The null distribution of the likelihood-ratio test for one or two outliers in a normal sample

For testingk upper or lower outliers in a normal sample, the sampling distribution of the likelihood-ratio statistic is still unknown in the literature except fork=1. In this paper, we find its exact distribution fork=2 and tabulate the extensive critical values, which are compared with the traditional simulated values. The research is partially supported by the National Sciences and Engineering Research Council of Canada.     

Piezoelectric Energy Harvesting Design Principles for Materials and Structures: Material Figure-of-Merit and Self-Resonance Tuning

Piezoelectric energy harvesters (PEHs) aim to generate sufficient power to operate targeting device from the limited ambient energy. PEH includes mechanical-to-mechanical, mechanical-to-electrical, and electrical-to-electrical energy conversions, which are related to PEH structures, materials, and circuits, respectively; these should be efficient for increasing the total power. This critical review focuses on PEH structures and materials associated with the two major energy conversions to improve PEH performance. First, the resonance tuning mechanisms for PEH structures maintaining continuous resonance, regardless of a change in the vibration frequency, are presented. Based on the manual tuning technique, the electrically- and mechanically-driven self-resonance tuning (SRT) techniques are introduced in detail. The representative SRT harvesters are summarized in terms of tunability, power consumption, and net power. Second, the figure-of-merits of the piezoelectric materials for output power are summarized based on the operating conditions, and optimal piezoelectric materials are suggested. Piezoelectric materials with large k_ij, d_ij, and g_ij values are suitable for most PEHs, whereas those with large k_ij and Q_m values should be used for on-resonance conditions, wherein the mechanical energy is directly supplied to the piezoelectric material. This comprehensive review provides insights for designing efficient structures and selection of proper piezoelectric materials for PEHs.     

On Jonckheere's k-Sample Test Against Ordered Alternatives

In this paper a k-sample non-parametric test for trend is considered. Given a sample of size n_i , i = 1, …, k respectively from each of k populations, the test rejects the hypothesis that the k populations are identical if S = Σ ^k _i=2 S_i ≥ S_i . Here Si is the Mann-Whitney statistic computed when each observation in the i-th sample is compared with the combined observations from the first (i – 1) populations. A recurrence formula is derived for computing the exact distribution of S. Tables of exact probabilities and critical values are given for nominal values of α = 0.5, 0.2, 0.1, 0.05, 0.025, 0.01, and 0.005 for k = 3 and all possible sample sizes from 2 to 8, and for equal sample sizes for values of n = 2(1)6, k = 4(1)6.     

On the outstanding elements and record values in the exponential and gamma populations

Summary Outstanding elements and record values are discussed in this paper as related to exponential and gamma populations. First, the problem of prediction is considered when there are available,k sets of independent observations from a general-type exponential distribution. In such a case, prediction of then _k -th record value in thek-th set is made in terms ofn _i -th (i=1, …,k−1) record values from other (k−1) sets. For this purpose a predictive distribution is obtained. Secondly, the distribution of the sum of record values as well as that of a linear combination of record values are obtained for the exponential case. Probability integrals of the sum of record values and the probability integral of the sum of outstanding, elements are suggested for all values. Then, the distribution of then-th record values in a gamma population is put in a closed form. Further, the distribution of the linear combination of the spacings of outstanding elements as well as that of the linear combination of outstanding elements themselves are obtained. Finally the distribution of a ratio of two record values is obtained.     

Investigation of Charge Transfer Kinetics of Polyaniline Supercapacitor Electrodes by Scanning Electrochemical Microscopy

Scanning Electrochemical Microscopy (SECM) is introduced as a promising technique to probe localized interfacial kinetics at the interface of electrolyte/supercapacitor electrode based on polyaniline (PANI) by measuring approach curves from which heterogeneous charge transfer rate constants (k _eff) are extracted. The values correlate with the effectiveness of the electrode material for supercapacitor application. Specifically, measurements on PANI films of different thicknesses show that potential‐dependent rate constants are observed only for PANI films of up to 5 μm thickness. In addition to the thickness of PANI, k _eff is also found to be affected by the applied potential and surface morphology of PANI electrodes. These findings correlate with the macroscopic electrochemical performance of PANI electrodes which shows enhanced specific charge storage ability when their thickness is below 5 μm. Under these conditions, they deliver a specific capacitance of 486 F g⁻¹ and a rate capability of 89%. The observed correlation between microscopic kinetic data determined by SECM and macroscopic device characteristics provides rational guidelines for the optimization of the physical and structural properties of high performance supercapacitor electrodes.     

