Riemann Zeta Based Surge Modelling of Continuous Real Functions in Electrical Circuits

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

A Novel Biomimetic Nanocomposite Exhibiting Petal Wetting Phenomenon: Fabrication and Experimental Investigations

A functional composite material that simultaneously exhibits hydrophobicity and water droplet adhesion has monumental potential in controlling fluid flow, studying phase separation, and biological research. This article reports the fabrication of a petal wetting biomimetic Boron Nitride Nanotubes (BNNTs) -Polydimethylsiloxane (PDMS) nanocomposite achieved by drop casting. The petal effect was investigated by non-destructive techniques. The nanotubes were synthesized by chemical vapor deposition at 1150 °C and were characterized by X-ray diffraction, scanning electron microscopy, and high-resolution transmission electron microscopy. The mean diameter of the nanotubes was found to be 70 nm. The nanocomposites had BNNT fillers ranging from 0.5 wt. % to 2 wt. %. Water contact angles for pure PDMS polymer was 94.7° and for the 2 wt. % BNNT-PDMS nanocomposite was 132.4°. The petal wetting nanocomposite displayed a characteristic trait of high contact angle hysteresis. The surface roughness parameters of the nanocomposites were determined by atomic force microscopy. Laser scanning confocal microscopy aided in analyzing the droplet penetration and in observing the trapped air between the water droplet and the nanocomposite surface. Based on surface observations, roughness parameters, and the extent of droplet penetration by the surface, we shed light on the Cassie impregnating wetting regime followed by the biomimetic nanocomposite. Such a surface would be beneficial in the study of the embryogenesis of cells and aid in moisture collection.