Dynamic properties of the Fisher–Kolmogorov–Petrovskii–Piscounov equation with the deviation of the spatial variable

We consider the problem of the density wave propagation of a logistic equation with the deviation of the spatial variable and diffusion (the Fisher–Kolmogorov equation with the deviation of the spatial variable). The Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of the wave propagation shows that, for a fairly small spatial deviation, this equation has a solution similar to that the classical Fisher–Kolmogorov equation. An increase in this spatial deviation leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase in the spatial deviation leads to the destruction of the traveling wave. This is expressed in the fact that undamped spatiotemporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is large enough we observe intensive spatiotemporal fluctuations in the whole area of wave propagation.     

Refinement of the thermodynamic modeling of the Nb–Ni system

The Nb–Ni system is reassessed on the basis of a critical literature review involving recent experimental data. These newly published experimental data include the phase relation associated with the NbNi₈ phase, phase transition temperatures resulting from selected alloys, all invariant reaction temperatures, and enthalpies of mixing of liquid, as well as the crystallographic data on the μ$\mu$ (Nb₇Ni₆) phase. A consistent thermodynamic data set for the Nb–Ni system is obtained by optimization of the selected experimental values. The calculated phase diagram, crystallographic properties and thermodynamic properties agree reasonably with the experimental data. Noticeable improvements have been made, compared with the previous thermodynamic optimizations.     

Thermodynamic evaluation of the thixoformability of Al–Si–Cu alloys

The aim of this work was to evaluate the thixoformability of Al-(2 to 7 wt%) Si–Cu alloys by differential thermal analysis (DTA), differential scanning calorimetry (DSC) and CALPHAD simulation. Thermoanalytical data were generated for exothermic (rheocasting) and endothermic (thixoforming) cycles under different kinetic conditions (heating and cooling rates of 5, 10, 15, 20 and 25 °C/min). The findings indicate that the SSM critical temperatures and liquid fractions are directly affected by the kinetic conditions, chemical composition and heat-flow direction and that the measured values of these critical temperatures and liquid fractions vary according to the thermodynamic evaluation technique used (Calphad simulation, DSC or DTA). The SSM working window (a) became smaller as the heating/cooling rates and Si content increased; (b) was larger for rheocasting (solidification) than for thixoforming (melting) operations; (c) was on average approximately 12 °C wider and covered a range of mass fractions approximately 0.12 greater for DSC measurements than for DTA measurements; and (d) had a low sensitivity for all the conditions analyzed, indicating the thermodynamic stability of the Al–Si–Cu system. CALPHAD simulation successfully predicted several transformations and the thermodynamic behavior of the temperatures and liquid fractions analyzed. The DTA and DSC techniques yielded discrepant results for some of the critical points investigated, such as the limits of the SSM working window. The majority of the DSC cycles were more sensitive to variations in kinetic conditions, heat-flow direction and chemical composition than the corresponding DTA cycles. Furthermore, the tertiary Al₂Cu phase transformation could not be identified in many of the DTA cycles. For these reasons, DTA should be used with caution when predicting the thermodynamic behavior of potential raw materials for SSM processing.     

Statistical moments of the solution of the random Burgers–Riemann problem

We solve Burgers’ equation with random Riemann initial conditions. The closed solution allows simple expressions for its statistical moments. Using these ideas we design an efficient algorithm to calculate the statistical moments of the solution. Our methodology is an alternative to the Monte Carlo method. The present approach does not demand a random numbers generator as does the Monte Carlo method. Computational tests are added to validate our approach.     

Experimental phase diagram study of the TiAl-rich part of the Ti–Al–Ni ternary system

Phase equilibria of the TiAl-rich part of the Ti–Al–Ni ternary system have been studied experimentally by scanning electron microscopy and electron probe micro-analysis of heat-treated alloys. Partial isothermal sections involving the liquid, β-Ti, α-Ti, α₂-Ti₃Al, γ-TiAl and τ₃-Al₃NiTi₂ phases were constructed between 1623 and 1273 K. Eight three-phase regions of the L + β + α, L + α + γ, L + β + γ, β + α + γ, L + β + τ₃, β + γ + τ₃, β + α₂ + τ₃ and α₂ + γ + τ₃ were derived. Extrapolations of these tie-triangles indicate the occurrence of three transition-type reactions; L + α = β + γ at around 1593 K, L + γ = β + τ₃ at around 1553 K, and β + γ = α + τ₃ at around 1393 K. The Ni solid solubility in the α and α₂ phase is extremely low, less than 1 at.% in all studied temperature ranges.     

