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We show that if f is an injection from an n-dimensional Boolean cube (considered as an additive group) to a group of prime order then the probability that f(x+y)=f(x)+f(y) is O(2^{-n/11})., 7 pages, added an example to section 2, corrected some errors

Medically Unexplained Symptoms (MUS) are bodily symptoms for which no organic cause has been identified, and which result in significant levels of psychological distress and functional impairment. MUS are thought to be highly prevalent in primary care settings, and have considerable costs to society. Despite evidence of overlapping psychological and physical presentations, MUS are not well understood or treated in culture that predominantly views the body through the lenses of dualism and mechanistic reductionism. An alternative ‘interactive’ view of the body as playing a more dynamic role is elaborated through George Kelly’s (1955) Personal Construct Psychology. The author draws upon Lin & Payne’s (2014) ‘frozen construing’ theory, and empirical literature on relationships between identity and MUS, to suggest that for people with MUS, the symptomatic body is distressing because the person is struggling to integrate its meaning with their identity. It is hypothesized that embodied processes, that may actually protect the self (and others who share a construct system with that person) from events which threaten to dramatically alter how the self is construed, are difficult to understand because of their preverbal nature. Hence symptoms, and the body itself, are dissociated from the person’s more elaborated verbal self-constructions. Several hypotheses relating to this suggestion were tested using a modified form of the repertory grid technique that was designed to explore construct systems of both mind and body, for self and others. Twenty participants with MUS, recruited from the community, completed the repertory grid interviews and measures of depression, anxiety and symptom severity, which were correlated with relevant repertory grid indices to test hypotheses. Findings indicated that symptom constructs, contrary to expectations, were well integrated into participants’ construct systems. The alleviation of psychological distress was significantly associated with increased perceived distance between the self in general and the self when symptoms are worst (a relationship which appeared to be independent of severity of symptoms), providing evidence of a process of dissociation that protected the current self from assimilating the undesirable characteristics that were associated with the symptom. The way in which the self when symptoms are worst is construed appeared to influence levels of distress, with more predictive power than several other indices. The study also found evidence for some participants of hypothesized relationships between desired aspects of the current self and symptoms, that would imply that symptom disappearance would actually threaten a desirable aspect of how the self is construed. Content analysis of these constructs revealed (as predicted) that such desirable aspects of self tended to relate to being responsible and sensitive to the needs of others, and were elaborated through bodily constructs in a way that suggested that they were not well integrated with the primary ways that these participants made sense of their identity. For these particular participants, discrepancies between the ideals that they had for themselves, and how they would like to be seen by others, were associated with increased depression. Several participants were identified whose constructions of self and others were dominated by constructs relating to both mental and physical strength and weakness. These participants appeared to be struggling to find coherent meaning for themselves as the result of symptoms, which were regarded as invalidating a pre-symptom construal of themselves as being ‘strong’. There seemed to be a continuum of being a ‘body for others’ on the one hand, a previously ‘strong person’ on the other, and a person who is ‘strong for others’ in the middle. Implications for clinical practice are discussed. Although the findings of the current study are limited by a small sample size, it appears that exploring the meaning of the body in the construction of self helps to elaborate the meaning of the body and symptoms in a verbal, expressible form. This process is likely to be helpful to those who struggle to find meanings for their symptoms both in their own construct systems and in a society that objectifies the body.

