Greedy sums of distinct squares

When a positive integer is expressed as a sum of squares, with each successive summand as large as possible, the summands decrease rapidly in size until the very end, where one may find two 's, or several 's. We find that the set of integers for which the summands are distinct does not have a natural density but that the counting function oscillates in a predictable way.

     

The equivalence between pointwise Hardy inequalities and uniform fatness

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.     

S-Structures for k-Linear Categories and the Definition of a Modular Functor

Ideas from string theory and quantum field theory have beenthe motivation for new invariants of knots and 3-dimensionalmanifolds which have been constructed from complex algebraicstructures such as Hopf algebras [17, 22], monoidal categorieswith additional structure [24], and modular functors [14, 23].These constructions are closely related. Here we take a unifyingcategorical approach based on a natural 2-dimensional generalisationof a topological field theory in the sense of Atiyah [1], andshow that the axioms defining these complex algebraic structuresare a consequence of the underlying geometry of surfaces.     

Alkene–Tetrazine Ligation for Imaging Cellular DNA

5‐Vinyl‐2′‐deoxyuridine (VdU) is the first reported metabolic probe for cellular DNA synthesis that can be visualized by using an inverse electron demand Diels–Alder reaction with a fluorescent tetrazine. VdU is incorporated by endogenous enzymes into the genomes of replicating cells, where it exhibits reduced genotoxicity compared to 5‐ethynyl‐2′‐deoxyuridine (EdU). The VdU–tetrazine ligation reaction is rapid (k≈0.02 M ^?1 s^?1) and chemically orthogonal to the alkyne–azide “click” reaction of EdU‐modified DNA. Alkene–tetrazine ligation reactions provide the first alternative to azide–alkyne click reactions for the bioorthogonal chemical labeling of nucleic acids in cells and facilitate time‐resolved, multicolor labeling of DNA synthesis.     

Novel Carboxyl Esterase Preparations for the Resolution of Linalyl Acetate

 Biocatalytic resolution of the tertiary terpene alcohol (±)-linalool was accomplished via hydrolysis of its corresponding acetate ester using two highly enantiospecific enzymes (E > 100). The latter were identified in a crude cell-free extract of Rhodococcus ruber DSM 43338 and could be separated by (partial) protein purification. Since they showed opposite enantiopreference, they were termed (R)- and (S)-linalyl acetate hydrolase (LAH). The activity and selectivity of the enzyme preparations was markedly dependent on the fermentation conditions.     

Formaldehyde Formation on Vanadium Oxide Surfaces V2O3(0001) and V2O5(001): How does the Stable Methoxy Intermediate Form?

Hydroxy‐mediated methoxy formation or stabilization is probably an important process in many methanol adsorption systems. Hydrogen atoms originating from the scission of the methanol O? H bond react with the substrate and form water. This process may result 1) in the production of additional surface defects as reactive centers for methoxy formation and 2) in the stabilization of methoxy groups by suppression of methanol formation.

     

Aluminiumorganyle mit pentakoordiniertem Aluminium-Atom: Synthese und Molekülstrukturen von [AlX2{2,6-(NEt2CH2)2C6H3}] (X = Cl,Et, H)

Aluminium Organyls with Pentacoordinate Aluminium: Syntheses and Molecular Structures of [AlX₂{2,6-(NEt₂CH₂)₂C₆H₃}] (X = Cl, Et, H) The reaction of [Li{2,6-(NEt₂CH₂)₂C₆H₃}]₂ with AlCl₃ or Et₂AlCl gives [AlX₂{2,6-(NEt₂CH₂)₂C₆H₃}] [X = Cl ( 1 ), Et ( 2 )] in good yield. 1 reacts with NaH in toluene to give [AlH₂{2,6-(NEt₂CH₂)₂C₆H₃}] ( 3 ). 1–3 were characterised spectroscopically (¹H, ¹³C, ²⁷Al n.m.r., i.r., mass spectroscopy). In solution at room temperature 1–3 exhibit dynamic behaviour. For 1 and 3 this can be frozen out below 278 K (¹H n.m.r.), indicating the presence of monomeric molecules with pentacoordinate Al at low temperature. Such species are also observed in the solid state as shown by an X-ray structure determination on 1 (monoclinic space group P2₁/n, a = 9.7325(14), b = 13.552(5), c = 28.858(7) Å, β = 99.57(2)°, V = 3753(2) ų, Z = 8, at 223(2) K) and 2 (monoclinic space group C2/c, a = 15.0045(12), b = 9.2986(8), c = 14.9955(12) Å, β =99.512(1)°, Z = 4, at 223(2) K).     

