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We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We consider the case when simple bounds (constraints) are also imposed on the curvature along the curve. The curvature at the endpoints may or may not be specified. We prove via optimal control theory that the optimal curve is some concatenation of Euler spiral arcs, circular arcs, and straight line segments. When the curvature is not constrained (or when the curvature constraint does not become active), an optimal curve is only made up of a concatenation of Euler spiral arcs, unless the oriented endpoints lie in a line segment or a circular arc of the prescribed length, in which case the whole curve is either a straight line segment or a circular arc segment, respectively. We propose numerical methods and illustrate these methods and the results by means of three example problems of finding such curves., Comment: 23 pages

This paper introduces in detail a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained problems and problems with non-binary variables. The algorithm returns optimal and near-optimal solutions from repeated preparation and measurement of an amplified state. The amplified state is prepared via repeated application of two unitaries; one which phase-shifts solution states dependent on objective function values, and the other which mixes phase-shifted probability amplitudes via a continuous-time quantum walk (CTQW) on a problem-specific mixing graph. The general interference process responsible for amplifying optimal solutions is derived in part from statistical analysis of objective function values as distributed over the mixing graph. The algorithm's versatility is demonstrated through its application to various problems: weighted maxcut, k-means clustering, quadratic assignment, maximum independent set and capacitated facility location. In all cases, efficient circuit implementations of the CTQWs are discussed. A penalty function approach for constrained problems is also introduced, including a method for optimising the penalty function. For each of the considered problems, the algorithm's performance is simulated for a randomly generated problem instance, and in each case, the amplified state produces a globally optimal solution within a small number of iterations., Comment: 60 pages, 37 figures

We consider the problem of finding curves of minimum pointwise-maximum curvature, i.e., curves of minimax curvature, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We reformulate the problem in terms of optimal control and use the maximum principle, as well as some geometrical arguments, to produce a classification of the types of solutions. Using the classification, we devise a numerical method which reduces the infinite-dimensional optimization problem to a finite-dimensional problem with just six variables. The solution types, together with some further observations on optimality, are illustrated via numerical examples.

The quality of the classroom environment, including ventilation, air quality and thermal conditions, has an important impact on children's health and academic achievements. The use of portable HEPA filter air cleaners is widely suggested as a strategy to mitigate exposure to particulate matter and airborne viruses. However, there is a need to quantify the relative benefits of such devices including the impacts on energy use. We present a simple coupled dynamic thermal and air quality model and apply it to naturally ventilated classrooms, representative of modern and Victorian era construction. We consider the addition of HEPA filters with, and without, reduced opening of windows, and explore concentrations of carbon dioxide (\co), \PM, airborne viral RNA, classroom temperature and energy use. Results indicate the addition of HEPA filters was predicted to reduce \PM~ by 40--60\% and viral RNA by 30--50\% depending on the classroom design and window opening behaviour. The energy cost of running HEPA filters is likely to be only 1\%--2\% of the classroom heating costs. In scenarios when HEPA filters were on and window opening was reduced (to account for the additional clean air delivery rate of the filters), the heating cost was predicted to be reduced by as much as -13\%, and these maximum reductions grew to -46\% in wintertime simulations. In these scenarios the HEPA filters result in a notable reduction in \PM~and viral RNA, but the \co\ concentration is significantly higher. The model provides a mechanism for exploring the relative impact of ventilation and air cleaning strategies on both exposures and energy costs, enabling an understanding of where trade-offs lie., Comment: 22 pages, 4 figures

This paper introduces a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained and non-binary problems. The algorithm leverages an engineered interference process achieved through repeated application of two unitaries; one inducing phase-shifts dependent on objective function values, and the other mixing phase-shifted probability amplitudes via a continuous-time quantum walk (CTQW) on a problem-specific graph. The algorithm's versatility is demonstrated through its application to various problems, namely those for which solutions are characterised by either a vector of binary variables, a vector of non-binary integer variables, or permutations (a vector of integer variables without repetition). An efficient quantum circuit implementation of the CTQW for each of these problem types is also discussed. A penalty function approach for constrained problems is also introduced, including a method for optimising the penalty function. The algorithm's performance is demonstrated through numerical simulation for randomly generated instances of the following problems (and problem sizes): weighted maxcut (18 vertices), maximum independent set (18 vertices), k-means clustering (12 datapoints, 3 clusters), capacitated facility location (12 customers, 3 facility locations), and the quadratic assignment problem (9 locations). For each problem instance, the algorithm finds a globally optimal solution with a small number of iterations., Comment: 11 pages, 15 figures, for inclusion in IEEE conference QCE24

As is well-known, numerical experiments show that Napoleon's Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere $S^2$. Spherical triangles for which an extension of Napoleon's Theorem holds are called ``Napoleonic'', and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon's Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations.

