Thermal Conductivity of Pyroclastic Soil (<Emphasis Type="BoldItalic">Pozzolana</Emphasis>) from the Environs of Rome

The paper reveals the experimental procedure and thermo-physical characteristics of a coarse pyroclastic soil (Pozzolana), from the neighborhoods of Rome, Italy. The tested samples are comprised of 70.7 % sand, 25.9 % silt, and 3.4 % clay. Their mineral composition contained 38 % pyroxene, 33 % analcime, 20 % leucite, 6 % illite/muscovite, 3 % magnetite, and no quartz content was noted. The effective thermal conductivity of minerals was assessed to be about \(2.14\,\hbox {W}{\cdot } \hbox {m}^{-1}{\cdot } \hbox {K}^{-1}\). A transient thermal probe method was applied to measure the thermal conductivity (\(\lambda \)) over a full range of the degree of saturation \((S_{\mathrm{r}})\), at two porosities (n) of 0.44 and 0.50, and at room temperature of about \(25\,^{\circ }\hbox {C}\). The \(\lambda \) data obtained were consistent between tests and showed an increasing trend with increasing \(S_{\mathrm{r}}\) and decreasing n. At full saturation (\(S_{\mathrm{r}}=1\)), a nearly quintuple \(\lambda \) increase was observed with respect to full dryness (\(S_{\mathrm{r}}=0\)). In general, the measured data closely followed the natural trend of \(\lambda \) versus \(S_{\mathrm{r}}\) exhibited by published data at room temperature for other unsaturated soils and sands. The measured \(\lambda \) data had an average root-mean-squared error (RMSE) of \(0.007\,\hbox {W}{\cdot } \hbox {m}^{-1}{\cdot } \hbox {K}^{-1}\) and \(0.008\,\hbox {W}{\cdot } \hbox {m}^{-1}{\cdot } \hbox {K}^{-1}\) for n of 0.50 and 0.44, respectively, as well as an average relative standard deviation of the mean at the 95 % confidence level \((\hbox {RSDM}_{0.95})\) of 2.21 % and 2.72  % for n of 0.50 and 0.44, respectively.     

Canadian Field Soils IV: Modeling Thermal Conductivity at Dryness and Saturation

The thermal conductivity data of 40 Canadian soils at dryness \((\lambda _{\mathrm{dry}})\) and at full saturation \((\lambda _{\mathrm{sat}})\) were used to verify 13 predictive models, i.e., four mechanistic, four semi-empirical and five empirical equations. The performance of each model, for \(\lambda _{\mathrm{dry}}\) and \(\lambda _{\mathrm{sat}}\), was evaluated using a standard deviation (SD) formula. Among the mechanistic models applied to dry soils, the closest \(\lambda _{\mathrm{dry}}\) estimates were obtained by MaxRTCM \((\textit{SD} = \pm ~0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})\), followed by de Vries and a series-parallel model (\(\hbox {S-}{\vert }{\vert }\)). Among the semi-empirical equations (deVries-ave, Advanced Geometric Mean Model (A-GMM), Chaudhary and Bhandari (C–B) and Chen’s equation), the closest \(\lambda _{\mathrm{dry}}\) estimates were obtained by the C–B model \((\pm ~0.022\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})\). Among the empirical equations, the top \(\lambda _{\mathrm{dry}}\) estimates were given by CDry-40 \((\pm ~0.021\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}\) and \(\pm ~0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}\) for18-coarse and 22-fine soils, respectively). In addition, \(\lambda _{\mathrm{dry}}\) and \(\lambda _{\mathrm{sat}}\) models were applied to the \(\lambda _{\mathrm{sat}}\) database of 21 other soils. From all the models tested, only the maxRTCM and the CDry-40 models provided the closest \(\lambda _{\mathrm{dry}}\) estimates for the 40 Canadian soils as well as the 21 soils. The best \(\lambda _{\mathrm{sat}}\) estimates for the 40-Canadian soils and the 21 soils were given by the A-GMM and the \(\hbox {S-}{\vert }{\vert }\) model.     

