Aliquot Sequence 3630 Ends After Reaching 100 Digits

Article dans revue scientifique avec comité de lecture.In this paper we present a new computational record: the aliquot sequence starting at $3630$ converges to~$1$ after reaching a hundred decimal digits. Also, we show the current status of all the aliquot sequences starting with a number under $10000$. In particular, we have reached at least $112$ digits for the so-called ``Lehmer five sequences'', and 101 digits for the ``Godwin twelve sequences''. Finally, we give a summary showing the number of aliquot sequences of unknown end starting with a number~$\le 10^6$

ALIQUOT SEQUENCE 3630 ENDS AFTER REACHING 100 DIGITS

Abstract. In this paper we present a new computational record: the aliquot sequence starting at 3630 converges to 1 after reaching a hundred decimal digits. Also, weshow thecurrent status of all thealiquot sequences starting with a number under 10000; we have reached at leat 95 digits for all of them. In particular, we have reached at least 112 digits for the so-called “Lehmer five sequences”, and 101 digits for the “Godwin twelve sequences”. Finally, we give a summary showing the number of aliquot sequences of unknown end starting with a number ≤ 10 6. For a positive integer n, letσ(n) denote the sum of its divisors (including 1 and n), and s(n) =σ(n) − n the sum of its proper divisors (without n). A perfect number is a number n such that s(n) =n, and an amicable pair of numbers (n, m) satisfies s(n) =m, s(m) =n. In a similar way, tuples of numbers (a1,a2,...,al) such that s(ai) =ai+1 for 1 ≤ i ≤ l − 1ands(al) =a1 are known as aliquot cycles or sociable numbers. Given n, the way to compute σ(n) (and then, s(n)) is as follows. We find the prime decomposition of n = p a1 1 ···pad d.Then (1) σ(p a

Nucleic acid detection using oligonucleotide cross-linked polymer composites

There has been much interest in recent years about the potential of microRNA as a new source of biomarkers for the diagnosis of disease. The delivery of new diagnostic tools based on this potential has been limited by shortcomings in current microRNA detection techniques. This thesis explores the development of a new method of microRNA detection through the incorporation of conductive particles into oligonucleotide-functionalised polymers to form oligonucleotide cross-linked polymer composites. Such composites could provide a simple, rapid, and low-cost means of microRNA detection that could be easily multiplexed, providing a valuable tool for point-of-care medical diagnostics. This work presents oligonucleotide-functionalised carbon/polyacrylamide composites which demonstrate a selective swelling response in the presence of analyte oligonucleotide sequences and for which the electrical conductivity decreases with swelling. The composites were synthesised via UV-initiated free-radical polymerisation of carbon/- monomer mixtures upon custom electrode devices, consisting of interdigitated platinum electrodes fabricated upon a silicon substrate. The optimal cross-linker density and carbon loading concentration were determined as well as the best means of dispersing the carbon particles within the polymer. Various types of carbon particles, with differing sizes and aspect ratios, were compared and their performances as conductive additives for polymer swelling transduction evaluated. The swelling behaviour of these composites was evaluated by analysing images of composite microdroplets as they swell. The electrical characteristics of the composites were determined by measuring either the two-terminal resistance or the complex impedance of composite microdroplets on the electrode devices. Alternating and direct current measurement techniques were compared to determine the best approach for the transduction of composite swelling. The volumetric and electrical responses of oligonucleotide-functionalised carbon/polyacrylamide composites were analysed in solutions of analyte oligonucleotide and non-complementary controls. It has been demonstrated that, using carbon nanopowder composites and a direct current two-terminal resistance measurement, it is possible to differentiate between analyte and control solutions to concentrations as low as 10 nM, with single-base precision, in less than three minutes. However, the inability to detect at concentrations below this value, difficulties in differentiating between different analyte concentrations and thermal instability mean that, in their current form, oligonucleotide cross-linked polymer composites are unsuitable for the detection of circulating microRNA at clinically relevant concentrations. Potential avenues of work to address these challenges are discussed. Also presented are collaborative results for oligonucleotide-responsive polymers functionalised with morpholino nucleic acid analogues, in what is believed to be the first example of such a material. These morpholino-functionalised polymers offer significant advantages, in terms of stability and sensitivity, over their nucleic acid equivalents for bio-responsive polymer applications

Sustainable Production in Food and Agriculture Engineering

This book is a collection of original research and review papers that report on the state of the art and recent advancements in food and agriculture engineering, such as sustainable production and food technology. Encompassed within are applications in food and agriculture engineering, biosystem engineering, plant and animal production engineering, food and agricultural processing engineering, storing industry, economics and production management and agricultural farms management, agricultural machines and devices, and IT for agricultural engineering and ergonomics in agriculture

Examining the relationships between Holocene climate change, hydrology, and human society in Ireland

This thesis explores human-environment interactions during the Mid-Late Holocene in raised bogs in central Ireland. The raised bogs of central Ireland are widely-recognised for their considerable palaeoenvironmental and archaeological resources: research over the past few decades has established the potential for such sites to preserve sensitive records of Holocene climatic variability expressed as changes in bog surface wetness (BSW); meanwhile archaeological investigations over the past century have uncovered hundreds of peatland archaeological features dating from the Neolithic through to the Post-Medieval period including wooden trackways, platforms, and deposits of high-status metalwork. Previous studies have attempted to explore the relationship between records of past environmental change and the occurrence of peatland archaeological sites reaching varying conclusions. More recently, environmentally-deterministic models of human-environment interaction in Irish raised bogs at the regional scale have been explicitly tested leading to the conclusion that there is no relationship between BSW and past human activity. These relationships are examined in more detail on a site-by-site basis in this thesis. To that end, testate amoebae-derived BSW records from nine milled former raised bogs in central Ireland were produced from sites with known and dated archaeological records. Relationships between BSW records and environmental conditions within the study area were explored through both the development of a new central Ireland testate amoebae transfer function and through comparisons between recent BSW records and instrumental weather data. Compilation of BSW records from the nine fossil study sites show evidence both for climate forcing, particularly during 3200-2400 cal BP, as well as considerable inter-site variability. Considerable inter-site variability was also evident in the archaeological records of the same sites. Whilst comparisons between BSW and archaeological records do not show a consistent linear relationship, examination of records on a site-by-site basis were shown to reveal interpretatively important contingent relationships. It is concluded therefore, that future research on human-environment interactions should focus on individual sites and should utilise theoretical approaches from the humanities in order to avoid the twin pitfalls of masking important local patterns of change, and of environmental determinism

Advancing Knowledge on Cyanobacterial Blooms in Freshwaters

Cyanobacterial blooms are a water quality problem that is widely acknowledged to have detrimental ecological and economic effects in drinking and recreational water supplies and fisheries. There is increasing evidence that cyanobacterial blooms have increased globally and are likely to expand in water resources as a result of climate change. Of most concern are cyanotoxins, along with the mechanisms that induce their release and determine their fate in the aquatic environment. These secondary metabolites pose a potential hazard to human health and agricultural and aquaculture products that are intended for animal and human consumption; therefore, strict and reliable control of cyanotoxins is crucial for assessing risk. In this direction, a deeper understanding of the mechanisms that determine cyanobacterial bloom structure and toxin production has become the target of management practices. This Special Issue, entitled “Advancing Knowledge on Cyanobacterial Blooms in Freshwaters”, aims to bring together recent multi- and interdisciplinary research, from the field to the laboratory and back again, driven by working hypotheses based on any aspect of mitigating cyanobacterial blooms, from ecological theory to applied research

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