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BackgroundA Treat-to-Target approach (T2T) is broadly considered to lead to better clinical outcomes and recommended in patients with RA. However, very few studies have analyzed the effect of T2T on radiographic progression, and any such studies have provided inconsistent results.ObjectivesTo investigate whether meticulously following a treat-to-target (T2T)-strategy in daily clinical practice leads to lower radiographic progression in RA.MethodsPatients from the multicenter RA-BIODAM cohort with ≥2 consecutive visits with radiographs available were included. In RA-BIODAM patients were enrolled as they were initiating a new csDMARD/bDMARD treatment were followed-up with the intention to benchmark and intensify treatment. The primary outcome of this analysis was the change in Sharp-van der Heijde score (SvdH, 0-448), assessed every 6 months, using average scores from 2 readers (scores with known chronological order). Following a DAS44-T2T remission strategy, which was defined at each 3-month visit, was the main variable of interest. Patients were categorized based on the proportion of visits in which T2T was followed according to our definition: very low (≤40% of the visits, low (>40%, 75%). Radiographic progression at 2 years was visualized across groups by cumulative probability plots. Per 3-month interval T2T could be followed zero, one or two times (in a total of 2 visits). Associations between the number of visits with T2T in an interval and radiographic progression, both in the same and in the subsequent 6-month interval, were analysed by generalised estimating equations, adjusted for age, gender, disease duration and country.ResultsIn total, 511 patients were included (mean (SD) age: 56 (13) years; 76% female). After 2 years, patients showed on average 2.2 (4.1) units progression (median:1 unit). Mean (SD) 2-year progression was not significantly different across categories of T2T: very low: 2.1 (2.7)-units; low: 2.8 (6.0); high: 2.4 (4.5), very high: 1.6 (2.2) (Figure 1). Meticulously following-up T2T in a 3-month interval neither reduced progression in the same 6-month interval (parameter estimates (for yes vs no): +0.15 units (95%CI: -0.04 to 0.33) for 2 vs 0 visits; and +0.08 units (-0.06;0.22) for 1 vs 0 visits) nor did it reduce progression in the subsequent 6-month interval (Table 1).Table 1.Effect of following DAS44-remission-T2T strategy on 6-month radiographic progression over 2 yearsChange in radiographic damage(regression coefficient (95% CI))N=506T2T during 3 months on radiographic progression in the same 6-month period 2 visits vs 0 followed0.15 (-0.04; 0.33) 1 visit vs 0 followed0.08 (-0.06; 0.22)T2T during 3 months on radiographic progression in the subsequent 6-month period 2 visits vs 0 followed-0.09 (-0.28; 0.10) 1 visit vs 0 followed-0.10 (-0.24; 0.05)Figure 1.Cumulative probability plot with 2-year radiographic progression according to the proportion of 3-monthly visits with T2T followedConclusionIn this daily practice cohort, more meticulously following T2T principles did not result in more reduction of radiographic progression than a somewhat more liberal attitude toward T2T. One possible interpretation of these results is that the intention to apply T2T already suffices and that a more stringent approach does not further improve outcome.AcknowledgementsBIODAM was financially supported by an unrestricted grant from AbbVieDisclosure of InterestsSofia Ramiro Speakers bureau: Eli Lilly, MSD, Novartis, UCB, Consultant of: AbbVie, Eli Lilly, MSD, Novartis, Pfizer, UCB, Sanofi, Grant/research support from: AbbVie, Galapagos, Novartis, Pfizer, UCB, Robert B.M. Landewé Speakers bureau: AbbVie, BMS, Gilead, Galapagos, GSK,Janssen, Lilly, Novartis, Pfizer, UCB, Consultant of: AbbVie, BMS, Gilead, Galapagos, GSK,Janssen, Lilly, Novartis, Pfizer, UCBDr Landewé owns Rheumatology Consultancy BV, Désirée van der Heijde Consultant of: AbbVie, Bayer, BMS, Cyxone, Eisai, Galapagos, Gilead, Glaxo-Smith-Kline, Janssen, Lilly, Novartis, Pfizer, UCB Pharma. Director of Imaging Rheumatology bv., Alexandre Sepriano Speakers bureau: Novartis, Consultant of: UCB, Oliver FitzGerald Speakers bureau: Biogen, Novartis, AbbVie, BMS, Pfizer, Grant/research support from: BMS, Novartis, UCB, Pfizer, Lilly, Janssen, Mikkel Østergaard Speakers bureau: Abbvie, BMS, Celgene, Eli-Lilly, Galapagos, Gilead, Janssen, Merck, Novartis, Orion, Pfizer, Roche and UCB, Consultant of: Abbvie, BMS, Boehringer-Ingelheim, Celgene, Eli-Lilly, Hospira, Janssen, Merck, Novartis, Novo, Orion, Pfizer, Regeneron, Roche, Sandoz, Sanofi and UCB, Grant/research support from: Abbvie, Amgen, BMS, Merck, Celgene and Novartis, Joanne Homik: None declared, Ori Elkayam Speakers bureau: Pfizer, Lilly, Novartis, Abbvie, BI, Janssen, Consultant of: Pfizer, Lilly, Novartis, Abbvie, BI, Janssen, Grant/research support from: Pfizer, Abbvie, Janssen, Carter Thorne Consultant of: Abbvie, Organon, Pfizer, Sandoz, Maggie Larché Speakers bureau: AbbVie, Actelion, Amgen, BMS, Boehringer-Ingelheim, Fresenius-Kabi, Gilead, Janssen, Mallinckrodt, Merck, Novartis, Pfizer, Roche, Sandoz, Sanofi, Sobi, UCB, Grant/research support from: Abbvie, BMS, Gianfranco Ferraccioli Speakers bureau: SOBI, Consultant of: Abbivie, Marina Backhaus: None declared, Gilles Boire Speakers bureau: Abbvie Canada, BMS Canada, Lilly Canada, Janssen Canada, Merck Canada, Pfizer Canada, Viatris, Consultant of: Abbvie Canada, Amgen Canada, BMS Canada, Celgene, GileadSciences, Janssen Canada, Lilly Canada, Merck Canada, Mylan Canada, Novartis Canada, Pfizer Canada, Roche Canada, Samsung Bioepis, Sanofi Canada, Teva, Grant/research support from: Lilly Canada, BMS Canada, Pfizer, Sandoz Canada, UCB Canada, Merck Canada, Novartis Canada, Roche Canada, Bernard Combe Speakers bureau: Abbvie, BMS,Celltrion,Galapgos-Gilead, Janssen, Lilly, MERCK, Pfizer,Roche-Chugai, Consultant of: Abbvie, Celltrion,Galapgos-Gilead, Janssen, Lilly, MERCK, Roche-Chugai, Grant/research support from: Pfizer, Roche-chugai, Thierry Schaeverbeke: None declared, Alain Saraux Speakers bureau: Abbvie, Lilly, Nordic, Novartis, Pfizer, Roche-Chugai, Sanofi, UCB, Consultant of: Abbvie, Lilly, Nordic, Novartis, Pfizer, Roche-Chugai, UCB, Grant/research support from: Novartis, Fresenius, Lilly, Maxime Dougados Consultant of: Pfizer, AbbVie, UCB, Merck, Lilly, Novartis, BMS, Galapagos, Biogen, Roche, Grant/research support from: Pfizer, AbbVie, UCB, Merck, Lilly, Novartis, BMS, Galapagos, Biogen, Roche, Maurizio Rossini Speakers bureau: Amgen, Abbvie, BMS, Eli-Lilly, Galapagos,MSD, Novartis, Pfizer, Sandoz, Theramex, UCB, Marcello Govoni Speakers bureau: Abbvie, Pfizer, Galapagos, BMS, Eli-Lilly, Paid instructor for: Pfizer, Consultant of: Abbvie, BMS, Novartis, Astrazeneca, Pfizer, Luigi Sinigaglia: None declared, Alain Cantagrel Speakers bureau: Abbvie, Amgen, Biogen, BMS, Janssen, Lilly France, Médac, MSD France, Nordic-Pharma, Novartis, Pfizer, Sanofi Aventis, UCB, Consultant of: BMS, Janssen, Lilly France, MSD France, Sandoz, Grant/research support from: MSD France, Novartis, Pfizer, Cornelia Allaart: None declared, Cheryl Barnabe Speakers bureau: Sanofi Genzyme, Pfizer, Fresenius Kabi, Janssen, Consultant of: Gilead, Celltrion Healthcare, Clifton Bingham Consultant of: AbbVie, BMS, Eli Lilly, Janssen, Moderna, Pfizer, Sanofi, Grant/research support from: BMS, Dirkjan van Schaardenburg: None declared, Hilde Berner Hammer Speakers bureau: AbbVie, Novartis, Lilly, Rana Dadashova: None declared, Edna Hutchings: None declared, Joel Paschke: None declared, Walter P Maksymowych Speakers bureau: Abbvie, Janssen, Novartis, Pfizer, UCB, Consultant of: Abbvie, Boehringer Ingelheim, Celgene, Eli-Lilly, Galapagos, Novartis, Pfizer, UCB, Grant/research support from: Abbvie, Novartis, Pfizer

