Bayes estimation of the general hazard rate model

In reliability theory and life testing models, the life time distributions are often specified by choosing a relevant hazard rate function. Here a general hazard rate function h(t)=a+bt^c−1, where c, a, b are constants greater than zero, is considered. The parameter c is assumed to be known. The Bayes estimators of (a,b) based on the data of type II/item-censored testing without replacement are obtained. A large simulation study using Monte Carlo Method is done to compare the performance of Bayes with regression estimators of (a,b). The criterion for comparison is made based on the Bayes risk associated with the respective estimator. Also, the influence of the number of failed items on the accuracy of the estimators (Bayes and regression) is investigated. Estimations for the parameters (a,b) of the linearly increasing hazard rate model h(t)=a+bt, where a, b are greater than zero, can be obtained as the special case, letting c=2.     

On using peak amplitude and rise time for AE source characterization

Acoustic Emission (AE) signals, which are electrical version of acoustic emissions, are usually analysed using a set of signal parameters. The major objective of signal analysis is to study the characteristics of the sources of emissions. Peak amplitude(P _a ) and rise time(R _t ) are two such parameters used for source characterization. In this paper, we theoretically investigate the efficiency ofP _a andR _t to classify and characterize AE sources by modelling the input stress pulse and transducer. Analytical expressions obtained forP _a andR _t clearly indicate their use and efficiency for source characterization. It is believed that these results may be of use to investigators in areas like control systems and signal processing also     

On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I

This paper deals with the problem of finding sets of mutually orthogonal Latin squares of order 4t (where 4t – 1 is a prime power) based on orthogonal mappings of a group. For the group G we take the module G(2, 2t) whose elements are vectors (a ₁, a ₂,) where a ₁ is a residue class (mod 2) and a ₂ is a residue class (mod 2t), the addition being defined by (a ₁, a ₂) + (b ₁, b ₂) = (c ₁, c ₂) where a ₁ + b ₁ = C ₁ (mod 2) and a ₂ + b ₂ = c ₂ (mod 2t). Then the search for orthogonal mappings is materially simplified by using a configuration based on the balanced incomplete block design with parameters v = b = 4t – 1, r = k = 2t – 1, λ = t – 1. Using this method, two sets of five mutually orthogonal Latin squares of order 12 were obtained.     

Mixed-Cation Thallium Phosphates Containing Di-, Tri-, and Pentavalent Cations

The mixed phosphates TlMn²⁺P₃O₉(a= 9.422 Å, b= 7.3199 Å, c= 12.227 Å sp. gr. Pnma), TlGaHP₃O₁₀(a= 12.01 Å, b= 8.471 Å, c= 9.098 Å, = 111.98° sp. gr. C2/c), and TlTa(PO₄)₂(a= 5.0308 Å, c= 8.497 Å sp. gr. P321) were obtained by precipitation from polyphosphoric-acid melts containing di-, tri-, and pentavalent cations in combination with Tl⁺. The crystal structures of TlMn²⁺P₃O₉and TlTa(PO₄)₂were determined by single-crystal x-ray diffraction.     

Setting expansion of gypsum-bonded investment in dental casting

The values of setting expansion of investment under uniaxial stress have been determined at conditions designed to obtain the difference of setting expansion between that parallel to the loading direction and that perpendicular to the loading axis. The setting expansion curves were represented by $$a(t) = a_0 (1 - P/E')[1 - exp( - kt)]$$ along the loading direction and $$a(t) = a_0 (1 + \nu 'P/E')[1 - exp( - kt)]$$ perpendicular to the loading direction, wherea(t) is a setting expansion,a ₀ = 0.009,ν′ = 0.2,E′ = 5 kg cm^?2,k = 0.032 min^?1,P applied stress, andt the time (min). On the basis of these results, a method to estimate the value of setting expansion under restrictive force was developed. According to this method, the setting expansion of the investment could be calculated by substituting?ε/?t forε, ka ₀ exp (?kt)/E′ for 1/E, ν′ forν, andka ₀ exp (?kt) forαT in the theory of elasticity.     