Bootstrap in semi-functional partial linear regression under dependence

This paper deals with the semi-functional partial linear regression model \(Y={{\varvec{X}}}^\mathrm{T}{\varvec{\beta }}+m({\varvec{\chi }})+\varepsilon \) under \(\alpha \)-mixing conditions. \({\varvec{\beta }} \in \mathbb {R}^{p}\) and \(m(\cdot )\) denote an unknown vector and an unknown smooth real-valued operator, respectively. The covariates \({{\varvec{X}}}\) and \({\varvec{\chi }}\) are valued in \(\mathbb {R}^{p}\) and some infinite-dimensional space, respectively, and the random error \(\varepsilon \) verifies \(\mathbb {E}(\varepsilon |{{\varvec{X}}},{\varvec{\chi }})=0\). Naïve and wild bootstrap procedures are proposed to approximate the distribution of kernel-based estimators of \({\varvec{\beta }}\) and \(m(\chi )\), and their asymptotic validities are obtained. A simulation study shows the behavior (on finite sample sizes) of the proposed bootstrap methodology when applied to construct confidence intervals, while an application to real data concerning electricity market illustrates its usefulness in practice.     

Further Estimates of $$(T-T_{90})$$ Close to the Triple Point of Water

Recent advances in primary acoustic gas thermometry (AGT) have revealed significant differences between temperature measurements using the International Temperature Scale of 1990, \(T_{90}\), and thermodynamic temperature, T. In 2015, we published estimates of the differences \((T-T_{90})\) from 118 K to 303 K, which showed interesting behavior in the region around the triple point of water, \(T_\mathrm{TPW}=273.16\) K. In that work, the \(T_{90}\) measurements below \(T_\mathrm{TPW}\) used a different ensemble of capsule standard platinum resistance thermometers (SPRTs) than the \(T_{90}\) measurements above \(T_\mathrm{TPW}\). In this work, we extend our earlier measurements using the same ensemble of SPRTs above and below \(T_\mathrm{TPW}\), enabling a deeper analysis of the slope \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) around \(T_\mathrm{TPW}\). In this article, we present the results of seven AGT isotherms in the temperature range 258 K to 323 K. The derived values of \((T-T_{90})\) have exceptionally low uncertainties and are in good agreement with our previous data and other AGT results. We present the values \((T-T_{90})\) alongside our previous estimates, with the resistance ratios W(T) from two SPRTs which have been used across the full range 118 K to 323 K. Additionally, our measurements show discontinuities in \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) at \(T_\mathrm{TPW}\) which are consistent with the slope discontinuity in the SPRT deviation functions. Since this discontinuity is by definition non-unique, and can take a range of values including zero, we suggest that mathematical representations of \((T-T_{90})\), such as those in the mise en pratique for the kelvin (Fellmuth et al. in Philos Trans R Soc A 374:20150037, 2016. doi: 10.1098/rsta.2015.0037), should have continuity of \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) at \(T_\mathrm{TPW}\).     

Influences of Detection Pinhole and Sample Flow on Thermal Lens Detection in Microfluidic Systems

Thermal lens microscopy (TLM), due to its high temporal (\({\sim }\mathrm{ms}\)) and spatial resolution (\({\sim }\upmu \mathrm{m}\)), has been coupled to lab-on-chip chemistry for detection of a variety of compounds in chemical or biological fields. Due to the very short optical path length (usually below 100 \(\upmu \mathrm{m}\)) in a microchip, the sensitivity of TLM is unfortunately still 10 to 100 times lower than conventional TLS with 1 cm sample length. Optimization of the TLM optical configuration was made with respect to different pinhole aperture-to-beam size ratios for the best signal-to-noise ratio. In the static mode, the instrumental noise comes mainly from the shot noise of the probe beam when the chopper frequency is over 1 kHz or from the flicker noise of the probe beam at low frequencies. In the flowing mode, the flow-induced noise becomes dominant when the flow rate is high. At a given flow rate, fluids with a higher density and/or a higher viscosity will cause larger flow-induced noise. As an application, a combined microfluidic flow injection analysis (\(\upmu \mathrm{FIA}\))–TLM device was developed for rapid determination of pollutants by colorimetric reactions. Hexavalent chromium [Cr(VI)] was measured as a model analyte. Analytical signals for 12 sample injections in 1 min have been recorded by the \({\upmu }\)FIA–TLM. For injections of sub-\(\upmu \)L samples into the microfluidic stream in a \(50\,\upmu \mathrm{m}\) deep microchannel, a limit of detection of \(4\,\mathrm{ng}{\cdot }\mathrm{mL}^{-1}\) was achieved for Cr(VI) in water at 60 mW excitation power.     