Given a real number x > 0 x>0 , we determine q s ( x ) ≔ inf ⁡ U ( x ) q_s(x)≔\operatorname {inf}{\mathscr {U}}(x) , where U ( x ) {\mathscr {U}}(x) is the set of all bases q ∈ ( 1 , 2 ] q\in (1,2] for which x x has a unique expansion of 0 0 ’s and 1 1 ’s. We give an explicit description of q s ( x ) q_s(x) for several regions of x x -values. For others, we present an efficient algorithm to determine q s ( x ) q_s(x) and the lexicographically smallest unique expansion of x x . We show that the infimum is attained for almost all x x , but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function q s q_s is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of q s q_s . A large part of the paper is devoted to the level sets L ( q ) ≔ { x > 0 : q s ( x ) = q } L(q)≔\{x>0:q_s(x)=q\} . We show that L ( q ) L(q) is finite for almost every q q , but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant q K L = min ⁡ U ( 1 ) ≈ 1.787 q_{KL}=\operatorname {min}{\mathscr {U}}(1)\approx 1.787 we prove that L ( q K L ) L(q_{KL}) has both infinitely many left- and infinitely many right accumulation points.

The primary goal of this letter is to study the impact of mobility in wireless sensor networks modeled as a leader-follower structure. Although it is intuitively known that mobility enhances the convergence rate of consensus-building in a sensor network, analytical reasoning for this intuition is not available in the literature. For filling the gap, this letter provides concrete proofs to demonstrate the benefits of introducing mobility in a sensor network with two leaders in terms of improved convergence rate in consensus building.

BackgroundLow-dose glucocorticoid (GC) therapy is widely used in RA but the true balance of benefit and harm is still unknown.ObjectivesWe studied the effects of prednisolone (5 mg/day, 2 years) in RA patients aged 65+, requiring adjustment of antirheumatic therapy (DAS28≥2.60).MethodsPragmatic double-blind placebo-controlled randomized trial; all co-treatments and changes therein were allowed during the trial except long-term open label GC; Ca/D supplementation was advised in all patients. Minimal exclusion criteria were tailored to seniors.Harm outcome: the number of patients with ≥1 serious adverse event (SAE), or ≥1 ‘other adverse event of special interest’ (other AESI). Other AESI comprised any AE (except worsening of RA) causing study discontinuation, and GC-specific events (Table 1).Table 1.Adverse events of special interest (AESI).*prednisolone (n=224)placebo (n=225)Events by protocol-defined categorySAEother AESISAEother AESI Infection261241691 Urinary tract449429 Pneumonia217213 Other20581049 Cardiovascular8260 Symptomatic fracture21146 New onset Hypertension1407 Diabetes mellitus0201 Cataract0726 Glaucoma0103 Other†43433526Total8019463140*AESI: Comprises serious adverse events (SAE) and other AESI, defined by protocol.†‘Other’ other AESI: non-serious AE outside of the above predefined categories, but associated with premature discontinuation.Benefit outcomes: improvement in disease activity (DAS28) and joint damage progression (Sharp/van der Heijde).Longitudinal mixed models analyzed the data. Given prior knowledge we report one-sided 95% confidence limit (95%CL) and statistical tests, performed only for the main outcomes.ResultsWe randomized 451 RA patients in 7 EU countries, 449 received the intervention; of these 63% prednisolone vs 61% placebo patients completed 2 years of follow up. Discontinuations were similar in both groups: for AE (14%) and active disease (4%); the remainder mostly for ‘trial fatigue’ and covid-related access issues (20%). Mean time on study drug was 19 (SD 8) months.70% of patients were female, mean age was 72 (max 88) years, RA duration 11 years; 67% were RF+, 56% ACPA+, 96% had joint damage on radiographs: mean score 20, median 8. Mean DAS28 was 4.5. Most patients (79%) were on current DMARD treatment, including 14% on biologics; 47% had previously used GC, 14% changed DMARD therapy at baseline. Patients had mean 2.1 active comorbidities, and used median 7 drugs.Benefit: Disease activity rapidly declined to stabilize after 1 year (Figure 1), and was lower on prednisolone (adjusted mean difference in DAS28 over 2 years: 0.37, 95%CL 0.23, pHarm: 60% prednisolone vs 49% placebo patients experienced the harm outcome: adjusted RR 1.24, 95%CL 1.04, p=0.02; number needed to harm 9.5 (Table 1). During the study 1 vs 2 patients died, and 3 vs 0 died within 5 months of discontinuation. Per 100 patient-years, AE totaled 278 in prednisolone vs 206 in placebo patients, and the difference was most marked for infections (Table 1); these were mostly mild or moderately severe. Other GC-specific AESI were rare without relevant differences.ConclusionAdd-on low dose prednisolone has beneficial long-term effects on disease activity and damage progression in senior RA patients on standard treatment. The tradeoff is a 24% increase in patients with mostly mild to moderate AE, suggesting a favorable balance of benefit and harm.AcknowledgementsTrial registration: NCT02585258 (clinicaltrials.gov).The trial is part of a larger project funded by the European Union’s Horizon 2020 research and innovation program under grant agreement No. 634886.Apart from the listed authors and centers, the GLORIA Trial Consortium comprises:L.M. Middelink, Middelinc BV The Netherlands, Operational Lead;V. Dekker, Amsterdam UMC, Vrije Universiteit, Financial Lead;Partners:Trial operations: N. van den Bulk, CR2O BV, The Netherlands;Study Medication (Development, Manufacturing & Supply): R.M.A. Pinto,Bluepharma – Indústria Farmacêutica, S.A., Portugal;Data management: L. Doerwald, Linical Netherlands BV, The Netherlands; S. Manger, Department of Epidemiology & Data Science, Amsterdam UMC, Vrije Universiteit, The Netherlands.Adherence monitoring: J. Redol, BeyonDevices LDA, Portugal;Safety monitoring: K. Prinsen, Clinfidence BV, The Netherlands;Patient partner: M. Scholte-Voshaar, Stichting Tools (Tools2Use), The Netherlands.Investigators (other recruiting centers):T.L.T.A. Jansen, VieCuri – location Venlo, The Netherlands;C. Codreanu, Clinical Center for Rheumatic Diseases, Bucarest, Rumania;R.M.Zandhuis-Mooij, MSc, Gelre Ziekenhuis, Apeldoorn, The Netherlands;E. Molenaar, Groene Hart Ziekenhuis, Gouda, The Netherlands;J.M. van Laar, UMC Utrecht, The Netherlands;Y.P.M. Ruiterman, Haga Ziekenhuis, Den Haag, The Netherlands;A.E.R.C.H. Boonen, MUMC, Maastricht, The Netherlands;M. Micaelo, Instituto Português de Reumatologia, Lisboa, Portugal;J. Costa, Hospital de Ponte Lima, Portugal;M. Sieburg, Rheumatologische Facharztpraxis Magdeburg, Germany;J.P.L. Spoorenberg, UMC Groningen, The Netherlands;U. Prothmann, Knappschaftsklinikum Saar GbmH, Puettlingen, Germany;M.J. Saavedra, Hospital de Santa Maria, Lisboa, Portugal;I. Silva, Hospital de Egas Moniz, Lisboa, Portugal;M.T. Nurmohamed, Reade, Amsterdam, The Netherlands;J.W.G. Jacobs, UMC Utrecht, The Netherlands; andS.W. Tas, Amsterdam UMC, University of Amsterdam, The Netherlands.Scientific Advisory Committee:J.W.J. Bijlsma, UMC Utrecht, The Netherlands;R. Christensen, The Parker Institute, Bispebjerg and Frederiksberg Hospital, Copenhagen, Denmark;Y.M. Smulders, Amsterdam UMC, VU University, The Netherlands; andS.H. Ralston, University of Edinburgh, Edinburgh, UK.Radiographic assessment:D.M.F.M. van der Heijde (Imaging Rheumatology BV, the Netherlands)coordinated the reading of the hand and foot x-rays.A.F. Marsman and W.F. Lems scored the spine X-rays.Patient panel:C. Rusthoven and M. Bakkers, The NetherlandsE. Frazão Mateus, and G. Mendes, PortugalC. Elling-Audersch and D. Borucki, GermanyA. Cardone, ItalyP. Corduta and O. Constantinescu, RomaniaP. Richards, United KingdomG. Aanerud, NorwayDisclosure of InterestsMaarten Boers Consultant of: Novartis, Linda Hartman: None declared, Daniela Opris-Belinski Consultant of: Abbvie, Pfizer, MSD, Novartis, Eli Lilly, Ewo Pharma, UCB, Reinhard Bos: None declared, Marc R Kok: None declared, José Antonio P. da Silva: None declared, Eduard N. Griep: None declared, Ruth Klaasen: None declared, Cornelia Allaart: None declared, Paul Baudoin: None declared, Hennie Raterman Consultant of: Abbvie, Pfizer, MSD, Novartis, Eli Lilly, Ewo Pharma, UCB, Zoltán Szekanecz: None declared, Frank Buttgereit Consultant of: Abbvie, AstraZeneca, Gruenenthal, Horizon Therapeutics, Mundipharma, Pfizer, Roche, Pavol MASARYK: None declared, Thomas Klausch: None declared, Sabrina Paolino: None declared, Annemarie M. Schilder Consultant of: Eli Lilly, Novartis, Genzyme, WIllem Lems Consultant of: Pfizer, Galapagos, Lilly, Amgen, UCB., Maurizio Cutolo: None declared

