A noncommutative extended de Finetti theorem

The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and K\"ummerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.Comment: 44 page

A Remote Interface for Live Interaction with OMNeT++ Simulations

Discrete event simulators, such as OMNeT++, provide fast and convenient methods for the assessment of algorithms and protocols, especially in the context of wired and wireless networks. Usually, simulation parameters such as topology and traffic patterns are predefined to observe the behaviour reproducibly. However, for learning about the dynamic behaviour of a system, a live interaction that allows changing parameters on the fly is very helpful. This is especially interesting for providing interactive demonstrations at conferences and fairs. In this paper, we present a remote interface to OMNeT++ simulations that can be used to control the simulations while visualising real-time data merged from multiple OMNeT++ instances. We explain the software architecture behind our framework and how it can be used to build demonstrations on the foundation of OMNeT++.Comment: Published in: A. Foerster, A. Udugama, A. Koensgen, A. Virdis, M. Kirsche (Eds.), Proc. of the 4th OMNeT++ Community Summit, University of Bremen - Germany - September 7-8, 201

Noncommutative Independence in the Infinite Braid and Symmetric Group

This is an introductory paper about our recent merge of a noncommutative de Finetti type result with representations of the infinite braid and symmetric group which allows to derive factorization properties from symmetries. We explain some of the main ideas of this approach and work out a constructive procedure to use in applications. Finally we illustrate the method by applying it to the theory of group characters

Analysis of Static and Dynamic Configurability of Existing Group Communication Systems

Active replication following the state machine replication (SMR) approach is a way to make existing systems and services more reliable and fault-tolerant. The additional communication overhead has a negative impact on the system's throughput and overall request latency. Today's systems should be highly optimized to their execution environment and usage scenario in order to remedy the performance loss introduced by such group communication systems (GCS). In addition to that, systems should be able to adapt to changing environmental conditions. This report analyzes the available configuration options of three existing GCSs. Therefore, it explains the available configuration parameters and describes the given reconfiguration mechanisms. The found parameters are then classified in a parameter scheme.Comment: Technical Report (38 pages

On the structure of non-commutative white noises

We consider the concepts of continuous Bernoulli systems and non-commutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum L{\'e}vy processes one can make quite precise statements about the time behaviour of their moments.Comment: 17 page

Tail algebras of quantum exchangeable random variables

We show that any countably generated von Neumann algebra with specified normal faithful state can arise as the tail algebra of a quantum exchangeable sequence of noncommutative random variables. We also characterize the cases when the state corresponds to a limit of convex combinations of free products states.Comment: The revised version includes more detailed exposition in section

Reliable Wireless Multi-Hop Networks with Decentralized Slot Management: An Analysis of IEEE 802.15.4 DSME

Wireless communication is a key element in the realization of the Industrial Internet of Things for flexible and cost-efficient monitoring and control of industrial processes. Wireless mesh networks using IEEE 802.15.4 have a high potential for executing monitoring and control tasks with low energy consumption and low costs for deployment and maintenance. However, conventional medium access techniques based on carrier sensing cannot provide the required reliability for industrial applications. Therefore, the standard was extended with techniques for time-slotted medium access on multiple channels. In this paper, we present openDSME, a comprehensive implementation of the Deterministic and Synchronous Multi-channel Extension (DSME) and propose a method for traffic-aware and decentralized slot scheduling to enable scalable wireless industrial networks. The performance of DSME and our implementation is demonstrated in the OMNeT++ simulator and on a physically deployed wireless network in the FIT/IoT-LAB. It is shown that in the given scenarios, twice as much traffic can be delivered reliably by using DSME instead of CSMA/CA and that the energy consumption can be reduced significantly. The paper is completed by presenting important trade-offs for parameter selection and by uncovering open issues of the current specification that call for further effort in research and standardization.Comment: 27 pages, 18 figure

Noncommutative continuous Bernoulli shifts

We introduce a non-commutative extension of Tsirelson-Vershik's noises, called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory. Such shifts are, in particular, capable of producing Arveson's product system of type I and type II. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (`classical') stationary flows for Tsirelson-Vershik's noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Levy processes. They lead, in particular, to units and `logarithms' of units in Arveson's product systems. Furthermore, we introduce (non-commutative) white noises, which are operator algebraic versions of Tsirelson's `classical' noises. We give examples coming from probability, quantum probability and from Voiculescu's theory of free probability. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: non-commutative extensions of stochastic Ito integration, stochastic logarithms and exponentials.Comment: 87 page

Semi-Cosimplicial Objects and Spreadability

To a semi-cosimplicial object (SCO) in a category we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid $\mathbb{B}^+_\infty$ and provide examples. In categories of (noncommutative) probability spaces SCOs correspond to spreadable sequences of random variables, hence SCOs can be considered as the algebraic structure underlying spreadability.Comment: 20 pages, minor changes (1.2, 2.9, 4.3) in (v3), to be published in: Rocky Mountain Journal of Mathematic

On Lehner's `free' noncommutative analogue of De Finetti's theorem

Inspired by Lehner's results on exchangeability systems we define `weak conditional freeness' and `conditional freeness' for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process.Comment: 9 page