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It is not clear to what extent associations between schizophrenia, cannabis use and cigarette use are due to a shared genetic etiology. We, therefore, examined whether schizophrenia genetic risk associates with longitudinal patterns of cigarette and cannabis use in adolescence and mediating pathways for any association to inform potential reduction strategies.
Associations between schizophrenia polygenic scores and longitudinal latent classes of cigarette and cannabis use from ages 14 to 19 years were investigated in up to 3925 individuals in the Avon Longitudinal Study of Parents and Children. Mediation models were estimated to assess the potential mediating effects of a range of cognitive, emotional, and behavioral phenotypes.
The schizophrenia polygenic score, based on single nucleotide polymorphisms meeting a training-set p threshold of 0.05, was associated with late-onset cannabis use (OR = 1.23; 95% CI = 1.08,1.41), but not with cigarette or early-onset cannabis use classes. This association was not mediated through lower IQ, victimization, emotional difficulties, antisocial behavior, impulsivity, or poorer social relationships during childhood. Sensitivity analyses adjusting for genetic liability to cannabis or cigarette use, using polygenic scores excluding the CHRNA5-A3-B4 gene cluster, or basing scores on a 0.5 training-set p threshold, provided results consistent with our main analyses.
Our study provides evidence that genetic risk for schizophrenia is associated with patterns of cannabis use during adolescence. Investigation of pathways other than the cognitive, emotional, and behavioral phenotypes examined here is required to identify modifiable targets to reduce the public health burden of cannabis use in the population.
The suboptimal provision of analgesia to children in the emergency department (ED) is well-described. A yet unexplored barrier is caregiver or child refusal of analgesia. We sought to evaluate the frequency of caregiver/child acceptance of analgesia offered in the ED.
We conducted a two-centre cross-sectional study of 743 caregivers of children 4–17 years presenting to the pediatric ED with an acutely painful condition using a survey and medical record review. The primary outcome was the proportion of children/caregiver pairs who accepted analgesia in the ED.
The median (IQR) age of children was 11 (7) years, and 339/743 (45.6%) were female. The overall survey response rate was 73% (743/1018). In the 24 hours preceding ED arrival, the median (IQR) maximal pain score rated by children and caregivers was 8/10 (4) and 5/10 (2), respectively, and 30.4% (226/743) of caregivers offered analgesia. In the ED, children reported a median (IQR) pain score of 8/10 (2) and 54.9% (408/743) were offered analgesia. When offered in the ED, analgesia was accepted by 91% (373/408). Overall, 55.7% (414/743) of children received some form of analgesia.
Most caregivers/children accept analgesia when offered by ED personnel, suggesting refusal is not a major barrier to optimal management of children’s pain and highlighting the importance of ED personnel in encouraging adequate analgesia. A large proportion of children in pain are not offered analgesia by caregivers or ED personnel. Educational strategies for recognizing and treating pain should be directed at children, caregivers, and ED personnel.
Variation in human cognitive ability is of consequence to a large number of health and social outcomes and is substantially heritable. Genetic linkage, genome-wide association, and copy number variant studies have investigated the contribution of genetic variation to individual differences in normal cognitive ability, but little research has considered the role of rare genetic variants. Exome sequencing studies have already met with success in discovering novel trait-gene associations for other complex traits. Here, we use exome sequencing to investigate the effects of rare variants on general cognitive ability. Unrelated Scottish individuals were selected for high scores on a general component of intelligence (g). The frequency of rare genetic variants (in n = 146) was compared with those from Scottish controls (total n = 486) who scored in the lower to middle range of the g distribution or on a proxy measure of g. Biological pathway analysis highlighted enrichment of the mitochondrial inner membrane component and apical part of cell gene ontology terms. Global burden analysis showed a greater total number of rare variants carried by high g cases versus controls, which is inconsistent with a mutation load hypothesis whereby mutations negatively affect g. The general finding of greater non-synonymous (vs. synonymous) variant effects is in line with evolutionary hypotheses for g. Given that this first sequencing study of high g was small, promising results were found, suggesting that the study of rare variants in larger samples would be worthwhile.
