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The row-column setting for comparing v experimental treatments is a p × q array of experimental units, with the p rows and q columns representing levels of two blocking factors. When p = q = v, the optimal design for estimating treatment contrasts is a Latin square. For a v ^t factorial treatment set in the same setting, t mutually orthogonal Latin squares are an optimal main-effects plan. Orthogonal collections of Latin squares arise when these results are generalized to b row-column layouts of size v × v, where b is greater than 1. This article studies these orthogonal collections, with special emphasis on small v and b.     

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.     

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.     

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.     

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.     

Privacy-Preserving Quantum Two-Party Geometric Intersection

Privacy-preserving computational geometry is the research area on the intersection of the domains of secure multi-party computation (SMC) and computational geometry. As an important field, the privacy-preserving geometric intersection (PGI) problem is when each of the multiple parties has a private geometric graph and seeks to determine whether their graphs intersect or not without revealing their private information. In this study, through representing Alice’s (Bob’s) private geometric graph G_A (G_B) as the set of numbered grids S_A (S_B), an efficient privacy-preserving quantum two-party geometric intersection (PQGI) protocol is proposed. In the protocol, the oracle operation O_A (O_B) is firstly utilized to encode the private elements of S_A =(a₀,a₁,…,a_M-1) (S_B =(b₀,b₁,…,b_N-1)) into the quantum states, and then the oracle operation O_f is applied to obtain a new quantum state which includes the XOR results between each element of S_A and S_B. Finally, the quantum counting is introduced to get the amount (t) of the states |a_i⊕b_j| equaling to |0|, and the intersection result can be obtained by judging t >0 or not. Compared with classical PGI protocols, our proposed protocol not only has higher security, but also holds lower communication complexity.     

Effect of Zinc Doping on the Phase-Transition Temperatures of Cu2Te

Structural transformations of Cu_1.50Zn_0.50Te were studied by high-temperature x-ray diffraction analysis. At 293 K, much like Cu₂Te, the Zn-substituted telluride consists of orthorhombic (a= 7.3192 Å, b= 22.2362 Å,c= 36.4581 Å) and hexagonal (a= 4.2478 Å, c= 7.2335 Å) phases. At 811 K, the orthorhombic phase transforms into the hexagonal one. At 970 K, the hexagonal-to-fcc phase transition occurs (a _fcc = 6.1187 Å).     

Load dependence of the apparent Knoop hardness of SiC ceramics in a wide range of loads

Silicon carbide (SiC) ceramics is a material with increasing use, due to its excellent mechanical properties, especially high hardness. In order to integrate this material into design process, we need to know its hardness as precise as possible. The Knoop hardness number (HK) is calculated using the expression: HK = α·F/d², where F is the applied load, d is the long diagonal of the resulting 1₀indentation and a is the Knoop indenter geometrical constant. In this paper, the Knoop hardness of SiC ceramics was measured in the applied load range from 4.9 to 98.07 N. For some materials measured “apparent” hardness value decreases with increasing applied test load (normal indentation size effect – ISE), while for some materials measured “apparent” hardness increases with increasing applied test load (reverse indentation size effect – RISE). Obtained results show the measured hardness exhibits the ISE. In the literature several models are given for the phenomenon explanation. We used the following models: Meyer's law (F = K·dⁿ), proportional specimen resistance – PSR (F = a₁·d + a₂·d²) and modified proportional specimen resistance – MPSR model (F = a₀ + a₁·d + a₂·d²). Results of regression analysis for all applied models show they can all be used for ISE analysis. “True” hardness was determined based on the PSR and MPSR model (HK_T = α·a₂). The obtained results were similar. If the specimen surface is carefully prepared and the range of loads is wide, the a₀ coefficient from MPSR model reaches small values and can be excluded. Therefore, for the calculation of SiC ceramics Knoop hardness, the simpler model (PSR) can be used.     

Hydrothermal synthesis and characterization of CsNiP and 2(LiZnHP2O7) crystals

CsNiP and 2(LiZnHP₂O₇) crystals were synthesized by hydrothermal technique at moderate P-T conditions. Solubility results of both the compound shown positive thermal coefficient and single crystal X-ray studies revealed, CsNiP crystallized in hexagonal system with cell parameters; a = 7.173(2), c = 5.944(9) Å, V = 264.87(7) ų and space group P63/mmc and 2(LiZnHP₂O₇) crystallized in orthorhombic system with cell parameters; a = 12.3636 Å, b = 27.5330 Å, c = 6.8647 Å and space group, Pca21 exhibiting ring type of cavities with open aperture in the structure. CsNiP is a frequency dependent paramagnetic and 2(LiZnHP₂O₇) is a diamagnetic.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

