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The EdResearch for Action "Overview Series" summarizes the research on key topics to provide K-12 education decision makers and advocates with an evidence base to ground discussions about how to best serve students. This overview brief discusses the central question: What is the evidence on strategies to build a sense of school connectedness and engagement among all students? Key insights are arranged in the following sections: (1) breaking down the issue; (2) evidence-based practices; and (3) practices to avoid. [The BARR Center and Network for College Success are additional collaborators for this report.]

We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius $l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved., Comment: arXiv admin note: substantial text overlap with arXiv:2107.07236

Motivated by the study of the non-parametric area $\mathcal A$ of the graph of the vortex map $u$ (a two-codimensional singular surface in $\mathbb R^4$) over the disc $\Omega \subset \mathbb R^2$ of radius $l$, we perform a careful analysis of the singular part of the relaxation of $\mathcal A$ computed at $u$. The precise description is given in terms of a area-minimizing surface in a vertical copy of $\mathbb R^3 \subset \mathbb R^4$, which is a sort of ``catenoid'' containing a segment corresponding to a radius of $\Omega$. The problem involves an area-minimization with a free boundary part; several boundary regularity properties of the minimizer are inspected., Comment: arXiv admin note: substantial text overlap with arXiv:2107.07236

Finding the solution to linear systems is at the heart of many applications in science and technology. Over the years a number of algorithms have been proposed to solve this problem on a digital quantum device, yet most of these are too demanding to be applied to the current noisy hardware. In this work, an original algorithmic procedure to solve the Quantum Linear System Problem (QLSP) is presented, which combines ideas from Variational Quantum Algorithms (VQA) and the framework of classical shadows. The result is the Shadow Quantum Linear Solver (SQLS), a quantum algorithm solving the QLSP avoiding the need for large controlled unitaries, requiring a number of qubits that is logarithmic in the system size. In particular, our heuristics show an exponential advantage of the SQLS in circuit execution per cost function evaluation when compared to other notorious variational approaches to solving linear systems of equations. We test the convergence of the SQLS on a number of linear systems, and results highlight how the theoretical bounds on the number of resources used by the SQLS are conservative. Finally, we apply this algorithm to a physical problem of practical relevance, by leveraging decomposition theorems from linear algebra to solve the discretized Laplace Equation in a 2D grid for the first time using a hybrid quantum algorithm.

We develop an optimized hardware strategy to mitigate errors in a noisy qubit. Our scheme builds on the physical principle of error filtration and exploits auxiliary qubits. Both signal and ancillas are subject to local noise, yet constructive interference (and in some cases post-selection) allows us to reduce the level of noise in the signal qubit. Seeking for the optimal unitary that makes the qubits interfere in the most effective way, we start with a set of universal gates and proceed by optimizing suitable functionals by gradient descent or stochastic approximation. We benchmark our approach against a number of figure of merits that correspond to different applications, including entanglement fidelity, quantum Fisher information (for applications in quantum sensing), and CHSH value (for applications in tests of non-locality and quantum cryptography), with one, two, and three ancillary qubits. With one and two ancillas we also provide explicit expressions from an ansatz for an optimal unitary., Comment: Comments are wellcome

We discuss the bicrossproduct structure of the quantum group $\varrho$-Poincar\'e and of the dual quantum universal enveloping algebra, expanding the construction to general Lie algebra-type deformations of Poincar\'e coming from classical $r$-matrices. We review the relation between different bases of the quantum universal enveloping algebra of $\varrho$-Poincar\'e and noncommutative $\star$-products defined on the $\varrho$-Minkowski spacetime, analysing some of their relevant features. Furthermore, we comment on the role of physical bases and introduce the $\varrho$-Poincar\'e quantum Lie algebra., Comment: License changed to CC-BY

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in \cite{DLSVG}. We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes $|\log \varepsilon|^a$, $a\geq 1$, where $\varepsilon$ denotes the linear size of the process zone near the defects. Further, we remove the \emph{a priori} restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for $\mathbb S^1$-valued $SBV^p$ functions, approximated through functions with essentially closed jump set, in the strong $BV$ norm.