Amphiphilic diacetylenes (DAs) can self-assemble into photopolymerizable liposomes that can be used to construct effective pathogen sensors. Here, modified commercial inkjet printers are used to disperse DAs into water, facilitating self-assembly. The liposomes are of similar size, but are significantly less polydisperse than liposomes formed using conventional sonication methods. The process is efficient, readily scalable and tolerant of structural modification. The derivitization of approximately 5% of the DA head groups and the incorporation of fluorophores into the hydrophobic bilayer allows for the preparation of novel multifluorophore PDA sensing systems that can provide enhanced bacterial discrimination in a single experiment by way of a fluorescent fingerprint.
Focus groups are a means of gathering qualitative data from a group of participants who discuss a given topic. This method has been used in health care research for the past 30 years, but has seen limited use in radiation therapy research. Focus group discussions are a useful tool for investigating a variety of educational, training and clinical issues from the perspective of practitioners, students and patients. This paper reviews the issues associated with using focus groups as a means of data collection. In particular, it addresses some of the decisions which have to be made about group composition and conduct of the discussions. The literature review is contextualised using a recent example of how the authors used focus groups to investigate fitness to practise in radiation therapy. Other challenges such as familiarity between participants and researchers, power relationships and anonymity are addressed. The paper concludes with a consideration of data analysis.
Folic acid (pteroylmonoglutamic acid) has historically been used as the reference folate in human intervention studies assessing the relative bioavailability of dietary folate. Recent studies using labelled folates indicated different plasma response kinetics to folic acid than to natural (food) folates, thus obviously precluding its use in single-dose experiments. Since differences in tissue distribution and site of biotransformation were hypothesised, the question is whether folic acid remains suitable as a reference folate for longer-term intervention studies, where the relative bioavailability of natural (food) folate is assessed based on changes in folate status. Healthy adults aged 18–65 years (n 163) completed a 16-week placebo-controlled intervention study in which the relative bioavailability of increased folate intake (453 nmol/d) from folate-rich foods was assessed by comparing changes in plasma and erythrocyte folate concentration with changes induced by an equal reference dose of supplemental (6S)-5-methyltetrahydrofolic acid or folic acid. The relative increase in plasma folate concentration in the food group was 31 % when compared with that induced by folic acid, but 39 % when compared with (6S)-5-methyltetrahydrofolic acid. The relative increase in erythrocyte folate concentration in the food group when compared with that induced by folic acid was 43 %, and 40 % when compared with (6S)-5-methyltetrahydrofolic acid. When recent published observations were additionally taken into account it was concluded that, in principle, folic acid should not be used as the reference folate when attempting to estimate relative natural (food) folate bioavailability in longer-term human intervention studies. Using (6S)-5-methyltetrahydrofolic acid as the reference folate would avoid future results' validity being questioned.
The purpose of the present paper is to review our current understanding of the chemistry and biochemistry of folic acid and related folates, and to discuss their impact on public health beyond that already established in relation to neural-tube defects. Our understanding of the fascinating world of folates and C1 metabolism, and their role in health and disease, has come a long way since the discovery of the B-vitamin folic acid by Wills (1931), and its first isolation by Mitchell et al. (1941). However, there is still much to do in perfecting methods for the measurement of folate bioavailability, and status, with a high extent of precision and accuracy. Currently, examination of the relationships between common gene polymorphisms involved in C1 metabolism and folate bioavailability and folate status, morbidity, mortality and longevity is evaluated as a series of individual associations. However, in the future, examination of the concurrent effects of such common gene polymorphisms may be more beneficial.
In the mathematical world, one occasionally encounters some rare individuals who seem to have the ability to hold an image in their minds of a four or higher-dimensional object. Even the ability to visualise three-dimensional geometry is a reasonably uncommon gift, and most of us have to get by with two (and sometimes much less) dimensional images in our heads.
The space of quasifuchsian once-punctured torus groups is described by two complex or four real parameters (the traces), which means it is one of these high-dimensional objects. Our approach to studying it has been to look at two-dimensional ‘slices’, meaning that we specify one of the complex trace parameters and then plot those values of the remaining parameter which correspond to quasifuchsian groups (or single cusps, in the case of the Maskit slice).
