A Simple Proof of Fast Polarization

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

A Simple Combinatorial Proof of

A Simple Proof of The

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß’ 2F1 summation and elementary series manipulations to give a simple proof of Dougall’s 2H2 summation. Similarly, we apply Rogers’ nonterminating 6φ5 summation and elementary series manipulations to give a simple proof of Bailey’s very-well-poised 6ψ6 summ...

A geometric proof of the polarization property

We analyze one version of the successive cancellation (SC) decoding that uses two random functions of the transmitted symbols: their likelihoods and variations of posterior probabilities. The first function increases its expected value on the upgrading channels, while the second does so on the degrading channels. We show that both quantities can be bounded by sin θ and cos θ of another random v...

designing a fast locking pll

a phase-locked loop (pll) based frequency synthesizer is an important circuit that is used in many applications, especially in communication systems such as ethernet receivers, disk drive read/write channels, digital mobile receivers, high-speed memory interfaces, system clock recovery and wireless communication system. other than requiring good signal purity such as low phase noise and low spu...