q-Identities from Lagrange and Newton Interpolation

Combining Newton and Lagrange interpolation, we give $q$-identities which generalize results of Van Hamme, Uchimura, Dilcher and Prodinger

Partition Analysis and Symmetrizing Operators

Using a symmetrizing operator, we give a new expression for the Omega operator used by MacMahon in Partition Analysis, and given a new life by Andrews and his coworkers. Our result is stated in terms of Schur functions.Comment: 5 page

Beyond the Min-Cut Bound: Deterministic Network Coding for Asynchronous Multirate Broadcast

In a single hop broadcast packet erasure network, we demonstrate that it is possible to provide multirate packet delivery outside of what is given by the network min-cut. This is achieved by using a deterministic non-block-based network coding scheme, which allows us to sidestep some of the limitations put in place by the block coding model used to determine the network capacity. Under the network coding scheme we outline, the sender is able to transmit network coded packets above the channel rate of some receivers, while ensuring that they still experience nonzero delivery rates. Interestingly, in this generalised form of asynchronous network coded broadcast, receivers are not required to obtain knowledge of all packets transmitted so far. Instead, causal feedback from the receivers about packet erasures is used by the sender to determine a network coded transmission that will allow at least one, but often multiple receivers, to deliver their next needed packet. Although the analysis of deterministic coding schemes is generally a difficult problem, by making some approximations we are able to obtain tractable estimates of the receivers' delivery rates, which are shown to match reasonably well with simulation. Using these estimates, we design a fairness algorithm that allocates the sender's resources so all receivers will experience fair delivery rate performance

Dynamic Rate Adaptation for Improved Throughput and Delay in Wireless Network Coded Broadcast

In this paper we provide theoretical and simulation-based study of the delivery delay performance of a number of existing throughput optimal coding schemes and use the results to design a new dynamic rate adaptation scheme that achieves improved overall throughput-delay performance. Under a baseline rate control scheme, the receivers' delay performance is examined. Based on their Markov states, the knowledge difference between the sender and receiver, three distinct methods for packet delivery are identified: zero state, leader state and coefficient-based delivery. We provide analyses of each of these and show that, in many cases, zero state delivery alone presents a tractable approximation of the expected packet delivery behaviour. Interestingly, while coefficient-based delivery has so far been treated as a secondary effect in the literature, we find that the choice of coefficients is extremely important in determining the delay, and a well chosen encoding scheme can, in fact, contribute a significant improvement to the delivery delay. Based on our delivery delay model, we develop a dynamic rate adaptation scheme which uses performance prediction models to determine the sender transmission rate. Surprisingly, taking this approach leads us to the simple conclusion that the sender should regulate its addition rate based on the total number of undelivered packets stored at the receivers. We show that despite its simplicity, our proposed dynamic rate adaptation scheme results in noticeably improved throughput-delay performance over existing schemes in the literature.Comment: 14 pages, 15 figure

Semi-Finite Forms of Bilateral Basic Hypergeometric Series

We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1\psi_1$ summation, Bailey's $_2\psi_2$ transformations, and Bailey's $_6\psi_6$ summation.Comment: 8 pages. accepted by Proc. Amer. Math. So