For a positive integer $M$ and a real base $q\in(1,M+1]$, let $\mathcal{U}_q$ denote the set of numbers having a unique expansion in base $q$ over the alphabet $\{0,1,\dots,M\}$, and let $\mathbf{U}_q$ denote the corresponding set of sequences in $\{0,1,\dots,M\}^{\mathbb{N}}$. Komornik et al. [Adv. Math. 305 (2017), 165--196] showed recently that the Hausdorff dimension of $\mathcal{U}_q$ is given by $h(\mathbf{U}_q)/\log q$, where $h(\mathbf{U}_q)$ denotes the topological entropy of $\mathbf{U}_q$. They furthermore showed that the function $H: q\mapsto h(\mathbf{U}_q)$ is continuous, nondecreasing and locally constant almost everywhere. The plateaus of $H$ were characterized by Alcaraz Barrera et al. [Trans. Amer. Math. Soc., 371 (2019), 3209--3258]. In this article we reinterpret the results of Alcaraz Barrera et al.~by introducing a notion of composition of fundamental words, and use this to obtain new information about the structure of the function $H$. This method furthermore leads to a more streamlined proof of their main theorem., 19 pages, 1 figure; only minor changes since last version

BackgroundOver 90% of patients with systemic sclerosis (SSc) experience Raynaud’s phenomenon (RP), which strongly influences quality of life. Therapeutic options of RP include drug treatment and general lifestyle measures such as smoking cessation and avoiding cold by wearing warm clothes and gloves including electrically heated gloves or silver fiber gloves. Clinical observations suggest an additional benefit of silver fiber gloves compared to normal gloves. Silver is thought to help by reflecting heat back into the hands allowing less heat to escape and has an antimicrobial effect. Despite its generalized use among SSc patients, no objective evidence regarding its superiority for RP over normal gloves exists.ObjectivesTo evaluate the added value of 8% silver fiber gloves compared to normal gloves in the treatment of patients with RP secondary to SSc.MethodsThis was a multicenter double-blind randomized cross-over trial in which 85 SSc patients were randomized in two sequences: 8% silver fiber gloves in period 1 and normal gloves in period 2 or vice versa; each period lasted six weeks. To reduce bias of interindividual differences and external factors (e.g. temperature), a cross-over design was performed in the Netherlands during the winter months. The primary outcome was the triweekly Raynaud Condition Score (RCS), a scale from 1 (no symptoms) to 10 (extreme symptoms). A linear mixed model was used with RCS as dependent and type of gloves as independent variable, adjusted for baseline RCS. Secondary outcome measures included number of RP attacks, RP attack duration, Health Assessment Questionnaire (HAQ-DI) and vascular complications. Secondary outcomes were also analyzed with linear mixed models. All analyses were performed and interpreted before unblinding.ResultsThe 85 included SSc patients had a mean age of 60 (SD:12), 80% were female, 60% had limited cutaneous SSc and 67% used vasoactive medication. Ten patients prematurely ended the study due to various reasons, most notable: allergic reaction to gloves (n=2). At baseline, mean RCS was 6.43 (SD 1.6), with silver fiber gloves the mean RCS decreased to 3.91 (SD 2.3) and with normal gloves to 3.90 (SD 2.3) (Figure 1). No statistically significant difference in RCS during follow-up was observed between the silver fiber gloves and normal gloves (β 0.067, 95% CI -0.006 to 0.19), meaning that on the 1-10 scale, silver fibre gloves gave only a 0.067 higher RCS compared to normal gloves (Table 1). For all other secondary outcome measures, we did not find a statistically significant difference between silver fiber gloves and normal gloves, except for the HAQ (β 0.036, 95% CI 0.026 to 0.046; Table 1), which is not clinically relevant. One vascular complication occurred in the silver fiber gloves, compared to three vascular in the normal gloves, which was not statistically significant different (OR:3.2, 95% CI 0.32 to 31.1).Table 1.Primary and secondary efficacy outcomesβ95% confidence intervalPrimary outcomeRaynaud Condition Score0.067-0.0059; 0.194Secondary outcomesRaynaud attacks frequency-0.480-1.215; 0.255Raynaud attacks duration39.80-36.051; 115.654VAS warmth hands-0.086-0.212; 0.041Impact Raynaud0.088-0.035; 0.211HAQ_DI0.0360.026; 0.046VAS: visual analogue scale; HAQ: Health Assessment QuestionnaireThe reference category was Normal gloves.Linear mixed models were performed with the primary and secondary outcomes as dependent variables, the type of gloves as independent variable, adjusted for baseline Raynaud Condition Score.Figure 1.Raynaud Condition Score during the study periodConclusionThis trial shows that wearing any type of glove decreases the RP burden in SSc patients, but no additional benefit from gloves containing 8% silver fibers compared to normal gloves could be demonstrated. Potentially, less vascular complications may arise in SSc patients wearing silver fiber gloves. Further confirmation of this potential benefit is necessary.AcknowledgementsThe authors would like to thank all participants of this study and Skafit for providing the gloves.Disclosure of InterestsNone declared

