B nuclei were studied. Experimental angular distributions with transitions to the states of the 11B nucleus (ground state (3/2−), 4.445 (5/2−) MeV, 6.743 (7/2−) MeV) and to the states of the 10B nucleus (ground state (3+), 0.718 MeV (1+), 1.74 MeV (0+, T = 1), 2.15 MeV (1+)) were analyzed by the coupled channel method. The value of the quadrupole deformation parameter β2= 0.80 ± 0.2 was extracted. The assumption of a direct mechanism for picking up a neutron in the reaction (d, t) made it possible to describe rather well the measured angular distributions for the states of the 10B nucleus. The values of the spectroscopic amplitudes for the transitions to these states have been extracted. It is shown that the mechanism of 8Be transfer both in the form of a whole cluster and in the sequential transfer of two α-particles does not play a significant role. It was found that the deformation of the 10B and 11B nuclei noticeably affects the calculated cross sections for the (d, t) reaction.

Exploring the ground state bulk and decay properties of the nuclei in superheavy island

The α-decay half-lives of 204 superheavy nuclei covering the range 114≤Z≤126 have been investigated using the relativistic mean-field model (RMF) for NL3⁎ parameter set. The ground state bulk properties such as binding energy, quadrupole deformation parameter (β2), and root-mean-square charge radii for these nuclei are analyzed. Four different semi-empirical formulae, namely, the universal decay law (UDL), the Viola-Seaborg (VSS) formula, the modified universal decay law (MUDL), and the modified Brown formula (MBrown), are used to obtain the α-decay half-lives for the considered nuclei. To examine the applicability of relativistic mean-field model within NL3⁎ parametrization, the α-decay energies, and the half-lives of a few known superheavy nuclei within the range 102 ≤ Z ≤ 118 are calculated and the results are compared with the experimental data along with the theoretical predictions. The α-decay energies (Q-values) are estimated from the binding energies of the parent, and daughter from the RMF (NL3⁎) parameter set. The calculated results are compared with macroscopic-microscopic Finite-Range-Droplet-Model (FRDM), Global Nuclear Mass Model (WS3, WS4), Weizsacker-Skyrme mass model (WS⁎) predictions, and the experimental data, wherever available. The possible standard deviations are also estimated for experimental and various theoretical predictions. We find a good consistency for the experimental-to-UDL, FRDM-to-UDL, and WS4-to-UDL estimates of the decay energy and corresponding half-life. The present analysis provides the theoretical predictions within the microscopic model for the upcoming experiments on the superheavy region.

ΛΛ pairing effects in multi-Λ hyperisotopes within a Skyrme-Hartree-Fock model

The ΛΛ pairing effects are investigated in multi-Λ hyperisotopes with the Skyrme-Hartree-Fock approach. The shell closure of hyperon occurrs at −S = 2, 8, 18, 20, 34, 58, 68, as well as 70 in spherical hypernuclei, which are attributed from the Λ hyperon potential being close to theWoods-Saxon potential. The more possible Λ hyperon magic numbers −S = 4, 6, 10, 14, 26, 30, and 32 in deformed hypernuclei are found based on the analysis of the single-particle energy levels, and are all sensitive to the quadrupole deformation parameter β2.

Proton inelastic scattering reveals deformation in 8He

A measurement of proton inelastic scattering of 8He at 8.25A MeV at TRIUMF shows a resonance at 3.54(6) MeV with a width of 0.89(11) MeV. The energy of the state is in good agreement with coupled cluster and no-core shell model with continuum calculations, with the latter successfully describing the measured resonance width as well. Its differential cross section analyzed with phenomenological collective excitation form factor and microscopic coupled reaction channels framework consistently reveals a large deformation parameter β2 = 0.40(3), consistent with no-core shell model predictions of a large neutron deformation. This deformed double-closed shell at the neutron drip-line opens a new paradigm.

