Uncertainties are inherent in the fabrication and operation of mechanisms. In terms of the four-bar linkage, uncertainties in the geometric parameters result in non-exact solutions for the coupler point. A design description which accounts for bounded uncertainties is termed an appropriate design. Appropriate analysis routines are analysis routines for guaranteeing that specific requirements will be satisfied, which is applicable to mechanisms described by an appropriate design. These routines rely on interval analysis to provide reliable results and are able to handle the uncertainties present in a mechanism. An appropriate analysis routine which is able to determine the worst-case bounds of the coupler curve resulting from uncertainties is proposed. As well, routines are introduced for evaluating the classification and assembly of mechanisms with uncertainties. Examples are presented to demonstrate the coupler curves of four-bar linkages described by an appropriate design which corresponds to each linkage classification, including folding linkages.

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Chromatin architecture, a key regulator of gene expression, can be inferred using chromatin contact data from chromosome conformation capture, or Hi-C. However, classical Hi-C does not preserve multi-way contacts. Here we use long sequencing reads to map genome-wide multi-way contacts and investigate higher order chromatin organization in the human genome. We use hypergraph theory for data representation and analysis, and quantify higher order structures in neonatal fibroblasts, biopsied adult fibroblasts, and B lymphocytes. By integrating multi-way contacts with chromatin accessibility, gene expression, and transcription factor binding, we introduce a data-driven method to identify cell type-specific transcription clusters. We provide transcription factor-mediated functional building blocks for cell identity that serve as a global signature for cell types.

The uncertainties arising from the fabrication and operation of mechanisms, specifically the four-bar linkage, significantly affect the expected performance of the mechanism. To accommodate uncertainties during the dimensional synthesis of the four-bar linkage, the desired coupler curve response is described by any number of precision point elements and trajectory elements, where each element is specified with an allowable error. A design description which accounts for bounded uncertainties is termed an appropriate design. An appropriate synthesis method is developed to synthesize the entire set of appropriate design solutions corresponding to the desired response. This method is able to completely explore the continuous design space and each synthesized appropriate design guarantees that the resulting four-bar linkage has a corresponding coupler point which lies inside each precision point response element, and a corresponding set of continuous coupler points which remain inside the trajectory response elements from start to finish.

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