BackgroundGlucocorticoids (GC) are widely used for the initial treatment of rheumatoid arthritis (RA), to induce rapid suppression of inflammation and clinical symptoms and thereby limit radiographic damage progression. There are concerns that GC use in the long term is associated with a dose and duration dependent risk of serious side effects. Therefore, international guidelines have recommended to start GC when initiating a csDMARD, but to discontinue GC as rapidly as clinically feasible, preferably within 3 months (bridging therapy). In contrast, due to the concerns of GC side effects, the ACR guidelines published in 2021 conditionally recommend to start csDMARD monotherapy without GC bridging therapy.ObjectivesWe aim to evaluate the success rate of GC discontinuation after using temporary GC as part of initial therapy (‘bridging’) both in observational cohorts and clinical trials in newly diagnosed RA patients.MethodsSystematic literature searches were conducted to identify observational cohorts (scoping search) and clinical trials (in-depth search) that included RA patients who were treated with initial GC bridging therapy. GC bridging was defined as oral or intramuscular GC treatment that was discontinued within one year, alongside conventional DMARD therapy. Patient percentages still or again using GC were considered to represent the reverse of successful discontinuation. Random-effects meta-analyses were performed stratified by time point.ResultsThe literature search on observational cohort studies could not identify any study answering the research question, since it remained unclear which patients had received GC as part of the initial treatment. The literature search for clinical trials identified 7160 abstracts, resulting in 10 included studies, with varying type and dose of GC and varying tapering schedules (Table 1). Of these included studies, 4 reported sufficient data on GC discontinuation or GC use after the bridging phase. The pooled proportion of patients who were still using GC was 22% (95% Confidence Interval (CI) 8; 37, based on 4 trials) at 12 months and 10% at 24 months (95% CI -1; 22, based on 2 trials) (Figure 1). Thus, the vast majority had stopped GC. Heterogeneity was substantial (I2 ≥ 65%).Table 1.Overview of included clinical trials.Study (publication year)Tapering schedule (mg/day)COBRA (1997)In 7 weeks to 7.5. Stop after 28 weeks.*BeSt (2005)In 7 weeks to 7.5. Stop in 8 weeks after week 28 if DAS persistently ≤2.4IDEA (2014)N.A.COBRA-light (2015)arm 1: in 7 weeks to 7.5 arm 2: in 9 weeks to 7.5 Stop after 32 weeks if DASIMPROVED (2014)In 7 weeks to 7.5. Stop after 20 weeks if DAS ARCTIC (2016)In 7 weeks to 0 if DAS tREACH (2013)In 10 weeks to 0.*CareRA (2017)- in 7 weeks to 7.5, further tapered from week 28, stop after 34 weeks.- Classic- in 6 weeks to 5, further tapered from week 28, stop after 34 weeks.- Slim- in 6 weeks to 5, further tapered from week 28, stop after 34 weeks.- Avant gardeAll if DAS28(CRP) ≤3.2.Hua et al. (2020)Tapering after 4 months to 5, stop after 6 months.*NORD-STAR (2020) - arm 1A (oral prednisolone)In 9 weeks to 5. Stop after 9 months.*DAS=disease activity score; mg=milligram; N.A.=not applicable.*GC tapered and stopped according to protocol, not depending on disease activity score.ConclusionThe success rate of GC discontinuation after bridging as part of initial treatment of RA has been described in a limited number of studies. Reports on observational cohorts did not answer the research question and in clinical trials reports, GC (dis)continuation data were also scarce. However, the available data show that GC can be discontinued successfully in a large majority of patients. The paucity of data also reveals that more efforts are needed to provide data towards identifying the optimal GC bridging and discontinuation strategy, combining Treatment to Target with Starting to Stop.AcknowledgementsWe would like to thank J.W. Schoones for his help and expertise in the systematic literature search.Disclosure of InterestsLotte van Ouwerkerk: None declared, Andriko Palmowski: None declared, Isabell Nevins: None declared, Frank Buttgereit Consultant of: Consultant of AstraZeneca, AbbVie, Grünenthal, Horizon Pharma, Pfizer, and Roche., Grant/research support from: Grant/research support from AbbVie, Horizon Pharma, Pfizer, and Roche., Patrick Verschueren Consultant of: Was consultant for ABBVIE, BMS, Celltrion, Eli Lilly, Galapagos, Gilead, Nordic Pharma, Pfizer and UCB., Employee of: Holds the Pfizer Chair Early Rheumatoid Arthritis Management at KU Leuven., Josef Smolen: None declared, Robert B.M. Landewé Shareholder of: Shareholder of: Director of Rheumatology Consultancy BV., Consultant of: Consultant of: Honoraria from AbbVie, AstraZeneca, BMS, Boehringer Ingelheim, Celgene, Galapagos, Gilead, Glaxo-Smith-Kline, Janssen, Eli-Lilly, Novartis, Pfizer, UCB Pharma., Hans Bijlsma Consultant of: Consultant for Galapagos, Lilly and Sun., Grant/research support from: Received study grants from AbbVie and Roche., Andreas Kerschbaumer: None declared, Rene Westhovens Consultant of: Was consultant for Celltrion, Galapagos and Gilead., Thomas Huizinga: None declared, Cornelia Allaart Grant/research support from: Received study grants for BeSt and IMPROVED from Centocor Inc. (now Janssen) and AbbVie, respectively., Sytske Anne Bergstra Grant/research support from: Received an ASPIRE grant from Pfizer.

BackgroundRheumatoid arthritis (RA) is the most common autoimmune disease, and requires long-term treatment to suppress inflammation. Currently, methotrexate is initiated as first-line treatment when arthritis becomes clinically apparent with joint swelling. However, disease processes begin long before and become clinically recognizable when patients develop symptoms. We hypothesized that the ‘at risk phase’ of symptoms and subclinical joint-inflammation is a therapeutic window to permanently modify the disease course.ObjectivesWe studied if intervention in the pre-arthritis phase of arthralgia and subclinical joint inflammation prevents the development of clinical arthritis or reduces the burden of disease.MethodsIn this randomised, double-blind, 2-year proof-of-concept trial, adults with arthralgia clinically suspected of progressing to RA and MRI-detected subclinical joint-inflammation, recruited from all rheumatology outpatient-clinics in the southwest-Netherlands, were randomly assigned (1:1) to a single intramuscular glucocorticoid injection (120 mg) and a one-year course of oral methotrexate (up to 25 mg/week), or placebo injection and placebo tablets. Subsequently, participants were followed for another year without study medication. The primary endpoint was the development of clinically detectable arthritis (fulfilling the 2010 RA-criteria or involving ≥2 joints) that persisted for at least 2 weeks. Patient reported physical functioning, along with symptoms and workability, were key secondary endpoints and measured 4-monthly. Additionally, the course of MRI-detected inflammation was studied (the sum of tenosynovitis, synovitis, osteitis, scored with the RA-MRI Scoring (RAMRIS) method). All participants entered the intention-to-treat analysis. We performed two prespecified subgroup analyses. Firstly, analyses were restricted in participants with high risk of clinical arthritis development (PPV ≥70%). Secondly, analyses were stratified for ACPA-status. The trial is registered with the Netherlands Trials Registry (NTR4853 trial NL4599).ResultsFrom April 16th, 2015 to September 11th, 2019, we randomly assigned 236 participants to treatment (n=119) or placebo (n=117). After 24 months, arthritis free survival was similar in both groups (80% versus 82%, HR 0.81 (95%CI 0.45, 1.48)). Physical functioning improved more in the treatment-group during the first months and remained better (mean between-group difference over two-years HAQ -0·1(-0·2,-0·03;p=0·004). Similarly, pain (-9 on scale 0-100: (95%CI -12,-4; pConclusionMethotrexate, the cornerstone treatment of RA, initiated at the pre-arthritis stage of joint symptoms and subclinical inflammation, did not prevent the development of clinical arthritis, but modified the disease course as measured by sustained improvement in MRI-detected inflammation, related symptoms and impairments. These findings of sustained disease modification may open up a new treatment landscape in a pre-arthritis phase of RA, where limitations can be just as severe as at the onset of clinical arthritis.Figure 1.AcknowledgementsWe thank Prof. dr. R. ten Cate, prof. dr. S. le Cessie and dr. A.M.J. Langers for their role in the Data Safety and Monitoring Board. We thank all participants, and all rheumatologist of the following hospitals: Albert Schweitzer Hospital, Alrijne Hospital, Erasmus Medical Center, Haven-policlinic Rotterdam, IJselland Hospital, Ikazia Hospital, Franciscus Gasthuis & Vlietland Hospital, Groene Hart Hospital, Haaglanden Medical Center (all locations), Haga Hospital, Langeland Hospital, Meander Medical Center, Maasstad, Hospital, Reinier de Graaf Gasthuis, Reumazorg Zuid-West Nederland and Spaarne Gasthuis. We acknowledge the team of treating rheumatologists and research nurses of the LUMC, in particular Dr F.J. van der Giesen. Our gratitude also goes to the PhD students who scored MRIs for trial screening, in particular dr. H.W. van Steenbergen, dr. W. Nieuwenhuis, dr. R.M. ten Brink, dr. D.M. Boeters, dr. L. Mangnus, X.M.E. Matthijssen and F. Wouters. We thank dr. M. Reijnierse, prof. dr. S.C. Cannegieter and prof. dr. D. van der Heijde for their advice, and dr. J. Schoones for his help with the systematic literature search. We acknowledge the funder of the study: NWO ZonMW grant (program ‘translationeel onderzoek’, project number 95104004).Disclosure of InterestsNone declared.

