Optimality in Quantum Data Compression using Dynamical Entropy

In this article we study lossless compression of strings of pure quantum states of indeterminate-length quantum codes which were introduced by Schumacher and Westmoreland. Past work has assumed that the strings of quantum data are prepared to be encoded in an independent and identically distributed way. We introduce the notion of quantum stochastic ensembles, allowing us to consider strings of quantum states prepared in a more general way. For any identically distributed quantum stochastic ensemble we define an associated quantum Markov chain and prove that the optimal average codeword length via lossless coding is equal to the quantum dynamical entropy of the associated quantum Markov chain

Tasting Misery Among Snakes: The Situation of Smiths in Anglo-Saxon Settlements

Published version. Article studying the location of smithing sites within Anglo-Saxon settlements and the significant insight their study provides to Anglo-Saxon society

D-spaces in infinite products of ordinals

William Fleissner and Adrienne Stanley showed that, in finite products of ordinals, the following are equivalent: 1. X is a D-space. 2. X is metacompact. 3. X is metalindel¨of. 4. X does not contain a closed subset which is homeomorphic to a stationary subset of a regular, uncountable cardinal. In this paper we construct a counterexample that shows that this equivalence does not extend to infinite products of ordinals. We also introduce a new property, club-separable, which we show implies D for subsets of ωω1. We hope that club-separable will be able to replace property (4) above in order to generalize the equivalence to infinite products of ordinals

Dynamical Entropy of Quantum Random Walks

In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina)

The archaeology of community on Mabuyag (Mabuiag) in the Western Torres Strait, Northeastern Australia

This paper provides new insights into the late Holocene history of Mabuyag in western Torres Strait. It addresses a question posed by McNiven et al. (2006:75): 'at what point [did] Mabuyag became [sic] a residential island and a separate people (i.e. th

'Middle Saxon’ Settlement and Society: The Changing Rural Communities of Central and Eastern England

This thesis explores the experiences of rural communities who lived between the seventh and ninth centuries in central and eastern England. Utilising archaeological evidence as the primary source for study, the central aim of this thesis is to demonstrate the ways in which settlement remains can provide a picture of contemporary social, economic and political conditions in ‘Middle Saxon’ England. Analysis of archaeological evidence from currently-occupied rural settlements represents a particularly unique and informative dataset to accomplish this central aim, and when combined with other forms of evidence illustrates that the seventh to ninth centuries was a period of fundamental social change, that impacted rural communities in significant and lasting ways. The transformation of settlement character was part of a more widespread process of landscape investment during the ‘Middle Saxon’ period, as rapidly stratifying social institutions began to manifest power and influence through new means. Such an analysis represents a significant departure from the prevailing scholarly outlook of the early medieval landscape, which continues to posit that the countryside of England remained largely unchanged until the development of historic villages from the ninth century onward. In this regard, the evidence presented by this thesis from currently-occupied rural settlements provides substantial backing to the idea that many historic villages emerged as part of a two-stage process which began during the ‘Middle Saxon’ period. Whilst it was only following subsequent change that recognisable later village plans began to take shape, key developments between the seventh and ninth centuries helped articulate the form and identity of rural centres, features that in many instances persisted throughout the medieval period and into the present day.Arts and Humanities Research Counci

The Colonisation of Palau: preliminary results from Angaur and Ulong

The prehistory of Palau and other parts of western Micronesia has recently become important to debates about the colonisation and pattern of cultural development in the west Pacific. The main reason for this has been the suggestion that the antiquity of human occupation there might be much earlier than has been thought (e.g. Masse 1990), ilnd well before the dispersal of Lapita culture from the Bismarck Archipelago to Samo,1, between 3300 and 2850 BP (Specht and Gosden 1998; Anderson and Clark 1999). Estimates for the settlement of the Marianas now start about -1800 years BP, with Palau occupied at 4500 BP and Yap probably before 3200 BP (Dodson <rnd Intoh 1999; Wickler 2001). These older than anticipated dates (e.g. Milsse 1990) are significant because they coincide approxunately with the spreild of a eolithic cultural complex in island South East Asia chilracterised by use of rice, pig and dog, manufacture of red-slipped or paddle-impressed cernmics, along with other distinctive portable artefacts that do not occur in pre-ceramic assemblc1ges of the region (Bellwood 2001 ). Direct evidence for the earliest settlement of the Marianas, Palau and Y<1p is, however, scarce, and has been largely inferred from the analysis of sediment cores which indicates anthropogenic activity eilrlier than the archaeological record In Palau these include the presence of charcoal particles, pollen from food plants like the giant swamp taro (Cyrtosperma clw111isso11is), and an increase in savannah plants at the expense of forest growth before 4000 BP (Athens and Ward 2001; Welch 2001). While the palaeoenv1ronmental results have furnished useful alternate colonisation chronologies there is a striking absence of early sites that allow us to identify either the origin and pattern of settlement m west Micronesia, or to investigate the colonisers' connection to early Austronesian movements in Island South East Asia and the Lapita dispersal in Near and Remote Oceania. This paper summarises recent investigations undertaken on the islands of Angaur and Ulong (Fig. l) aimed at recovering early cultural materials from Palau's sequence to clarify the archipelago's colonisation history. The earliest securely d,1ted and adequately reported cultural deposits from Palau date to rn. 2300 BP (Welch 2001), and several reasons for an absence of sites older than 2500 BP have been proposed

Further radiocarbon dates from Dabangay, a mid- to late Holocene settlement site in western Torres Strait

Dabangay , on the island of Mabuyag, is one of only two known mid-Holocene sites in Torres Strait. Eleven new radiocarbon dates, combined with nine previous determinations, clarify its site formation processes and settlement history. The sequence shows two sustained settlement periods between 7239-3211 cal. BP and 1815 cal. BP-present, with little evidence for use during the intervening period. This differs from Badu 15, approximately 15 km south of Mabuyag, where human activity became sporadic after 6500 cal. BP. There is no evidence for a settlement expansion at 2500 BP as observed at other sites in the western Torres Strait. These differences suggest varied human responses to post-glacial marine transgression and the subsequent sea-level high stand in western Torres Strait