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The paper determines and classifies the spectrum of a non-self-adjoint operator pencil generated by the time-harmonic Maxwell problem with a nonlinear dependence on the frequency for the case of two homogeneous materials joined at a planar interface. We study spatially one-dimensional and two-dimensional reductions in the whole space $\mathbb{R}$ and $\mathbb{R}^2$. The dependence on the spectral parameter, i.e. the frequency, is in the dielectric function and we make no assumptions on its form. These function values determine the spectral sets. In order to allow also for non-conservative media, the dielectric function is allowed to be complex, yielding a non-self-adjoint problem. The whole spectrum consists of eigenvalues and the essential spectrum, but the various standard types of essential spectra do not coincide in all cases. The main tool for determining the essential spectra are Weyl sequences., Comment: 45 pages, 4 figures (Lorentz model as an example added; proof. of Lemma 5.13 corrected; new sections 6 and 7; notation changes (V,W not used any more); many new remarks; several new references)

We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity $\varepsilon$, permeability $\mu$ and conductivity $\sigma$, on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil $\mathrm{div}((\omega\varepsilon + i \sigma) \nabla\,\cdot\,)$, and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwell's and related systems., Comment: 22 pages

This report profiles key trends and emerging technologies and practices shaping the future of teaching and learning and envisions a number of scenarios and implications for that future. It is based on the perspectives and expertise of a global panel of leaders from across the higher education landscape. Starting in 2020, COVID-19 pandemic has transformed the higher education landscape. Though it remains to be seen whether those transformations have taken root and will persist into the future, it is not hard to imagine that higher education may never be the same in some important ways. Indeed, the potential lasting effects of the pandemic loomed large in the Horizon panelists' discussions and nominations as they identified the trends, technologies, and practices shaping the future of teaching and learning. This current edition summarizes the results of those discussions and nominations and serves as one vantage point on where the post-pandemic future may be headed. [The report was written with Aras Bozkurt, Steven Crawford, Laura Czerniewicz, Rob Gibson, Katie Linder, Jon Mason, and Victoria Mondelli. For the 2019 edition, see ED607329.]

In assuming ownership of the Horizon Report, EDUCAUSE recognized the challenges of anticipating the future. The authors have, in this first major revision of the report's methodology, structure, and content, striven to break the mold of the classic Horizon Report without losing its essential purpose. The report begins with a scan of the current environment to identify the major trends that are shaping global higher education and teaching and learning. The Horizon Expert Panel named fifteen social, technological, economic, higher education, and political trends that signal departures from the past, that are influencing the present, and that will almost certainly help shape the future. For educational technologies, the report moves away from the time-to-adoption structure, which implied a prediction precision that the project was unable to achieve. In its place, the new report offers evidence, data, and scenarios. The report includes evidence for the trends, as well as panelists' quantitative ratings of factors that often temper actual adoption of emerging technologies and practices in higher education. These factors include impact on learning outcomes, level of risk in adoption, faculty receptiveness, issues of equity and inclusion, and required level of spending. The final choice in reimagining the Horizon Report was to provide more-helpful, richer resources to assist the community in considering choices and formulating action plans. In addition to identifying trends and emerging technologies and practices, the authors offer scenarios for how the future could play out. ["2020 EDUCAUSE Horizon Report: Teaching and Learning Edition" was written with Bryan Alexander, Maha Bali, Stephanie Bulger, Shawna Dark, Nicole Engelbert, Kevin Gannon, Adrienne Gauthier, David Gibson, Rob Gibson, Brigitte Lundin, George Veletsianos, and Nicole Weber. This report was sponsored by Oracle. For "EDUCAUSE Horizon Report: 2019 Higher Education Edition," see ED603350.]

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator.

A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is proven. No special geometrical condition is imposed on the inaccessible part of the boundary of the domain, apart from imposing that the boundary of the domain is $C^{1,1}$. The coefficients are assumed to coincide on a neighbourhood of the boundary, a natural property in applications., Comment: New correct proof of Proposition 5.2

What is on the five-year horizon for higher education institutions? Which trends and technology developments will drive educational change? What are the critical challenges and how can we strategize solutions? These questions regarding technology adoption and educational change steered the discussions of 71 experts to produce this report. Six key trends, six significant challenges, and six developments in educational technology profiled in this higher education report are likely to impact teaching, learning, and creative inquiry in higher education. The three sections of this report constitute a reference and technology planning guide for educators, higher education leaders, administrators, policymakers, and technologists. [For the "NMC Horizon Report: 2017 Higher Education Edition," see ED582134.]

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, guided mode spectrum can be created by arbitrarily "small" perturbations. Secondly we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum $k_x$ the perturbation generates at most finitely many new eigenvalues inside the gap., Comment: 19 pages, 1 figure

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-L\"{u}st type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -- which together indicate that the abstract results are probably best possible., Comment: 21 pages minor amendments, extra references

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map., Comment: 16 pages, submitted to LMS; v2: including more references and some minor corrections; changed presentation in section 2, added theorem 4.4, modified example in section 5; v3: includes new references