Thermal Corrections to R\'enyi entropies for Free Fermions

We calculate thermal corrections to R\'{e}nyi entropies for free massless fermions on a sphere. More specifically, we take a free fermion on $\mathbb{R}\times\mathbb{S}^{d-1}$ and calculate the leading thermal correction to the R\'{e}nyi entropies for a cap like region with opening angle $2\theta$. By expanding the density matrix in a Boltzmann sum, the problem of finding the R\'{e}nyi entropies can be mapped to the problem of calculating a two point function on an $n$ sheeted cover of the sphere. We follow previous work for conformal field theories to map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the R\'{e}nyi entropies.Comment: 15 pages, 5 figure

Tracing Through Scalar Entanglement

As a toy model of a gapped system, we investigate the entanglement entropy of a massive scalar field in 1+1 dimensions at nonzero temperature. In a small mass m and temperature T limit, we put upper and lower bounds on the two largest eigenvalues of the covariance matrix used to compute the entanglement entropy. We argue that the entanglement entropy has exp(-m/T) scaling in the limit m >> T. We comment on the relation between our work and the Ryu-Takayanagi proposal for computing the entanglement entropy holographically.Comment: 17 pages, 11 figures; v2 ref added, typos fixed; v3 refs added, minor clarifications, version to appear in PR

Understanding Teacher Leadership in Middle School Mathematics: A Collaborative Research Effort

We report findings from a collaborative research effort designed to examine how teachers act as leaders in their schools. We find that teachers educated by the Math in the Middle Institute act as key sources of advice for colleagues within their schools while drawing support from a network consisting of other teachers in the program and university-level advisors. In addition to reporting on our findings, we reflect on our research process, noting some of the practical challenges involved, as well as some of the benefits of collaboration

Teacher Learning and Instructional Change: How Formal and On-the-Job Learning Opportunities Predict Change in Elementary School Teachers\u27 Practice

Recent education reform has emphasized the importance of teacher learning in improving classroom instruction and raising student achievement. This article focuses on teachers\u27 learning opportunities, including formal professional development and on-the-job learning that occurs through interactions with colleagues. Using data from 30 elementary schools in a mid-sized urban school district, the authors concurrently explore the relationships between teachers\u27 formal professional development and on-the-job learning opportunities and instructional change. Results suggest that formal professional development and on-the-job opportunities to learn are both significantly associated with changes in teachers\u27 instructional practice in mathematics and English language arts

Meaningful & Sustainable School Improvement with Distributed Leadership

School leadership is broadly acknowledged to be the lynchpin for school success. Yet, amongst the countless demands that school leaders face, making wise leadership choices is increasingly challenging. On what should leaders focus their attention and how should they prioritize their improvement efforts? How can they identify, understand, and make headway on the difficult challenges that will substantially enhance the educational experiences of their students, and how can they bring their faculty together with commitment around these improvement efforts? In this essay we lay out a research-informed framework for advancing meaningful school improvement using a distributed leadership approach. This report was funded by the Bill & Melinda Gates Foundation. Opinions in this paper reflect those of the authors and do not necessarily reflect those of the Consortium for Policy Research in Education (CPRE), the University of Pennsylvania Graduate School of Education, that of the funder

Managing to Lead: Reframing School Leadership and Management

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Infrastructure Redesign and Instructional Reform in Mathematics: Formal Structure and Teacher Leadership

Designing infrastructures to support instruction remains a challenge in educational reform. This article reports on a study of one school system\u27s efforts to redesign its infrastructure for mathematics instruction by promoting teacher leadership. Using social network and interview data from 12 elementary schools, we explore how the district\u27s infrastructure redesign efforts were internally coherent with and built upon existing infrastructure components. We then explore relations between infrastructure and school practice as captured in the instructional advice- and information-seeking interactions among school staff, finding that teacher leaders emerged as central actors and brokers of advice and information about mathematics within and between schools. Further, changes in school advice and information networks were associated with shifts in teachers\u27 beliefs about and practices in mathematics toward inquiry-oriented approaches consistent with district curriculum. We argue that the district\u27s redesign efforts to support teacher leadership coupled district curriculum and school and classroom practice in mathematics

Don't Thrash: How to Cache Your Hash on Flash

This paper presents new alternatives to the well-known Bloom filter data structure. The Bloom filter, a compact data structure supporting set insertion and membership queries, has found wide application in databases, storage systems, and networks. Because the Bloom filter performs frequent random reads and writes, it is used almost exclusively in RAM, limiting the size of the sets it can represent. This paper first describes the quotient filter, which supports the basic operations of the Bloom filter, achieving roughly comparable performance in terms of space and time, but with better data locality. Operations on the quotient filter require only a small number of contiguous accesses. The quotient filter has other advantages over the Bloom filter: it supports deletions, it can be dynamically resized, and two quotient filters can be efficiently merged. The paper then gives two data structures, the buffered quotient filter and the cascade filter, which exploit the quotient filter advantages and thus serve as SSD-optimized alternatives to the Bloom filter. The cascade filter has better asymptotic I/O performance than the buffered quotient filter, but the buffered quotient filter outperforms the cascade filter on small to medium data sets. Both data structures significantly outperform recently-proposed SSD-optimized Bloom filter variants, such as the elevator Bloom filter, buffered Bloom filter, and forest-structured Bloom filter. In experiments, the cascade filter and buffered quotient filter performed insertions 8.6-11 times faster than the fastest Bloom filter variant and performed lookups 0.94-2.56 times faster.Comment: VLDB201