The samples we have given, Maskit's slice and the trace 3 slice, offer an impression of a object with a somewhat pointy boundary, which is however not terribly complicated otherwise. In recent years, as more detailed plots have emerged, this simple picture has begun to change. Exactly how the slices all fit together is a puzzle of very current interest.
the end of which, as is said, is as refined as it is useless.
Ars Magna, Girolamo Cardano
The intricate fractal shapes we are aiming to draw are based on the algebra and geometry of complex numbers. Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them ‘nature's numbers’. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution. When complex numbers were first dreamed of in the Renaissance, they were treated as an esoteric, almost mystical, concept. This aura of mystery persisted well into the twentieth century – the senior author's aunt Margaret Silcock (née Mumford), who studied mathematics at Girton College Cambridge in 1916, liked to describe them as a ‘delightful fiction’. In fact we still use the term ‘imaginary numbers’ to this day. Modern scientists, however, take complex numbers for granted, FORTRAN makes them a predefined data type, and they are standard toolkit for any electronic engineer. Perhaps the most remarkable fact about complex numbers is that they are absolutely essential to modern physics. In the theory of quantum mechanics, not only can the universe exist probabilistically in two states at once, but the uncertain composite state is constructed by adding the two simple states together with complex coefficients, introducing a complex ‘phase’.
He [AI Gore] was captivated by the metaphorical power of a phenomenon scientists have called the ‘edge of chaos’.
John F. Harris, Washington Post
Our progression through the book has been the investigation of more and more remarkable ways in which two Möbius maps a and b can dance together. Figure 9.1 shows another level of complexity, an array of interlocking spirals which literally took our breath away when we first drew it. It results from creating a double cusp group in which the generator b and the word a15B are both parabolic. Surely it cannot be coincidence that there are exactly 16 coloured circles forming a chain across the centre of the picture? Let's pick apart the dynamics of a and b, using the diagrammatic version Figure 9.2 for notation. In particular, let's try to see from the picture why a15B is parabolic and where its fixed point is located.
The action of b is quite easy. It is parabolic with fixed point at the bottom of the picture at —i. It pushes points out from its fixed point along clockwise circular trajectories. (You may like to compare with Figure 8.4 on p. 233 to help follow this.) One trajectory lies along the boundary of the outer unit circle framing the picture (note b(—1) = +1), and another is the boundary of the white circle tangent to the unit circle at —i.
Figure 8.14 shows two pictures in which one of the two traces is 2 and the other is 3. In other words, one of the two generators a and b is parabolic but the other is not. Both pictures are rather like the gasket picture in frame (vi) of Figure 8.2, but on the left only the circles Ca and CA have come together with an extra point of tangency, while on the right the tangency is between the circles Cb and CB. This may be easier to see if you compare the left frame of Figure 8.14 to Figure 8.4. See how the fixed points of a have come together pinching off the lefthand part of the picture from the right. If, on the other hand, we fix ta and send tb to 2, the upper and lower pincers come together resulting in the righthand frame of Figure 8.14.
The myriad small circles in these pictures appear for exactly the same reason as they did in the last chapter. If for example a is parabolic, then so is bAB, and so also is abAB. Thus the two elements a,bAB generate a subgroup conjugate to the modular group, which as we know means we expect to see circles in the limit set. Well, here they are!
Groups like these in which one element is parabolic are called cusp groups, because they can be explained in terms of pinching points on surfaces to cusps. Some groups, like the ones in our pictures here, have one ‘extra’ parabolic element (in this case b) and so are called single cusps.
Thus far, our symmetry groups have always gone hand in hand with a tiling; either the bathroom floor tiles in the picture on p. 17 in Chapter 1, or the ring-shaped tiles associated to the group generated by a single loxodromic transformation in Chapter 3. As we already hinted, there is still a set of symmetrical tiles hidden amid the circles in the Schottky array. It is a bit harder to spot them, because their shape is a rather more exotic than those we have met thus far. Perhaps a volunteer could help us. Dr. Stickler, would you do us the favour of guiding us around?