Nonparametric density and survival function estimation in the multiplicative censoring model

Consider the multiplicative censoring model given by \(Y_i=X_iU_i\), \(i=1, \ldots ,n\) where \((X_i)\) are i.i.d. with unknown density f on \({\mathbb {R}}\), \((U_i)\) are i.i.d. with uniform distribution \({\mathcal {U}}([0,1])\) and \((U_i)\) and \((X_i)\) are independent sequences. Only the sample \((Y_i)\) is observed. We study nonparametric estimators of both the density f and the corresponding survival function \(\bar{F}\). First, kernel estimators are built. Pointwise risk bounds for the quadratic risk are given, and upper and lower bounds for the rates in this setting are provided. Then, in a global setting, a data-driven bandwidth selection procedure is proposed. The resulting estimator has been proved to be adaptive in the sense that its risk automatically realizes the bias-variance compromise. Second, when the \(X_i\)s are nonnegative, using kernels fitted for \({\mathbb {R}}^+\)-supported functions, we propose new estimators of the survival function which are also adaptive. By simulation experiments, we check the good performances of the estimators and compare the two strategies.     

The Influence of Impurities on the Zinc Fixed Point

Impurities are considered to be the most significant source of uncertainty for the realization of the International Temperature Scale of 1990 by means of metal fixed points. The determination and further reduction in this uncertainty require a traceable chemical analysis of dissolved impurities in the fixed-point metal and accurate knowledge of the specific temperature change caused by impurities (slope of the liquidus line). We determined the slope of the liquidus line for three binary systems and present results and conclusions from the chemical analysis of zinc with a nominal purity of 7N. For the Fe–Zn system, we determined a liquidus slope of (\(-0.91\pm 0.14\)) mK / (\(\upmu \hbox {g}{\cdot }\hbox { g}^{-1}\)) from the evaluation of freezing plateaus and (\(-0.76~\pm 0.20\)) mK / (\(\upmu \hbox {g}{\cdot }\hbox { g}^{-1}\)) from the evaluation of melting plateaus; for the Pb–Zn system, the corresponding results are (\(-0.27~\pm 0.05\)) mK / (\(\upmu \hbox {g}{\cdot }\hbox { g}^{-1}\)) and (\(-0.26~\pm 0.05\)) mK / (\(\upmu \hbox {g}{\cdot }\hbox { g}^{-1}\)). Although for the Sb–Zn system, we determined a liquidus slope of about \(-0.8\) mK / (\(\upmu \hbox {g}{\cdot }\hbox { g}^{-1}\)), our investigations showed that a correction of the influence of antimony is highly questionable because antimony can be found in zinc in a fully dissolved state or precipitated as an insoluble compound. Iron is the only impurity where a correction of the fixed-point temperature was possible. For the realization of the zinc fixed point at PTB, this correction is between 2 \(\upmu \)K and 16 \(\upmu \)K depending on the batch of zinc used. The influence of the sum of all impurities was estimated by means of the OME method. The resulting uncertainty contribution is between 12 \({\upmu }\hbox {K}\) and 48 \({\upmu }\hbox {K}\).     

The parameters governing the coefficient of dispersion of cubes in rotating cylinders

Axial dispersion of cubic particles in horizontal, rotating cylinders was investigated using discrete element modelling simulations. We found that, similar to the behavior of spheres, the axial dispersion coefficient of cubes depends on (1) the rotational speed of the cylinder \({\omega }\), (2) the acceleration due to gravity g and (3) the particle size d, satisfying the relationship \({D}_\mathrm {ax}\propto {\omega }^{1-2{\lambda }}{g}^{{\lambda }}{d}^{2-{\lambda }}\) with \({\lambda }\approx 0.15\) (\({\lambda }\approx 0.1\) for beds of spheres) (Third et al. in Powder Technol 203:510–517, 2010). This observation suggested that, although particle shape influences significantly the rate of axial dispersion (cubes disperse almost twice as fast as spheres of equal volume), the parameters controlling the coefficient of dispersion are independent of particle shape.     

