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The linear response of non-Hermitian resonant systems demonstrates various intriguing features such as the emergence of non-Lorentzian lineshapes. Recently, we have developed a systematic theory to understand the scattering lineshapes in such systems and, in doing so, established the connection with the input/output scattering channels. Here, we follow up on that work by presenting a different, more transparent derivation of the resolvent operator associated with a non-Hermitian system under general conditions and highlight the connection with the structure of the underlying eigenspace decomposition. Finally, we also present a simple solution to the problem of self-orthogonality associated with the left and right Jordan canonical vectors and show how the left basis can be constructed in a systematic fashion. Our work provides a unifying mathematical framework for studying non-Hermitian systems such as those implemented using dielectric cavities, metamaterials, and plasmonic resonators.

 

Abstract Understanding the linear response of any system is the first step towards analyzing its linear and nonlinear dynamics, stability properties, as well as its behavior in the presence of noise. In non-Hermitian Hamiltonian systems, calculating the linear response is complicated due to the non-orthogonality of their eigenmodes, and the presence of exceptional points (EPs). Here, we derive a closed form series expansion of the resolvent associated with an arbitrary non-Hermitian system in terms of the ordinary and generalized eigenfunctions of the underlying Hamiltonian. This in turn reveals an interesting and previously overlooked feature of non-Hermitian systems, namely that their lineshape scaling is dictated by how the input (excitation) and output (collection) profiles are chosen. In particular, we demonstrate that a configuration with an EP of order M can exhibit a Lorentzian response or a super-Lorentzian response of order M s with M s  = 2, 3, …,  M , depending on the choice of input and output channels. 

We develop a linear theory for non-Hermitian optical systems having exceptional points. In contrast to previous studies, our analysis results in an exact expression for the resolvent operator without the need to use perturbation expansions. 

We introduce a new class of photonic resonators with resonant modes that feature hybrid standing-travelling waves. 

Anthropogenic climate change remains divisive in the United States, where skepticism of the scientific consensus is associated with conservative worldviews, resulting in political polarization. This study considers three hypotheses regarding U.S. polarization over climate change that have emerged from social psychology research and applies them to science education by showing how these hypotheses could relate to adolescents' science learning. We then test each hypothesis within an experimental educational intervention designed to study the influence of worldview, mechanistic knowledge, and quantitative reasoning on students' written arguments about climate change. We used mixed methods to analyze the results of this individually randomized trial with clustering involving 357 participants in grades 9–11 from 5 U.S. sites. Findings show that: (a) exposure to mechanistic knowledge about climate change increased odds of receptivity toward climate change; (b) increasingly conservative worldviews were associated with decreased odds of receptivity; (c) worldview and quantitative reasoning interacted, resulting in an amplified effect of worldview for students with greater quantitative reasoning. Results also suggest that the influence of worldview and mechanistic knowledge on receptivity work independently from one another in our dataset. This study demonstrates the value of teaching mechanistic understandings of climate change, yet also demonstrates the influence of worldview on receptivity to climate change for adolescents, as well as complex interactions between quantitative reasoning (something school science aims to develop) and worldview. It shows that moving the U.S. public toward the scientific consensus is complex and involves confronting ideologically motivated reasoning within science education.

 

This article reports an investigation of a professional development program to enhance elementary teachers’ ability to engage their students in argument from evidence in science. Using a quasi-experimental approach, three versions were compared: Version A—a 1-week summer institute with a 2-week summer practicum experience and 8 follow-up days (four per year), Version B without the practicum experience, and Version C—a revision of Version A in Year 3. All teachers were videoed twice each year, and the videos were rated using an instrument to measure the quality of discourse. All versions led to a significant improvement in teachers’ facilitation of classroom discourse. Neither the practicum nor the revised program had an additional effect. Implications for the field are discussed.

 

When people are exposed to information that leads them to overestimate the actual amount of genetic difference between racial groups, it can augment their racial biases. However, there is apparently no research that explores if the reverse is possible. Does teaching adolescents scientifically accurate information about genetic variation within and between US census races reduce their racial biases? We randomized 8^thand 9^thgrade students (n = 166) into separate classrooms to learn for an entire week either about the topics of (a) human genetic variation or (b) climate variation. In a cross‐over randomized trial with clustering, we demonstrate that when students learn about genetic variation within and between racial groups it significantly changes their perceptions of human genetic variation, thereby causing a significant decrease in their scores on instruments assessing cognitive forms of prejudice. We then replicate these findings in two computer‐based randomized controlled trials, one with adults (n = 176) and another with biology students (n = 721, 9^th–12^thgraders). These results indicate that teaching about human variation in the domain of genetics has potentially powerful effects on social cognition during adolescence. In turn, we argue that learning about the social and quantitative complexities of human genetic variation research could prepare students to become informed participants in a society where human genetics is invoked as a rationale in sociopolitical debates.