Physical laws manifest themselves through the amalgamation of mathematical symbols, numbers, functions, geometries, and relationships. These intricate combinations unfold within a mathematical model devised to capture and represent the ``objective reality'' of the system under examination. In this symbiotic relationship between physics and mathematics, the language of mathematics becomes a powerful tool for describing and predicting the behavior of the physical world. The language used and the associated concepts are in a perpetual state of evolution, mirroring the ongoing expansion of the phenomena accessible to our scientific understanding. In this contribution, written in honor of Richard Kerner, we delve into fundamental, at times seemingly elementary, elements of the mathematical language inherent to the physical sciences, guided by the overarching principles of symmetry and group theory. Our focus turns to the captivating realm of spheres, those strikingly symmetric entities that manifest prominently within our geometric landscape. By exploring the interplay between mathematical abstraction and the tangible beauty of symmetry, we seek to deepen our understanding of the underlying structures that govern our interpretation of the physical world.

Semiconductor photocatalytic performances can be modulated through morphology modification. Herein porous hierarchical BiOBr microspheres (BiOBr-MS) of ~3 m was firstly self-assembled without the assistance of a template via a facile solvothermal synthesis in triethylene glycol (TEG) at 150 oC for 3 h. KBrO3 was exploited as a bromine source, which slowly provided bromide ions upon reduction in TEG and controlled the growth and self-assembly of primary BiOBr nanoplates. The addition of PVP during solvothermal synthesis of BiOBr-MS reduced the particle size by about three-fold to generate BiOBr sub-microspheres (BiOBr-sMS) of <1 m. BiOBr-sMS exhibited significantly higher photocatalytic activity than BiOBr-MS for aerobic photooxidation of benzyl alcohol (BzOH) to benzaldehyde (BzH) under simulated sunlight irradiation (conversions of BzOH (50 mM) over BiOBr-sMS and BiOBr-MS were respectively 51.3% and 29.6% with 100% selectivity to BzH after Xenon illumination for 2 h at 25 oC). The photogenerated holes and •O2- were found to be main reactive species for the BzOH oxidation over BiOBr spheres by scavenging tests and spin-trapping EPR spectra. The higher photocatalytic activity of BiOBr-sMS was attributed to its more open hierarchical structure, efficient charge separation, more negative conduction band position and generation of larger amounts of •O2-.

The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, Δs, between the excess entropy per particle (relative to an ideal gas with the same temperature and density), s_ex, and the pair-correlation contribution, s₂. Thus, the RMPE represents the net contribution to s_ex due to spatial correlations involving three, four, or more particles. A heuristic “ordering” criterion identifies the vanishing of the RMPE as an underlying signature of an impending structural or thermodynamic transition of the system from a less ordered to a more spatially organized condition (freezing is a typical example). Regardless of this, the knowledge of the RMPE is important to assess the impact of non-pair multiparticle correlations on the entropy of the fluid. Recently, an accurate and simple proposal for the thermodynamic and structural properties of a hard-sphere fluid in fractional dimension 1 < d < 3 has been proposed [Santos, A.; López de Haro, M. Phys. Rev. E 2016, 93, 062126]. The aim of this work is to use this approach to evaluate the RMPE as a function of both d and the packing fraction ϕ. It is observed that, for any given dimensionality d, the RMPE takes negative values for small densities, reaches a negative minimum Δs_min at a packing fraction ϕ_min, and then rapidly increases, becoming positive beyond a certain packing fraction ϕ₀. Interestingly, while both ϕ_min and ϕ₀ monotonically decrease as dimensionality increases, the value of Δs_min exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality d ≃ 2.38. A plot of the scaled RMPE Δs/|Δs_min| shows a quasiuniversal behavior in the region −0.14 ≲ ϕ − ϕ₀ ≲ 0.02.

Three-dimensional V₂O₅ hollow structures have been prepared through a simple synthesis strategy combining solvothermal treatment and a subsequent thermal annealing. The V₂O₅ materials are composed of microspheres 2–3 μm in diameter and with a distinct hollow interior. The as-synthesized V₂O₅ hollow microspheres, when evaluated as a cathode material for lithium-ion batteries, can deliver a specific capacity as high as 273 mAh·g⁻¹ at 0.2 C. Benefiting from the hollow structures that afford fast electrolyte transport and volume accommodation, the V₂O₅ cathode also exhibits a superior rate capability and excellent cycling stability. The good Li-ion storage performance demonstrates the great potential of this unique V₂O₅ hollow material as a high-performance cathode for lithium-ion batteries.

Hierarchical micro-/mesoporous graphitic carbon spheres (HGCS) with a uniform diameter of ~0.35 μm were synthesized by Fe-catalyzed graphitization of amorphous carbon spheres resultant from hydrothermal carbonization. The HGCS resultant from 3 h at 900 °C with 1.0 wt% Fe catalyst had high graphitization degree and surface area as high as 564 m²/g. They also exhibited high specific capacitance of 140 F/g at 0.2 A/g, good electrochemical stability with 94% capacitance retention after consecutive 2500 cycles. The graphitization degree of the HGCS contributed about 60% of their specific capacitance, and their specific capacitance per unit surface area was as high as 0.2 F/m² which was much higher than in the most cases of porous amorphous carbon materials reported before. In addition, the HGCS showed a high adsorption capacity of 182.8 mg/g for methylene blue (MB), which was 12 times as high as that in the case of carbon spheres before graphitization.

In case of the maximally rotating Black Holes (BH) through Kerr-Neumann frames, or as described in Boyer-Lindquist coordinates metrics, the rotation axis of the BHs inputs a frame dragging effect i.e., relativistically a Lens-Thirring Precession that accelerates the photon trajectories oscillates with a shaped induced rotations through the ring singularity, between alternate universes, as a means of an induced geodesics that takes a sharp turning points back and forth provided, in the prograde photon sphere, due to magnetorotational instability, the path tracing of a photons circulates as a smooth fiber bundles over the event horizon curves, that when gets interpolate between mixed trajectories behaves as a geodesics and thus forms a smooth Jacobi-fields through Jacobi-lines by mutual intersection of geodesics over the photon sphere which when somehow gets leaked inside the event horizon, then gets sucked in with not sufficient escape velocity for retardation and trapped in the compact singularity, oscillating back and forth through alternate universes.

Starting from Thomas Friedrich’s work “Weak Spin(9) structures on 16-dimensional Riemannian manifolds”, we review several interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry, the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.