BackgroundMucosal surfaces may be involved in the pathophysiology of rheumatoid arthritis (RA) (1). IgA is the most abundant class of immunoglobulin at mucosal sites. Therefore, it is worthwhile to study this isotype in RA patients in more detail. Humans have two IgA subclasses, IgA1 and IgA2, which are not evenly distributed. IgA1 is dominant in serum, whereas IgA1 and IgA2 are more balanced at mucosal surfaces (2). Besides these differences in location, IgA2 has also been ascribed pro-inflammatory properties (3).ObjectivesAs IgA subclasses might provide new insights into mucosal involvement and potential pro-inflammatory mechanisms, we investigated total and autoantibody-specific IgA subclasses responses in sera of rheumatoid arthritis patients.MethodsSera from two cohorts of RA patients, the IMPROVED (baseline visit) and the EAC (1-year visit), were selected based on previous autoantibody measurements. All patients fulfilled the 1987 (EAC) or 2010 (IMPROVED) ACR criteria for RA. Total IgA subclasses and IgA subclasses of rheumatoid factor (RF) and anti-citrullinated protein antibodies (ACPA) were measured using (in-house) ELISA’s, and compared to healthy donors. Associations between these IgA subclass levels and markers of inflammation (CRP and disease activity score (DAS)) were investigated using Spearman’s rank correlation. Mann–Whitney U tests were performed to investigate the association between IgA1 and IgA2 levels and smoking, a proxy for chronic mucosal inflammation. To correct for confounders, a multivariate linear model including age, gender, CRP and smoking was used.ResultsTotal IgA1 and IgA2 levels were increased in RA patients compared to healthy donors in both cohorts (Figure 1A-C, data IMPROVED). This increase was more pronounced in seropositive RA versus seronegative RA. Both total IgA subclasses were raised to the same extent, since the percentage of IgA2 of total IgA in serum did not differ between patients and healthy donors. In seropositive patients, RF and anti-CCP2 IgA1 and IgA2 could be detected, but measurements of anti-CCP2 IgA2 levels proved challenging due to interference of RF IgA. Although IgA2 has been postulated to be more proinflammatory, no correlations were found between total, RF and ACPA IgA subclass levels and DAS. An association between CRP and RF IgA2 was observed, but the effect size was small and did not remain significant after correction for multiple testing in the EAC. In smoking seropositive RA patients, a trend towards a selective increase in total IgA2 and RF IgA1 and IgA2 was observed (Figure 1D, data IMPROVED seropositive RA).Figure 1.ConclusionSeropositive RA patients have raised IgA1 and IgA2 levels and can also harbor RF and ACPA IgA subclasses. No shift towards IgA2 was observed, indicating that the increase in total IgA is not due to translocation of mucosal IgA into the bloodstream. However, chronic mucosal inflammation might be one of the mechanisms involved in the raise in IgA(2) levels in RA, given the association between smoking and total IgA2 levels. Despite its’ pro-inflammatory properties, no strong associations between IgA2 and markers of inflammation were found, which suggests that IgA2 does not play a essential role in the ongoing pro-inflammatory processes in RA patients.References[1]Holers VM, Demoruelle MK, Kuhn KA, et al. Rheumatoid arthritis and the mucosal origins hypothesis: protection turns to destruction. Nature Reviews Rheumatology. 2018;14(9):542-57.[2]Woof JM, Russell MW. Structure and function relationships in IgA. Mucosal Immunol. 2011;4(6):590-7.[3]Steffen U, Koeleman CA, Sokolova MV, et al. IgA subclasses have different effector functions associated with distinct glycosylation profiles. Nat Commun. 2020;11(1):120.Disclosure of InterestsNone declared

BackgroundGuidelines suggest glucocorticoids (GC) should be used as bridge therapy in rheumatoid arthritis (RA), but many patients are treated chronically with low doses. The effects of withdrawal in such patients has not been studied extensively.ObjectivesTo study disease activity score (DAS28), disease flares and signs of adrenal insufficiency after withdrawal of blinded trial medication (prednisolone 5 mg/day or placebo for 2 years).MethodsThe 2-year, double-blind GLORIA trial evaluated the long-term benefits and harms of low dose GC added to standard care (see main GLORIA trial abstract). Senior RA patients (≥ 65 years) were randomly assigned to prednisolone 5 mg/day or placebo.After the final trial visit study medication was linearly tapered to zero in 3 months by adding a stop day every two weeks, and patients were reassessed. Those who successfully completed the trial and did not receive open-label GC during the 4 weeks after the final trial visit were included in this follow-up study.The primary outcome was change in DAS28 at follow-up compared to the final trial visit. Secondary outcomes included the occurrence of disease flares (DAS28 increase > 0.6 or open-label GC between week 4 and 12 of the taper phase) and signs of adrenal insufficiency, assessed by 9 items selected from the 57-symptom list from the MDHAQ questionnaire (1) and hypotension (systolic RR < 90 or diastolic RR < 60). In a subset of patients from 3 Dutch centres, cortisol and ACTH were measured in spot serum samples during the follow-up visit.Analysis of covariance assessed the change in DAS28. Linear regression and chi-square test were used for the remaining outcomes.Results278 participants completed the GLORIA study, 21 received GC within 4 weeks after the end of the trial, 58 had missing data, leaving 199 patients eligible for this study.34 patients received open label GC after 4 weeks and were excluded for the primary analysis. In the remaining 165 patients (80 prednisolone, 85 placebo), mean (SD) DAS28 was higher on placebo: 3.14 (1.04) vs 2.92 (1.13) prednisolone at the final trial visit. After tapering, disease activity increased significantly (p=0.02) in the prednisolone group to 3.18 (1.20) but was stable in placebo (3.14). The difference in the increase of DAS28 between the groups was 0.21 (95%CI –0.05;0.47; p=0.11).For signs of adrenal insufficiency, 33 out of 165 had missing data, leaving 60 in the prednisolone group and 72 in placebo (Table 1). Mean (SD) number of signs for prednisolone was 1.1 (1.1) versus 0.9 (1.3) for placebo at final trial visit and 0.8 (1.2) versus 0.8 (1.0) at follow-up. Difference in the change of the number of signs was –0.1 (95%CI –0.4;0.3; p=0.66).Table 1.Adrenal insufficiency signs and symptoms.prednisolone (n=60)placebo(n=72)end of trialchange after 3 monthsend of trialchange after 3 monthsFatigue (unusual)15113–1Appetite loss5–144Muscle weakness7–26–2Dizziness32101Stomach pain3431Muscle pain19–619–1Nausea5–322Vomiting1001Diarrhoea5–23–2Hypotension*2–14–2Sum**1.1 (1.1)–0.2 (1.3)0.9 (1.3)0.0 (1.3)* Systolic RR < 90 or diastolic RR < 60.**Mean (SD)No differences were seen in ACTH or cortisol levels: mean (SD) ACTH was 5.8 (4.1) in 23 prednisolone patients, and 5.1 (3.7) in 24 placebo patients; cortisol 296 (113) v 310 (166), cortisol/ACTH 67 (40) v 77 (54). Two prednisolone and one placebo patient had cortisol levels below 80. None developed clinical hypoadrenalism during further follow-up.199 patients qualified for the disease flares sample, 99 prednisolone and 100 placebo; 44 patients flared on prednisolone tapering vs 31 on placebo, relative risk 1.43 (95%CI 0.99; 2.07; p=0.07).ConclusionTapering prednisolone moderately increases disease activity to placebo levels (mean still at low disease activity levels) and numerically increases the risk of flare without any evidence of adrenal insufficiency. This suggests that withdrawal of low dose prednisolone is feasible after 2 years of administration.References[1]DeWalt DA et al. Clin Exp Rheumatol. 2004;22:453-61.AcknowledgementsThe GLORIA trial is registered at clinicaltrials.gov under NCT02585258.The GLORIA project is funded by the European Union’s Horizon 2020 research and innovation programme under the topic ‘’Personalizing Health and Care’’, grant agreement No 634886.Disclosure of InterestsAbdullah Almayali: None declared, Maarten Boers Consultant of: Novartis, Linda Hartman: None declared, Daniela Opris-Belinski Consultant of: Abbvie, Pfizer, MSD, Novartis, Eli Lilly, Ewo Pharma, UCB, Reinhard Bos: None declared, Marc R Kok: None declared, José Antonio P. da Silva: None declared, Eduard N. Griep: None declared, Ruth Klaasen: None declared, Cornelia Allaart: None declared, Paul Baudoin: None declared, Hennie Raterman Consultant of: AbbVie, Amgen, Celgene, Roche, Sandoz, Sanofi Genzyme and UCB, Zoltán Szekanecz: None declared, Frank Buttgereit Consultant of: Abbvie, AstraZeneca, Gruenenthal, Horizon Therapeutics, Mundipharma, Pfizer, Roche, Pavol MASARYK: None declared, WIllem Lems Consultant of: Pfizer, Galapagos, Lilly, Amgen, UCB., Maurizio Cutolo: None declared, Marieke ter Wee: None declared

