Over the past decade, public libraries have shifted from quiet repositories of knowledge to raucous centers of public engagement. Public libraries seek to fill the educational and social gaps left by other informal education organizations (such as museums and science centers) that target specific populations or require paid access for their resources. These gaps are filled by hiring social workers, providing accessible makerspaces, developing English language learner (ELL) programs, facilitating hands-on STEM activities, providing information about community resources and social services, providing summer meals, and much more. But what are the next steps to continue this high level of engagement? By utilizing a Community Dialogue Framework (Dialogues), libraries have engaged with new members of their communities to reach groups not currently benefiting from library services, provided equitable access to new resources, engaged with new partners, and - in the time of COVID - began to address the digital divide in their communities. An examination of forty public libraries’ engagement with and learning from Dialogues was conducted using a qualitative approach and reflexive thematic analysis. An account from a librarian who hosted multiple Dialogues is also presented as a first-person narrative describing their methods and successes using the tool. Benefits and practical considerations for conducting Dialogues are discussed in the results section, followed by limitations and recommendations for further research in this area.
The Apollonian circle packing, generated from three mutually-tangent circles in the plane, has inspired over the past half-century the study of other classes of space-filling packings, both in two and in higher dimensions. Recently, Kontorovich and Nakamura introduced the notion of crystallographic sphere packings, n-dimensional packings of spheres with symmetry groups that are isometries of H n+1 . There exist at least three sources which give rise to crystallographic packings, namely polyhedra, reflective extended Bianchi groups, and various higher dimensional quadratic forms. When applied in conjunction with the Koebe-Andreev-Thurston Theorem, Kontorovich and Nakamura's Structure Theorem guarantees crystallographic packings to be generated from polyhedra in n = 2. The Structure Theorem similarly allows us to generate packings from the reflective extended Bianchi groups in n = 2 by applying Vinberg's algorithm to obtain the appropriate Coxeter diagrams. In n > 2, the Structure Theorem when used with Vinberg's algorithm allows us to explore whether certain Coxeter diagrams in H n+1 for a given quadratic form admit a packing at all. Kontorovich and Nakamura's Finiteness Theorem shows that there exist only finitely many classes of superintegral such packings, all of which exist in dimensions n 20. In this work, we systematically determine all known examples of crystallographic sphere packings.
Sarnak's golden mean conjecture states that (m+1)dϕ(m) 1+ 2 √ 5for all integers m 1, where ϕ is the golden mean and d θ is the discrepancy function for m + 1 multiples of θ modulo 1. In this paper, we characterize the set S of values θ that share this property, as well as the set T of those with the property for some lower bound m M . Remarkably, S mod 1 has only 16 elements, whereas T is the set of GL 2 (Z)-transformations of ϕ. arXiv:2002.03092v1 [math.NT] 8 Feb 2020 * d θ (m) = d 1−θ (m), so θ ∈ S if and only if 1 − θ ∈ S, which is why only 8 values are specified in the table.† It is elementary that θ must have a continued fraction and that it cannot be finite.
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