Measurement Uncertainty Budget of the PMV Thermal Comfort Equation

Fanger’s predicted mean vote (PMV) equation is the result of the combined quantitative effects of the air temperature, mean radiant temperature, air velocity, humidity activity level and clothing thermal resistance. PMV is a mathematical model of thermal comfort which was developed by Fanger. The uncertainty budget of the PMV equation was developed according to GUM in this study. An example is given for the uncertainty model of PMV in the exemplification section of the study. Sensitivity coefficients were derived from the PMV equation. Uncertainty budgets can be seen in the tables. A mathematical model of the sensitivity coefficients of \(T_{\mathrm{a}}\), \(h_{\mathrm{c}}\), \(T_{\mathrm{mrt}}\), \(T_{\mathrm{cl}}\), and \(P_{\mathrm{a}}\) is given in this study. And the uncertainty budgets for \(h_{\mathrm{c}}\), \(T_{\mathrm{cl}}\), and \(P_{\mathrm{a}}\) are given in this study.     

High-pressure studies of superconductivity in $$\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}$$

\(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\) crystallizes in tetragonal CeOBiS\(_{2}\) structure (S. G. P4/nmm). We have investigated the effect of pressure on magnetization measurements. Our studies suggest improved superconducting properties in polycrystalline samples of \(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\). The \(T_{\mathrm{c}}\) in our sample is 5.3 K, at ambient pressure, which is marginal but definite enhancement over \(T_{\mathrm{c}}\) reported earlier (= 5.1 K). The upper critical field \(H_{\mathrm{c}2}\)(0) is greater than 3 T, which is higher than earlier report on this material. As determined from the M–H curve, both \(H_{\mathrm{c}2}\) and \(H_{\mathrm{c}1}\) decrease under external pressure P (0 \(\le P \le \) 1 GPa). We observe a decrease in critical current density and transition temperature on applying pressure in \(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\).     

Thermodynamic Properties at Saturation Derived from Experimental Two-Phase Isochoric Heat Capacity of 1-Hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide

New measurements are reported for the isochoric heat capacity of the ionic liquid substance 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide ([C6mim][NTf2]). These measurements extend the ranges of our earlier study (Polikhronidi et al. in Phys Chem Liq 52:657, 2014) by 5 % of the compressed liquid density and by 75 K. An adiabatic calorimeter was used to measure one-phase \((C_{\mathrm{V1}})\) liquid and two-phase \((C_{\mathrm{V2}})\) liquid + vapor isochoric heat capacities, densities \((\rho _s)\), and phase-transition temperatures \((T_s)\) of the ionic liquid (IL) substance. The combined expanded uncertainty of the density \(\rho \) and isochoric heat capacity \(C_\mathrm{V}\) measurements at the 95 % confidence level with a coverage factor of \(k = 2\) is estimated to be 0.15 % and 3 %, respectively. Measurements are concentrated in the immediate vicinity of the liquid + vapor phase-transition curve, in order to closely observe phase transitions. The present measurements and those of our earlier study are analyzed together and are presented in terms of thermodynamic properties \((T_s\), \(\rho _s\), \(C_{\mathrm{V1}}\) and \(C_{\mathrm{V2}})\) evaluated at saturation and in terms of key-derived thermodynamic properties \(C_\mathrm{P}\), \(C_\mathrm{S}\), \(W_\mathrm{S}^{{\prime }}\), \(K_{\mathrm{TS}}^{{\prime }}\), \(\left( {\partial P/\partial T} \right) _{\mathrm{V}}^{\prime }\), and \(\left( {\partial V/\partial T} \right) _\mathbf{P}^{\prime })\) on the liquid + vapor phase-transition curve. A thermodynamic relation by Yang and Yang is used to confirm the internal consistency of measured two-phase heat capacities \(C_{\mathrm{V2}} \), which are observed to fall perfectly on a line as a function of specific volume at a constant temperature. The observed linear behavior is exploited to evaluate contributions to the quantity \(C_{\mathrm{V2}} = f(V, T)\) from chemical potential \(C_{{\mathrm{V}\upmu }} =-T\frac{\mathrm{d}^{{2}}\mu }{\mathrm{d}T^{2}}\) and from vapor pressure \(C_{\mathrm{VP}} =VT\frac{\mathrm{d}^{2}P_{\mathrm{S}} }{\mathrm{d}T^{2}}\). The physical nature and specific details of the temperature and specific volume dependence of the two-phase isochoric heat capacity and some features of the other derived thermodynamic properties of IL at liquid saturation curve are considered in detail.     