A Note on Estimation from a Type I Extreme-Value Distribution

If the random variable T has the ta-o-parameter Weibull distribution with cumulative distribution function F(t; θ, K) = 1 – exp[–(t/θ) ^k ], where θ is the scale parameter and K is the shape parameter, then the random vatiable X = In T has the Type I extreme-value distribution of smallest values with cumulative distribution function F(x; u, b) = 1 – exp {–exp [(x – u)/b}, where u = In θ is the location parameter (mode) and b = 1/K is the scale parameter. It is therefore possible to obtain the maximum-likelihood estimator û _mn | b of u, based on the first m order statistics of a sample of size n, when b is known, by a simple transformation of the corresponding estimator of θ when K is known. Use is made of the fact that û _mn | b = In _mn | K, where 2m( _mn | K) ^k /θ ^k has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on u. The probability density function of û _mn | b which for given m is the same for any n ≥ m, is obtained by a simple transformation of that of _mn | K. Integration yields expressions, involving digamma and trigamma functions, for the bias E = E[(û _mn |b) – u] and the variance V = V(û _mn | b). By subtracting the bias E](û _mn |b) – u] from û _mn |b, one obtains an unbiased estimator û|b which has the same variance as the maximum-likelihood estimator. Values of E/b(6DP) and of V/b ²(6DP) are tabulated for m = 1(1)100. The use of the table is discussed and illustrated by a numerical example.     

Synthesis and properties of structural analogs of the mineral bournonite

Rare-earth analogs of the mineral bournonite, PbCuSbS₃, have been synthesized for the first time and their physicochemical properties have been studied. PbCuSbS₃, EuCuSbS₃, YbCuSbS₃, PbCuLaS₃, and PbCuNdS₃ are isostructural with each other and crystallize in orthorhombic symmetry with the following unit-cell parameters: a = 8.176, b = 8.660, c = 7.796 Å (PbCuSbS₃); a = 8.156, b = 8.68, c = 7.786 Å (EuCuSbS₃); a = 8.15, b = 8.64, c = 7.76 Å (YbCuSbS₃); a = 8.26, b = 8.84, c = 7.96 Å (PbCuLaS₃); a = 8.20, b = 8.80, c = 7.92 Å (PbCuNdS₃) (Z = 4, sp. gr. Pmn2₁).     

A linear representation of the diffusion of solvents into insoluble polymers

The amount,q, of a solvent taken up by a sample of an insoluble polymer during a time,t, can be represented accurately by the relationq=t/(at + b), which has a very simple linear equivalent: (t/q)=at+b, wherea andb are constants. The same behaviour is also shown by the exchange of ions in an ion-exchange resin and in the case of a swell process, whereq may be measured as a volume increase. The kinetic relationq=k(Q−q) ², where Q=a⁻¹ andk=Q ⁻²b⁻¹, is compared with Fick's first law of diffusion and the dependence of the diffusion coefficient and the concentration gradient onq is discussed.     

Thermal expansion of NZP-family alkali-metal (Na, K) zirconium phosphates

The thermal expansion of the A _x Zr_2.25-0.25x(PO₄)₃ phosphates with A = Na(x = 0.5,1.0,2.0,3.0,4.0, 5.0) and K(x = 1.0, 3.0, 5.0), crystallizing in structures of the NaZr₂(PO₄)₃ type (sp. gr.R3c or C2/c), was studied by high-temperature x-ray powder diffraction in the range 20–700‡C. The lattice parametersa and c and thea- andc-axis thermal expansion coefficients (α_a and α_c) were determined. The thermal expansion of the phosphates was found to be highly anisotropic (α_a < 0, α_c > 0). The strongest anisotropy was found in NaZr₂(PO₄)₃ (α_a = -4.89 x 10^-6 K^-1, α_c = 22.02 x 10^-6 K^-1), KZr₂(PO₄)₃ (α_a =-5.30 x 10^-6 K^-1, α_c = 5.41 x 10^-6 K^-1), and Na₅Zr(PO₄)₃ (α_a = -5.82 x 10^-6 K^-1, α_c = 20.73 x 10^-6 K^-1). K₅Zr(PO₄)₃ exhibited the smallest thermal expansion and weak anisotropy (α_a = -2.14 x 10^-6 K^-1, α_c = 2.65 x 10^-6 K^-1). The effects of M(l) and M(2) site occupancies on α_a, α_c,a, and c were assessed. The relative magnitudes of crystal-chemical and thermal expansion in the Na and K compounds were analyzed.     

