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We study an analog of Serre's modularity conjecture for projective representations [Formula: see text], where K is a totally real number field. We prove cases of this conjecture when [Formula: see text].

Objective: Proportion Of suboptimal Disease Control And Strategy of Treatment in IBD (PODCAST-IBD) was an international real-world study which aimed to quantify disease control in IBD using STRIDE-II recommendations. Design/Method: Cross-sectional assessment of IBD patients attending routine clinic appointments in four UK centers October 2022 to January 2023. Clinician-reported outcomes, patient-reported outcomes and retrospective data from medical chart review were used to assess IBD control against red flags aligned to STRIDE-II. Results: Data were available from 198 UK patients. IBD was suboptimally controlled in 52.4% (54/103) of patients with Crohn's disease (CD) and 45.3% (43/95) with ulcerative colitis (UC). Impaired quality of life (QOL), defined as Short inflammatory bowel disease questionnaire (SIBDQ) score <50, was the main contributor to suboptimal disease control. Suboptimal disease control has a detrimental impact on fatigue and disability with significantly lower mean Functional Assessment of Chronic Illness Therapy – Fatigue (FACIT-F) score in suboptimally controlled disease (CD: 81.5 vs 125, UC: 87.4 vs 122.8) and IBD Disk. Suboptimal disease control results in higher health care resource use (HCRU) (CD: £4,746 vs £1,924; UC: £2,428 vs £1,121) and higher rates of work productivity loss (CD: 41.7% vs 11.9%, UC: 38.0% vs 22.6%). Conclusion: IBD was suboptimally controlled in around one-half of patients. Impaired QOL was the most common contributor (64%, 62/97) to suboptimal control. Suboptimal control had a considerable economic impact; HCRU more than doubled and productivity fell. Physicians could consider regular QOL assessments to prompt timely disease monitoring to enable identification of early active disease and appropriate treatment. PLAIN LANGUAGE SUMMARY: Inflammatory bowel disease (IBD), which encompasses Crohn's disease (CD) and ulcerative colitis (UC), is a life-long, painful and debilitating disease. Symptoms include abdominal pain, diarrhea and extreme tiredness (fatigue) and may also affect the eyes, joints and skin. People with IBD have periods of time where their symptoms are not controlled (known as relapse), with minimal symptoms (known as remission) at other times. This paper reports on people from the UK who participated in the wider international PODCAST-IBD study. The PODCAST-IBD study used information from people with IBD, their doctors and their medical notes to assess how well IBD was controlled and the impact of suboptimal disease control on their lives and use of healthcare. Overall, IBD was suboptimally controlled in around one-half of the people with IBD: 52.4% (54/103) of those with CD and 45.3% (43/95) with UC. Reduced quality of life (QOL) was the most common contributor to suboptimal control with almost two-thirds of people reporting impaired QOL. Suboptimal control of IBD impacts on people's everyday life – resulting in fatigue and disability, reducing QOL and making it difficult to work. Suboptimal control of IBD also has a considerable economic impact since it results in increased healthcare use. It might be helpful for doctors to consider regular QOL assessments to help to identify those people whose IBD is not well controlled to ensure that they receive appropriate treatment to control disease and improve their lives. [ABSTRACT FROM AUTHOR]

Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p(Q^3 \ell n+\beta)\equiv 0\pmod \ell where \ell and Q are prime and 5\leq \ell \leq 31; these lie in two natural families distinguished by the square class of 1-24\beta \pmod \ell. In recent decades much work has been done to understand congruences of the form p(Q^m\ell n+\beta)\equiv 0\pmod \ell. It is now known that there are many such congruences when m\geq 4, that such congruences are scarce (if they exist at all) when m=1, 2, and that for m=0 such congruences exist only when \ell =5, 7, 11. For congruences like Atkin's (when m=3), more examples have been found for 5\leq \ell \leq 31 but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime \ell \geq 5, there are infinitely many congruences like Atkin's in the first natural family which he discovered and that for at least 17/24 of the primes \ell there are infinitely many congruences in the second family. [ABSTRACT FROM AUTHOR]

We study an analog of Serre's modularity conjecture for projective representations ...:Gal(.../K)→PGL2(k), where K is a totally real number field. We prove cases of this conjecture when k=F5. [ABSTRACT FROM AUTHOR]

We revisit the paper [Automorphy lifting for residually reducible $l$ -adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents. [ABSTRACT FROM AUTHOR]