Moser-Trudinger inequality on conformal discs

We show that the Moser-Trudinger inequality holds in a conformal disc if and only if the metric is bounded from above by the Hyperbolic metric. We also find a necessary and sufficient condition for the Moser-Trudinger inequality to hold in an unbounded subset of the two dimensional Euclidean space

Multilevel Polarization of Polar Codes Over Arbitrary Discrete Memoryless Channels

It is shown that polar codes achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes. It is shown that in general, channel polarization happens in several, rather than only two levels so that the synthesized channels are either useless, perfect or "partially perfect". Any subset of the channel input alphabet which is closed under addition, induces a coset partition of the alphabet through its shifts. For any such partition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which the channel input belongs. By a slight modification of the encoding and decoding rules, it is shown that perfect transmission of certain information symbols over partially perfect channels is possible. Our result is general regarding both the cardinality and the algebraic structure of the channel input alphabet; i.e we show that for any channel input alphabet size and any Abelian group structure on the alphabet, polar codes are optimal. It is also shown through an example that polar codes when considered as group/coset codes, do not achieve the capacity achievable using coset codes over arbitrary channels

Expressing the Behavior of Three Very Different Concurrent Systems by Using Natural Extensions of Separation Logic

Separation Logic is a non-classical logic used to verify pointer-intensive code. In this paper, however, we show that Separation Logic, along with its natural extensions, can also be used as a specification language for concurrent-system design. To do so, we express the behavior of three very different concurrent systems: a Subway, a Stopwatch, and a 2x2 Switch. The Subway is originally implemented in LUSTRE, the Stopwatch in Esterel, and the 2x2 Switch in Bluespec

Design of PID Controller for Higher Order Discrete Systems Based on Order Reduction Employing ABC Algorithm

This paper proposes a new computational simple scheme for Model Order Reduction to design a discrete PID controller for higher order linear time invariant discrete systems. Artificial Bee Colony (ABC) optimization algorithm is employed for both order reduction and controller design. First a successful reduced order model is obtained for original higher order discrete system using ABC optimization algorithm which is based on the minimization of integral square error between the original and reduced order models pertaining to step input. Then a PID controller is designed for reduced order model, based on the minimization of integral square error between the desired response and actual response, pertaining to a unit step input using ABC algorithm. Finally the designed PID controller is connected to the original higher order discrete system to get the desired specifications. The validity of the proposed method is illustrated through a numerical example. Keywords: Discrete system, Model order reduction, PID controller, Integral square error, Artificial Bee Colony algorithm