Thermodynamic modeling of the Co–Fe–O system

As a part of the research project aimed at developing a thermodynamic database of the La–Sr–Co–Fe–O system for applications in Solid Oxide Fuel Cells (SOFCs), the Co–Fe–O subsystem was thermodynamically re-modeled in the present work using the CALPHAD methodology. The solid phases were described using the Compound Energy Formalism (CEF) and the ionized liquid was modeled with the ionic two-sublattice model based on CEF. A set of self-consistent thermodynamic parameters was obtained eventually. Calculated phase diagrams and thermodynamic properties are presented and compared with experimental data. The modeling covers a temperature range from 298 K to 3000 K and oxygen partial pressure from 10⁻¹⁶ to 10² bar. A good agreement with the experimental data was shown. Improvements were made as compared to previous modeling results.     

Thermodynamic assessment of the Al–Cu–Er system

The Cu–Er binary system had been thermodynamically assessed with the CALPHAD approach. The solution phases including Liquid, Fcc and Hcp were treated as substitutional solution phases, of which the excess Gibbs energies were formulated with the Redlich–Kister polynomial function. All the binary intermetallic compounds were treated as stoichiometric phases. Combining with the thermodynamic parameters of the Al–Cu and Al–Er binary systems cited from the literature, the Al–Cu–Er ternary system was thermodynamically assessed. The calculated phase equilibria were in good agreement with the experimental data.     

Thermodynamic description of the Al–Fe–Zr system

The Fe–Zr and Al–Fe–Zr systems were critically assessed by means of the CALPHAD technique. The solution phases, liquid, face-centered cubic, body-centered cubic and hexagonal close-packed, were described by the substitutional solution model. The compounds with homogeneity ranges, hex.- Fe₂Zr, Fe₂Zr, FeZr₂ and FeZr₃ in the Fe–Zr system, were described by the two-sublattice model in formulas such as hex.- Fe₂(Fe,Zr), (Fe,Zr)₂(Fe,Zr), (Fe,Zr)Zr₂ and (Fe,Zr)(Fe,Zr)₃ respectively. The compounds Al_mZr_n except Al₂Zr in the Al–Zr system were treated as line compounds (Al,Fe)_mZr_n in the Al–Fe–Zr system. The compounds FeZr₂ and FeZr₃ in the Fe–Zr system were treated as (Al,Fe,Zr)Zr₂ and (Al,Fe,Zr)(Fe,Zr)₃ in the Al–Fe–Zr system, respectively. All compounds in the Al–Fe system and hex.- Fe₂Zr in the Fe–Zr system have no solubilities of the third components Zr or Al, respectively, in the Al–Fe–Zr system. The ternary compounds _λ1${\lambda }_1$ with C14 structure and _λ2${\lambda }_2$ with C15 structure in the Al–Fe–Zr system were treated as _λ1${\lambda }_1$- (Al,Fe,Zr)₂(Fe,Zr) with Al₂Zr in the Al–Zr system and _λ2${\lambda }_2$- (Al,Fe,Zr)₂(Fe,Zr) with Fe₂Zr in the Fe–Zr system, respectively. And the ternary compounds _τ1${\tau }_1$, _τ2${\tau }_2$ and _τ3${\tau }_3$ in the Al–Fe–Zr system were treated as (Al,Fe)₁₂Zr, Fe(Al,Zr)₂Zr₆ and Fe₇Al₆₇Zr₂₆, respectively. A set of self-consistent thermodynamic parameters of the Al–Fe–Zr system was obtained.     

On the cost–effectiveness of PRAMs

We introduce a formalism which allows to treat computer architecture as a formal optimization problem. We apply this to the design of shared memory parallel machines. While present parallel computers of this type only support the programming model of a shared memory but often process simultaneous access by several processors to the shared memory sequentially, theoretical computer science offers solutions for this problem that are provably fast and asymptotically optimal. But the constants in these constructions seemed to be too large to let them be competitive. We modify these constructions under engineering aspects and improve the price/performance ratio by roughly a factor of 6. The resulting machine has surprisingly good price/performance ratio even if compared with distributed memory machines. For almost all access patterns of all processors into the shared memory, access is as fast as the access of only a single processor. Received: 29 June 1993 / 22 June 1999     

Optimality proof of the Kise–Ibaraki–Mine algorithm

Kise, Ibaraki and Mine (Operations Research 26:121–126, 1978) give an O(n ²) time algorithm to find an optimal schedule for the single-machine number of late jobs problem with agreeable job release dates and due dates. Li, Chen and Tang (Operations Research 58:508–509, 2010) point out that their proof of optimality for their algorithm is incorrect by giving a counter-example. In this paper we give a correct proof of optimality for their algorithm.     

Computation of the Interval of Stability of Runge–Kutta–Nyström Methods

We present a Mathematica package to compute the interval of stability of Runge–Kutta–Nystrom methods fory">=f(t,y).