The degree of experimentally attainable nonlocality, as gauged by the amount of loophole-free violation of Bell inequalities, remains severely limited due to inefficient detectors. We address an experimentally motivated question: Which quantum strategies attain the maximal loophole-free nonlocality in the presence of inefficient detectors? For any Bell inequality and any specification of detection efficiencies, the optimal strategies are those that maximally violate a tilted version of the Bell inequality in ideal conditions. In the simplest scenario, we demonstrate that the quantum strategies that maximally violate the tilted versions of Clauser-Horne-Shimony-Holt inequality are unique up to local isometries. However, self-testing via the standard sum of squares decomposition method turns out to be analytically intractable since even high levels of the Navascu\'es--Pironio--Ac\'in hierarchy are insufficient to saturate the maximum quantum violation of these inequalities. Instead, we utilize a novel Jordan's lemma-based proof technique to obtain robust analytical self-testing statements for the entire family of tilted-Bell inequalities. These results allow us to unveil intriguing aspects of the effect of inefficient detectors and the complexity of characterizing the set of quantum correlations, in the simplest Bell scenario., Comment: First draft, comments are welcome

This paper studies the problem of action model learning with full observability. Following the learning by search paradigm by Mitchell, we develop a theory for action model learning based on version spaces that interprets the task as search for hypothesis that are consistent with the learning examples. Our theoretical findings are instantiated in an online algorithm that maintains a compact representation of all solutions of the problem. Among these range of solutions, we bring attention to actions models approximating the actual transition system from below (sound models) and from above (complete models). We show how to manipulate the output of our learning algorithm to build deterministic and non-deterministic formulations of the sound and complete models and prove that, given enough examples, both formulations converge into the very same true model. Our experiments reveal their usefulness over a range of planning domains.

Securitization is a financial process where the cash flows of income-generating assets are sold to institutional investors as securities, liquidating illiquid assets. This practice presents persistent challenges due to the absence of a comprehensive mathematical framework for structuring asset-backed securities. While existing literature provides technical analysis of credit risk modeling, there remains a need for a definitive framework detailing the allocation of the inbound cash flows to the outbound positions. To fill this gap, we introduce the PEAL Method: a 10-step mathematical framework to streamline the securitization structuring across all time periods. The PEAL Method offers a rigorous and versatile approach, allowing practitioners to structure various types of securitizations, including those with complex vertical positions. By employing standardized equations, it facilitates the delineation of payment priorities and enhances risk characterization for both the asset and the liability sides throughout the securitization life cycle. In addition to its technical contributions, the PEAL Method aims to elevate industry standards by addressing longstanding challenges in securitization. By providing detailed information to investors and enabling transparent risk profile comparisons, it promotes market transparency and enables stronger regulatory oversight. In summary, the PEAL Method represents a significant advancement in securitization literature, offering a standardized framework for precision and efficiency in structuring transactions. Its adoption has the potential to drive innovation and enhance risk management practices in the securitization market.

Powerful domain-independent planners have been developed to solve various types of planning problems. These planners often require a model of the acting agent's actions, given in some planning domain description language. Manually designing such an action model is a notoriously challenging task. An alternative is to automatically learn action models from observation. Such an action model is called safe if every plan created with it is consistent with the real, unknown action model. Algorithms for learning such safe action models exist, yet they cannot handle domains with conditional or universal effects, which are common constructs in many planning problems. We prove that learning non-trivial safe action models with conditional effects may require an exponential number of samples. Then, we identify reasonable assumptions under which such learning is tractable and propose SAM Learning of Conditional Effects (Conditional-SAM), the first algorithm capable of doing so. We analyze Conditional-SAM theoretically and evaluate it experimentally. Our results show that the action models learned by Conditional-SAM can be used to solve perfectly most of the test set problems in most of the experimented domains.

