Subword complexity of the Fibonacci–Thue–Morse sequence: The proof of Dekking’s conjecture

Recently F. M. Dekking conjectured the form of the subword complexity function for the Fibonacci–Thue–Morse sequence. In this note we prove his conjecture by purely computational means, using the free software Walnut.     

Molecular signal suppression by in situ microextraction in nuclear magnetic resonance spectroscopy

The detailed characterization of complex mixtures by NMR is often hampered by the presence of signals from uninformative compounds, the resonances of which overlap with those of the molecules of interest. We provide here a proof of principle for an approach to NMR signal suppression in complex samples using Molecularly Imprinted Polymers (MIPS). Addition of a few milligrams of polymer to a solution traps the target molecule in typical micromolar to millimolar concentration, thus achieving in situ signal suppression, without altering any other spectral features. This method minimized any manipulation or perturbation of the spectrum and was applied to a complex mixture of known compounds and to a plant extract, in both cases spiked with a compound (bisphenol A), which was subsequently removed by selective binding to a complementary MIP. What is described in this report is comparable with microextraction and may in due course be applied to a large number of analytical challenges. Copyright © 2014 John Wiley & Sons, Ltd.     

Chitosan/Ce(IV) redox polymerization‐based amplification for detection of DNA point mutation

Polymerization‐based signal amplification, a technique developed for use in rapid diagnostic tests, hinges on the ability to localize initiators as a function of interfacial binding events. We report here a new DNA detection method in which polymer growth in redox‐polymerization is used as a means to amplify detection signals. The introduction of biotin‐labeled chitosan (biotin‐CS) with highly dense amino groups into the polymerization amplification as macromolecular reducing agent, beneficially simplifies amplification operation, as well as, provides a large amount of initiation points to improve the sensitivity of detection. DNA hybridization, SA and biotin binding reactions led to the attachment of CS on a solid surface where specific DNA sequences were located. With the addition of the mixture containing monomer AM, crosslinker PEGDA and oxidant CAN onto the CS location, the growth of polymer films was triggered to render the corresponding spots readily distinguishable to the naked eye. Direct visualization of 0.21 fmol target DNA molecules of interest was demonstrated. Non‐small cell lung cancer p53 sequence was further selected as a proof‐of‐principle to detect DNA point mutation. The proposed method exhibited an efficient amplification performance for molecule detection, and paved a new way for visual diagnosis of biomolecules. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 1929–1937     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

Koszulity of some path categories

We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.