Background: Juvenile Arthritis Disease Activity Score (JADAS) is a 4 variable composite disease activity (DA) score for JIA (including active 10, 27 or 71 joint count (AJC), physician global (PGA), parent/child global (PGE) and ESR). The validity of JADAS for all ILAR subtypes in the routine clinical setting is unknown. We investigated the construct validity of JADAS in the clinical setting in all subtypes of JIA through application to a prospective inception cohort of UK children presenting with new onset inflammatory arthritis. Methods: JADAS 10, 27 and 71 were determined for all children in the Childhood Arthritis Prospective Study (CAPS) with complete data available at baseline. Correlation of JADAS 10, 27 and 71 with single DA markers was determined for all subtypes. All correlations were calculated using Spearman's rank statistic. Results: 262/1238 visits had sufficient data for calculation of JADAS (1028 (83%) AJC, 744 (60%) PGA, 843 (68%) PGE and 459 (37%) ESR). Median age at disease onset was 6.0 years (IQR 2.6-10.4) and 64% were female. Correlation between JADAS 10, 27 and 71 approached 1 for all subtypes. Median JADAS 71 was 5.3 (IQR 2.2-10.1) with a significant difference between median JADAS scores between subtypes (p < 0.01). Correlation of JADAS 71 with each single marker of DA was moderate to high in the total cohort (see Table 1). Overall, correlation with AJC, PGA and PGE was moderate to high and correlation with ESR, limited JC, parental pain and CHAQ was low to moderate in the individual subtypes. Correlation coefficients in the extended oligoarticular, rheumatoid factor negative and enthesitis related subtypes were interpreted with caution in view of low numbers. Conclusions: This study adds to the body of evidence supporting the construct validity of JADAS. JADAS correlates with other measures of DA in all ILAR subtypes in the routine clinical setting. Given the high frequency of missing ESR data, it would be useful to assess the validity of JADAS without inclusion of the ESR. Disclosure statement: All authors have declared no conflicts of interest. Table 1Spearman's correlation between JADAS 71 and single markers DA by ILAR subtype ILAR Subtype Systemic onset JIA Persistent oligo JIA Extended oligo JIA Rheumatoid factor neg JIA Rheumatoid factor pos JIA Enthesitis related JIA Psoriatic JIA Undifferentiated JIA Unknown subtype Total cohort Number of children 23 111 12 57 7 9 19 7 17 262 AJC 0.54 0.67 0.53 0.75 0.53 0.34 0.59 0.81 0.37 0.59 PGA 0.63 0.69 0.25 0.73 0.14 0.05 0.50 0.83 0.56 0.64 PGE 0.51 0.68 0.83 0.61 0.41 0.69 0.71 0.9 0.48 0.61 ESR 0.28 0.31 0.35 0.4 0.6 0.85 0.43 0.7 0.5 0.53 Limited 71 JC 0.29 0.51 0.23 0.37 0.14 -0.12 0.4 0.81 0.45 0.41 Parental pain 0.23 0.62 0.03 0.57 0.41 0.69 0.7 0.79 0.42 0.53 Childhood health assessment questionnaire 0.25 0.57 -0.07 0.36 -0.47 0.84 0.37 0.8 0.66 0.47

We show that if G is a finite group and A is a subset of G with no non-trivial solutions to xz=yy then |A| < |G|/(log log |G|)^c., 27pp

Let $\mathbb{F}$ be a fixed finite field, and let $A \subset \mathbb{F}^n$. It is a well-known fact that there is a subspace $V \leq \mathbb{F}^n$, $\mbox{codim} V \ll_{\delta} 1$, and an $x$, such that $A$ is $\delta$-uniform when restricted to $x + V$ (that is, all non-trivial Fourier coefficients of $A$ restricted to $x + V$ have magnitude at most $\delta$). We show that if $\mathbb{F} = \mathbb{F}_2$ then it is possible to take $x = 0$; that is, $A$ is $\delta$-uniform on a subspace $V \leq \mathbb{F}^n$. We give an example to show that this is not necessarily possible when $\mathbb{F} = \mathbb{F}_3$. ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners showed us a rather short argument he had found in 2013, giving a better bound for our main theorem. We do not, therefore, intend to publish this note. The example over $\mathbb{F}_3$ may still be of interest to some readers and so we will not withdraw the paper from the arxiv., Comment: 4 pages

ECM survey article discussing the structure of subsets of Abelian groups which behave `a bit like' cosets (of subgroups)., 14pp

We prove the following result due to Hamidoune using an analytic approach. Suppose that A is a subset of a finite group G with |AA^{-1}| \leq (2-\varepsilon)|A|. Then there is a subgroup H of G and a set X of size O_\varepsilon(1) such that A \subset XH., 6pp

ILSI Europe Concise Monograph series Concise Monograph Series : Bone and musculoskeletal health; International audience

E-thesis pagination differs from approved hard bound copy, Cambridge University Library classmark: PhD.30726, This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L^1-norm of the Fourier transform, and the closely related idempotent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable. Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above. The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem.

