A 2-dimensional framework (G, p) is a graph G = (V, E) together with a map p : V → R 2 . We view (G, p) as a straight line realization of G in R 2 . Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u, v} is globally linked in G if the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [12] by characterizing globally linked pairs in M -connected graphs, an important family of rigid graphs. As a by product we simplify the proof of a result of Connelly [5] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M -connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].

An update on the JPEG XL standardization effort: JPEG XL is a practical approach focused on scalable web distribution and efficient compression of high-quality images. It will provide various benefits compared to existing image formats: significantly smaller size at equivalent subjective quality; fast, parallelizable decoding and encoding configurations; features such as progressive, lossless, animation, and reversible transcoding of existing JPEG; support for high-quality applications including wide gamut, higher resolution/bit depth/dynamic range, and visually lossless coding. Additionally, a royalty-free baseline is an important goal. The JPEG XL architecture is traditional block-transform coding with upgrades to each component. We describe these components and analyze decoded image quality.

Enzymatic synthesis and hydrolysis of nucleoside phosphate compounds play a key role in various biological pathways, like signal transduction, DNA synthesis and metabolism. Although these processes have been studied extensively, numerous key issues regarding the chemical pathway and atomic movements remain open for many enzymatic reactions. Here, using the Mason–Pfizer monkey retrovirus dUTPase, we study the dUTPase-catalyzed hydrolysis of dUTP, an incorrect DNA building block, to elaborate the mechanistic details at high resolution. Combining mass spectrometry analysis of the dUTPase-catalyzed reaction carried out in and quantum mechanics/molecular mechanics (QM/MM) simulation, we show that the nucleophilic attack occurs at the α-phosphate site. Phosphorus-31 NMR spectroscopy (31P-NMR) analysis confirms the site of attack and shows the capability of dUTPase to cleave the dUTP analogue α,β-imido-dUTP, containing the imido linkage usually regarded to be non-hydrolyzable. We present numerous X-ray crystal structures of distinct dUTPase and nucleoside phosphate complexes, which report on the progress of the chemical reaction along the reaction coordinate. The presently used combination of diverse structural methods reveals details of the nucleophilic attack and identifies a novel enzyme–product complex structure.