BackgroundPreviously we showed that joint inflammation in rheumatoid arthritis (RA) tends to recur in the same joint, suggesting local factors contributing to joint inflammation. In the same study population, we now investigated whether cumulative local joint inflammation is associated with local radiographic damage progression.ObjectivesTo investigate whether there is an association between long-term cumulative joint swelling and progression of radiographic damage in treated-to-target patients with RA.MethodsData from the BeSt study were used, in which newly diagnosed patients with RA (ACR 1987 criteria) were treated-to-target (DAS ≤2.4) during 10 years. Local joint swelling (yes/no) was determined by clinical evaluation by trained nurses of all hand and foot joints at 3-monthly study visits. Yearly radiographs of hand and feet were scored for radiographic joint damage (Sharp-van der Heijde method) in random order by two independent readers who were blind for clinical results. Per joint, damage was expressed as the percentage of the maximum possible damage score, to account for differences in maximal scores per joint. Missing values were imputed using the last observation carried forward method. A generalized linear mixed model was used to assess the association between local joint swelling over time (i.e., percentage of study visits with observed local joint swelling) and degree of joint damage at the end of follow-up. Joints were clustered within patients. The model was adjusted for baseline damage and follow-up duration. To test the association between cumulative local joint swelling and joint damage as a local or a general inflammation effect, we did two analyses. First, we additionally adjusted the primary analysis for the mean disease activity score (DAS) over time. Second, we did a permutation test to study whether joint damage progression was better predicted by joint swelling in the joint itself than by joint swelling in randomly selected other joints, which is indicated by a p-value of ResultsOf the 16,150 joints of 475 patients with at least one year follow-up with both radiographic and joint swelling assessment available, 16% (2,564) had radiographic joint damage (damage score ≥ 0.5) at the end of follow-up. Median (IQR) follow-up time was 10 (6-10) years. Of the joints with damage at the end of follow-up, 46% (1,163) was swollen at baseline, versus 36% (4,818) of the joints without damage. The median (IQR) percentage of visits at which joint swelling was observed was 6 (0-17) and 3 (0-8) for joints with and without joint damage respectively.We found a β of 0.13 (95% CI 0.12 to 0.14) for the association between cumulative local joint swelling and local progression, that is, with each 1% increase in the number of visits with local joint swelling, local radiographic joint damage progression on average increased with 0.13 percent. In an analysis with 10-years completers only (both baseline and year 10 damage score available, n = 9,520) we also found an association between cumulative local joint swelling and local radiographic damage (β 0.24, 95% CI 0.22 to 0.26). The association was also found in a subset of joints that were swollen at least once (β 0.20, 95% CI 0.18 to 0.22), indicating that joint damage is not only associated with ever-occurrence but also with the frequency of joint swelling.This association was found for both erosions (β 0.07, 95% CI 0.07 to 0.08) and joint space narrowing (β 0.21, 95% CI 0.19 to 0.22). The results of the primary analysis did not change after adjustment for DAS over time. The permutation test showed that local joint damage progression was better predicted by the frequency of joint swelling of that joint, than by joint swelling frequency of other joints (pConclusionCumulative local joint swelling over time is associated with joint damage progression in the same joint in treated-to-target (DAS ≤2.4) patients with RA. Our results indicate that this is a local effect rather than an effect of general disease activity.Disclosure of InterestsSascha Heckert: None declared, Sytske Anne Bergstra: None declared, Yvonne Goekoop-Ruiterman: None declared, Melek Güler-Yüksel: None declared, WIllem Lems: None declared, M. van Oosterhout: None declared, Thomas Huizinga: None declared, Cornelia Allaart Grant/research support from: The original BeSt study was funded by a research grant from the Dutch College of Health Insurances with additional funding from Schering-Plough BV and Centocor Inc.

BackgroundUse of prednisone in rheumatoid arthritis has been questioned because it may trigger side effects such as hyperglycemia and diabetes.ObjectivesTo assess whether in RA the use of prednisone is associated with the development of hyperglycemia and diabetes.MethodsThe BeSt study is a multicenter, assessor-blinded randomized controlled 10-years follow-up trial in 508 non-diabetic early RA patients. Patients were randomised to 4 dynamic DMARD treatment strategy groups: 1) sequential monotherapy, 2) step-up combination therapy, 3) initial combination therapy including prednisone (60 mg/day, tapered to 7.5 mg/day in 7 weeks) and 4) initial combination therapy with infliximab. In groups 1, 2 and 4, prednisone had a maximum dose of 7.5 mg/day by protocol. Treatment was steered at disease activity score (DAS) ≤2.4. We performed a GEE over time to assess whether current prednisone use or cumulative prednisone dose were associated with hyperglycemia (glucose levels ≥7.8) and cox regression analyses to investigate the relationship between cumulative prednisone dose, previous prednisone use and diabetes (defined as either use of anti-diabetic medication or two instances of a glucose ≥ 11.1), assessed at 3-monthly visits. All analyses were adjusted for potential confounders.ResultsIn total, 33/508 patients (6.5%) developed diabetes during the trial; 12 of these (36%) had received prior treatment with prednisone (any dose). Median (IQR) duration of prednisone use in all 508 patients was 9 (15) months and cumulative doses ranged from 0 to 27942 mg. The mean cumulative dose ranged from 55.5 mg in group 1 to 6170.0 mg in group 3. Previous prednisone use nor cumulative prednisone dose was associated with hyperglycemia or diabetes, with effect sizes ranging from a hazard ratio of 0.588 (95% CI 0.285; 1.21) for the association between any prednisone dose and diabetes to an odds ratio of 1.04 (95% CI 0.978; 1.13) for the association between cumulative prednisone dose and diabetes (Table 1). To identify potential causes for these results, we investigated the relationship between DAS and the same outcomes. We found a higher DAS was significantly associated with development of diabetes, but not with hyperglycemia.Table 1.The relationship between prednisone dose, DAS and glucose levels, hyperglycemia and diabetesGEEhyperglycemia*OR95% CIAny prednisone dose10.9490.805; 1.12Cumulative dose11.04**0.978; 1.13DAS21.240.842; 1.85Cox Regressiondiabetes (any of the definitions)HR95% CIAny prednisone dose30.5880.285; 1.21Cumulative dose30.996**0.960; 1.03DAS21.601.13; 2.26CI: confidence interval, GEE: Generalized Estimating Equations, OR: odds ratio, HR: hazard ratio, DAS: disease activity* hyperglycemia: glucose level above 7.8 mmol/L; diabetes: random glucose level above 11.1 mmol/L at at least two time points** odds ratio per 500mg cumulative prednisone increase: 1adjusted for DAS, age, diabetes and BMI: 2adjusted for cumulative prednisone dose, age, gender and BM: 3adjusted for DAS, age and BMIConclusionIn early RA patients, cumulative dose nor any previous prednisone use was associated with the risk of hyperglycemia or diabetes. A higher DAS was significantly associated with increased risk of developing diabetes. Potential risks of prednisone may have been mitigated by suppression of DAS.Disclosure of InterestsJoy van der Pol: None declared, Sytske Anne Bergstra Grant/research support from: Pfizer, Thomas Huizinga: None declared, Cornelia Allaart Grant/research support from: The BeSt study was supported by a government grant from the Dutch Insurance Companies, with additional funding from Schering-Plough B.V. and Janssen B.V.

This paper studies a large class of continuous functions f : [ 0 , 1 ] → R d whose range is the attractor of an iterated function system { S 1 , … , S m } consisting of similitudes. This class includes such classical examples as Polya's space-filling curves, the Riesz–Nagy singular functions and Okamoto's functions. The differentiability of f is completely classified in terms of the contraction ratios of the maps S 1 , … , S m . Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) f is nowhere differentiable; (ii) f is non-differentiable almost everywhere but with uncountably many exceptions; or (iii) f is differentiable almost everywhere but with uncountably many exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and (iii) above is calculated, and more generally, the complete multifractal spectrum of f is determined.

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number (base) $1, Comment: 25 pages. Slightly extended the results and added more references

LMM for Poznanski, BA and BMD adjusted for age and/or symptom duration. (DOCX 14 kb)

A Sensitivity analysis of patients with wrist arthritis, without nâ =â 6 with never wrist arthritis. B Sensitivity analysis of patients with polyarticular JIA. C LMM for mean JADAS10 score in relation to Poznanski, BA and BMD. (DOCX 804 kb)

Bland-Altman plots with 95% limits of agreement. (DOCX 245 kb)

Flow chart of patient selection process for the Poznanski-score. (DOCX 23 kb)

