“Bio”‐Macromolecules: Polymer‐Protein Conjugates as Emerging Scaffolds for Therapeutics

Polymer‐protein conjugates are biohybrid macromolecules derived from covalently connecting synthetic polymers with polypeptides. The resulting materials combine the properties of both worlds: chemists can engineer polymers to stabilize proteins, to add functionality, or to enhance activity; whereas biochemists can exploit the specificity and complexity that Nature has bestowed upon its macromolecules. This has led to a wealth of applications, particularly within the realm of biomedicine. Polymer‐protein conjugation has expanded to include scaffolds for drug delivery, tissue engineering, and microbial inhibitors. This feature article reflects upon recent developments in the field and discusses the applications of these hybrids from a biomaterials standpoint.

     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

Eine neuartige Fragmentierung cyclischer α, ß-ungesättigter Carbonylsysteme; Synthese von Exalton und rac-Muscon aus Cyclododecanon Vorläufige Mitteilung

The conversion of bicyclo[10.3.0]-1,12-epoxy-13-pentadecanone into 4-cyclopenta-decyne-1-one, effected by p-tolylsulfonylhydrazine under mild conditions and in high yield, provides an example of a new fragmentation reaction.     

α-Chlor-nitrone IV: Zur Stereochemie der λ-Lacton-Synthese und eine Methode zum Aufbau von α-Methyliden-λ-lactonen aus Olefinen. Über synthetische Methoden, 8. (vorläufige) Mitteilung

The Ag⁺-induced reaction between N-cyclohexyl-α-chloro-propionaldonitrone and the two diastereomeric 2-butenes in liquid SO₂ is a stereospecific cis-addition. The use of N-cyclo-hexyl-α,β-dichloro-propionaldonitrone in this type of reaction provides a preparative route from olefines to α-methylidene-butyrolactones.     

Ammonia Borane,NH3BH3: A Threshold Photoelectron–Photoion Coincidence Study of a Potential Hydrogen-Storage Material

We have investigated the photoionization of ammonia borane (AB) and determined adiabatic ionization energy to be 9.26±0.03 eV for the X^{+ 2}E←X ¹A₁ transition. Although the threshold photoelectron spectrum appears at first glance to be similar to the one of the isosteric ethane, the electronic situation differs markedly, due to different orbital energies. In addition, an appearance energy AE_0K(NH₃BH₃, NH₃BH₂⁺)= 10.00±0.03 eV has been determined, corresponding to the loss of a hydrogen atom at the BH₃-site. From the data, a 0 K bond dissociation energy for the B−H bond in the cation of 71.5±3 kJ mol⁻¹ was derived, whereas the one in the neutral compound has been estimated to be 419±10 kJ mol⁻¹.     

A Survey on Some Anisotropic Hardy-Type Function Spaces

Let $A$ be a general expansive matrix on $\mathbb{R}^n$. The aims of this article are twofold. The first one is to give a survey on the recent developments of anisotropic Hardy-type function spaces on $\mathbb{R}^n$, including anisotropic Hardy–Lorentz spaces, anisotropic variable Hardy spaces and anisotropic variable Hardy–Lorentz spaces as well as anisotropic Musielak–Orlicz Hardy spaces. The second one is to correct some errors and seal some gaps existing in the known articles. Some unsolved problems are also presented.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

An Approach to Wavelet Isomorphisms of Function Spaces Via Atomic Representations

Both wavelet and atomic decomposition techniques are essential tools in the study of function spaces nowadays, but they both have their advantages and disadvantages. The celebrated bridge between both concepts was given by the compactly supported Daubechies wavelets which can be interpreted as atoms. In this paper we deal with the converse direction, that is, we present a fairly general approach how to construct compactly supported wavelets when an atomic decomposition is known already. The main idea is Taylor’s expansion combined with our new, so-called \(\varkappa \)-convergence assumption in the admitted sequence spaces. We finally exemplify our main result and collect some known and new settings where such a wavelet decomposition is obtained, e.g., in spaces of Besov or Triebel–Lizorkin type with a doubling weight.