Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that .

Fast median calculation method

The ever increasing demand for high image quality requires fast and efficient methods for noise reduction. The best-known order-statistics filter is the median filter. A method is presented to calculate the median on a set of N W-bit integers in W/B time steps. Blocks containing B-bit slices are used to find B-bits of the median; using a novel quantum-like representation allowing the median to be computed in an accelerated manner compared to the best-known method (W time steps). The general method allows a variety of designs to be synthesised systematically. A further novel architecture to calculate the median for a moving set of N integers is also discussed.

In-situ comparison of the NOy instruments flown in MOZAIC and SPURT

Two aircraft instruments for the measurement of total odd nitrogen (NOy) were compared side by side aboard a Learjet A35 in April 2003 during a campaign of the AFO2000 project SPURT (Spurengastransport in der Tropopausenregion). The instruments albeit employing the same measurement principle (gold converter and chemiluminescence) had different inlet configurations. The ECO-Physics instrument operated by ETH-Zürich in SPURT had the gold converter mounted outside the aircraft, whereas the instrument operated by FZ-Jülich in the European project MOZAIC III (Measurements of ozone, water vapour, carbon monoxide and nitrogen oxides aboard Airbus A340 in-service aircraft) employed a Rosemount probe with 80 cm of FEP-tubing connecting the inlet to the gold converter. The NOy concentrations during the flight ranged between 0.3 and 3 ppb. The two data sets were compared in a blind fashion and each team followed its normal operating procedures. On average, the measurements agreed within 7%, i.e. within the combined uncertainty of the two instruments. This puts an upper limit on potential losses of HNO3 in the Rosemount inlet of the MOZAIC instrument. Larger transient deviations were observed during periods after calibrations and when the aircraft entered the stratosphere. The time lag of the MOZAIC instrument observed in these instances is in accordance with the time constant of the MOZAIC inlet line determined in the laboratory for HNO3.

An experiment on spatial competition with endogenous pricing

Hotelling's (1929) principle of minimum differentiation and the alternative prediction that firms will maximally differentiate from their rivals in order to relax price competition have not been explicitly tested so far. We report results from experimental spatial duopolies designed to address this issue. The levels of product differentiation observed are systematically lower than predicted in equilibrium under risk neutrality and compatible with risk aversion. The observed prices are consistent with collusion attempts. Our main findings are robust to variations in three experimental conditions: automated vs. human market sharing rule for ties, individual vs. collective decision making, and even vs. odd number of locations.

Software-based study of a non-sorting method for median calculation on a set of integer numbers

This paper presents a software-based study of a hardware-based non-sorting median calculation method on a set of integer numbers. The method divides the binary representation of each integer element in the set into bit slices in order to find the element located in the middle position. The method exhibits a linear complexity order and our analysis shows that the best performance in execution time is obtained when slices of 4-bit in size are used for 8-bit and 16-bit integers, in mostly any data set size. Results suggest that software implementation of bit slice method for median calculation outperforms sorting-based methods with increasing improvement for larger data set size. For data set sizes of N > 5, our simulations show an improvement of at least 40%.

Median architecture by accumulative parallel counters

The time to process each of W/B processing blocks of a median calculation method on a set of N W-bit integers is improved here by a factor of three compared to the literature. Parallelism uncovered in blocks containing B-bit slices are exploited by independent accumulative parallel counters so that the median is calculated faster than any known previous method for any N, W values. The improvements to the method are discussed in the context of calculating the median for a moving set of N integers for which a pipelined architecture is developed. An extra benefit of smaller area for the architecture is also reported.

