The Relative Efficiency of Educational Systems: A Cross Country Prescriptive Analysis

How does a country achieve the most efficient education system possible? We examine the efficiency of educational systems across the world using internationally comparable performance of secondary school pupils. We use OECD aggregate panel data on 39 countries across 17 years to model the most efficient combination of teacher salaries, class size and capital resources to secure the maximum level of students’ achievement. In the face of constrained education budgets prescriptive policy evaluations are provided to suggest how countries could make a significant improvement to their educational outcomes by adjusting their educational resource inputs. We also ask if there is a trade-off between the efficiency and equity of educational provision.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

The efficiency index: which education systems deliver the best value for money?

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A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed for weighted Bergman spaces of analytic functions in the upper half-plane. The result is applied to show that the invariant subspaces of the shift operator on the standard Bergman space of the unit disk can be identified with those of a convolution Volterra operator on the space L2 (R+, (1/t)dt).Plan Nacional I+D (Ministerio de Ciencia y Tecnología)Junta de AndalucíaSecretaría de Estado de Educación y Universidade

Rate region boundary of the SISO Z-interference channel with improper signaling

This paper provides a complete characterization of the boundary of an achievable rate region, called the Pareto boundary, of the single-antenna Z interference channel (Z-IC), when interference is treated as noise and users transmit complex Gaussian signals that are allowed to be improper. By considering the augmented complex formulation, we derive a necessary and sufficient condition for improper signaling to be optimal. This condition is stated as a threshold on the interference channel coefficient, which is a function of the interfered user rate and which allows insightful interpretations into the behavior of the achievable rates in terms of the circularity coefficient (i.e., degree of impropriety). Furthermore, the optimal circularity coefficient is provided in closed form. The simplicity of the obtained characterization permits interesting insights into when and how improper signaling outperforms proper signaling in the single-antenna Z-IC. We also provide an in-depth discussion on the optimal strategies and the properties of the Pareto boundary.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under grant SCHR 1384/6-1. The work of I. Santamaría was supported by the Ministerio de Economía y Competitividad (MINECO), Spain, under projects RACHEL (TEC2013-47141-C4-3-R) and CARMEN (TEC2016-75067-C4-4-R)

Analysis of maximally improper signaling schemes for underlay cognitive radio networks

In this paper, the impact of improper Gaussian signaling is studied for an underlay cognitive radio (CR) scenario comprised of a primary user (PU), which has a rate constraint, and a secondary user (SU), both single-antenna. We first derive expressions for the achievable rate of the SU when it transmits proper and maximally improper Gaussian signals (assuming that the SU is solely limited by the CR constraint). These expressions depend on the channel gains to and from the SU through a single variable. Thereby, we observe that improper signaling is beneficial whenever the SU rate is below a threshold, which depends on the signal-to-noise ratio (SNR) and rate requirement of the PU. Furthermore, we provide bounds on the achievable gain that also depend only on the PU parameters. Then, the achievable rate is studied from a statistical viewpoint by deriving its cumulative distribution function considering a constant received SNR at the PU. In addition, we specialize this expression for the Z interference channel, for which the expected achievable rate is also derived. Numerical examples illustrate our claims and show that the SU may significantly benefit from using improper signaling.C. Lameiro and I. Santamaría have received funding from the Spanish Government (MICINN) under projects CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS), TEC2013-47141-C4-3-R (RACHEL) and FPU Grant AP2010-2189; and also from the Deutscher Akademischer Austauschdienst (DAAD) under its programm “Research grants for doctoral candidates and young academics and scientists”. Peter Schreier receives financial support from the Alfried Krupp von Bohlen und Halbach foundation, under its program “Return of German scientists from abroad”

Benefits of improper signaling for underlay cognitive radio

In this letter we study the potential benefits of improper signaling for a secondary user (SU) in underlay cognitive radio networks. We consider a basic yet illustrative scenario in which the primary user (PU) always transmit proper Gaussian signals and has a minimum rate constraint. After parameterizing the SU transmit signal in terms of its power and circularity coefficient (which measures the degree of impropriety), we prove that the SU improves its rate by transmitting improper signals only when the ratio of the squared modulus between the SU-PU interference link and the SU direct link exceeds a given threshold. As a by-product of this analysis, we obtain the optimal circularity coefficient that must be used by the SU depending on its power budget. Some simulation results show that the SU benefits from the transmission of improper signals especially when the PU is not highly loaded.C. Lameiro and I. Santamaría have received funding from the Spanish Government (MICINN) under projects CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS), TEC2013-47141-C4-3-R (RACHEL) and FPU Grant AP2010-2189; and also from the Deutscher Akademischer Austauschdienst (DAAD) under its programm ”Research grants for doctoral candidates and young academics and scientists”. P. Schreier receives financial support from the Alfried Krupp von Bohlen und Halbach foundation, under its program ”Return of German scientists from abroad”