Ground-State Properties of Even–Even Nuclei from He (Z = 2) to Ds (Z = 110) in the Framework of Skyrme–Hartree–Fock–Bogoliubov Theory

The ground-state properties of 2094 even–even nuclei with $$2\le Z\le 110$$ over a wide range of isotopes have been calculated and studied in the framework of the Hartree–Fock–Bogoliubov (HFB) theory with the SKI3 Skyrme type effective nucleon–nucleon interaction. The calculated ground-state values of the binding energy per nucleon (B.E/A), quadrupole deformation parameter (β2), two-neutron separation energy (S2n), charge radii (RC), proton rms radii, neutron rms radii and neutron pairing gap have been compared with the available experimental data of the atomic mass evaluation (AME2016), the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny effective nucleon–nucleon interaction and predictions of some nuclear models such as the finite range droplet model (FRDM2012) and relativistic mean-field model. Based on the HFBTHO results, the parameters of the Skyrme SKI3-HFB method can be successfully used for describing the ground-state properties of the investigated nuclei, in particularly the neutron-rich nuclei and the exotic nuclei near the neutron drip-line.

Invariant Gibbs dynamics for the dynamical sine-Gordon model

In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.

Effect of temperature on the volume and surface contributions in the symmetry energy of rare earth nuclei

The effect of temperature on the volume and surface contributions in the nuclear symmetry energy and their ratio in the isotopic chains of rare earth Nd, Sm, Gd, and Dy nuclei with N = 82-126 is analyzed in the framework of coherent density fluctuation model (CDFM). The weight function of nuclei, within CDFM, are calculated by using the densities from the temperature-dependent relativistic mean-field (RMF) model. Firstly, we discuss the temperature-dependence of bulk properties of nuclei, within RMF model, such as binding energy, deformation parameter, charge radius and isotopic shift along with comparison with the available experimental data at temperature T=0 MeV. Further, we discuss the thermal evolution of symmetry energy and its volume and surface components. At T=0 MeV, the persistence of a peak in the symmetry energy and components at neutron number N=100 shows the manifestation of deformed shell closure in consonance with an earlier study by one of us [L. Satpathy, S.K. Patra, J. Phys. G 30 (2004) 771; S.K. Ghorui, et al., Phys. Rev. C 85 (2012) 064327]. However, the scenario changes with rise in temperature and the magnitude of peak decreases at higher temperatures. At T=3 MeV, the peak disappears which may be due to shape change in addition to quenching of shell effects since the quadrupole deformation parameter β2 decreases with an increase in temperature and nuclei become spherical at T=3 MeV. It indicates that behavior of symmetry energy is closely related to the deformation/shape of the nuclei. We have also discussed the values of volume symmetry energy, surface symmetry energy and their ratio which are in consonance with available experimental data.

Giant dipole resonance and shape evolution in Nd isotopes within TDHF method

The isovector giant dipole resonance (IVGDR) in even–even Nd isotopes from A = 124 to A = 160 is studied in the framework of time-dependent Hartree–Fock (TDHF) with Skyrme forces SkI3, SVbas, SLy5 and SLy6. The dipole strength is calculated and compared with the experimental data on photon absorption cross section σγ. An overall agreement between them is obtained. The dipole strengths in 124–140Nd and 152–160Nd are predicted. In addition, the correlation between the quadrupole deformation parameter β2 and the splitting of the giant dipole resonance (GDR) spectra is studied. The results confirm that is proportional to β2. Shape phase transition in Nd isotopes is also investigated in the light of IVGDR.

Estimation of Parameters of Two Non-linear Regression Models Using Assumed Values: Reciprocal Power Regression Models

Fitting nonlinear models is not a single-step procedure but it involved a process that requires careful examination of each individual step. Depending on the objective and the application domain, different priorities are set when fitting nonlinear models; these include obtaining acceptable parameter estimates and a good model fit while meeting standard assumptions of statistical models. We propose steps in fitting nonlinear models in this research work. Two reciprocal power regression models were considered with a non-linear data set. Then, the following steps are considered (i) fit the models to the data collected using iterative steps, (ii) to develop a linear model to estimate the parameter β1 and β2 when the initial value (or growth rate β3) lies between -1.0 ≤ β3 ≤1.0 ); using the transform models of the reciprocal power regression model (iii) to find the “best” model between the two models using R2, AIC and BIC. The results show Model B is better than Model A, using the model selection criteria.