A UNIFIED TIMOSHENKO BEAM B-SPLINE RAYLEIGH–RITZ METHOD FOR VIBRATION AND BUCKLING ANALYSIS OF THICK AND THIN BEAMS AND PLATES

First, the shear-locking phenomenon in the wψB_kSRRM^1–3 is investigated and the shear-locking terms are identified in both one-dimensional beam and two-dimensional plate analyses. Subsequently the shear-locking free conditions are proposed and under the guidance of these conditions the Timoshenko beam B-spline Rayleigh–Ritz method, designated as TB_kSRRM, is formulated for vibration analysis of beams based on Timoshenko beam theory and vibration and buckling analysis of isotropic plates or fibre-reinforced composite laminates based on the first-order shear deformation plate theory (SDPT). In TB_kSRRM the number of degrees of freedom is exactly the same as that when the Bernoulli–Euler beam theory or classical plate theory (CPT) is used. However, the TB_kSRRM includes the through-thickness shearing and rotary inertia effects in full. Several numerical applications are presented and they show that this unified approach is extremely efficient for both thick and thin beams and plates. © 1997 by John Wiley & Sons, Ltd.     

Photon Statistics of an m-photon Laser with k-photon Losses

With the help of an approximation scheme valid for n ? 1 and Δ ⁿ² ?n ^-2 (where n is the mean value and Δ ⁿ² the variance of the photon number), the photon statistics of an m-photon laser with k-photon losses (m,k = 1, 2, …) is studied theoretically for arbitrary values of the atomic population inversion s produced by an external pumping source. For m>k, m = k and m < k, the dependences of the steady-state values of the mean photon number and of the quantity Δ ⁿ² / n on gain and saturation are investigated. In the limit of strong saturation (well above the threshold for m > k) the photon statistics are characterized by the steady-state value Δ ⁿ² / n = (1/2) (1 + m/ks).     

Distributions in Statistics: Continuous Univariate Distributions-1 and 2

The density of a symmetric statistic T = g(X ₁, X ₂, …, X_n ), for a random sample from a mixed population with density f(x) = pf ₁(x) + pf ₂), is a binomial mixture of the densities of the statist.ics T_k = g(X_k1 , X_k2 , X_kn ), k = 0, 1, … n. where X_ki 's are independent with density f ₁(x) if i ≤ k and density f ₂(x) if i > k. It is shown how to find the distributions of some important symmetric statistics like sample mean, sample variance, and order statistics by using T_k 's. The results are applied to normal and exponential mixtures.     

The Prediction of the Dissolution Rate Constant by Mixing Rules: The Study of Acetaminophen Batches