Benefits of improper signaling for overlay cognitive radio

This paper considers improper Gaussian signaling (IGS) in an overlay cognitive radio scenario. We follow a protocol in which the secondary user (SU) uses part of its power to relay the message for the primary user (PU) and consider a simple yet illustrative 2-user scenario. We analyze two communication schemes depending on whether or not the PU cooperates with the SU and derive closed-form expressions for the optimal transmission parameters that maximize the SU rate while ensuring a specified minimum performance of the PU. Our numerical results show that IGS may significantly outperform proper signaling and that, interestingly, the cooperative approach provides negligible performance gains over its non-cooperative counterpart.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under grants SCHR 1384/6-1 and LA 4107/1-1. The work of I. Santamaría was supported by the Ministerio de Economía y Competitividad (MINECO) and AEI/FEDER funds of the UE, Spain, under project CARMEN (TEC2016-75067-C4-4-R)

Improper Gaussian signaling for multiple-access channels in underlay cognitive radio

This paper considers an unlicensed multiple-access channel (MAC) that coexists with a licensed point-to-point user, following the underlay cognitive radio paradigm. We assume that every transceiver except the secondary base station has one antenna and that the primary user (PU) is protected by a minimum rate constraint. In contrast to the conventional assumption of proper Gaussian signaling, we allow the secondary users to transmit improper Gaussian signals, which are correlated with their complex conjugate. When the secondary base station performs zero-forcing, we show that improper signaling is optimal if the sum of the interference channel gains (in an equivalent canonical model) is above a certain threshold. Additionally, we derive an efficient algorithm to compute the transmission parameters that attain the rate region boundary for this scenario. The proposed algorithm exploits a single-user representation of the secondary MAC along with new results on the optimality of improper signaling in the single-user case when the PU is corrupted by an improper noise.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under grants SCHR 1384/6-1 and LA 4107/1-1. The work of I. Santamaría was supported by the Ministerio de Economía y Competitividad (MINECO) and AEI/FEDER funds of the UE, Spain, under projects RACHEL (TEC2013-47141-C4-3-R) and CARMEN (TEC2016-75067-C4-4-R)

Performance analysis of maximally improper signaling for multiple-antenna systems

The transmission of improper Gaussian signals, instead of the conventional proper ones, has been shown to improve the performance in interference-limited networks. In this work we analyze the performance of a multiple-antenna user that transmits maximally improper signals and whose transmit covariance matrix satisfies a set of constraints that limit the harmfulness of the interference caused by this user. As opposed to the single-antenna case, there are different possible improper spatial signatures, which provide different performance. We first obtain new results for maximally improper random vectors based on majorization theory. We then apply these results to derive the improper spatial signatures that either maximize or minimize the performance. Numerical examples show that the performance difference between these two extreme cases can be surprisingly large.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under grants SCHR 1384/6-1 and LA 4107/1-1. The work of I. Santamaría was supported by the Ministerio de Economía y Competitividad (MINECO) and AEI/FEDER funds of the UE, Spain, under projects RACHEL (TEC2013-47141-C4-3-R) and CARMEN (TEC2016-75067-C4-4-R)

Improper Gaussian signaling for the two-user broadcast channel treating interference as noise

Improper Gaussian signaling (IGS) has been shown to enlarge the rate region achievable by conventional proper Gaussian signaling (PGS) schemes in several interference-limited multiuser networks. In this work, we consider the 2-user broadcast channel (BC) when treating interference as noise “TIN” at every receiver. For this scenario, we derive a closed-form characterization of the rate region boundary with IGS. The Pareto-optimal points are achieved when at least one of the users employs maximally improper (rectilinear) signals. Differently from other interference-limited networks, our results show that IGS always outperforms PGS for the 2-user BC with TIN. Furthermore, IGS also enlarges the PGS rate region with time-sharing for this scenario.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under grants SCHR 1384/6-1 and LA 4107/1-1. The work of I. Santamar´ıa was supported by the Ministerio de Economía y Competitividad (MINECO) and AEI/FEDER funds of the UE, Spain, under project CARMEN (TEC2016-75067-C4-4-R)