Star-Shaped Ruthenium Complexes as Prototypes of Molecular Gears

The design and synthesis of two families of molecular-gear prototypes is reported, with the aim of assembling them into trains of gears on a surface and ultimately achieving controlled intermolecular gearing motion. These piano-stool ruthenium complexes incorporate a hydrotris(indazolyl)borate moiety as tripodal rotation axle and a pentaarylcyclopentadienyl ligand as star-shaped cogwheel, equipped with five teeth ranging from pseudo-1D aryl groups to large planar 2D paddles. A divergent synthetic approach was followed, starting from a pentakis(p-bromophenyl)cyclopentadienyl ruthenium(II) complex as key precursor or from its iodinated counterpart, obtained by copper-catalyzed aromatic Br/I exchange. Subsequent fivefold cross-coupling reactions with various partners allowed high structural diversity to be reached and yielded molecular-gear prototypes with aryl-, carbazole-, BODIPY- and porphyrin-derived teeth of increasing size and length.     

One-dimensional displacement of active matter on curved substrates

We analytically find the diffusion of overdamped active Brownian particles (ABPs) constrained to move along curved one-dimensional channels. The autonomous motion of these particles is achieved by a projection of their internal propulsion force along the channels' long section. In particular, the diffusion of ABPs moving on one-dimensional channels with a form of a circle, an ellipse, and a limacon of second order is analysed. To characterise the effect of substrate's geometry and self-propulsion on their diffusion, analytical expressions for the ABPs short- and long-time variances, as well as their steady angular probability density functions are offered. Curvature effects are found to reduce the time an ABP reaches its steady state. Our theoretical results are validated using Brownian dynamics simulations. This model may be relevant for experiments dealing with catalytic driven systems, bacteria, and tumour cell dispersion in one-dimensional channels.     

Ab initio study of Fe-doped SnO: Local structure and hyperfine interactions at the Fe nucleus

In this work we perform an ab initio study of the electric field gradient (EFG) at the nucleus of Fe impurities in crystalline SnO. The Augmented Plane Waves plus Local Orbitals method is used to obtain the electronic structure of the doped system and the atomic relaxations introduced by the impurities in the SnO host in a fully self-consistent way. Most calculations are performed assuming that Fe ions replace the Sn atoms of the structure, in some cases including oxygen vacancies in order to discuss their role in the hyperfine interactions and in determining the local structure around Fe impurities. The case of interstitial Fe sites is also considered. Our predictions are compared with available Müssbauer spectroscopy results and also with theoretical and experimental results obtained for rutile SnO₂ and TiO₂.     

Invariant convex sets in polar representations

We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar representations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ? p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted1 momentum map.     

The movement towards a more experimental approach to problem solving in mathematics using coding

Motivated by a problem proposed in a coding competition for secondary students, I will show on this paper how coding substantially changed the problem-solving process towards a more experimental approach.     

Free differential algebras: Their use in field theory and dual formulation

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p>1). We study here the algebra of FDA transformations. To every p-form in the FDA, we associate an extended Lie derivative l generating a corresponding gauge transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's.     

X‐ray absorption study of the local structure at the NiO/oxide interfaces

This work reports an X‐ray absorption near‐edge structure (XANES) spectroscopy study at the Ni K‐edge in the early stages of growth of NiO on non‐ordered SiO₂, Al₂O₃ and MgO thin films substrates. Two different coverages of NiO on the substrates have been studied. The analysis of the XANES region shows that for high coverages (80 Eq‐ML) the spectra are similar to that of bulk NiO, being identical for all substrates. In contrast, for low coverages (1 Eq‐ML) the spectra differ from that of large coverages indicating that the local order around Ni is limited to the first two coordination shells. In addition, the results also suggest the formation of cross‐linking bonds Ni—O—M (M = Si, Al, Mg) at the interface.     

Complete minimal hypersurfaces in quaternionic hyperbolic space

We construct new examples of embedded, complete minimal hypersurfaces in quaternionic hyperbolic space and also some minimal foliations. We introduce fans and construct analytic deformations of bisectors.     

Exact recession velocity and cosmic redshift based on cosmological principle and Yang-Mills gravity

Based on the cosmological principle and quantum Yang-Mills gravity in the super-macroscopic limit, we obtain an exact recession velocity and cosmic redshift z, as measured in an inertial frame F ≡ F(t, x, y, z). For a matter-dominated universe, we have the effective cosmic metric tensor G_(μν)(t) =(B~2(t),-A~2(t),-A~2(t),-A~2(t)),A ∝ B ∝ t~(1/2), where t has the operational meaning of time in F frame. We assume a cosmic action S ≡ S cos involving Gμν(t) and derive the ‘Okubo equation' of motion, G μν(t)?μS ?νS-m2= 0, for a distant galaxy with mass m. This cos-√mic equation predicts an exact recession velocity, ■, where H = A˙(t)/A(t) and Co = B/A, as observed in the inertial frame F. For small velocities, we have the usual Hubble's law r≈ rH for recession velocities. Following the formulation of the accelerated Wu-Doppler effect, we investigate cosmic redshifts z as measured in F. It is natural to assume the massless Okubo equation, G μν(t)?μψe?νψe= 0, for light emitted from accelerated distant galaxies. Based on the principle of limiting continuation of physical laws, we obtain a transformation for covariant wave 4-vectors between and inertial and an accelerated frame, and predict a relationship for the exact recession velocity and cosmic redshift, z = [(1 + V_r)/(1-V_r~2)~(1/2)]-1, where Vr= r˙/Co 1, as observed in the inertial frame F. These predictions of the cosmic model are consistent with experiments for small velocities and should be further tested.     

Trace level vanadium determination using thermal lens spectrophotometry

Trace level vanadium determination is reported using a dual beam thermal lens spectrometer. The thermal lens was generated using an argon ion beam laser (pump beam) which was focused into a sample cuvette. The thermal lens signal (TLs) was monitored with a He-Ne laser beam and a photodiode detector. Multichannel averager software was developed for processing the transient TLs. The optimal set up, ensuring maximum sensitivity and linear calibration graphs was obtained using experimental design techniques. Under optimized conditions, the detection limits for aqueous and ethanol-water (2+3 v/v) and (4+1 v/v) vanadium complex solutions were, respectively, 0.0071 mg/l, 0.0065 and 0.0039 mg/l.