On Infinite Series of Bessel functions of the First Kind: ∑νJNν+p(x),∑ν(−1)νJNν+p(x)\sum_\nu J_{N\nu+p}(x), \sum_\nu (-1)^\nu J_{N\nu+p}(x)

Infinite series of Bessel function of the first kind, $\sum_\nu^{\pm\infty} J_{N\nu+p}(x)$, $\sum_\nu^{\pm\infty} (-1)^\nu J_{N\nu+p}(x)$, are summed in closed form. These expressions are evaluated by engineering a Dirac comb that selects specific sequences within the Bessel series

Nanoscale assembly processes revealed in the nacroprismatic transition zone of Pinna nobilis mollusc shells

Intricate biomineralization processes in molluscs engineer hierarchical structures with meso-, nano-, and atomic architectures that give the final composite material exceptional mechanical strength and optical iridescence on the macroscale. This multiscale biological assembly inspires new synthetic routes to complex materials. Our investigation of the prism-nacre interface reveals nanoscale details governing the onset of nacre formation using high-resolution scanning transmission electron microscopy. A wedge polishing technique provides unprecedented, large-area specimens required to span the entire interface. Within this region, we find a transition from nanofibrillar aggregation to irregular early-nacre layers, to well-ordered mature nacre suggesting the assembly process is driven by aggregation of nanoparticles (~50-80 nm) within an organic matrix that arrange in fiber-like polycrystalline configurations. The particle number increases successively and, when critical packing is reached, they merge into early-nacre platelets. These results give new insights into nacre formation and particle-accretion mechanisms that may be common to many calcareous biominerals.Comment: 5 Figure