Newman’s identity and infinite families of congruences modulo 7 for broken 3-diamond partitions

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu presented some conjectures on congruences modulo 7 for \(\Delta _3(n)\) which were proved by Jameson and Xiong based on the theory of modular forms. Very recently, Xia proved several infinite families of congruences modulo 7 for \(\Delta _3(n)\) using theta function identities. In this paper, many new infinite families of congruences modulo 7 for \(\Delta _3(n)\) are derived based on an identity of Newman and the (p; k)-parametrization of theta functions due to Alaca, Alaca and Williams. In particular, some non-standard congruences modulo 7 for \(\Delta _3(n)\) are deduced. For example, we prove that for \(\alpha \ge 0\), \(\Delta _3\left( \frac{14\times 757^{\alpha }+1}{3}\right) \equiv 6 -\alpha \ (\mathrm{mod}\ 7)\).     

Effect of pH on H2O2 production in the radiolysis of water

The yields of hydrogen peroxide have been measured in the radiolysis of aqueous solutions of acrylamide, bromide, nitrate, and air in the pH range of 1-13. Hydrogen peroxide is the main stable oxidizing species formed in the radiolysis of water, and its long-term yield is found to be very sensitive to the system used in the measurements. Experiments with γ-irradiation combined with model calculations show that the primary yields of hydrogen peroxide are nearly independent of pH in the range of 2-12. Slightly higher primary yields are suggested at very low pH in particular when O(2) is present, while the yields seem to decrease at very high pH. Irradiations were performed with 5 MeV H ions, 5 MeV He ions, and 10 MeV C ions to evaluate the intratrack and homogeneous kinetic contributions to H(2)O(2) formation with different ions. Many of the trends in hydrogen peroxide yields with pH observed with γ-irradiations are observed with irradiation by the heavy ions. The lower yields of radicals in the homogeneous phase with the heavier ions tend to minimize the effects of radicals on the hydrogen peroxide yields at long times.     

A Water Molecule Triggers Guest Exchange at a Mono-Zinc Centre Confined in a Biomimetic Calixarene Pocket: a Model for Understanding Ligand Stability in Zn Proteins

In this study, the ligand exchange mechanism at a biomimetic Zn^II centre, embedded in a pocket mimicking the possible constrains induced by a proteic structure, is explored. The residence time of different guest ligands (dimethylformamide, acetonitrile and ethanol) inside the cavity of a calix[6]arene-based tris(imidazole) tetrahedral zinc complex was probed using 1D EXchange SpectroscopY NMR experiments. A strong dependence of residence time on water content was observed with no exchange occurring under anhydrous conditions, even in the presence of a large excess of guest ligand. These results advocate for an associative exchange mechanism involving the transient exo-coordination of a water molecule, giving rise to 5-coordinate Zn^II intermediates, and inversion of the pyramid at the Zn^II centre. Theoretical modelling by DFT confirmed that the associative mechanism is at stake. These results are particularly relevant in the context of the understanding of kinetic stability/lability in Zn proteins and highlight the key role that a single water molecule can play in catalysing ligand exchange and controlling the lability of Zn^II in proteins.     

Thermodynamics and Spectroscopy of Halogen- and Hydrogen-Bonded Complexes of Haloforms with Aromatic and Aliphatic Amines

Similarities and differences of halogen and hydrogen bonding were explored via UV–Vis and ¹H NMR measurements, X-ray crystallography and computational analysis of the associations of CHX₃ (X=I, Br, Cl) with aromatic (tetramethyl-p-phenylenediamine) and aliphatic (4-diazabicyclo[2,2,2]octane) amines. When the polarization of haloforms was taken into account, the strengths of these complexes followed the same correlation with the electrostatic potentials on the surfaces of the interacting atoms. However, their spectral properties were quite distinct. While the halogen-bonded complexes showed new intense absorption bands in the UV–Vis spectra, the absorptions of their hydrogen-bonded analogues were close to the superposition of the absorption of reactants. Additionally, halogen bonding led to a shift in the NMR signal of haloform protons to lower ppm values compared with the individual haloforms, whereas hydrogen bonding of CHX₃ with aliphatic amines resulted in a shift in the opposite direction. The effects of hydrogen bonding with aromatic amines on the NMR spectra of haloforms were ambivalent. Titration of all CHX₃ with these nucleophiles produced consistent shifts in their protons’ signals to lower ppm values, whereas calculations of these pairs produced multiple hydrogen-bonded minima with similar structures and energies, but opposite directions of the NMR signals’ shifts. Experimental and computational data were used for the evaluation of formation constants of some halogen- and hydrogen-bonded complexes between haloforms and amines co-existing in solutions.     

An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

Let \(\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}\) be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let \(\mathcal {X}\) be a certain Banach function space of X-valued functions. We prove that the growth bound of the family \(\mathcal {U}\) is less than or equal to \(-\frac{1}{c(\mathcal {U}, \mathcal {X})}\) provided that the convolution operator \(f\mapsto \mathcal {U}*f\) acts on \(\mathcal {X}.\) It is well known that under the latter assumption, the convolution operator is bounded and then \(c(\mathcal {U}, \mathcal {X})\) denotes (ad-hoc) its norm in \(\mathcal {L}(\mathcal {X}).\) As a consequence, we prove that if \(\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,\) then \(\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.\) Finally, we give an example showing that the accuracy of the estimates may be quite accurate.     

Scientific investigations of the Imperial Gates from the Petrindu wooden church,Salaj County,Romania

In order to preserve and restore the Imperial Gates from the Petrindu wooden church, Salaj County, Romania, (map location—latitude, longitude: 46.97, 23.19), the scientific investigation of the wooden support and painting materials (ground and pigments) was performed, employing Fourier‐transform infrared spectroscopy (FTIR), X‐ray fluorescence spectroscopy, and 3D scanning. FTIR spectroscopy offered information about the wooden degradation stage, whereas X‐ray fluorescence and FTIR spectroscopic methods were employed for structural painting materials characterization. The structural data can be correlated with the artistic, theological, and historical analysis of this religious patrimony object. After obtaining information about the wooden support and painting materials, the Imperial Gates were 3D digitized using state of the art laser scanning technology. The digital 3D model obtained was restored in a virtual environment and converted into an interactive 3D model that can be used for Romanian cultural heritage digital dissemination.     

Implicit Regularization and Renormalization of QCD

We apply the Implicit Regularization (IR) technique in a non-abelian gauge theory. We show that IR preserves gauge symmetry as encoded in relations between the renormalization constants required by the Slavnov–Taylor identities at the one-loop level of QCD. Moreover, we show that the technique handles divergencies in massive and massless QFT on equal footing.     

The strong perfect graph conjecture holds for diamonded odd cycle-free graphs

In this paper, we show that the strong perfect graph conjecture holds for a new class of graphs which we call diamonded odd cycle-free graphs. The class of diamonded odd cycle-free graphs contains the classes of threshold graphs and K₄\e-free graphs.