The Dependence of the Deformation of Exotic Nuclei on the Half-Life

A correlation between the parameter β2 and a half-life T1/2 for oblate nuclei with sign β2 0 are found. These analytical expressions for the function β2(T1/2) make it possible to calculate the shape parameters of β2 nuclei if we know the half-lives T1/2 for exotic nuclei for which this value is most accurately measured.

Modelling of Short Rate (β2) Parameter Diebold-Li Model Using Vasicek Stochastic Differential Equations

Investment is the activity of investing or allocating money to earn some profit. There are several type of investments, one of which is bond. The interest rate of the bond is called a yield. A yield is considered as a representation of market expectations depends on interest rate movements according to market price at a certain time. The yield’s value will change following a stochastic process. Estimation of a short rate parameter (β2) of a Diebold-Li model is conducted by applying a least square method. Then, the yield modeling is developed based on the estimated short rate parameter Diebold-li model (β2) using Vasicek Stochastic Differential Equation. The final result is that modelling of short rate (β2) parameter Diebold-Li model using Vasicek Stochastic Differential Equation generates prediction values that have a high level of accuracy by MAPE prediction value of 8.65%.

2D numerical contributions for the study of non-cohesive sediment transport beneath tidal bores

2D numerical simulations of tidal bores were obtained using the OpenFOAM CFD software to solve the Navier–Stokes equations by means of the Finite Volume Method by applying a LES turbulence model. The trajectories of non-cohesive sediment particles beneath tidal bores were estimated using a tracker method. Using the fourth order Runge–Kutta scheme, the tracker method solves the Maxey and Riley equations, which requires the knowledge of the velocity field at time t. From 2D numerical simulations of tidal bores, we proposed a classification of tidal bores with respect to the Froude number Fr (or r the ratio of water depths). For a Froude number 1<Fr<1.43 (1<r<1.57), the tidal bore is undular. For a Froude number 1.43<Fr<1.57 (1.57<r<1.75), the tidal bore is partially breaking, which is similar to the transitional tidal bore defined by Furgerot (2014). And for a Froude number Fr>1.57 (r>1.75), the tidal bore is totally breaking. The numerical results of trajectories of non-cohesive sediment particles are similar to the type of trajectories given by the analytical model proposed by Chen et al. (2012) with some modifications to take into account the effects of gravity, elevation, and attenuation. The parameters of modified Chen's model, β1, β2 and β3, are linearly proportional to the Froude number Fr. This is because the level of turbulence for undular tidal bores is low. The flow induced by an undular tidal bore is not complex. This physical phenomenon is quasi linear. The parameter β1, related to the front celerity of the undular tidal bore, decreases when the Froude number Fr increases. The parameter β2, related to the elevation, increases when the Froude number Fr increases. And the parameter β3, related to the attenuation of the secondary waves, increases when the Froude number Fr increases.

Quadrupole Moment & Deformation Parameter for Even-Even 38Sr (A=76-102) Nuclide

In this work, the electric quadrupole moment (Qo) and deformation parameter (β2) have been studied for 38Sr isotopes which has neutron magic number (N=50).The aim of this work is to study the effect of electric quadrupole moment (charge distribution) on the deformation of the nucleus by calculating the deformation parameter depending on the value of electric quadrupole moment.Also electric quadrupole moment (Qo) and deformation parameter (β2) have been plotted as a function to neutron number to study the effect of neutron magic number (N=50) on them.The results show that the shapes of nuclei approach to spherical shape due to its charge distribution when the nuclei be closer to the neutrons magic number.