Long-term variations in the magnetic fields of the Sun and the heliosphere: their origin, effects, and implications

Recent studies of the variation of geomagnetic activity over the past 140 years have quantified the "coronal source" magnetic flux F-s that leaves the solar atmosphere and enters the heliosphere and have shown that it has risen, on average, by an estimated 34% since 1963 and by 140% since 1900. This variation of open solar flux has been reproduced by Solanki et al. [2000] using a model which demonstrates how the open flux accumulates and decays, depending on the rate of flux emergence in active regions and on the length of the solar cycle. We here use a new technique to evaluate solar cycle length and find that it does vary in association with the rate of change of F-s in the way predicted. The long-term variation of the rate of flux emergence is found to be very similar in form to that in F-s, which may offer a potential explanation of why F-s appears to be a useful proxy for extrapolating solar total irradiance back in time. We also find that most of the variation of cosmic ray fluxes incident on Earth is explained by the strength of the heliospheric field (quantified by F-s) and use observations of the abundance of the isotope Be-10 (produced by cosmic rays and deposited in ice sheets) to study the decrease in F-s during the Maunder minimum. The interior motions at the base of the convection zone, where the solar dynamo is probably located, have recently been revealed using the helioseismology technique and found to exhibit a 1.3-year oscillation. This periodicity is here reported in observations of the interplanetary magnetic field and geomagnetic activity but is only present after 1940, When present, it shows a strong 22-year variation, peaking near the maximum of even-numbered sunspot cycles and showing minima at the peaks of odd-numbered cycles. We discuss the implications of these long-term solar and heliospheric variations for Earth's environment.

The tensor-kinetic field in nuclear collisions

The role of the tensor terms in the Skyrme interaction is studied for their effect in dynamic calculations where non-zero contributions to the mean-field may arise, even when the starting nucleus, or nuclei are even-even and have no active time-odd potentials in the ground state. We study collisions in the test-bed 16O-16O system, and give a qualitative analysis of the behaviour of the time-odd tensor-kinetic density, which only appears in the mean field Hamiltonian in the presence of the tensor force. We find an axial excitation of this density is induced by a collision.

Bit-index sort: a fast non-comparison integer sorting algorithm for permutations

This paper describes a fast integer sorting algorithm, herein referred as Bit-index sort, which is a non-comparison sorting algorithm for partial per-mutations, with linear complexity order in execution time. Bit-index sort uses a bit-array to classify input sequences of distinct integers, and exploits built-in bit functions in C compilers supported by machine hardware to retrieve the ordered output sequence. Results show that Bit-index sort outperforms in execution time to quicksort and counting sort algorithms. A parallel approach for Bit-index sort using two simultaneous threads is included, which obtains speedups up to 1.6.

Pipelined median architecture

The core processing step of the noise reduction median filter technique is to find the median within a window of integers. A four-step procedure method to compute the running median of the last N W-bit stream of integers showing area and time benefits is proposed. The method slices integers into groups of B-bit using a pipeline of W/B blocks. From the method, an architecture is developed giving a designer the flexibility to exchange area gains for faster frequency of operation, or vice versa, by adjusting N, W and B parameter values. Gains in area of around 40%, or in frequency of operation of around 20%, are clearly observed by FPGA circuit implementations compared to latest methods in the literature.

The heliospheric Hale cycle over the last 300 years and its implications for a “lost” late 18th century solar cycle

