Modular class primes in the Sundaram sieve

The purpose of this paper is to consider analogues of the twin-prime conjecture in various classes within modular rings.     

On the role of kinesthetic thinking in computational geometry

Computational geometry is a new (about 30 years) and rapidly growing branch of knowledge in computer science that deals with the analysis and design of algorithms for solving geometric problems. These problems typically arise in computer graphics, image processing, computer vision, robotics, manufacturing, knot theory, polymer physics and molecular biology. Since its inception many of the algorithms proposed for solving geometric problems, published in the literature, have been found to be incorrect. These incorrect algorithms rather than being ‘purely mathematical’ often contain a strong kinesthetic component. This paper explores the relationship between computational geometric thinking and kinesthetic thinking, the effect of the latter on the correctness and efficiency of the resulting algorithms, and their implications for education.     

Dispersion and localisation in structured Rayleigh beams

This paper brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, our main focus is on the analysis of dispersive waves, band-gaps and localised waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler–Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations.     

Structure and Polymorphism of M(thd)3 (M = Al,Cr, Mn,Fe, Co,Ga, and In)

Formation, crystal structure, polymorphism, and transition between polymorphs are reported for M(thd)₃, (M = Al, Cr, Mn, Fe, Co, Ga, and In) [(thd)^– = anion of H(thd) = C₁₁H₂₀O₂ = 2, 2, 6, 6‐tetramethylheptane‐3, 5‐dione]. Fresh crystal‐structure data are provided for monoclinic polymorphs of Al(thd)₃, Ga(thd)₃, and In(thd)₃. Apart from adjustment of the M–O_k bond length, the structural characteristics of M(thd)₃ complexes remain essentially unaffected by change of M. Analysis of the M–O_k, O_k–C_k, and C_k–C_k distances support the notion that the M–O_k–C_k–C_k–C_k–O_k– ring forms a heterocyclic unit with σ and π contributions to the bonds. Tentative assessments according to the bond‐valence or bond‐order scheme suggest that the strengths of the σ bonds are approximately equal for the M–O_k, O_k–C_k, and C_k–C_k bonds, whereas the π component of the M–O_k bonds is small compared with those for the O_k–C_k, and C_k–C_k bonds. The contours of a pattern for the occurrence of M(thd)₃ polymorphs suggest that polymorphs with structures of orthorhombic or higher symmetry are favored on crystallization from the vapor phase (viz. sublimation). Monoclinic polymorphs prefer crystallization from solution at temperatures closer to ambient. Each of the M(thd)₃ complexes subject to this study exhibits three or more polymorphs (further variants are likely to emerge consequent on systematic exploration of the crystallization conditions). High‐temperature powder X‐ray diffraction shows that the monoclinic polymorphs convert irreversibly to the corresponding rotational disordered orthorhombic variant above some 100–150 °C (depending on M). The orthorhombic variant is in turn transformed into polymorphs of tetragonal and cubic symmetry before entering the molten state. These findings are discussed in light of the current conceptions of rotational disorder in molecular crystals.     

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo–Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber–Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies–Gaffney estimates.     

Gap formulas for closed linear relations

A classical formula for the gap between two densely defined closed operators in a Hilbert space is extended to the case of linear relations. Copyright © 2014 John Wiley & Sons, Ltd.     

How to Obtain the Maximum Properties Flexibility of 3D Printed Ketoprofen Tablets Using Only One Drug-Loaded Filament?

The flexibility of dose and dosage forms makes 3D printing a very interesting tool for personalized medicine, with fused deposition modeling being the most promising and intensively developed method. In our research, we analyzed how various types of disintegrants and drug loading in poly(vinyl alcohol)-based filaments affect their mechanical properties and printability. We also assessed the effect of drug dosage and tablet spatial structure on the dissolution profiles. Given that the development of a method that allows the production of dosage forms with different properties from a single drug-loaded filament is desirable, we developed a method of printing ketoprofen tablets with different dose and dissolution profiles from a single feedstock filament. We optimized the filament preparation by hot-melt extrusion and characterized them. Then, we printed single, bi-, and tri-layer tablets varying with dose, infill density, internal structure, and composition. We analyzed the reproducibility of a spatial structure, phase, and degree of molecular order of ketoprofen in the tablets, and the dissolution profiles. We have printed tablets with immediate- and sustained-release characteristics using one drug-loaded filament, which demonstrates that a single filament can serve as a versatile source for the manufacturing of tablets exhibiting various release characteristics.