Thermodynamic assessment of the Eu–Pb and Lu–Pb systems

The thermodynamic assessment of the Eu–Pb and Lu–Pb binary systems has been carried out by using the Calculation of Phase Diagrams (CALPHAD) method based on the available literature data including the phase equilibria and thermodynamic properties. The Gibbs free energies of the liquid, bcc, fcc and hcp phases were described by the subregular solution model with Redlich–Kister formula, and those of the intermetallic compounds (Eu₂Pb, Eu₅Pb₃, βEuPb, αEuPb, EuPb₃, Lu₅Pb₃, βLu₅Pb₄, αLu₅Pb₄, Lu₆Pb₅, and LuPb₂) in the binary systems were described by the two-sublattice model. A consistent set of the thermodynamic parameters leading to a reasonable agreement between the calculated results and literature data was obtained.     

Thermodynamic optimization of the Gd–Pb system using random solution and associate models

The Gd–Pb system was critically modeled by means of the CALPHAD technique on the basis of experimental data in the literature. Given the asymmetric shape of the liquidus in the Gd–Pb phase diagram, the associate model for the liquid phase was tested and compared with the substitutional solution model. The results of the optimization show that a better agreement with the available experimental data is obtained by means of the associate model than the substitutional solution model. The solution phases (liquid, bcc, fcc, hcp) were treated with the Redlich–Kister equation. The intermetallic compounds Gd₅Pb₃, αGd₅Pb₄, βGd₅Pb₄, Gd₁₁Pb₁₀, Gd₆Pb₇, GdPb₂, GdPb₃ were treated as stoichiometric compounds. Two sets of self-consistent thermodynamic parameters of the Gd–Pb system were obtained.     

Thermodynamic optimization of the binary PbO–CaO and ternary PbO–CaO–SiO2 systems

Liquidus phase equilibrium data from the recent study for the PbO–CaO and the PbO–CaO–SiO₂ systems (as a part of research program on the characterization of the multicomponent PbO–ZnO–FeO–Fe₂O₃-“Cu₂O”-CaO-SiO₂ system), combined with phase equilibrium and thermodynamic data from the literature, have been used to obtain a self-consistent set of parameters of the thermodynamic models for all phases: liquid, (Ca,Pb)₂SiO₄, (Ca,Pb)₃SiO₅, (Ca,Pb)SiO₃ (wollastonite and pseudowollastonite), Pb₃(Ca,Pb)₂Si₃O₁₁ (ganomalite) solutions, SiO₂ (quartz, tridymite, cristobalite), Ca₃Si₂O₇ (rankinite), CaO (lime), PbSiO₃ (alamosite), Pb₂SiO₄, Pb₁₁Si₃O₁₇, Pb₅SiO₇ lead silicates, PbO (massicot), Ca₂PbO₄, Pb₈CaSi₆O₂₁ (barysilite), PbCa₂Si₃O₉ (margarosanite) and Pb₃Ca₁₂Si₅O₂₅ compounds. Analysis of available data has shown the lack of data in the two immiscible liquids range over cristobalite, where several new experiments were done to support the model. The modified quasichemical model is used to describe the liquid slag phase. From these model parameters, the optimized ternary phase diagram is back calculated.     

Thermodynamic assessments of the Er–Sb and Sb–Tm systems

By using the calculation of phase diagrams (CALPHAD) method, thermodynamic assessments of the Er–Sb and Sb–Tm systems were carried out based on the available experimental data including thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, hcp, and rhombohedral phases in the Er–Sb and Sb–Tm systems were modeled by the substitutional solution model with the Redlich–Kister formula, and the intermetallic compounds (Er₅Sb₃, αErSb, βErSb, ErSb₂, Sb₂Tm, αSbTm, βSbTm, α Sb₃Tm₅, and β Sb₃Tm₅ phases) in these two binary systems were described by the sublattice model. An agreement between the present calculated results and experimental data was obtained.     