Field emission or dark current from RF electron guns is problematic for accelerators for several reasons, and can be more difficult to predict, measure and understand, compared to the main beam. The work presented here is a study of the dark current emitted from the CLARA 3GHz 2.5 cell electron gun with exchangeable photocathode, which is a type of gun used for single pass high quality low emittance beams for uses such as Free Electron Lasers. The main purpose of these studies was to identify sources of dark current emission within the gun, by examining the dark current distribution and patterns on scintillating screens. It is shown that the features of the patterns can be reproduced qualitatively, and to some extent quantitatively, by finite element simulations. Thus, certain surfaces within the gun and cathode apparatus which contribute most significantly to problematic dark current can be identified, and further development and preparation of the guns and their cathodes can be informed to minimise dark current.

Let $K$ and $L$ be two disjoint unions of strictly convex obstacles contained within a Riemannian manifold with boundary $S$ of dimension $m\geq 2$. The sets of travelling times $\mathcal{T}_K$ and $\mathcal{T}_L$ of $K$ and $L$, respectively, are composed of triples $(x,y,t)\in\partial S\times\partial S\times\mathbb{R}^+$ where $t$ is the length of a billiard trajectory with endpoints $x$ and $y$ that reflects elastically on $K$ (or $L$ for $(x,y,t)\in\mathcal{T}_L$). It has been shown (arXiv:2309.11141) that (under some natural curvature bounds on $S$) if $\mathcal{T}_K=\mathcal{T}_L$ and $K$ and $L$ were equivalent up to tangency then $K = L$. In this paper we remove this requirement for $K$ and $L$, and show that if $\mathcal{T}_K = \mathcal{T}_L$ then $K = L$ whenever $m\geq 3$.

Suppose that $K$ and $L$ are two disjoint unions of strictly convex obstacles with the same set of travelling times, contained in an $n$-dimensional Riemannian manifold $M$ (where $n\geq2$). Under some natural curvature conditions on $M$, and provided that no geodesic intersects more than two components in $K$ or $L$, we show that $K = L$.

Noakes and Stoyanov (2021) introduced a method of recovering strictly convex planar obstacles from their set of travelling times. We provide an extension of this construction for obstacles on Riemannian surfaces under some general curvature conditions. It is required that no smooth geodesic intersect more than two obstacles.

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.

The MITRE ATT&CK framework, a comprehensive knowledge base of adversary tactics and techniques, has been widely adopted by the cybersecurity industry as well as by academic researchers. Its broad range of industry applications include threat intelligence, threat detection, and incident response, some of which go beyond what it was originally designed for. Despite its popularity, there is a lack of a systematic review of the applications and the research on ATT&CK. This systematization of work aims to fill this gap. To this end, it introduces the first taxonomic systematization of the research literature on ATT&CK, studies its degree of usefulness in different applications, and identifies important gaps and discrepancies in the literature to identify key directions for future work. The results of this work provide valuable insights for academics and practitioners alike, highlighting the need for more research on the practical implementation and evaluation of ATT&CK.

The European Workshop on Photocathodes for (particle) Accelerator Applications (EWPAA) brings together experts in the field of photocathode based electron sources for use in particle accelerators, with the aim of sharing their knowledge and latest research and development progress in this crucial field of particle accelerator science. The workshop is convened every other year, and is thus complementary to the P3 workshop (Photocathode Physics for Particle accelerators) run in the USA. Consequently, there is a workshop focusing on photocathodes for particle accelerator applications convened every year, either in Europe or the USA. The EWPAA 2022 is the 4th meeting in this workshop series. The event was hosted by the INFN LASA Institute in Milan in collaboration with University of Milano between September 20th and 22nd. The programme was organised with 7 working groups, with each oral contribution assigned to the most appropriate group. This report presents, summarised by the work group conveners, the innovative ideas, the challenges and the main points raised by each of the speakers., Comment: 11 pages, 1 figure Workshop website: https://agenda.infn.it/event/ewpaa2022

Learning sciences research uncovers how people learn best and has the potential to enable technology to build powerful learning opportunities. This report explores the ways in which several edtech products focus their efforts on an intended learning purpose, consult and integrate research in their design, and strive to achieve their articulated goal. The five case studies included in the report provide examples of and best practices for leveraging research at multiple phases of product development, including acquisition, establishing cross-functional teams to collaboratively leverage learning sciences research, and using research to understand the best ways to measure success.