Figure 4.7 shows a time elapsed photograph of Dr. Stickler's journey. Starting boldly at the centre of the picture, you see him shrinking and turning as he progresses ever more deeply into the Schottky array. A new Dr. Stickler appears for each transformation in the group. The most striking feature of the picture is that the arrows exactly reflect the pattern in the word tree. Each Dr. Stickler is at a node, and if you copied over the labelling from the word tree you would find each copy labelled by the exact transformation which gets him from his central position to the given location. Of course, we only show a finite number of levels, but it is not hard to imagine the whole infinite tree of words extending outwards, the edges getting ever shorter until we eventually reach a limit point at the infinite end of the branch.
‘I could spin a web if I tried,’ said Wilbur, boasting. ‘I've just never tried.’
‘Let's see you do it,’ said Charlotte…
‘OK,’ replied Wilbur. ‘You coach me and I'll spin one. It must be a lot of fun to spin a web. How do I start?’
As any mathematician who has revealed his (or her) occupation to a neighbour on a plane flight has discovered, most people associate mathematics with something akin to the more agonizing forms of medieval torture. It seems indeed unlikely that mathematics would be done at all, were it not that a few people discover the play that lies at its heart. Most published mathematics appears long after the play is done, cloaked in lengthy technicalities which obscure the original fun. The book in hand is unfortunately scarcely an exception. Never mind; after a fairly detailed introduction to the art of creating tilings and fractal limit sets out of two very carefully chosen Möbius maps, we are finally set to embark on some serious mathematical play. The greatest rewards will be reaped by those who invest the time to set up their own programs and join us charting mathematical territory which is still only partially explored.
All the limit sets we have constructed thus far began from a special arrangement of four circles, the Schottky circles, grouped into two pairs. For each pair, we found a Möbius map which moved the inside of one circle to the outside of the other.
Early in this chapter, we sailed down into the deeper southern reaches of the Maskit boundary by carefully avoiding the jutting rocky promontories where the cusp groups sit. To reach Jørgensen's degenerate group at the furthermost point, we had to head through the slice holding a tight almost due south course without very much room for straying. However the cusp groups, by virtue of their location on promontories, can either be approached straight on heading south from the interior or by laying up and following up northwards along the edge of the rocky promontories which lead to the cusps. The coastline route, not unlike the coast of Maine, has surprises in store.
Let's return to the most prominent cusps of Maskit's slice corresponding to 0/1, at the value μ = 2i. As we have seen, the limit set of this group is none other than the Apollonian gasket. A direct approach from the interior of Maskit's slice would be to sail due south down the imaginary axis, that is, through groups with μ = ti with real numbers t decreasing to 2. We have already shown several pictures of this kind of direct ‘pinching’, where gaps are slowly closed to pinch points (for example, see Figure 7.4). Nothing terribly exciting happens during these trips.
In Chapter 8, we played (rather irresponsibly) with traces around 2. One especially spiralliferous spot was near trace 1.91+0.05i, exactly the complex conjugate of the trace corresponding to the μ-value 0.05+1.91i, the red dot in Figure 9.12. (Complex conjugate traces, remember, give mirror image groups.) Looking back at Figure 8.17, perhaps you can see how we might have predicted that the μ-value for this group should be near the 1/9 and 1/10 cusps. The spiral head is largest around words with prefix ab10 and ab9, and in fact the most extreme point seems to be about ab9ab10. The roles of a and b are reversed because of the mirror symmetry, which more or less explains why in our present set-up it is a good idea to declare a10B a special word.
Figure 9.12 is like a road map, delineating the boundary between order and chaos. How about using it to drive right up to one of these two nearby cusps? With any luck, it should exhibit both the beautiful spirals of Figure 8.17 and the delicate lacework of the Apollonian gasket. Imagine how those two effects will be combined. The results are – wait one moment, let's not get ahead of ourselves. Perhaps we should look very closely at the boundary near μ = 0.05 + 1.91i. Zoom in to the red dot in Figure 9.12 to get the very small (0.03 × 0.03) frame in Figure 9.14.