The average amount of information lost in multiplication

We show that if X and Y are integers independently and uniformly distributed in the set {1, ..., N}, then the information lost in forming their product (which is given by the equivocation H(X,Y | XY)), is of order log log N. We also prove two extremal results regarding cases in which X and Y are not necessarily independently or uniformly distributed. First, we note that the information lost in multiplication can of course be 0. We show that the condition H(X,Y | XY) = 0 implies that 2log_2 N - H(X, Y) is of order at least log log N. Furthermore, if X and Y are independent and uniformly distributed on disjoint sets of primes, it is possible to have H(X,Y | XY) = 0 with log_2 N - H(X) and log_2 N - H(Y) each of order at most log log N. Second, we show that however X and Y are distributed, H(X,Y | XY) is of order at most log N/log log N. Furthermore, there are distributions (in which X and Y are independent and uniformly distributed over sets of numbers having only small and distinct prime factors) for which H(X,Y | XY) is of order log log N.Comment: i+11 p

In search of ‘lost’ knowledge and outsourced expertise in flood risk management

This paper examines the parallel discourses of ‘lost’ local flood expertise and the growing use of commercial consultancies to outsource aspects of flood risk work. We critically examine the various claims and counter-claims about lost, local and external expertise in flood management, focusing on the aftermath of the 2007 floods in East Yorkshire, England. Drawing on interviews with consultants, drainage engineers and others, we caution against claims that privilege ‘local’ floods knowledge as ‘good’ and expert knowledge as somehow suspect. This paper urges carefulness in interpreting claims about local knowledge, arguing that it is important always to think instead of hybrid knowledge formations. We conclude by arguing that experiments in the co-production of flood risk knowledge need to be seen as part of a spectrum of ways for producing shared knowledge

Information Loss in Quantum Gravity Without Black Holes

We use the weak field approximation to show that information is lost in principle in quantum gravity.Comment: 14pp, Late

Analysis of Large Unreliable Stochastic Networks

In this paper a stochastic model of a large distributed system where users' files are duplicated on unreliable data servers is investigated. Due to a server breakdown, a copy of a file can be lost, it can be retrieved if another copy of the same file is stored on other servers. In the case where no other copy of a given file is present in the network, it is definitively lost. In order to have multiple copies of a given file, it is assumed that each server can devote a fraction of its processing capacity to duplicate files on other servers to enhance the durability of the system. A simplified stochastic model of this network is analyzed. It is assumed that a copy of a given file is lost at some fixed rate and that the initial state is optimal: each file has the maximum number $d$ of copies located on the servers of the network. Due to random losses, the state of the network is transient and all files will be eventually lost. As a consequence, a transient $d$-dimensional Markov process $(X(t))$ with a unique absorbing state describes the evolution this network. By taking a scaling parameter $N$ related to the number of nodes of the network. a scaling analysis of this process is developed. The asymptotic behavior of $(X(t))$ is analyzed on time scales of the type $t\mapsto N^p t$ for $0\leq p\leq d{-}1$. The paper derives asymptotic results on the decay of the network: Under a stability assumption, the main results state that the critical time scale for the decay of the system is given by $t\mapsto N^{d-1}t$. When the stability condition is not satisfied, it is shown that the state of the network converges to an interesting local equilibrium which is investigated. As a consequence it sheds some light on the role of the key parameters $\lambda$, the duplication rate and $d$, the maximal number of copies, in the design of these systems

Increased salinity improves the thermotolerance of mesophilic nitrification

Nitrification is a well-studied and established process to treat ammonia in wastewater. Although thermophilic nitrification could avoid cooling costs for the treatment of warm wastewaters, applications above 40 A degrees C remain a significant challenge. This study tested the effect of salinity on the thermotolerance of mesophilic nitrifying sludge (34 A degrees C). In batch tests, 5 g NaCl L-1 increased the activity of aerobic ammonia-oxidizing bacteria (AerAOB) by 20-21 % at 40 and 45 A degrees C. For nitrite-oxidizing bacteria (NOB), the activity remained unaltered at 40 A degrees C, yet decreased by 83 % at 45 A degrees C. In a subsequent long-term continuous reactor test, temperature was increased from 34 to 40, 42.5, 45, 47.5 and 50 A degrees C. The AerAOB activity showed 65 and 37 % higher immediate resilience in the salt reactor (7.5 g NaCl L-1) for the first two temperature transitions and lost activity from 45 A degrees C onwards. NOB activity, in contrast to the batch tests, was 37 and 21 % more resilient in the salt reactor for the first two transitions, while no difference was observed for the third temperature transition. The control reactor lost NOB activity at 47.5 A degrees C, while the salt reactor only lost activity at 50 A degrees C. Overall, this study demonstrates salt amendment as a tool for a more efficient temperature transition for mesophilic sludge (34 A degrees C) and eventually higher nitrification temperatures

A treasure lost : the Portocarrero collection of scientific instruments and interest in the sciences in hospitaller Malta

One of the little-known treasures acquired by the Order of St John during the eighteenth century and brought over to Malta consisted of a collection of scientific and mathematical instruments. This impressive collection, which was added to during the succeeding years, eventually vanished without a trace. Further evidence of scientific pursuits in areas such as aerostatics, astronomy, climatic observations and natural history confirm a growing local interest in the various sciences, itself reflecting contemporary developments in Europe.peer-reviewe

Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them.Comment: Final version, some typos were corrected. 21 pages, uses svmult.cl

Asymptotic behavior of odd-even partitions

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions

Scaling: Lost in the smog

In this commentary we discuss the validity of scaling laws and their relevance for understanding urban systems and helping policy makers. We show how the recent controversy about the scaling of CO2 transport-related emissions with population size, where different authors reach contradictory conclusions, is symptomatic of the lack of understanding of the underlying mechanisms. In particular, we highlight different sources of errors, ranging from incorrect estimate of CO2 to problems related with the definition of cities. We argue here that while data are necessary to build of a new science of cities, they are not enough: they have to go hand in hand with a theoretical understanding of the main processes. This effort of building models whose predictions agree with data is the prerequisite for a science of cities. In the meantime, policy advice are, at best, a shot in the dark.Comment: 2 pages, 1 figur