Algebraic Riccati equation and J-spectral factorization for estimation

In this paper we investigate on the existence of the stabilizing solution of the algebraic Riccati equation (ARE) related to the filtering problem with a prescribed attenuation level γ. It is well known that such a solution exists and is positive definite for γ larger than a certain γ_F and it does not exist for γ smaller than a certain γ₀. We consider the intermediate case γ(γ₀,γ_F] and show that in this interval the stabilizing solution does exist, except for a finite number of values of γ. We show how the solution of the ARE may be employed to obtain a minimum-phase J-spectral factor of the J-spectrum associated with the filtering problem.     

A thermodynamic reassessment of the La–Sn system

Thermodynamic modelling of the La–Sn binary system was carried out with the help of the CALPHAD (CALculation of PHAse Diagram) method. The liquid phase has been described with the association solution model with a ‘ La₁Sn₁’ associated complex. The intermetallic compounds were treated as stoichiometric phases. The calculated phase diagram and the thermodynamic properties of the system are in satisfactory agreement with the majority of the experimental data.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

Experimental investigation and thermodynamic assessment of the Cu-Sn-Ti ternary system

The Cu-Sn-Ti ternary system has been studied via experiments and thermodynamic modelling. In the experimental section, the composition of the alloys was selected based on the preliminary calculations and available literature data. Metallography, scanning electron microscopy and electron probe microanalysis were employed to analyse alloy samples prepared by arc-melting after annealing at 800 °C for 760 h. Solid phase relations at 800 °C were established. In contrast to earlier reports, the CuSn₃Ti₅ phase was interpreted as a binary intermetallic compound (Sn₃Ti₅) with extended Cu solubility. In the modelling section, three binary sub-systems were critically evaluated and updated according to the new experimental data and theoretical calculations reported in literature. According to their crystal structures and homogeneity ranges, appropriate sublattice models were proposed for SnTi₃,SnTi₂,Sn₃Ti₅ and Sn₅Ti₆. A set of self-consistent thermodynamic parameters for the Cu-Sn-Ti ternary system was obtained by considering the present experimental results and reported experimental information. The calculated results compare with the available experimental data to validate the present thermodynamic assessment.     

Thermodynamic assessment of the Te-X (X = As,Si, Co) systems

Thermodynamic assessment of the Te-X (X = As, Si, Co) binary systems has been carried out using the CALculation of Phase Diagrams (CALPHAD) method. The thermodynamic parameters were evaluated based on the available experimental data in the literature. Thermodynamic models were constructed for all the phases in the three systems. The substitutional solution model was employed to describe the thermodynamic properties of solution phases including liquid, (Te), (As), (Si), and (Co). Due to the existence of associate species Si₂Te₃ and the difficulty of describing the liquid phase by the substitutional solution model, the associate model was adopted to describe the liquid phase in the Te-Si system. The intermetallic compounds in the Te-As and Te-Si systems were treated as stoichiometric phases and the compounds βCoTe and γCoTe₂ in the Te-Co system were modeled by using sublattice model due to the homogeneity range. A set of self-consistent thermodynamic parameters for the Te-X (X = As, Si, Co) systems was obtained. Comparisons between the calculated results and experimental data available in the literature show that almost all the reliable experimental information can be satisfactorily accounted for by the present modeling.     

On Parallel Integer Merging

The problem of merging two sorted arrays A = (a₁, a₂, ..., a_n1) and B = (b₁, b₂, ..., b_n2) is considered. For input elements that are drawn from a domain of integers [1...s] we present an algorithm that runs in O(log log log s) time using n/log log log s CREW PRAM processors (optimal speed-up) and O(ns^ε) space, where n = n₁ + n₂. For input elements that are drawn from a domain of integers [1...n] we present a second algorithm that runs in O(α(n)) time (where α(n) is the inverse of Ackermann′s function) using n/α(n) CREW PRAM processors and linear space. This second algorithm is non-uniform; however, it can be made uniform at a price of a certain loss of speed, or by using a CRCW PRAM.     