The purpose of this article is to promote two simple and scalable methods to accelerate the formulation development of formulated granules using acetaminophen as a model system. In method I, formulated granules made from the batch of small particle–sized acetaminophen (1) by ball milling the batch of large particle–sized acetaminophen (2), and the mixture of the two batches at equal weights (mix) gave the dissolution rate constants (k) of k₁?=?0.43 ± 0.15 minutes^?1, k₂?=?0.18 ± 0.01 minutes^?1, and k_mix?=?0.30 ± 0.03 minutes^?1 for 75 wt percent formulation; k₁?=?0.75 ± 0.01 minutes^?1, k₂?=?0.18 ± 0.01 minutes^?1, and k_mix?=?0.34 ± 0.03 minutes^?1 for 62 wt percent formulation; and k₁?=?0.28 ± 0.01 minutes^?1, k₂?=?0.16 ± 0.01 minutes^?1, and k_mix?=?0.22 ± 0.02 minutes^?1 for 30 wt percent formulation. In method II, the mixture of the formulated granules produced by mixing the formulated granules from the two batches at equal weights gave dissolution rate constants of k_mix?=?0.30 ± 0.03 minutes^?1, 0.30 ± 0.02 minutes^?1, and 0.22 ± 0.01 minutes^?1 for 75 wt percent, 62 wt percent, and 30 wt percent formulations, respectively. After fitting the three data points of k₁, k₂, and k_mix to the 10 mixing rules in materials science—series mixing rule, Hashin and Shtrikman upper bound, logarithmic mixing, Looyenga mixing rule, effective media approximation (EMA), three-point lower bound, Torquato approximation, three-point upper bound, Maxwell mixing rule, and parallel mixing rule—we found that the selection of the best suited mixing rules based on k₁, k₂, and k_mix was solely dependent on the formulations under a given operating condition and regardless of whether the system was a powder mixture or a granular mixture. The values of k₁, k₂, and k_mix in both the 75 wt percent and 30 wt percent formulations were enveloped by the parallel mixing rule and Maxwell mixing rule, whereas the values of k₁, k₂, and k_mix for the 62 wt percent formulation were encompassed by the logarithmic mixing rule, Hashin and Shtrikman upper bound, and the series mixing rule. Apparently, the best suited mixing rules could be used to predict the right proportions of either the powder mixture (Method I) or the granular mixture (Method II) for obtaining any other desired dissolution rate constant, k_mix, whose value fell in between the values of k₁ and k₂.     

Notices

This paper presents the small sample optimum choice of the k(≤ r ₂) order statistics for the best linear unbiased estimate (BLUE) of the parameters μ and σ or σ alone (μ known) when the sample is Type II censored on the right. For n = 2(1)10, k = 1(1)r ₂ and Tr ₂ = ([.504n]+1}(1)n, the optimum ranks, the coefficients of the BLUES have been presented in Table I.     

Diffusion Coefficients Estimated by Dynamic Fluorescence Quenching at High Pressure: Pyrene, 9,10-Dimethylanthracene, and Oxygen in n-Hexane

The fluorescence quenching of pyrene (PY) by carbon tetrabromide (CBr₄) at pressures of up to 400 MPa in n-hexane was investigated. It was found that the fluorescence quenching is not fully, but nearly, diffusion-controlled. From the pressure-induced solvent viscosity dependence, the quenching rate constant, k _q , was separated into the contributions of the bimolecular rate constant in the solvent cage, k _bim, and that for diffusion, k _diff. Using the values of k _diff separated, together with those of the diffusion coefficient of CBr₄, the diffusion coefficients of PY were successfully estimated. This analysis was applied to the quenching systems of 9,10-dimethylanthracene (DMEA)/CBr₄ and of PY/O₂ and DMEA/O₂ that were studied previously. Using the values of k _diff for these systems, together with those of the corresponding diffusion coefficients of the fluorophore or quencher, the diffusion coefficients of DMEA and O₂ were also evaluated. Based on the results, the pressure-induced solvent viscosity, , dependence on the diffusion coefficients is discussed.     

Estimation of the Parameters of the Gamma Distribution by Sample Quantiles

This paper deals with large sample estimation of the location parameter (α₁ and the scale parameter α₂ in the gamma distribution with known shape parameter. Best linear unbiased estimates based on k sample quantiles are used. For a given k, the optimum spacings of the sample quantiles can be replaced by simpler “nearly optimum” spacings at virtually no loss of asymptotic efficiency. The theory behind the nearly optimum spacings is briefly reviewed. The major part of the paper concerns estimation of α₂ when α₂ is known. Nearly optimum spacings together with the coefficients to be used in computing the estimates are presented in a number of tables for k = 1(1) 10, and various values of the shape parameter. The paper also contains brief discussions of estimation of α₁, when α₂ is known, and simultaneous estimation of α₁ and α₂.     