Picosecond multidimensional fluorescence spectroscopy: a tool to measure real-time protein dynamics during function

Advanced multidimensional time-correlated single photon counting (mdTCSPC) and picosecond time-resolved fluorescence in combination with site-directed fluorescence labeling are valuable tools to study the properties of membrane protein surface segments on the pico- to nanoseconds time scale. Time-resolved fluorescence anisotropy changes of protein bound fluorescent probes reveal changes in protein dynamics and steric restriction. In addition, the change in fluorescence lifetime and intensity of the covalently bound fluorescent dye is indicative of environmental changes at the protein surface. In this study, we have measured the changes in fluorescence lifetime traces of the fluorescent dye fluorescein covalently bound to the first cytoplasmic loop of bacteriorhodopsin (bR) after light activation of protein function. The fluorescence is excited by a picosecond laser pulse. The retinylidene chromophore of bR is light-activated by a 10 ns laser pulse, which in turn triggers recording of a sequence of fluorescence lifetime traces in the mdTCSPC-module. The fluorescence decay changes upon protein function occur predominantly in the 100 ps time range. The kinetics of these changes shows two transitions between three intermediate states in the second part of the bR photocycle. Correlation with photocycle kinetics allows for the determination of reaction intermediates at the proteins surface which are coupled to changes in the retinal binding pocket.     

Synthesis and crystal structure of CuZrTiO5—A new crystal structure type

A new compound, CuZrTiO₅, was synthesized as strongly pleochroic green crystals from the oxides between 995 and 1010 °C, 1 atm. Its crystal structure was determined by single crystal XRD, resulting in R (F²>2σ(F²))=0.032 and wR (all data)=0.079). CuZrTiO₅ is orthorhombic, space group P2₁2₁2₁, a=3.5871(3) Å, b=6.6968(4) Å, c=14.6679(9) Å, V=352.35(4) Å³, Z=4. The structure is topologically similar to In₂TiO₅ but differs in space group and cation coordination. CuZrTiO₅ has relatively regular TiO₆ polyhedra, but coordination is 7+1 for Zr, and 4+2 for Cu due to the Jahn-Teller effect. Ordering of the long Cu-O bonds causes reduction in symmetry relative to In₂TiO₅. Layers of Cu alternate with Ti+Zr on (001), giving rise to a distinct cleavage. Bond valence sums on Ti and Zr are far from ideal, which appears due to the limited ability of this structural topology to avoid close next-nearest neighbour distances.     

Quantitative evaluation of mechanosensing of cells on dynamically tunable hydrogels

Thin hydrogel films based on an ABA triblock copolymer gelator [where A is pH-sensitive poly(2-(diisopropylamino)ethyl methacrylate) (PDPA) and B is biocompatible poly(2-(methacryloyloxy)ethyl phosphorylcholine) (PMPC)] were used as a stimulus-responsive substrate that allows fine adjustment of the mechanical environment experienced by mouse myoblast cells. The hydrogel film elasticity could be reversibly modulated by a factor of 40 via careful pH adjustment without adversely affecting cell viability. Myoblast cells exhibited pronounced stress fiber formation and flattening on increasing the hydrogel elasticity. As a new tool to evaluate the strength of cell adhesion, we combined a picosecond laser with an inverted microscope and utilized the strong shock wave created by the laser pulse to determine the critical pressure required for cell detachment. Furthermore, we demonstrate that an abrupt jump in the hydrogel elasticity can be utilized to monitor how cells adapt their morphology to changes in their mechanical environment.