Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which expresses a...

Ring-opening metathesis polymerization or ROMP involves strained cycloalkenes as starting materials. The mechanism of ROMP proceeds by reacting cycloalkene with Grubbs catalyst to give metallacyclobutane intermediate which undergoes a ring-opening reaction to form new carbene. The new carbene reacts with another molecule of cycloalkene. Repetition of these steps leads to the formation of an unsaturated open-chain polymer product. All these steps are reversible, however, relieving the ring...

Acyclic diene metathesis polymerization or ADMET polymerization involves cross-metathesis of terminal dienes, such as 1,8-nonadiene, to give linear unsaturated polymer and ethylene. As ADMET is a reversible process, the formed ethylene gas must be removed from the reaction mixture to complete the polymerization process.
Similar to cross-metathesis, ADMET also involves the formation of metallacyclobutane intermediate by [2+2] cycloaddition of one of the double bonds of a terminal diene with...

Recently, the development of olefin metathesis polymerization advanced the field of polymer synthesis. Simply put, the reorganization of substituents on their double bonds between two olefins in the presence of a catalyst is known as the olefin metathesis reaction. The use of metathesis reaction for polymer synthesis is called olefin metathesis polymerization.
Ruthenium-based Grubbs catalyst is the most commonly used catalyst for olefin metathesis polymerization. Grubbs catalyst consists of a...

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...

Alkenes can be obtained from amines via an E2 elimination. The amine is first converted into a good leaving group, such as a quaternary ammonium salt. This is accomplished by treating the amine with an excess of alkyl halide, which results in a halide salt. Next, the halide salt is transformed into a hydroxide salt that functions as a base to enable elimination.
Under thermal conditions, the hydroxide can abstract a proton from the β carbon; this generates an alkene with the simultaneous...