Exploring students’ procedural flexibility in three countries

Background In this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies. Results Our results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees. Conclusions Although it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning.

Simple and Robust Algorithm for Multiphase Equilibrium Computations at Temperature and Volume Specifications

Summary Two-phase and three-phase equilibria are frequently encountered in a variety of industrial processes, such as carbon dioxide (CO2) injection for enhanced oil recovery in oil reservoirs, multiphase separation in surface separators, and multiphase flow in wellbores and pipelines. Simulation and engineering design of these processes using isothermal/isochoric (VT) multiphase equilibrium algorithms are sometimes more convenient than that using the conventional isothermal/isobaric (PT) algorithms. This work develops a robust algorithm for VT multiphase equilibrium calculations using a nested approach. The proposed algorithm is simple because a robust PT multiphase equilibrium algorithm is used in the inner loop without any further modifications, while an effective equation-solving method (i.e., Brent’s method; Brent 1971) is applied in the outer loop to solve the pressure corresponding to a given volume/temperature specification. The robustness of the VT algorithm is safeguarded by using a highly efficient trust-region-method-based PT algorithm. We demonstrate the good performance of the newly developed algorithm by applying it to calculate the isochores of fluid mixtures that exhibit both two-phase and three-phaseequilibria.

Analog signal processing through space-time digital metasurfaces

In the quest to realize analog signal processing using subwavelength metasurfaces, in this paper, we present the first demonstration of programmable time-modulated metasurface processors based on the key properties of spatial Fourier transformation. Exploiting space-time coding strategy enables local, independent, and real-time engineering of not only amplitude but also phase profile of the contributing reflective digital meta-atoms at both central and harmonic frequencies. Several illustrative examples are demonstrated to show that the proposed multifunctional calculus metasurface is capable of implementing a large class of useful mathematical operators, including 1st- and 2nd-order spatial differentiation, 1st-order spatial integration, and integro-differential equation solving accompanied by frequency conversions. Unlike the recent proposals based on the Green’s function (GF) method, the designed time-modulated signal processor effectively operates for input signals containing wide spatial frequency bandwidths with an acceptable gain level. Proof-of-principle simulations are also reported to demonstrate the successful realization of image processing functions like edge detection. This time-varying wave-based computing system can set the direction for future developments of programmable metasurfaces with highly promising applications in ultrafast equation solving, real-time and continuous signal processing, and imaging.

Practical Quantum Computing

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.