Material design of plasma-enhanced chemical vapour deposition SiCH films for low-k cap layers in the further scaling of ultra-large-scale integrated devices-Cu interconnects

Abstract

Cap layers for Cu interconnects in ultra-large-scale integrated devices (ULSIs), with a low dielectric constant (k-value) and strong barrier properties against Cu and moisture diffusion, are required for the future further scaling of ULSIs. There is a trade-off, however, between reducing the k-value and maintaining strong barrier properties. Using quantum mechanical simulations and other theoretical computations, we have designed ideal dielectrics: SiCH films with Si–C₂H₄–Si networks. Such films were estimated to have low porosity and low k; thus they are the key to realizing a cap layer with a low k and strong barrier properties against diffusion. For fabricating these ideal SiCH films, we designed four novel precursors: isobutyl trimethylsilane, diisobutyl dimethylsilane, 1, 1-divinylsilacyclopentane and 5-silaspiro [4,4] noname, based on quantum chemical calculations, because such fabrication is difficult by controlling only the process conditions in plasma-enhanced chemical vapor deposition (PECVD) using conventional precursors. We demonstrated that SiCH films prepared using these newly designed precursors had large amounts of Si–C₂H₄–Si networks and strong barrier properties. The pore structure of these films was then analyzed by positron annihilation spectroscopy, revealing that these SiCH films actually had low porosity, as we designed. These results validate our material and precursor design concepts for developing a PECVD process capable of fabricating a low-k cap layer.     

Sound Speeds and Excess Isentropic Compressibilities of Ternary Mixtures of Tetrahydropyran and Aromatic Hydrocarbons at 308.15 K

Speeds of sound, u _ijk , of tetrahydropyran (THP) (i) + benzene (j) + toluene or o- or p-xylene (k), and tetrahydropyran (i)+toluene (j) + o- or p-xylene (k) ternary mixtures have been measured over the entire mole fraction range at 308.15 K and atmospheric pressure. The speed-of-sound data have been used to calculate isentropic compressibilities, (k_S)_ijk{(\kappa_S)_{ijk}} , and excess isentropic compressibilities, (k_S^E)_ijk{(\kappa_S^{\rm E})_{ijk}} . The (k_S^E)_ijk{(\kappa_S^{\rm E})_{ijk}} values for the investigated mixtures are correlated with the Redlich–Kister equation to estimate ternary adjustable parameters and standard deviations. The Moelwyn–Huggins concept (Huggins, Polymer 12:357, 1971) of interaction between the surfaces of components of binary mixtures has been extended to predict excess isentropic compressibilities of ternary mixtures by employing the concept of connectivity parameters of the third degree of a molecule (which in turn depends on its topology). It has been observed that (k_S)_ijk{(\kappa_S)_{ijk}} values predicted by the Moelwyn–Huggins concept compare well with corresponding experimental values.     

Order Statistics from the Gamma Distribution

This paper deals with order statistics from a gamma or χ² (Pearson Type III) distribution. Expressions are derived for the moments of an order statistic and for the covariance between two order statistics. A table of moments (about the origin as well as the mean) is presented. Equations are derived which are used to solve for the percentage points of the order statistics and a table of the percentage points is given. The problem of the best linear unbiased estimate of the scale parameter of the distribution which is based on order statistics is, briefly, discussed for the censored (type II) case. The modal value of an order statistic is derived and a table of these values is given for the cases of the smallest and the largest order statistics. Applications ‘to life-tests, extreme values, reliability and maintenance are described and illustrated in some cases.     

Study on the Interaction Coefficients in PR Equation with vdW Mixing Rules for HFC and HC Binary Mixtures

The Peng-Robinson equation of state with the van der Waals mixing rules was used to correlate vapor–liquid equilibrium (VLE) data for HFC/HC, HFC/HFC, and HC/HC binary mixtures. The interaction parameter k _ij was obtained for every binary mixture. It was assumed that k _ij has contributions from the two components, and each component has its own constant contribution factor k _i for the mixture, and the values of k _ij indicate the degree in difference of properties between the two components. Therefore, the interaction parameters k _ij is proposed as: k _ij  = k _i  − k _j . The values of the mixing factor k _i for Hydrofluorocarbons (HFCs) and Hydrocarbons (HCs), including propane, isobutane, n-butane, R23, R32, R125, R143a, R134a, R152a, R227ea R236fa, R236ea, and R245fa, were obtained by least-square fitting. In total, 39 refrigerant binary mixtures were analyzed on the basis of this method, and the results showed good agreement with experimental data. The overall average absolute deviations of pressure and vapor mole fraction are 1.3 % and 0.0089, respectively.     