Synthesis and Characteristics of Double Np(VI) Potassium and Pu(VI) Potassium Silicates K[(NpO2)(SiO3OH)]·H2O and K[(PuO2)(SiO3OH)]·H2O

K[(NpO₂)(SiO₃OH)]·H₂O (I) and K[(PuO₂)(SiO₃OH)]·H₂O (II) were prepared by hydrothermal synthesis at 120°C, and their IR and near-IR spectra were measured. The unit cell parameters of the compounds were determined from powder X-ray patterns [a = 7.068(1), b = 7.064(2), c = 6.640(1) Å. = 106.68(1)°, V = 317.6 ų (I); a = 7.102(1), b = 7.100(2), c = 6.637(1) Å, = 107.62(1)°, V = 318.9 ų (II)]. These compounds are isostructural with potassium boltwoodite K[(UO₂)(SiO₃OH)]·H₂O.     

Fatigue of Kaowool reinforced aluminum: Effect of shot particle wall thickness

The fatigue performance of Kaowool fiber reinforced 339 aluminum composites at 300°C is limited by spherical thin walled hollow Kaowool shot particles. These act as crack initiation sites particularly when located at the surface. This problem does not occur for thick walled particles or particles filled with the aluminum matrix. The effect of wall thickness (t) is evaluated from finite element analysis of both 2D and 3D models, with and without plasticity. Both models predict that hollow thin walled particles act as defects, while thick walled particles act as reinforcements, this transition being defined by a critical wall thickness (t _c). The 3D model is preferred in that it predicts more accurate and smaller values of t _c. Specifically, the 3D elastic/plastic model predicts that the largest stress concentration occurs for a fractional surface particle and that in this condition t _c = 0.18a, where a is the particle radius. This value agrees with our experimental observation that particles with t > 0.2a do not initiate failures.     

Estimating the Degrees of Freedom for Linear Combinations of Mean Squares by Satterthwaite's Formula

In calculating the variance of a mean or in constructing an approximate F-test, it is frequently necessary, particularly with unbalanced data, to form a linear function of mean squares, MS = Σa _i, where the a_i are known constants. Satterthwaite [1] suggests that MS is approximately distributed as x ² _f E(MS)/f_t where the degrees of freedom is estimated by, f = (MS)²/Σ[(a_iMS_i )²/f_i ], where f_i is the degrees of freedom associated with MS_i . Satterthwaite remarks that caution should be used in applying this formula where some of the a_i are negative. The primary purpose of this paper is to investigate the validity of Satterthwaite's procedure when some MS_i are subtracted. Due to the mathematical complexity of the distribution of MS, the problem is studied by a computer simulation.     

Reduction of perovskite oxide LnCoO3(Ln = La-Eu) in a hydrogen atmosphere

The reduction of LnCoO₃(Ln = La-Eu) in a hydrogen atmosphere of 2×10⁶ Pa at 25 to 600° C was investigated by X-ray diffraction and thermogravimetric analysis, which were carried out to pursuein situ the reaction of reduction of LnCoO₃. The amount of lattice oxygen consumed at 600° C increases on going from LaCoO₃ to EuCoO₃. It was shown that the reduction process proceeded through the formation of a series of oxygen-deficient structures in keeping the perovskite structures; for example, the final X-ray powder diffraction pattern of NdCoO_3–x (x=1.1) could be indexed on a cubic cell (a = 0.39 nm) and an orthorhombic cell (a₀=0.522 nm,b ₀ = 0.559nm, c₀ = 0.795nm), the relationship with the cubic cell beinga ₀ 2^1/2 a _c,b ₀ 2^1/2 a _c,c 2a_c.     

Crystal Structure of Rb[UO2(SO4)F]

Single crystal of Rb[UO₂(SO₄)F] (I) was studied by X-ray diffraction [rhombic system, space group Pca2 ₁, Z = 12, unit cell parameters: a = 25.393(3), b = 6.735(1), c = 11.496(3) Å]. The main structural units of crystals of I are [UO₂(SO₄)F]^- layers belonging to the crystallochemical group AT³M² (A = UO₂ ^{2 +}) of uranyl complexes and combined into three-dimensional framework by electrostatic interactions involving interlayer rubidium ions.     

Synthesis and Structure of ErNi7.9B2

The new boride ErNi_7.9B₂ is synthesized, and its crystal structure is determined by single-crystal x-ray diffraction: sp. gr. C2/c, new structure type, a= 1.6626(2) nm,b = 0.9525(1) nm,c = 1.0686(1) nm, = 117.76(1)°, R _F = 0.0399. The crystal-chemical features of ErNi_7.9B₂ are discussed.