Fast matrix exponent for deterministic or random excitations

The solution of ż=Az is z(t)=exp(At)z₀=E_tz₀, z₀=z(0). Since z(2t)=E_2tz₀=Ez₀, z(4t)=E_4tz₀=Ez₀, etc., one function evaluation can double the time step. For an n‐degree‐of‐freedoms system, A is a 2n matrix of the nth‐order mass, damping and stiffness matrices M, C and K. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean‐square response P to a white noise random force with intensity W(t) is governed by the Lyapunov differential equation: Ṗ=AP+PA^T+W. The solution of the homogeneous Lyapunov equation is P(t)=exp(At) P₀ exp(A^Tt), P₀=P(0). One function evaluation can also double the time step. If W(t) is given as piecewise polynomials, the mean‐square response can also be obtained analytically. In fact, exp(At) consists of the impulsive‐ and step‐response functions and requires no special treatment. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non‐stationary response is resulted immediately without integration. The method is further extended to modulated noise excitations. The method gives analytical mean‐square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean‐square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non‐white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means‐square response matrix of a simply supported beam by finite element method. Copyright © 2001 John Wiley & Sons, Ltd.     

A variant of the heat balance integral method and a new proof of the exponentially fast asymptotic behavior of the solutions in heat conduction problems with absorption

We give a new and explicit estimate for the asymptotic behavior of the solutions of the problem u_t − u_xx + u₊^P = 0, x > 0, t> 0, with conditions u(0, t) = 1, t > 0 and u(x, 0) = U₀(x) ≥ 0, x > 0, for a class of functions U₀ and parameter 0 < p < 1. We use an approximate solution given by the heat balance integral method with the innovation property which fixes appropriately the asymptotic limit of the corresponding approximate free boundary.     

Investigation of the gate-bias induced instability for InGaZnO TFTs under dark and light illumination

Mechanism of the instability for indium–gallium–zinc oxide thin film transistors caused by gate-bias stress performed in the dark and light illumination was investigated in this paper. The parallel V_t shift with no degradation of subthreshold swing (S.S) and the fine fitting to the stretched-exponential equation indicate that charge trapping model dominates the degradation behavior under positive gate-bias stress. In addition, the significant gate-bias dependence of V_t shift demonstrates that electron trapping effect easily occurs under large gate-bias since the average effective energy barrier of electron injection decreases with increasing gate bias. Moreover, the noticeable decrease of threshold voltage (V_t) shift under illuminated positive gate-bias stress and the accelerated recovery rate in the light indicate that the charge detrapping mechanism occurs under light illumination. Finally, the apparent negative V_t shift under illuminated negative gate-bias stress was investigated in this paper. The average effectively energy barrier of electron and hole injection were extracted to clarify that the serious V_t degradation behavior comparing with positive gate-bias stress was attributed to the lower energy barrier for hole injection.     

The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations

We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential type partial differential equation in the space of (x,t,τ). When the group-preserving scheme is used to integrate these equations under a semi-discretization of u(x,t,τ) at the spatial-temporal grid points, we can achieve rather accurate solutions.     

Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.     

Synthesis and properties of LiZr2(AsO4)3 and LiZr2(AsO4) x (PO4)3 − x

The LiZr₂(AsO₄)₃ arsenate and LiZr₂(AsO₄) _x (PO₄)_{3 ? x} solid solutions have been prepared through precipitation followed by heat treatment, and characterized by X-ray diffraction, X-ray structure analysis, IR spectroscopy, and impedance spectroscopy. We have established conditions for the crystallization of the arsenate and a continuous series of arsenate phosphate solid solutions (0 ≤ x ≤ 3), which have been obtained as two polymorphs: monoclinic and hexagonal. Using the Rietveld method, we have refined the crystal structures of the polymorphs of LiZr₂(AsO₄)₃ (sp. gr. P2₁/n, a = 9.1064(2), b = 9.1906(2), c = 12.7269(3) Å, β = 90.844(2)°, V =1065.03(5) ų, Z = 4; sp. gr. R $\bar 3$ c, a = 9.1600(4), c = 22.9059(13) Å, V = 1664.44(14) Å, Z = 6) and LiZr₂(AsO₄)_1.5(PO₄)_1.5. Their structural frameworks are built up of AsO₄ tetrahedra—or (As,P)O₄ tetrahedra occupied by arsenic and phosphorus atoms at random—and ZrO₆ octahedra, with the lithium atoms in between. The ionic conductivity of the materials has been measured. The cation conductivity of monoclinic LiZr₂(AsO₄) _x (PO₄)_{3 ? x} with 0 ≤ x ≤ 1 has been shown to exceed the conductivity of lithium zirconium phosphate.     

