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1. Quantifying Bound Proteins on Pegylated Gold Nanoparticles Using Infrared Spectroscopy

   https://doi.org/10.1021/acsabm.4c00012  

   Handali, Paul R; Webb, Lauren J (February 2024, ACS Applied Bio Materials)

   Protein−nanoparticle (NP) complexes are nanomaterials that have numerous potential uses ranging from biosensing to biomedical applications such as drug delivery and nanomedicine. Despite their extensive use quantifying the number of bound proteins per NP remains a challenging characterization step that is crucial for further developments of the conjugate, particularly for metal NPs that often interfere with standard protein quantification techniques. In this work, we present a method for quantifying the number of proteins bound to pegylated thiol-capped gold nanoparticles (AuNPs) using an infrared (IR) spectrometer, a readily available instrument. This method takes advantage of the strong IR bands present in proteins and the capping ligands to quantify protein−NP ratios and circumvents the need to degrade the NPs prior to analysis. We show that this method is generalizable where calibration curves made using inexpensive and commercially available proteins such as bovine serum albumin (BSA) can be used to quantify protein−NP ratios for proteins of different sizes and structures. 

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2. An effective Chabauty–Kim theorem

   https://doi.org/10.1112/S0010437X19007243  

   Balakrishnan, Jennifer S.; Dogra, Netan (June 2019, Compositio Mathematica)

   The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem. 

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3. Winding Numbers on Discrete Surfaces

   https://doi.org/10.1145/3592401  

   Feng, Nicole; Gillespie, Mark; Crane, Keenan (July 2023, ACM Transactions on Graphics)

   In the plane, thewinding numberis the number of times a curve wraps around a given point. Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven valuable for geometry processing tasks ranging from surface reconstruction to mesh booleans. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. By processing the derivatives of such functions, we can robustly filter out components of the input that do not bound any region. Ultimately, our algorithm yields (i) a closed, completed version of the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region. The main computational cost is solving a standard Poisson equation, or for surfaces with nontrivial topology, a sparse linear program. We also introduce special basis functions to represent singularities that naturally occur at endpoints of open curves. 

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4. Five-body recombination of identical bosons

   https://doi.org/10.1073/pnas.2503390122  

   Higgins, Michael D; Greene, Chris H (May 2025, Proceedings of the National Academy of Sciences)

   This work treats resonant collisions between five identical ultracold bosons in the framework of the adiabatic hyperspherical representation. The five-body recombination rate coefficient is quantified using a semiclassical description in conjunction with an analysis of the lowest five-body hyperspherical adiabatic potential curves in a scattering length regime with no universal weakly bound tetramers, trimers, or dimers. A comparison is made between these results and the only existing experimental measurement of five-body loss in an ultracold gas of bosonic cesium atoms and with the lone theoretical estimation of the loss rate. The recombination rate for the processB+B+B+B+B→B₄+Bis also computed in a different regime of scattering lengths where there is one universal bound tetramer by implementing a few-channel quantum scattering calculation based on five-body hyperspherical potential curves and nonadiabatic couplings. Our calculations predict regions where five-body recombination can cause decay of the atom cloud in an ultracold gas that is even faster than 3-body and 4-body recombination, which can ideally be tested by using the current generation of box traps having nearly uniform density. 

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5. Fréchet Distance for Uncertain Curves

   https://doi.org/10.1145/3597640  

   Buchin, Kevin; Fan, Chenglin; Löffler, Maarten; Popov, Aleksandr; Raichel, Benjamin; Roeloffzen, Marcel (July 2023, ACM Transactions on Algorithms)

   In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [ 5 ] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models. On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe–Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets. 

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