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Background Bridging factors are relational ties, formal arrangements, and processes that connect outer system and inner organizational contexts. They may be critical drivers of evidence-based practice (EBP) implementation and sustainment. Yet, the complex interplay between outer and inner contexts is often not considered. Bridging factors were recently defined in the updated Exploration, Preparation, Implementation, Sustainment (EPIS) framework. Further identification and specification of this construct will advance implementation models, measures, and methods. Our goal is to advance bridging factor research by identifying relevant dimensions and exemplifying these dimensions through illustrative case studies. Methods We used a multiple case study design. Each case (n = 10) represented different contexts, EBPs, and bridging factor types. Inclusion criteria were the presence of clearly distinguishable outer and inner contexts, identifiable bridging factor, sufficient information to describe how the bridging factor affected implementation, and variation from other cases. We used an iterative qualitative inquiry process to develop and refine a list of dimensions. Case data were entered into a matrix. Dimensions comprised the rows and case details comprised the columns. After a review of all cases, we collectively considered and independently coded each dimension as function or form. Results We drew upon the concepts of functions and forms, a distinction originally proposed in the complex health intervention literature. Function dimensions help define the bridging factor and illustrate its purpose as it relates to EBP implementation. Form dimensions describe the specific structures and activities that illustrate why and how the bridging factor has been customized to a local implementation experience. Function dimensions can help researchers and practitioners identify the presence and purpose of bridging factors, whereas form dimensions can help us understand how the bridging factor may be designed or modified to support EBP implementation in a specific context. We propose five function and three form bridging factor dimensions. Conclusions Bridging factors are described in many implementation models and studies, but without explicit reference or investigation. Bridging factors are an understudied and critical construct that requires further attention to facilitate implementation research and practice. We present specific recommendations for a bridging factors research agenda.

We demonstrate a purely arithmetic construction of the Eisenstein and Gaussian topographs of Bestvina and Savin. Influenced by Conway's approach, we recover these topographs as incidence geometries over sets of ''generalized lax bases". We use the results of Johnson and Weiss to identify our constructions with the Coxeter geometries arising from projective general linear groups.

I have considered two main questions in my research. First, which foliations on a manifold are compatible with a particular symmetry group. Second, which contact structures are preserved by which symmetry groups. The existence of both foliations and contact structures have been studied for over a half-century with no consideration for symmetry groups. I have looked thus far at finite symmetry groups. Historically, the study of manifolds and symmetry groups goes back to Riemann's formulation of the manifold concept in his 1854 lecture on the foundations of geometry.An equivalence class of smooth 1-forms L on a manifold M consists of all hL where h is any smooth function M to R. Given a smooth finite group action G on M, preserving this equivalence class means that there is a homomorphism E from G to {-1,1} and a L' in [L] such that for any g in G, g*L' = E(g)L'. In other words, there is a representative form L' such that g*L'=+-L'.There are mostly negative results for orientation reversing group actions. Let M be a smooth oriented (4n-1)-manifold. If Z_2k, k>1 acts smoothly but not orientation preservingly on M then there is no contact form on M which is compatible with this group action. For foliations, there are no compatible codimension 1 foliations on S^3 with a group action containing an orientation reversing subgroup that is isomorphic to Z_2k, for k>1.An equivalent condition for a smooth n-manifold M to have a G-invariant codimension 1 foliation is that each connected component of each fixed point set has Euler characteristic zero, where G is an odd order group that acts smoothly on M, with isotropy groups linearly ordered by inclusion.The last result is constructive. Let M be a closed smooth oriented 3-manifold with a smooth orientation-preserving G-action, where G is a group of prime order p. Then there is a G-invariant contact form T on M. This form is constructed from an open-book decomposition, and a branched covering.