Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets Λ⊂ℝ ^d with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. For our main result, we introduce the notion of algebraic Delone sets Λ⊂ℝ² and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.     

Investigation of Water Interaction with Polymer Matrices by Near-Infrared (NIR) Spectroscopy

The interaction of water with polymers is an intensively studied topic. Vibrational spectroscopy techniques, mid-infrared (MIR) and Raman, were often used to investigate the properties of water–polymer systems. On the other hand, relatively little attention has been given to the potential of using near-infrared (NIR) spectroscopy (12,500–4000 cm⁻¹; 800–2500 nm) for exploring this problem. NIR spectroscopy delivers exclusive opportunities for the investigation of molecular structure and interactions. This technique derives information from overtones and combination bands, which provide unique insights into molecular interactions. It is also very well suited for the investigation of aqueous systems, as both the bands of water and the polymer can be reliably acquired in a range of concentrations in a more straightforward manner than it is possible with MIR spectroscopy. In this study, we applied NIR spectroscopy to investigate interactions of water with polymers of varying hydrophobicity: polytetrafluoroethylene (PTFE), polypropylene (PP), polystyrene (PS), polyvinylchloride (PVC), polyoxymethylene (POM), polyamide 6 (PA), lignin (Lig), chitin (Chi) and cellulose (Cell). Polymer–water mixtures in the concentration range of water between 1–10%(w/w) were investigated. Spectra analysis and interpretation were performed with the use of difference spectroscopy, Principal Component Analysis (PCA), Median Linkage Clustering (MLC), Partial Least Squares Regression (PLSR), Multivariate Curve Resolution Alternating Least Squares (MCR-ALS) and Two-Dimensional Correlation Spectroscopy (2D-COS). Additionally, from the obtained data, aquagrams were constructed and interpreted with aid of the conclusions drawn from the conventional approaches. We deepened insights into the problem of water bands obscuring compound-specific signals in the NIR spectrum, which is often a limiting factor in analytical applications. The study unveiled clearly visible trends in NIR spectra associated with the chemical nature of the polymer and its increasing hydrophilicity. We demonstrated that changes in the NIR spectrum of water are manifested even in the case of interaction with highly hydrophobic polymers (e.g., PTFE). Furthermore, the unveiled spectral patterns of water in the presence of different polymers were found to be dissimilar between the two major water bands in NIR spectrum (ν_s + ν_as and ν_as + δ).     

Computing Arithmetic Functions Using Immobilised Enzymatic Reaction Networks

Living systems use enzymatic reaction networks to process biochemical information and make decisions in response to external or internal stimuli. Herein, we present a modular and reusable platform for molecular information processing using enzymes immobilised in hydrogel beads and compartmentalised in a continuous stirred tank reactor. We demonstrate how this setup allows us to perform simple arithmetic operations, such as addition, subtraction and multiplication, using various concentrations of substrates or inhibitors as inputs and the production of a fluorescent molecule as the readout.