Thermodynamic modeling of B–Mo–Nb ternary system with modified B–Nb description

Based on a critical evaluation of the literature data, the B–Mo–Nb ternary system has been assessed by means of the CALPHAD (CALculation of PHAse Diagrams) technique. There is no ternary compound reported in this system. According to their crystal structures, the substitutional solution is adopted to model the ternary liquid, and the sublattice models are used to describe the Mo₂B, αMoB, (Mo,Nb)B, (Mo,Nb)B₂, Mo₂B₅, MoB₄, Nb₃B₂, Nb₅B₆, Nb₃B₄, Nb₂B₃, (Mo,Nb) and (B) phases. The modeling covers the whole composition and temperature ranges. Due to the same crystal structure, the sublattice model of the NbB phase in the B–Nb binary system is adjusted as (Me,Va)_0.5(B,Va)_0.5, in order to be consistent with the βMoB phase in the B–Mo binary system. The obtained thermodynamic parameters of the B–Mo–Nb system are demonstrated to be self-consistent and reasonable by adequately comparing the calculated and reported thermodynamic information and phase diagram. Based on the present thermodynamic parameters, the liquidus projection and reaction scheme of B–Mo–Nb system are presented.     

Thermodynamic assessment of the Ce-Pt system

The thermodynamic assessment of the Ce-Pt binary system has been carried out by using the CALPHAD (Calculation of Phase Diagrams) method based on the available literature data including the phase equilibria and thermodynamic properties. The Gibbs free energies of the liquid, bcc_A2 and fcc_A1 phases were described by the subregular solution model with the Redlich-Kister formula, and those of the intermetallic compounds (Ce₇Pt₃, Ce₃Pt₂, Ce₅Pt₄, CePt, Ce₃Pt₄, CePt_x and CePt₅) in the Ce-Pt binary system were described by the sublattice model, and two intermetallic compounds (Ce₅Pt₃,CePt₂) in the Ce-Pt binary system were described by the two-sublattice model. A consistent set of thermodynamic parameters leading to a reasonable agreement between the calculated results and literature data was obtained.     

Suboptimal robust synthesis for MIMO plant under coprime factor perturbations

Suboptimal robust synthesis for MIMO nominal system under coprime factor perturbations is considered in classical and non-classical statements. In the classical statement, weights of perturbations and upper bound on magnitude bounded exogenous disturbance are assumed to be known to controller designer. Suboptimal synthesis within ε tolerance is reduced to the solution of log₂(1/ε) standard mixed sensitivity problems of ℓ₁ optimization. In the non-classical statement, the upper bounds on perturbations and exogenous disturbance are to be estimated from measurement data and suboptimal synthesis is reduced to the solution of 1/ε mixed sensitivity problems.     

Thermodynamic modeling of the Sn–V binary system

The Sn–V binary system was thermodynamically modeled using the CALPHAD approach combined with first-principles calculations. The predicted Gibbs free energy of the end-members in sublattice models of “ V ₃Sn” phase by the first-principles calculations was used to describe the lattice stabilities. A set of thermodynamic parameters for the Sn–V system was obtained using the PanOptimizer program in Pandat software. The calculated phase diagram and thermodynamic properties agree well with the reported experimental data.     

An accurate and reliable solution procedure for the EOQ model with deteriorating items under supplier credits linked to the ordering quantity

The inventory problem consists of two parts: (1) the modelling and (2) the solution procedure. The modelling can provide insight to solve the inventory problem and the solution procedure involves the implementation of the inventory model. Basically, the modelling and the solution procedure to the inventory problem are equally important. Chang et al.’s inventory model [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996] is correct and interesting. However, they ignore the explorations of the functional behaviors of the annual total relevant cost to locate the optimal solutions such that proofs of their solution procedures are not perfect from the viewpoint of logic. The main purposes of this paper are to provide accurate and reliable solution procedures to improve [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996].