Background:Patients with an immune mediated inflammatory disorder or post solid organ transplantation (IMIDT), are at risk for infectious complications especially if they are treated with immunosuppressive drugs (imeds). There is still great uncertainty whether these IMIDT patients are more susceptible to COVID-19 than controls, and/or should be advised to avoid taking their immunosuppressive treatment.Objectives:To evaluate whether patients with IMIDT are more at risk for CLS than controls.Methods:The IENIMINI study is a prospective cohort study in patients with IMIDT and controls (healthy or no IMIDT) who were identified based on the registration database of the Leiden University Medical Center. Over time, participants registered COVID-like symptoms (CLS) as they occurred, and filled in additional questionnaires. Univariate and multivariate regression analyses were done to identify variables associated with having CLS.Results:Of the 8670 individuals approached, 2110 with IMIDT and 1067 controls agreed to participate. In March and April, 454 (22%) of IMIDT patients and 242 (23%) of controls recorded to have CLS, mostly mild with a median (IQR) duration of seven (3-14) days in the IMIDT group and six days (4-11) in the control group. Eleven (5%) of the IMIDT patients with immunosuppressive medication (imed), 6 (3%) of IMIDTs without imed and 2 (1%) of controls were hospitalized with CLS (p=0.04). In May and June, fewer episodes overall were recorded. Being female (OR 1.45 95%CI 1.15;1.82), having a lung disease (OR 1.50 95%CI 1.20;1.88) and wearing a face mask (OR 1.42 95%CI 1.13-1.77) were independently associated with a higher risk, while higher age (OR 0.96 95%CI 0.96;0.97) and having an IMIDT with immunosuppressive medication use (OR 0.68 95%CI 0.51;0.91) were independently associated with a lower risk (see Table 1). Similar results were found after data imputation.Table 1.Univariate & multivariate analysis of variables associated with having CLS or not (OR with 95% CI)n0UnivariateMultivariate*Sex, female25461.89 (1.58;2.25)1.45 (1.15;1.82)BMI23910.99 (0.97;1.01)1.00 (0.98;1.03)Age25460.97 (0.96;0.97)0.96 (0.96;0.97)IMIDT without imed†25461.00 (0.82;1.23)0.94 (0.72;1.24)IMIDT with imed †25460.79 (0.65;0.97)0.68 (0.51;0.91)Smoking (current)24631.35 (1.02;1.78)1.05 (0.74;1.50)Physical contact with family**22201.47 (1.22;1.78)1.22 (0.98;1.53)Visiting other people (not family)22051.26 (1.05;1.51)0.96 (0.77;1.20)Wearing a face mask21961.46 (1.20;1.76)1.42 (1.13;1.77)Close contact (at work)21801.65 (1.34;2.03)1.27 (0.97;1.66)Self-reported Diabetes Mellitus23810.69 (0.50;0.96)0.89 (0.58;1.36)Self-reported lung disease23961.30 (1.09;1.54)1.50 (1.20;1.88)Self-reported heart disease23990.85 (0.69;1.04)1.09 (0.83;1.43)Daily alcohol use24160.84 (0.71;1.00)1.20 (0.96;1.50)Influenza vaccination***24150.71 (0.60;0.84)0.96 (0.76;1.21)Solid organ transplantation25460.74 (0.54;1.03)0.79 (0.47;1.35)Good adherence to lockdown rules22451.17 (0.41;3.29)2.46 (0.65;9.38)Use of oral corticosteroids25460.84 (0.66;1.06)1.44 (0.95;2.20)Working outside the house24351.39 (1.16;1.68)0.92 (0.71;1.20)Abbreviations: BMI=body mass index; CI= confidence intervals; CLS=Covid like symptoms; IMIDT= with immune mediated inflammatory disorders or transplant organ; n0=number of observations; OR=odds ratio.* number of observations: 1835** physical contact specified as ‘holding/shaking hands, hugging etcetera’*** in autumn 2019† control group = reference groupConclusion:Between March and July 2020, IMIDT patients, whether or not taking imeds, did not show an increased risk of reported COVID-like symptoms compared to controls. Continuing immunosuppressant drugs as long as not ill, while following the Dutch COVID rules, appears to be safe.Disclosure of Interests:Lotte van Ouwerkerk: None declared., Andrea van der Meulen Speakers bureau: Dr. van der Meulen reports personal fees from Janssen, grants and personal fees from Takeda, personal fees from Galapogos, grants from Nestle, grants from Norgine, outside the submitted work., Grant/research support from: Dr. van der Meulen reports personal fees from Janssen, grants and personal fees from Takeda, personal fees from Galapogos, grants from Nestle, grants from Norgine, outside the submitted work., Maarten Ninaber: None declared., Y.K. Onno Teng: None declared., Thomas Huizinga: None declared., Cornelia Allaart: None declared.

Background:It is unknown whether in the disease course of rheumatoid arthritis (RA), inflammation recurs in the same joints over time or is more variable in joint locations. Joint involvement patterns over time might provide clues about the underlying mechanisms causing local joint inflammation.Objectives:The aim of this study is to assess if local joint inflammation at presentation of RA tends to recur or persist in the same joints.Methods:Data from the BeSt study were used, a treat-to-target (DAS≤2.4) trial in newly diagnosed RA (ACR 1987 criteria) patients. During 10 years, for each patient 68 joints were assessed three-monthly (41 visits) by trained nurses for swelling (yes/no) and tenderness.We analyzed the association between local joint swelling at baseline and later swelling of the same joint using a multilevel mixed-effects logistic regression model. Models were adjusted for joint location and for timepoint, with joints clustered within patients. A sensitivity analysis was done for the 25% most affected joints (MCP 1-3, PIP 2-3, wrists and MTP 2-4).To investigate whether later swelling of a joint is predicted by baseline swelling of that same joint specifically, rather than by baseline swelling in general, a permutation test with 1000 permutations was performed. A p-value In a separate model, with an interaction term between baseline swelling and previous visit swelling (yes/no), we evaluated if the association between baseline swelling and later local swelling was influenced by whether later swelling was persistent (swelling at both the current and previous visit) or recurrent (swelling at current visit but not at the previous visit).Results:The 508 patients had a median (IQR) follow-up duration of 10 (6-10) years. At baseline, 8,137/34,423 (24%) assessed joints were scored as swollen. Baseline swelling was subsequently persistent in 21% of the joints with a median (IQR) duration of 1 (1-2) visit (± 3 months after baseline). In addition, after resolution of initial swelling, swelling recurred at least once in 46% of the joints with baseline swelling.Baseline swelling was significantly associated with swelling in the same joint during follow-up (OR 2.37, 95% CI 2.30-2.43). A sensitivity analysis of the most affected joints showed similar results (OR 2.10 [95% CI 2.03-2.19]).The permutation test showed a significant result with pThe association between baseline swelling and later local swelling was weaker in case of persistent swelling than in case of recurrent swelling (interaction term baseline swelling * swelling at previous timepoint ‘yes’: OR 0.80 [95% CI 0.75-0.85]).Conclusion:In newly diagnosed RA, over median 10 years of treatment to target DAS≤2.4, baseline swelling persisted in 21% of the joints, for median 3 months after baseline. Local recurrence after initial resolution occurred in 46% of the joints. Baseline joint swelling was significantly associated with local joint swelling during follow-up, even when taking into account the higher a priori chance of swelling in the joints that are most often affected, and joint swelling during follow-up was better predicted by baseline swelling of that particular joint than by baseline swelling of other joints. Local persistence and recurrence of joint swelling despite DAS≤2.4 steered treatment adjustments suggest that local joint conditions or even joint memory play a role in mechanisms of joint inflammation.Acknowledgements:We would like to thank all patients for their contribution as well as the rheumatologists who participated in the BeSt study group. We would also like to thank all other rheumatologists and trainee rheumatologists who enrolled patients in these studies, and all research nurses for their contributions.Disclosure of Interests:Sascha Heckert: None declared, Sytske Anne Bergstra: None declared, Xanthe Matthijssen: None declared, Yvonne Goekoop-Ruiterman: None declared, F. Fodili: None declared, Cornelia Allaart Grant/research support from: The original BeSt study was supported by a government grant from the Dutch insurance companies, with additional funding from Schering-Plough B.V. and Janssen B.V., Thomas Huizinga: None declared

This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These functions satisfy a functional equation of the form $\phi(a_k x+b_k)=c_k x+d_k\phi(x)+e_k$, for $k=1,2,\dots,r$ and $x\in[0,1]$. They include the Takagi function, the Riesz-Nagy singular functions, Okamoto's functions, and many other well-known examples. It is shown that the multifractal spectrum of $\phi$ is given by the multifractal formalism when $|d_k|\geq |a_k|$ for at least one $k$, but the multifractal formalism may fail otherwise, depending on the relationship between the shear parameters $c_k$ and the other parameters. In the special case when $a_k>0$ for every $k$, an exact expression is derived for the pointwise H\"older exponent at any point. These results extend recent work by the author [Adv. Math. 328 (2018), 1-39] and S. Dubuc [Expo. Math. 36 (2018), 119-142]., Comment: 40 pages, 3 figures. The Introduction has been reorganized somewhat

Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper investigates how the set $\mathscr{U}$ is distributed over the interval $(1,M+1)$ by determining the limit $$f(q):=\lim_{\delta\to 0}\dim_H\big(\mathscr{U}\cap(q-\delta,q+\delta)\big)$$ for all $q\in(1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in\overline{\mathscr{U}}\backslash\mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called {\emph relative bifurcation sets}, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear in Acta Arithmetica}, 2018]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [{\emph Adv. Math.}, 308:575--598, 2017] about strongly univoque sets., Comment: 31 pages and 1 figure