This work introduces a novel NISQ-friendly procedure for estimating spectral gaps in quantum systems. By leveraging Adiabatic Thermalization, we are able to create the Spectral Gap Superposition state, a newly defined quantum state exhibiting observable fluctuations in time that allow for the accurate estimation of any energy gap. Our method is tested by estimating the energy gap between the ground and the first excited state for the 1D and 2D Ising model, the Hydrogen molecule H2 and Helium molecule He2. Despite limiting our circuit design to have at most 40 Trotter steps, our numerical experiments of both noiseless and noisy devices for the presented systems give relative errors in the order of $10^{-2}$ and $10^{-1}$. Further experiments on the IonQ Aria device lead to spectral gap estimations with a relative error of $10^{-2}$ for a 4-site Ising chain, demonstrating the validity of the procedure for NISQ devices and charting a path towards a new way of calculating energy gaps.

The impact of formation flying on interferometry is growing over the years for the potential performance it could offer. However, it is still an open field, and many studies are still required. This article presents the basic principles behind interferometry focusing first on a single array and secondly on a formation of satellites. A sensitivity analysis is carried out to evaluate how the performance of the interferometry is affected by an error in the relative position in the formation geometry. This is estimated by computing the loss of the performance in terms of percentage deviation due to a non-nominal relative trajectory, including two-dimensional errors and defining a payload index. The main goal of this study is to estimate whether some errors in the relative state are more impacting than others. The final objective is to compute the link between a position error and a specific loss of performance, to foresee the origin of the the error. Furthermore, a dynamical model is developed to describe the relative motion in the Low Earth Orbit environment, considering both the unperturbed and the J2 and drag contributions. A Proportional, Integral and Derivative controller is implemented for the position control of a multiple satellite formation flying, considering a low thrust control profile. The Formation Flying L-band Aperture Synthesis study is taken as the case scenario, analysing both nominal and non-nominal configurations. This study serves as a starting point for the development of a combined tool to assess the performance of the interferometry and the control on the relative state for future remote sensing studies involving relative motion.

Given any $\Gamma=\gamma(\mathbb{S}^1)\subset\mathbb{R}^2$, image of a Lipschitz curve $\gamma:\mathbb{S}^1\rightarrow \mathbb{R}^2$, not necessarily injective, we provide an explicit formula for computing the value of \[ \mathcal A(\gamma):=\inf\left\{\left. \int_{B_1(0)}|\mathrm{det}(\nabla u)| \mathrm{d} x \ \right| \ u=\gamma \text{ on }\mathbb{S}^1\right\}, \] where the infimum is evaluated among all Lipschitz maps $u:B_1(0)\rightarrow \mathbb{R}^2$ having boundary datum $\gamma$. This coincides with the area of a minimal disk spanning $\Gamma$, i.e., a solution of the Plateau problem of disk type for the oriented contour $\Gamma$. The novelty of the results relies in the fact that we do not assume the curve $\gamma$ to be injective and our formula allows for any kind of self-intersections

Starting from the Hamiltonian representation of the dynamics in \cite{rosengren2015chaos,colombo2019long}, this work proposes an innovative procedure to design fully-analytical maneuvers for post-mission disposal of HEOs satellites, exploiting the third-body perturbations. The Hamiltonian representation has been selected to include the external perturbing effects and to obtain a phase space representation. Notably, the orbit evolution can be described through the variation of double-averaged orbital elements over the orbital periods of the spacecraft and the perturbing bodies around the central planet. this work conveys a two-dimensional Hamiltonian representation under the third-body perturbations and the central planet's oblateness. The effect of solar radiation pressure has been neglected in this analysis.

We investigate two possible techniques to authenticate the q-digest data structure, along with a worst-case study of the computational complexity both in time and space of the proposed solutions, and considerations on the feasibility of the presented approaches in real-world scenarios. We conclude the discussion by presenting some considerations on the information complexity of the queries in the two proposed approaches, and by presenting some interesting ideas that could be the subject of future studies on the topic., Comment: 14 pages, LaTeX; corrected layout problem in figure 4 caused by migration from LuaLaTex to pdfLaTex