Stable and Holomorphic Implementation of Complex Functions Using a Unit Circle-Based Transform

A stable and holomorphic implementation of complex functions in ℂ plane making use of a unit circle-based transform is presented in this paper. In this method, any complex number or function can be represented as an infinite series sum of progressive products of a base complex unit and its conjugate only, where both are defined inside the unit circle. With each term in the infinite progression lying inside the unit circle, the sum ultimately converges to the complex function under consideration. Since infinitely large number of terms are present in the progression, the first element of which may be deemed as the base unit of the given complex number, it is addressed as complex baselet so that the complex number or function is termed as the complex baselet transform. Using this approach, various fundamental operations applied on the original complex number in ℂ are mapped to equivalent operations on the complex baselet inside the unit circle, and results are presented. This implementation has unique properties due to the fact that the constituent elements are all lying inside the unit circle. Out of numerous applications, two cases are presented: one of a stable implementation of an otherwise unstable system and the second case of functions not satisfying Cauchy–Riemann equations thereby not holomorphic in ℂ plane, which are made complex differentiable using the proposed transform-based implementation. Various lemmas and theorems related to this approach are also included with proofs.

Picosecond semiconductor generator for capacitive sensors calibration

The paper describes a semiconductor picosecond pulse generator that can be used to calibrate capacitive high voltage sensors of MV range. The generator is designed as a base unit, to which external pulse converters are connected. In the base unit, semiconductor devices – first a semiconductor opening switch (SOS) and then a semiconductor sharpener (SS) – generate an output pulse with a rise time of 220 ps and a subsequent flat-top of 2 ns in duration. The pulse amplitude is around 1 kV across 50 Ω load. An external diode sharpener generates a pulse with 120 ps rise time and 500-ps flat-top at the amplitude of 850 V. To switch the semiconductor sharpeners to the conducting state, the shock-ionization wave mode is used. Additional pulse converters make it possible to generate output pulses across 50 Ω load with the rise time of 70-150 ps, the pulse duration of 135-310 ps, and the amplitude of 130–480 V. The electrical diagram of the generator and waveforms of the output pulses are presented. An example of the calibration of capacitive sensors of a multi-gigawatt picosecond generator is also shown.

An efficient water-soluble fluorescent chemosensor based on furan Schiff base functionalized PEG for the sensitive detection of Al3+ in pure aqueous solution

An efficient reversible fluorescent chemosensor, PEGFB, based on polyethylene glycol bearing a furan Schiff base unit has been successfully developed to sensitively detect Al3+ in pure aqueous solution.