A Hale cycle, one complete magnetic cycle of the Sun, spans two complete Schwabe cycles (also referred to as sunspot and, more generally, solar cycles). The approximately 22-year Hale cycle is seen in magnetic polarities of both sunspots and polar fields, as well as in the intensity of galactic cosmic rays reaching Earth, with odd- and even-numbered solar cycles displaying qualitatively different waveforms. Correct numbering of solar cycles also underpins empirical cycle-to-cycle relations which are used as first-order tests of stellar dynamo models. There has been much debate about whether the unusually long solar cycle 4 (SC4), spanning- 1784–1799, was actually two shorter solar cycles combined as a result of poor data coverage in the original Wolf sunspot number record. Indeed, the group sunspot number does show a small increase around 1794–1799 and there is evidence of an increase in the mean latitude of sunspots at this time, suggesting the existence of a cycle ‘‘4b’’. In this study, we use cosmogenic radionuclide data and associated reconstructions of the heliospheric magnetic field (HMF) to show that the Hale cycle has persisted over the last 300 years and that data prior to 1800 are more consistent with cycle 4 being a single long cycle (the ‘‘no SC4b’’ scenario). We also investigate the effect of cycle 4b on the HMF using an open solar flux (OSF) continuity model, in which the OSF source term is related to sunspot number and the OSF loss term is determined by the heliospheric current sheet tilt, assumed to be a simple function of solar cycle phase. The results are surprising; Without SC4b, the HMF shows two distinct peaks in the 1784–1799 interval, while the addition of SC4b removes the secondary peak, as the OSF loss term acts in opposition to the later rise in sunspot number. The timing and magnitude of the main SC4 HMF peak is also significantly changed by the addition of SC4b. These results are compared with the cosmogenic isotope reconstructions of HMF and historical aurora records. These data marginally favour the existence of SC4b (the ‘‘SC4b’’ scenario), though the result is less certain than that based on the persistence of the Hale cycle. Thus while the current uncertainties in the observations preclude any definitive conclusions, the data favour the ‘‘no SC4b’’ scenario. Future improvements to cosmogenic isotope reconstructions of the HMF, through either improved modelling or additional ice cores from well-separated geographic locations, may enable questions of the existence of SC4b and the phase of Hale cycle prior to the Maunder minimum to be settled conclusively.

Hydrocarbon biodegradation and soil microbial community response to repeated oil exposure

Bioremediation strategies continue to be developed to mitigate the environmental impact of petroleum hydrocarbon contamination. This study investigated the ability of soil microbiota, adapted by prior exposure, to biodegrade petroleum. Soils from Barrow Is. (W. Australia), a class A nature reserve and home to Australia’s largest onshore oil field, were exposed to Barrow production oil (50 ml/kg soil) and incubated (25 °C) for successive phases of 61 and 100 days. Controls in which oil was not added at Phase I or II were concurrently studied and all treatments were amended with the same levels of additional nutrient and water to promote microbial activity. Prior exposure resulted in accelerated biodegradation of most, but not all, hydrocarbon constituents in the production oil. Molecular biodegradation parameters measured using gas chromatography–mass spectrometry (GC–MS) showed that several aromatic constituents were degraded more slowly with increased oil history. The unique structural response of the soil microbial community was reflected by the response of different phospholipid fatty acid (PLFA) sub-classes (e.g. branched saturated fatty acids of odd or even carbon number) measured using a ratio termed Barrow PLFA ratio (B-PLFAr). The corresponding values of a previously proposed hydrocarbon degrading alteration index showed a negative correlation with hydrocarbon exposure, highlighting the site specificity of PLFA-based ratios and microbial community dynamics. B-PLFAr values increased with each Phase I and II addition of production oil. The different hydrocarbon biodegradation rates and responses of PLFA subclasses to the Barrow production oil probably relate to the relative bioavailability of production oil hydrocarbons. These different effects suggest preferred structural and functional microbial responses to anticipated contaminants may potentially be engineered by controlled pre-exposure to the same or closely related substrates. The bioremediation of soils freshly contaminated with petroleum could benefit from the addition of exhaustively bioremediated soils rich in biota primed for the impacting hydrocarbons.

Generalised golden ratios over integer alphabets

It is a well known result that for β ∈ (1,1+√52) and x ∈ (0,1β−1) there exists uncountably many (ǫi)∞i=1 ∈ {0,1}N such that x = P∞i=1ǫiβ−i. When β ∈ (1+√52,2] there exists x ∈ (0,1β−1) for which there exists a unique (ǫi)∞i=1 ∈ {0,1}N such that x=P∞i=1ǫiβ−i. In this paper we consider the more general case when our sequences are elements of {0, . . . , m}N. We show that an analogue of the golden ratio exists and give an explicit formula for it.

Gál-type GCD sums beyond the critical line

We prove that ∑k,ℓ=1N(nk,nℓ)2α(nknℓ)α≪N2−2α(logN)b(α) holds for arbitrary integers 1≤n1<⋯