Study of optical properties of potassium permanganate ($$\hbox {KMnO}_{4}$$) doped poly(methylmethacrylate) (PMMA) composite films

We have developed films of pure polymethylmethacrylate (PMMA) (0.5, 1, 2 and 5%) and potassium permanganate \((\hbox {KMnO}_{4})\)-doped PMMA composite films of thickness (\(\sim 100\, \upmu \hbox {m}\)) using the solution-cast technique. To identify the possible change that happen to the PMMA films due to doping, the optical properties were investigated for different concentrations of \(\hbox {KMnO}_{4}\) by recording the absorbance (A) and transmittance (\(T\%\)) spectra of these films using UV–Vis spectrophotometer in the wavelength range of 300–1100 nm. From the data obtained from the optical parameters viz. absorption coefficient (\(\alpha \)), extinction coefficient (\(\kappa \)), finesse coefficient (F), refractive index (\(\eta \)), real and imaginary parts of dielectric constant (\(\varepsilon _{\mathrm{r}}\) and \(\varepsilon _{\mathrm{i}})\) and optical conductivity (\(\sigma \)) were calculated for the prepared films. The indirect optical band gap for the pure and the doped-PMMA films were also estimated.     

Evidence for Argon Content in Pure Oxygen from Thermal Data

Since many years it is known that argon impurities in oxygen change the temperature of the oxygen triple point by \(+12 \hbox { K}{\cdot }\mathrm{mol}^{-1}\) (positive, while most impurities decrease the temperature) without any effect on the melting range of this transition, for the impurity concentrations usually encountered in nominally pure gases. It has been hypothesized that thermal measurements on the \(\alpha -\beta \) solid-to-solid transition at 23.8 K or the \(\beta -\gamma \) solid-to-solid transition at 43.8 K may give reliable evidence regarding the argon content. However, such measurements require very long times for full completion of each transition (with prohibitive costs if liquid helium is used) and for the \(\alpha -\beta \) solid-to-solid transition the heat pulse method cannot be applied reliably. The availability of closed-cycle refrigerators permits the first obstacle to be surmounted. The automatic system earlier developed at INRIM during the EU Multicells project and used extensively for the project on the isotopic effect in neon is perfectly suited for such measurements. Thus, the uncertainties of the temperature measurements are similar to those obtained previously (of the order of 0.1 mK or less). Three argon-in-oxygen mixtures were prepared gravimetrically and certified at KRISS, just as was previously done for the work on the neon isotopic compositions. Results of continuous-current measurements on the \(\alpha -\beta \) solid-to-solid transition, along with the triple-point data obtained with the heat pulse method, are presented for one cell with a known doped argon content, to be compared with similar data from a cell with oxygen of very high purity. In addition, some preliminary data for the \(\beta -\gamma \) solid-to-solid transition are given. The measurements on the mixture with the highest argon content, about \(1002\, \upmu \hbox {mol}{\cdot } \mathrm{mol}^{-1}\), imply a (linear) sensitivity of \((116 \pm 7) \hbox {K}{\cdot }\mathrm{mol}^{-1}\) for the \(\alpha -\beta \) transition. This sensitivity may be different at much lower argon contents, and follow-up measurements with the other (smaller) mixtures will shed light on the linearity of this dependence.     

Magnetism in Solid Oxygen Studied by High-Pressure Neutron Diffraction

A systematic modification of the entropy trajectory (\(S_\mathrm{m}(T)\)) is observed at very low temperature in magnetically frustrated systems as a consequence of the constraint (\(S_\mathrm{m}\ge 0\)) imposed by the Nernst postulate. The lack of magnetic order allows to explore and compare new thermodynamic properties by tracing the specific heat (\(C_\mathrm{m}\)) behavior down to the sub-Kelvin range. Some of the most relevant findings are: (i) a common \(C_\mathrm{m}/T|_{T\rightarrow 0} \approx 7\) J/mol K\(^2\) ‘plateau’ in at least five Yb-based very-heavy-fermions (VHF) compounds; (ii) quantitative and qualitative differences between VHF and standard non-Fermi-liquids; (iii) entropy bottlenecks governing the change of \(S_\mathrm{m}(T)\) trajectories in a continuous transition into alternative ground states. A comparative analysis of \(S_\mathrm{m}(T\rightarrow 0)\) dependencies is performed in compounds suitable for adiabatic demagnetization processes according to their \(\partial ^2 S_\mathrm{m}/\partial T^2\) derivatives.     