Current dietary recommendations advise reducing the intake of saturated fatty acids (SFAs) to reduce coronary heart disease (CHD) risk, but recent findings question the role of SFAs. This expert panel reviewed the evidence and reached the following conclusions: the evidence from epidemiologic, clinical, and mechanistic studies is consistent in finding that the risk of CHD is reduced when SFAs are replaced with polyunsaturated fatty acids (PUFAs). In populations who consume a Western diet, the replacement of 1% of energy from SFAs with PUFAs lowers LDL cholesterol and is likely to produce a reduction in CHD incidence of ≥2-3%. No clear benefit of substituting carbohydrates for SFAs has been shown, although there might be a benefit if the carbohydrate is unrefined and has a low glycemic index. Insufficient evidence exists to judge the effect on CHD risk of replacing SFAs with MUFAs. No clear association between SFA intake relative to refined carbohydrates and the risk of insulin resistance and diabetes has been shown. The effect of diet on a single biomarker is insufficient evidence to assess CHD risk. The combination of multiple biomarkers and the use of clinical endpoints could help substantiate the effects on CHD. Furthermore, the effect of particular foods on CHD cannot be predicted solely by their content of total SFAs because individual SFAs may have different cardiovascular effects and major SFA food sources contain other constituents that could influence CHD risk. Research is needed to clarify the role of SFAs compared with specific forms of carbohydrates in CHD risk and to compare specific foods with appropriate alternatives.

This report summarises a workshop convened by the UK Food Standards Agency (FSA) on 14 October 2008 to discuss current FSA-funded research on carbohydrates and cardiovascular health. The objective of this workshop was to discuss the results of recent research and to identify any areas which could inform future FSA research calls. This workshop highlighted that the FSA is currently funding some of the largest, well-powered intervention trials investigating the type of fat and carbohydrate, whole grains and fruit and vegetables, on various CVD risk factors. Results of these trials will make a substantive contribution to the evidence on diet and cardiovascular risk.

Our objective in this study was to develop and implement an effective intervention strategy to manipulate the amount and composition of dietary fat and carbohydrate (CHO) in free-living individuals in the RISCK study. The study was a randomized, controlled dietary intervention study that was conducted in 720 participants identified as higher risk for or with metabolic syndrome. All followed a 4-wk run-in reference diet [high saturated fatty acids (SF)/high glycemic index (GI)]. Volunteers were randomized to continue this diet for a further 24 wk or to 1 of 4 isoenergetic prescriptions [high monounsaturated fatty acids (MUFA)/high GI; high MUFA/low GI; low fat (LF)/high GI; and LF/low GI]. We developed a food exchange model to implement each diet. Dietary records and plasma phospholipid fatty acids were used to assess the effectiveness of the intervention strategy. Reported fat intake from the LF diets was significantly reduced to 28% of energy (%E) compared with 38%E from the HM and LF diets. SF intake was successfully decreased in the HM and LF diets toor =10%E compared with 17%E in the reference diet (P = 0.001). Dietary MUFA in the HM diets was approximately 17%E, significantly higher than in the reference (12%E) and LF diets (10%E) (P = 0.001). Changes in plasma phospholipid fatty acids provided further evidence for the successful manipulation of fat intake. The GI of the HGI and LGI arms differed by approximately 9 points (P = 0.001). The food exchange model provided an effective dietary strategy for the design and implementation across multiple sites of 5 experimental diets with specific targets for the proportion of fat and CHO.

We show that if [math] contains no three-term arithmetic progressions in which all the elements are distinct then [math] .

We provide further explanation of the significance of a construction in a recent paper of Wolf [Israel J. Math. 179 (2010), 253-278] in the context of the problem of finding large subspaces in sumsets., 4pp. Updated references. Corrected typos

We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n)., Comment: 24 pp. Corrected typos. Updated references. Minor revisions.

We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\log |A|) such that |A \cap Span(L)| >> c^{1/3}|A|., 5 pp. Included refinement from c^{1/2} to c^{1/3}, independently discovered by Shkredov and Yekhanin in arXiv:1004.2294. Corrected typos

We prove a theorem claimed in math.CA/0605519 which asserts that if A is a subset of a compact abelian group G with density of a particular (natural, although technical) form then the A(G)-norm (that is the sum of the absolute values of the Fourier transform) of the characteristic function of A cannot be too small.

Suppose that A is a subset of F_2^n of density as close to 1/3 as possible. We show that the A(F_2^n)-norm (that is the sum of the absolute values of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times log n as n tends to infinity., Comment: 13 pp. Corrected typos. Updated references.