In the face of a worldwide pandemic, educators rose to the challenge of engaging students and families remotely to not only continue learning but also to provide as much normalcy as possible at a time of chaos and uncertainty. Many educators were looking for quick, meaningful ways to learn new skills to effectively teach remotely and connect with learners from afar. The Edcamp Community by Digital Promise saw this as an opportunity to further demonstrate how the Edcamp model of professional learning paired with learning sciences research could quickly meet the needs of educators and create spaces for shared learning and networking. The Edcamp team created the Edcamp: Design for Learning (DfL) series. Supported by the Chan Zuckerberg Initiative and in partnership with Transcend, a national nonprofit organization focused on innovation in school design, we refined our model for online Edcamps to emphasize bringing together learning sciences expertise within Digital Promise and educator context and pedagogy expertise from the Edcamp Community to co-design a series of four online Edcamps. This report shares the key lessons learned from that series. [This report was supported by the Chan Zuckerberg Initiative DAF, an advised fund of Silicon Valley Community Foundation.]

Tidal features in the outskirts of galaxies yield unique information about their past interactions and are a key prediction of the hierarchical structure formation paradigm. The Vera C. Rubin Observatory is poised to deliver deep observations for potentially of millions of objects with visible tidal features, but the inference of galaxy interaction histories from such features is not straightforward. Utilising automated techniques and human visual classification in conjunction with realistic mock images produced using the NEWHORIZON cosmological simulation, we investigate the nature, frequency and visibility of tidal features and debris across a range of environments and stellar masses. In our simulated sample, around 80 per cent of the flux in the tidal features around Milky Way or greater mass galaxies is detected at the 10-year depth of the Legacy Survey of Space and Time (30-31 mag / sq. arcsec), falling to 60 per cent assuming a shallower final depth of 29.5 mag / sq. arcsec. The fraction of total flux found in tidal features increases towards higher masses, rising to 10 per cent for the most massive objects in our sample (M*~10^{11.5} Msun). When observed at sufficient depth, such objects frequently exhibit many distinct tidal features with complex shapes. The interpretation and characterisation of such features varies significantly with image depth and object orientation, introducing significant biases in their classification. Assuming the data reduction pipeline is properly optimised, we expect the Rubin Observatory to be capable of recovering much of the flux found in the outskirts of Milky Way mass galaxies, even at intermediate redshifts (z<0.2)., Comment: 29 pages, 25 figures; accepted for publication in MNRAS following minor corrections

In our hierarchical structure-formation paradigm, the observed morphological evolution of massive galaxies -- from rotationally-supported discs to dispersion-dominated spheroids -- is largely explained via galaxy merging. However, since mergers are likely to destroy discs, and the most massive galaxies have the richest merger histories, it is surprising that any discs exist at all at the highest stellar masses. Recent theoretical work by our group has used a cosmological, hydrodynamical simulation to suggest that extremely massive (M* > 10^11.4 MSun) discs form primarily via minor mergers between spheroids and gas-rich satellites, which create new rotational stellar components and leave discs as remnants. Here, we use UV-optical and HI data of massive galaxies, from the SDSS, GALEX, DECaLS and ALFALFA surveys, to test these theoretical predictions. Observed massive discs account for ~13% of massive galaxies, in good agreement with theory (~11%). ~64% of the observed massive discs exhibit tidal features, which are likely to indicate recent minor mergers, in the deep DECaLS images (compared to ~60% in their simulated counterparts). The incidence of these features is at least four times higher than in low-mass discs, suggesting that, as predicted, minor mergers play a significant (and outsized) role in the formation of these systems. The empirical star-formation rates agree well with theoretical predictions and, for a small galaxy sample with HI detections, the HI masses and fractions are consistent with the range predicted by the simulation. The good agreement between theory and observations indicates that extremely massive discs are indeed remnants of recent minor mergers between spheroids and gas-rich satellites., Comment: Accepted in MNRAS, 9 pages, 7 figures

Today, more is known than ever about how people learn through advances in learning sciences research. It is also known, through research, that people learn differently. By incorporating learning sciences research findings in the design and development of edtech products, developers can create powerful learning experiences for the full spectrum of learners. Digital Promise Global spoke with more than 50 stakeholders across the country, representing district and school leaders, educators, classroom-based coaches, staff from education nonprofits and procurement platforms, and edtech product teams. From these conversations, three pervasive challenges stakeholders face in the current edtech industry were identified. This report discusses these challenges, and potential solutions.