Fracture of pre-cracked thin metallic conductors due to electric current induced electromagnetic force

We investigated propagation of a sharp crack in a thin metallic conductor with an edge crack due to electric current induced electromagnetic forces. Finite element method (FEM) simulations showed mode I crack opening in the edge-cracked conductor due to the aforementioned (i.e., self-induced) electromagnetic forces. Mode I stress intensity factor due to the self-induced electromagnetic forces, \(K_{\mathrm{IE},}\) was evaluated numerically as \(K_{\mathrm{IE}}=\upmu l^{2}j^{2}(\uppi a)^{0.5}f(a/w)\), where \(\upmu \) is the magnetic permeability, l is the length of the conductor, a is the crack length, j is the current density, w is the width of the sample and f(a / w) is a geometric factor. Effect of dynamic electric current loading on edge-cracked conductor, incorporating the effects of induced currents, was also studied numerically, and dynamic stress intensity factor, \(K_{\mathrm{IE,d}}\), was observed to vary as \(K_{\mathrm{IE,d}} \sim f_{d}(a/w)j^{2}(\uppi a)^{1.5}\). Consistent with the FEM simulation, experiments conducted using \(12\,\upmu \hbox {m}\) thick Al foil with an edge crack showed propagation of sharp crack due to the self-induced electromagnetic forces at pulsed current densities of \(\ge \) \(1.85\times 10^{9}\,\hbox {A/m}^{2}\) for \(a/w = 0.5\). Further, effects of current density, pulse-width and ambient temperature on the fracture behavior of the Al foil were observed experimentally and corroborated with FEM simulations.     

First-Principles Study of the Structural,Optical, Dynamical and Thermodynamic Properties of BaZn$$\hbox {O}_{2}$$ Under Pressure

The structural, optical, dynamical, and thermodynamic properties of BaZn\(\hbox {O}_{2}\) under pressure are studied based on the density functional theory. The calculated structural parameters are consistent with the available experimental data. In the ground state, the electronic band structure and density of states indicate that BaZn\(\hbox {O}_{2}\) is an insulator with a direct gap of 2.2 eV. The Mulliken charges are also analyzed to characterize the bonding property. After the structural relaxation, the optical properties are studied. It is found that the dielectric function of E \(\Vert x\hbox { and }E\Vert y\) are isotropic, whereas the \(E\Vert x\) and \(E\Vert z\) are anisotropic. The effect of pressure on the energy-loss function in the ultraviolet region becomes more obvious as the pressure increases. Furthermore, the dynamical properties under different pressures are investigated using the finite displacement method. We find that the \(P3_{1}21\) phase of \(\hbox {BaZn}\hbox {O}_{2}\) is dynamically stable under the pressure ranging from 0 GPa to 30 GPa. The phonon dispersion curves, phonon density of states, vibrational modes and atoms that contribute to these vibrations at \({{\varvec{\Gamma }}}\) point under different pressures are also reported in this work. Finally, by employing the quasi-harmonic approximation, the thermodynamic properties such as the temperature dependence of the thermal expansion coefficient, specific heat, entropy and Gibbs free energy under different pressures are investigated. It is found that the influences of the temperature on the heat capacity are much more significant than that of the pressure on it.     

A hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element

A quadratically convergent Newton-type iterative scheme is proposed for approximating the generalized Drazin inverse \(b{^\mathrm d}\) of the Banach algebra element b. Further, its extension into the form of the hyperpower iterative method of arbitrary order \(p\ge 2\) is presented. Convergence criteria along with the estimation of error bounds in the computation of \(b{^\mathrm d}\) are discussed. Convergence results confirms the high order convergence rate of the proposed iterative scheme.