Kozeny–Carman permeability relationship with disintegration process predicted from early dissolution profiles of immediate release tablets

This study was oriented toward the disintegration profiling of the diclofenac sodium (DS) immediate-release (IR) tablets and development of its relationship with medium permeability k_perm based on Kozeny–Carman equation. Batches (L1–L9) of DS IR tablets with different porosities and specific surface area were prepared at different compression forces and evaluated for porosity, in vitro dissolution and particle-size analysis of the disintegrated mass. The k_perm was calculated from porosities and specific surface area, and disintegration profiles were predicted from the dissolution profiles of IR tablets by stripping/residual method. The disintegration profiles were subjected to exponential regression to find out the respective disintegration equations and rate constants k_d. Batches L1 and L2 showed the fastest disintegration rates as evident from their bi-exponential equations while the rest of the batches L3–L9 exhibited the first order or mono-exponential disintegration kinetics. The 95% confidence interval (CI_95%) revealed significant differences between k_d values of different batches except L4 and L6. Similar results were also spotted for dissolution profiles of IR tablets by similarity (f₂) test. The final relationship between k_d and k_perm was found to be hyperbolic, signifying the initial effect of k_perm on the disintegration rate. The results showed that disintegration profiling is possible because a relationship exists between k_d and k_perm. The later being relatable with porosity and specific surface area can be determined by nondestructive tests.     

Fundamental study of thermal conduction in dry soils

The thermal conductivity of the different soil components—mineral, liquids and air—varies across two orders of magnitude. Two studies are implemented to explore the role of contacts in heat conduction in dry granular materials. The first set of experiments is designed to elucidate heat transfer at contacts, and it is complemented with a numerically based inversion analysis for different local and boundary conditions to extract proper material parameters. Then, the thermal conductivity of dry soils is measured at different packing densities to address the relevance of coordination number and particle shape effects. Together, both studies confirm the prevailing effect of contact quality and number of contacts per unite volume on heat conduction in granular materials. Interparticle contacts and the presence of liquids in pores play a critical role in heat transfer, and determine the ordered sequence of typical thermal conductivity values: k _air < k _dry-soil < k _water < k _{saturated-soil} < k _mineral.     

New weld toe magnification factors for semi‐elliptical cracks in plate‐to‐plate butt‐welded joints

British Standard 7910:2013 includes a single set of equations for estimating the weld toe magnification factor (M_k) of different types of welded joints with a semi‐elliptical surface crack. In this study, extensive finite element analyses are performed to determine the M_k of plate‐to‐plate butt‐welded joints subjected to axial and bending loading. The M_k values at the crack deepest points and the crack ends are determined. It is found that the M_k results of butt welds are different with those of single T‐butt joints with percentage difference in values being as high as 63.8% and 97.4% for axial and bending loading cases, respectively. Percentage differences of 60.4% and 358.2% for axial and bending loading cases are observed between butt welds and X‐joints. New M_k equations for butt welds are proposed using multiple regression analyses.     

Acoustic Resonator Experiments at the Triple Point of Water: First Results for the Boltzmann Constant and Remaining Challenges

We report on two sets of isothermal acoustic measurements made with argon close to the triple point of water using a 50 mm radius, thin-walled, diamond-turned quasisphere. Our two isotherms yielded values for the Boltzmann constant, k _B, which differ by 0.9 parts in 10⁶, and have an average value of k _B = (1.380 649 6 ± 0.000 004 3) × 10⁻²³J · K⁻¹. The relative uncertainty is 3.1 parts in 10⁶, and the average value is 0.58 parts in 10⁶ below the 2006 CODATA value (Mohr et al. Rev Mod Phys 80:633, 2008), and so the values are consistent within their combined (k = 1) uncertainties.