Random binning is a widely used technique in information theory with diverse applications. In this paper, we focus on the output statistics of random binning (OSRB) using the Tsallis divergenceTα. We analyze all values of α ∈ (0,∞)∪{∞} and consider three scenarios: (i) the binned sequence is generated i.i.d., (ii) the sequence is randomly chosen from an ϵ-typical set, and (iii) the sequence originates from an ϵ-typical set and is passed through a non-memoryless virtual channel. Our proofs cover both achievability and converse results. To address the unbounded nature ofT∞, we extend the OSRB framework via Rényi’s divergence with order infinity, denotedD∞. As part of our exploration, we analyze a specific form of Rényi’s conditional entropy and its properties. Additionally, we demonstrate the application of this framework in deriving achievability results for the wiretap channel, where Tsallis divergence serves as a security measure. The secure rate we obtain through the OSRB analysis matches the secure capacity for α ∈ (0, 2] ∪ {∞} and serves as a potential candidate for the secure capacity when α ∈ (2,∞).

Random binning is a widely utilized tool in information theory, particularly for proving achievability bounds. In this paper, we investigate the output statistics of random binning (OSRB) for two cases: where the binned sequence is i.i.d. generated, and randomly chosen from an$\epsilon$-typical set using the Tsallis divergence$T_{\alpha}$measure for all values of$\alpha\in(0, \infty)$. For$\alpha=\infty$, due to the unbounded nature of$T_{\infty}$, we analyze the OSRB framework using Rényi's divergence criterion with the order of infinity, denoted as$D_{\infty}$. While exploring OSRB, we encounter a specific form of Renyi's conditional entropy and delve into its properties. Additionally, we demonstrate the effectiveness of this framework in establishing achievability results for wiretap channels, where Tsallis divergence serves as a security measure. The secure rate we obtain is equal to the capacity for$\alpha\in(0.2]$.

Thr design of a new algorithm for the simplification of expressions which are polynomial in the Riemann tensor [11] and its covariant derivatives is described.The simplification of such indicial expressions is complex and should be based on nondeterministic algorithms, Due to the large number ofpossible terms the nondeterministic algorithms should be complemented by stochastic methods to control the size of the search space, We show that appropriately implemented genetic algorithms can provide solutions to these problems.Thealgorithms have been implemented inthe MapleTensor system for performing indicial and component tensor calculations using symbolic computation.

Progress on the development of a new system for performing indicial and component tensor calculations is described.The system uses some of the facilities of the Maple Computer Algebra System.We discuss our approach to the representation of tensors and the capabilities of the system in handling tensorial quantities.Some of the unique features of the system are the natural and straightforward representations of tensors, the existence of a tensor database with some heuristic methods to browse the database and the flexibility of the knowledge representation in the database in order to handle different gravitational theories.