A Note on Selection of the Greatest Exceedance Probability

The purpose of this paper is to give a sequential procedure for selecting a best normal population from among several. Suppose that p(i, a) is tb.e probability that an observation from population pi is greater than a given constant a. Population π _r , is better than population π₈ if p(r, a) > p(s, a).     

FATIGUE CRACK GROWTH IN D16T A1-ALLOY SHEET SUBJECTED TO BIAXIAL OUT-OF-PHASE LOADING

Abstract— Fractographic peculiarities of fatigue crack development are studied in cruciform specimens of D16T aluminium alloy under out-of-phase biaxial tension and tension-compression. In the range of the biaxial load ratios λ from ?0.5 to +0.5 and an R-ratio of 0.3, fatigue striation formation took place beyond a crack growth rate near to 4 × 10^?8 m/cycle. The striation spacing and the crack growth rate increase as the φ-angle of the out-of-phase biaxial loads increases in the range of φ-angles from 0° to 180°. The ratio between the increment of crack growth, da/dN, and the striation spacing, δ, is approximately 1 to 1 when da/dN is greater than 4 × 10^?8 m/cycle. The relationship between the number of cycles from the beginning of a test up to the growth rate of 10^?6 m/cycle (N_d), and the crack growth period, N_P, from when the crack initiates up to the instant when that growth rate is reached, was determined for different λ ratios and φ angles. The value of N_d decreases as the φ angle is increased in the range from 0° to 1807deg;. Cycle loading parameters must be taken into account in order to describe the crack growth period when using a unified method that involves an equivalent stress intensity factor K_e=K_IF₁(λ, R)F₂(φ). The values of F₂(φ) were determined. The calculated fatigue crack growth period, N_c, applicable up to and including the stage of fatigue striation formation (predicted by using both of the F₁(λ, R) and F₂(φ) functions) is correlated with the experimental data and the error is of the order of 15%.     

Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity

We introduce a two-parameter family of ‘partially hyperbolic’ skew products (G_{a, t})_{a > 0, t ∈ [ ? ε, ε]} maps with one dimensional centre direction. In this family, the parameter a models the central dynamics and the parameter t the unfolding of cycles (that occurs for t = 0). The parameter a also measures the ‘central distortion’ of the systems: for small a, the distortion of the systems is small and it increases and goes to infinity as a → ∞. The family (G_{a, t}) displays some of the main characteristic properties of the unfolding of heterodimensional cycles as intermingled homoclinic classes of different indices and secondary bifurcations via collision of hyperbolic homoclinic classes. For a ∈ (0, log?2), the dynamics of (G_{a, t}) is always non-hyperbolic after the unfolding of the cycle. However, for a > log?4 intervals of t-parameters corresponding to hyperbolic dynamics appear and turn into totally prevalent as a → ∞ (the density of ‘hyperbolic parameters’ goes to 1 as a → ∞). The dynamics of the maps G_{a, t} is described using a family of iterated function systems modelling the dynamics in the one-dimensional central direction.     

Adaptive Distribution-free Regression Methods and their Applications

We consider the model Y = βx + Z, where the random variable Z has a continuoustype distribution that can be badly skewed, contaminated, or censored. To test the hypothesis H ₀ : β = β₀, we use the distribution-free statistic K(β₀) = Σc(Q _i )a(R _i ), where c(·) and a(·) are increasing score functions and Q _i and R _i are the respective ranks of x _i and y _i – β₀ x _i . The score functions c(·) and a(·) can be adapted or chosen after observing the data without destroying the distribution-free nature of the test. A Monte Carlo study is presented which illustrates the excellent performance of an adaptive test when a wide range of distributions is considered for the residuals. Interval and point estimates of sβ can be found by employing the “inverse” of the testing procedure. These results are used to find estimates of the percentile lines. Two examples are given which involve lifetimes of electric motor insulation and grade point averages of beginning university students, respectively.     