A construction is given for the recovery of a disjoint union of strictly convex smooth planar obstacles from travelling-time information. The obstacles are required to be such that no Euclidean line meets more than two of them., Comment: 14 pages, 3 figures

The risk of long range, herein `airborne', infection needs to be better understood and is especially urgent during the current COVID-19 pandemic. We present a method to determine the relative risk of airborne transmission that can be readily deployed with either modelled or monitored CO$_2$ data and occupancy levels within an indoor space. For spaces regularly, or consistently, occupied by the same group of people, e.g. an open-plan office or a school classroom, we establish protocols to assess the absolute risk of airborne infection of this regular attendance at work or school. We present a methodology to easily calculate the expected number of secondary infections arising from a regular attendee becoming infectious and remaining pre/asymptomatic within these spaces. We demonstrate our model by calculating risks for both a modelled open-plan office and by using monitored data recorded within a small naturally ventilated office. In addition, by inferring ventilation rates from monitored CO$_2$ we show that estimates of airborne infection can be accurately reconstructed; thereby offering scope for more informed retrospective modelling should outbreaks occur in spaces where CO$_2$ is monitored. Our modelling suggests that regular attendance at an office for work is unlikely to significantly contribute to the pandemic but only if relatively quiet desk-based work is carried out in the presence of adequate ventilation (i.e. at least 10\,l/s/p following UK guidance), appropriate hygiene controls, distancing measures, and that all commuting presents minimal infection risk. Crucially, modelling even moderate changes to the conditions within the office, or basing estimates for the infectivity of the SARS-CoV-2 variant B1.1.7 current data, typically results in the prediction that for a single infector within the office the airborne route alone gives rises to more than one secondary infection., Comment: Submitted to Indoor Air for review: 43 pages, 6 figures

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.

Digital Promise's Marketplace Initiative, which has collaborated with district leaders across the country to develop tools and resources that support their edtech evaluation efforts, wanted to understand how tools can support education decision makers use evidence to inform procurement and adoption decisions. In 2018, Digital Promise administered a survey to further contribute to the research that exists on research and evidence use in schools and districts. Most research on this topic only considers the perspectives of superintendents and principals. This new survey, however, intended to incorporate perceptions and attitudes around research and evidence use across a more expansive list of education roles, including educators, coaches, and instructional specialists. To guide future work, the survey also sought to understand self-reported research and evidence use across Digital Promise networks to understand who needs coaching and support going forward and in what areas. Through the survey, findings uncover differences between education leaders' perceptions of the significance of research and evidence to inform program adoption and procurement. Moreover, the survey finds interesting differences in the types of research and evidence used by various education roles to inform decisions around the adoption and procurement of innovative educational programs.

Research findings suggest that instructional technology coaching may be a critical lever in closing the gap in the usage of technology, sometimes referred to as the digital use divide. In the 2017-2018 school year, we provided 50 schools in 20 school districts across five states, with a grant to support an onsite, full-time instructional technology coach (called a Dynamic Learning Project (DLP) coach). Our data shows that after one year of working with their DLP coach, teachers are using technology more frequently and in more powerful ways. DLP teachers report significant increases in using technology for both teaching content and pedagogy--in other words, teachers are using technology to support what they are teaching, as well as how they are teaching it.

In ideal MHD, the magnetic flux is advected by the plasma motion, freezing flux-surfaces into the flow. An MHD equilibrium is reached when the flow relaxes and force balance is achieved. We ask what classes of MHD equilibria can be accessed from a given initial state via smooth incompressible ideal motion. It is found that certain boundary displacements are formally not supported. This follows from yet another investigation of the Hahm--Kulsrud--Taylor (HKT) problem, which highlights the resonant behaviour near a rational layer formed by a set of degenerate critical points in the flux-function. When trying to retain the mirror symmetry of the flux-function with respect to the resonant layer, the vector field that generates the volume-preserving diffeomorphism vanishes at the identity to all order in the time-like path parameter., Comment: 9 pages, 3 figures

Magnetostatics defines a class of boundary value problems in which the topology of the domain plays a subtle role. For example, representability of a divergence-free field as the curl of a vector potential comes about because of homological considerations. With this in mind, we study gauge-freedom in magnetostatics and its effect on the comparison between magnetic configurations through key quantities such as the magnetic helicity. For this, we apply the Hodge decomposition of $k$-forms on compact orientable Riemaniann manifolds with smooth boundary, as well as de Rham cohomology, to the representation of magnetic fields through potential $1$-forms in toroidal volumes. An advantage of the homological approach is the recovery of classical results without explicit coordinates and assumptions about the fields on the exterior of the domain. In particular, a detailed construction of a minimal gauge and a formal proof of relative helicity formulae are presented., Comment: 15 pages