Some Remarks Concerning the Scattering Theory for the Sturm–Liouville Operator

Journal of Dynamics and Differential Equations - We start to discuss some aspects of the scattering theory for the Sturm–Liouville operator $$L:\dfrac{1}{y}\left[ -D^2+q\right] $$ . In...     

A Stable Porphyrin Functionalized Graphite Electrode Used at the Oxygen Evolution Reaction Potential

Many researches have been devoted to rechargeable power generators that can store (but also release) energy. This availability is ensured through (e. g.) the oxygen evolution reaction (OER). However, (i) large values of the overpotentials and (ii) a progressive detriment of the anode (graphite) electrode limit the ultimate device. In view of enhancing the electrode performances, graphite was protected by following different strategies, which oblige to follow precise preparation protocols. Here, we prove that a thin layer of free-base porphyrin molecules is able to protect the underneath graphite electrode from detriment even if many (about 100) electrochemical cycles are performed.     

An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2-Dimensional Case

Using a direct approach, we prove a two-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension 2, the C^1,α regularity of the free boundary in the scalar one-phase and double-phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc.     

Repdigits as sums of three Pell numbers

In this paper, all base 10 repdigits expressible as sums of three Pell numbers are found.     

Synthesis and Biological Evaluation of RGD Peptidomimetic–Paclitaxel Conjugates Bearing Lysosomally Cleavable Linkers

Two small‐molecule–drug conjugates (SMDCs, 6 and 7 ) featuring lysosomally cleavable linkers (namely the Val–Ala and Phe–Lys peptide sequences) were synthesized by conjugation of the α_vβ₃‐integrin ligand cyclo[DKP–RGD]‐CH₂NH₂ ( 2 ) to the anticancer drug paclitaxel (PTX). A third cyclo[DKP–RGD]–PTX conjugate with a nonpeptide “uncleavable” linker ( 8 ) was also synthesized to be tested as a negative control. These three SMDCs were able to inhibit biotinylated vitronectin binding to the purified α_Vβ₃‐integrin receptor at nanomolar concentrations and showed good stability at pH 7.4 and pH 5.5. Cleavage of the two peptide linkers was observed in the presence of lysosomal enzymes, whereas conjugate 8 , which possesses a nonpeptide “uncleavable” linker, remained intact under these conditions. The antiproliferative activities of the conjugates were evaluated against two isogenic cell lines expressing the integrin receptor at different levels: the acute lymphoblastic leukemia cell line CCRF‐CEM (α_Vβ₃?) and its subclone CCRF‐CEM α_Vβ₃ (α_Vβ₃+). Fairly effective integrin targeting was displayed by the cyclo[DKP–RGD]–Val–Ala–PTX conjugate ( 6 ), which was found to differentially inhibit proliferation in antigen‐positive CCRF‐CEM α_Vβ₃ versus antigen‐negative isogenic CCRF‐CEM cells. The total lack of activity displayed by the “uncleavable” cyclo[DKP–RGD]–PTX conjugate ( 8 ) clearly demonstrates the importance of the peptide linker for achieving the selective release of the cytotoxic payload.     

Solubility and Crystallizability: Facile Access to Functionalized π‐Conjugated Compounds with Chlorendylimide Protecting Groups

Functional π‐conjugated molecules are relevant for the preparation of new organic electronic materials with improved performance. However, their synthesis is often rendered difficult by their inherently low solubility, and the permanent attachment of solubilizing groups may change the properties of the material. Here, we introduced the chlorendylimidyl moiety as a new temporary protecting group for the straightforward large‐scale synthesis of protected quarter‐, sexi‐, octathiophene, and perylene bisimide diamine and dicarboxylic acid derivatives. The obtained chlorendylimides and chlorendylimidyl active esters were highly soluble in organic solvents, and optical spectroscopy confirmed the low tendency of the compounds to aggregate in solution. At the same time, they could be conveniently purified by recrystallization or precipitation. Single‐crystal X‐ray structures obtained for most compounds showed supramolecular motifs highlighting the role of the rigid, polychlorinated chlorendyl moieties in their crystallization. The obtained protected diamine and dicarboxylic acid derivatives were easily deprotected and converted into various amide‐substituted oligothiophenes and perylene bisimides that are of interest as new functional materials for organic electronic thin film or nanowire devices.     

Word problems and ceers

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.