New lead vanadium phosphate with langbeinite-type structure: Pb1.5V2(PO4)3

The new lead vanadium phosphate Pb_1.5V₂(PO₄)₃ was synthesized by solid state reaction and characterized by X-ray powder diffraction, electron microscopy, and magnetic susceptibility measurements. The crystal structure of Pb_1.5V₂(PO₄)₃ (a = 9.78182(8) Å, S.G. P2₁3, Z = 4) was determined from X-ray powder diffraction data and belongs to the langbeinite-type structures. It is formed by corner-linked V³⁺O₆ octahedra and tetrahedral phosphate groups resulting in a three-dimensional framework. The lead atoms are situated in the structure interstices and only partially occupy their positions. An electron microscopy study confirmed the structure solution. Magnetic susceptibility measurements revealed Curie-Weiss (CW) behavior for Pb_1.5V₂(PO₄)₃ at high temperature whereas at around 14 K an abrupt increase on the susceptibility was observed.     

An analysis of the electrical conductivity in BaSO4-added Ag2SO4 solid electrolyte system

The compositions (1 −x)Ag₂SO₄−(x)BaSO₄, wherex=0·01 to 0·6, were prepared by slow cooling of the melt. The extent of the solid solubility of Ba²⁺ in Ag₂SO₄ was determined by X-ray powder diffraction and scanning electron microscopy. The bulk conductivity of each sample was obtained using a detailed impedance analysis. The partial substitution of Ba²⁺ results in the enhancement of conductivity in compliance with the classical aliovalent doping theory. A simplistic model based on lattice distortion (expansion) due to partial substitution of Ag⁺ by the bigger Ba²⁺ has been considered to explain enhanced conductivity. Beyond solid-solubility limit (5·27 mole%) the BaSO₄-dispersed Ag₂SO₄ conductivity follows the usual trend seen in binary systems. An increase in conductivity in this case is discussed in the light of interfacial reactions and surface defect chemistry. The maximum conductivity in 20 mole% BaSO₄ dispersed Ag₂SO₄ is due to percolation threshold.     

X-ray dosimetric properties of vapor-grown CdGa<Subscript>2</Subscript>S<Subscript>4</Subscript> single crystals

CdGa₂S₄ single crystals have been grown from a presynthesized source material by closed-tube iodine vapor transport, and their X-ray dosimetric properties have been studied. Their X-ray sensitivity coefficient K ranges from K = 1.26 × 10⁻¹¹ to 1.39 × 10⁻¹⁰ A min/(V R) at effective X-ray hardnesses V _a = 25–50 keV and dose rates E = 0.75–78.05 R/min, and increases with X-ray dose. The K(V _a) curve has a negative slope, in contrast to the K(E) curve. The photocurrent-dose curves of the CdGa₂S₄ single crystals demonstrate that the steady-state X-ray photocurrent is a power-law function of X-ray dose rate: ΔI _E,0 ∼ E ^α . With increasing V _a, the slope of the curves sharply decreases and α approaches unity.     

Maximum-Likelihood Estimation of the Parameters of a Four-Parameter Generalized Gamma Population from Complete and Censored Samples

Consider the four-parameter generalized Gamma population with location parameter c, scale parameter a, shape/power parameter b, and power parameter p (shape parameter d = bp) and probability density function f(x; c, a, b, p) = p(x — c) ^bp–1 exp {–[(x – c)/a] ^p }/a ^bp Γ(b), where a, b, p > 0 and x ≥ c ≥ 0. The likelihood equations for parameter estimation are obtained by equating to zero the first partial derivatives, with respect to each of the four parameters, of the natural logarithm of the likelihood function for a complete or censored sample. The asymptotic variances and covariances of the maximum-likelihood estimators are found by inverting the information matrix, whose components are the limits, as the sample size n → ∞, of the negatives of the expected values of the second partial derivatives of the likelihood function with respect to the parameters. The likelihood equations cannot be solved explicitly, but an iterative procedure for solving them on an electronic computer is described. The results of applying this procedure to samples from Gamma, Weibull, and half-normal populations are tabulated, as are the asymptotic variances and covariances of the maximum-likelihood estimators.