In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.

The electrochemical CO2 reduction reaction (CO2RR) to yield high-value added fuels and chemicals provides a promising approach towards global carbon neutrality. Constant endeavors have been devoted the exploration of high-efficiency catalyst with rapid kinetics, low energy input, high selectivity. In addition maximum metal atomic utilization unique catalytic performance single-atom (SAC), dual-...

The morphological features of α-synuclein (AS) amyloid aggregation in vitro and in cells were elucidated at the nanoscale by far-field subdiffraction fluorescence localization microscopy. Labeling AS with rhodamine spiroamide probes allowed us to image AS fibrillar structures by fluorescence stochastic nanoscopy with an enhanced resolution at least 10-fold higher than that achieved with convent...

Atomic Bose-Einstein condensates confined to a dual-ring trap support Josephson vortices as topologically stable defects in the relative phase. We propose a test of the scaling laws for defect formation by quenching a Bose gas to degeneracy in this geometry. Stochastic Gross-Pitaevskii simulations reveal a -1/4 power-law scaling of defect number with quench time for fast quenches, consistent wi...

Since 1990, the integer quantum Hall effect has provided the electrical resistance standard, and there has been a firm belief that the measured quantum Hall resistances are described only by fundamental physical constants--the elementary charge e and the Planck constant h. The metrological application seems not to rely on detailed knowledge of the microscopic picture of the quantum Hall effect;...