In Robbins' problem of minimizing the expected rank, a finite sequence of $n$ independent, identically distributed random variables are observed sequentially and the objective is to stop at such a time that the expected rank of the selected variable (among the sequence of all $n$ variables) is as small as possible. In this paper we consider an analogous problem in which the observed random variables are the steps of a symmetric random walk. Assuming continuously distributed step sizes, we describe the optimal stopping rules for the cases $n=2$ and $n=3$ in two versions of the problem: a "full information" version in which the actual steps of the random walk are disclosed to the decision maker; and a "partial information" version in which only the relative ranks of the positions taken by the random walk are observed. When $n=3$, the optimal rule and expected rank depend on the distribution of the step sizes. We give sharp bounds for the optimal expected rank in the partial information version, and fairly sharp bounds in the full information version., Comment: 18 pages, 2 tables

Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is well-known that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \[ \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. \] Note that $\mathcal B$ is contained in the set $\mathcal{U}^R$ of bases $q\in(1,M+1]$ such that $1\in\mathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash\mathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $\mathcal B\backslash\mathcal{U}^R$ is $\frac{\log 2}{3\log \lambda^*}\approx 0.368699$, where $\lambda^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$., Comment: 28 pages, 1 figures and 1 table. To appear in J. Fractal Geometry

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach., Comment: 18 pages

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\em $\beta$-expansions}) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto's function (itself a generalization of the well-known functions of Perkins and Katsuura) to a two-parameter family $\{F_{N,a}: N\in\mathbb{N}, a\in(0,1)\}$. We first show that for each $x$, $F_{N,a}'(x)$ is either $0$, $\pm\infty$, or undefined. We then extend Okamoto's theorem by proving that for each $N$, depending on the value of $a$ relative to a pair of thresholds, the set $\{x: F_{N,a}'(x)=0\}$ is either empty, uncountable but Lebesgue null, or of full Lebesgue measure. We compute its Hausdorff dimension in the second case. The second result is a characterization of the set $\mathcal{D}_\infty(a):=\{x:F_{N,a}'(x)=\pm\infty\}$, which enables us to closely relate this set to the set of points which have a unique expansion in the (typically noninteger) base $\beta=1/a$. Recent advances in the theory of $\beta$-expansions are then used to determine the cardinality and Hausdorff dimension of $\mathcal{D}_\infty(a)$, which depends qualitatively on the value of $a$ relative to a second pair of thresholds., Comment: 20 pages, 4 figures

Let T be Takagiʼs continuous but nowhere-differentiable function. It is known that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. We show that the most common cardinality of the level sets of T is two, and investigate in detail the set of ordinates y such that the level set at level y has precisely two elements. As a by-product, we obtain a simple iterative procedure for solving the equation T ( x ) = y . We show further that any positive even integer occurs as the cardinality of some level set, and conjecture that all even cardinalities occur with positive probability if an ordinate y is chosen at random from the range of T. The key to the results is a system of set equations for the level sets, which are derived from the partial self-similarity of T. These set equations yield a system of linear relationships between the cardinalities of level sets at various levels, from which all the results of this paper flow.

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands., Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figures

This paper considers a parametrized family of generalized Takagi functions f_p with parameter p. Tabor and Tabor [J. Math. Anal. Appl. 356 (2009), 729-737] recently proved that for p in [1,2], f_p is (1,p)-midconvex. We give a simpler proof of this result by developing an explicit expression for f_p at dyadic rational points and showing that (1,p)-midconvexity of f_p reduces to a simple inequality for weighted sums of binary digits., Comment: 11 pages, included more detail in the proofs and added a new Corollary 4

Optimal stopping rules are developed for the perpetual American call option and the Russian option under a correlated random walk model. The optimal rules are of twin threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Depending on the choice of parameter values, one of the thresholds may be infinite. Precise expressions for the thresholds and optimal expected returns are given both in the positively and the negatively correlated case, for problems both with and without discounting. The optimal rules are illustrated by several numerical examples.

Let X 1,…, X n be independent, identically distributed random variables that are nonnegative and integrable, with known continuous distribution. These random variables are observed sequentially, and the goal is to maximize the expected X value at which one stops. Let V n denote the optimal expected return of a player who can observe at time j only whether X j is a relative record (j = 1,…, n), and W n that of a player who observes at time j the actual value of X j . It is shown that V n > a n W n , where , and this inequality is sharp.

This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable function. Let F(x):=∑ =1 ∞ $$ (\tfrac{1} {2})^{n - 1} $$ e n ϕ(2 n−1 x), x ∈ R, where ɛ 1, ɛ 2, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ 1 = 1), M:= max {F(x): x ∈ R}, and $$ \mathcal{M} $$ := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ n }, and it is shown that the probability distribution µ of M is purely atomic if p < $$ \tfrac{1} {2} $$ , and is singular continuous if p ≧ $$ \tfrac{1} {2} $$ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set $$ \mathcal{M} $$ is finite almost surely if p < $$ \tfrac{1} {2} $$ , and is topologically equivalent to a Cantor set almost surely if p ≧ $$ \tfrac{1} {2} $$ . The distribution of the cardinality of $$ \mathcal{M} $$ is determined in the first case, and the almost-sure Hausdorff dimension of $$ \mathcal{M} $$ is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of $$ \mathcal{M} $$ is also given. Finally, some of the results are extended to the more general functions Σa n − 1 ɛ n ϕ(2 n − 1 x), where 0 < a < 1.

Correlated random walks provide an elementary model for processes that exhibit directional reinforcement behavior. This paper develops optimal multiple stopping strategies - buy/sell rules - for correlated random walks. The work extends previous results given in Allaart and Monticino (2001) by considering random step sizes and allowing possibly negative reinforcement of the walk's current direction. The optimal strategies fall into two general classes - cases where conservative buy-and-hold type strategies are optimal and cases for which it is optimal to follow aggressive trading strategies of successively buying and selling the commodity depending on whether the price goes up or down. Simulation examples are given based on a stock index fund to illustrate the variation in return possible using the theoretically optimal stop rules compared to simpler buy-and-hold strategies.

The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.

This paper considers a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if $X_1,\dots,X_n$ are independent random variables with known continuous distributions and $V_n(X_1,\dots,X_n):=\sup_\tau P(X_\tau=M_n)$, where $M_n:=\max\{X_1,\dots,X_n\}$ and the supremum is over all stopping times adapted to $X_1,\dots,X_n$, then $$V_n(X_1,\dots,X_n)\geq \left(1-\frac{1}{n}\right)^{n-1},$$ and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of two-valued random variables, and then applying Bruss' sum-the-odds theorem (2000). In order to obtain a sharp bound for each $n$, we improve Bruss' lower bound (2003) for the sum-the-odds problem., Comment: 13 pages

Okamoto's one-parameter family of self-affine functions $F_a: [0,1]\to[0,1]$, where $01/2$. For all $a$, we determine the Hausdorff dimension of the sets of points where: (i) $F_a'=0$; and (ii) $F_a$ has neither a finite nor an infinite derivative. The upper and lower densities of the digit $1$ in the ternary expansion of $x\in[0,1]$ play an important role in the analysis, as does the theory of $\beta$-expansions of real numbers., Comment: 26 pages; more figures were added and Theorem 2.6 now includes additional statements

Let X 1, X 2,… be any sequence of [0,1]-valued random variables. A complete comparison is made between the expected maximum E(max j≤n Y j ) and the stop rule supremum sup t E Y t for two types of discounted sequences: (i) Y j = b j X j , where {b j } is a nonincreasing sequence of positive numbers with b 1 = 1; and (ii) Y j = B 1… B j−1 X j , where B 1, B 2,… are independent [0,1]-valued random variables that are independent of the X j , having a common mean β. For instance, it is shown that the set of points {(x, y): x = sup t E Y {(x, y): x=sup t E Y and y = E(max j≤n Y j ), for some sequence X 1,…,X n and Y j = b j X j }, is precisely the convex closure of the union of the sets {(b j x, b j y): (x, y) ∈ C j }, j = 1,…,n, where C j = {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ x[1 + (j − 1)(1 − x 1/(j−1))]} is the prophet region for undiscounted random variables given by Hill and Kertz [8]. As a special case, it is shown that the maximum possible difference E(max j≤n β j−1 X j ) − sup t E(β t−1 X t ) is att...

Let X 1, X 2… and B 1, B 2… be mutually independent [0, 1]-valued random variables, with EB j = β > 0 for all j. Let Y j = B 1 … sB j−1 X j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y 1,…,Y n ):=sup{EY τ:τ is a stopping rule for Y 1,…,Y n } and E(max 1≤j≤n Y j ). It is shown that the set of ordered pairs {(x, y):x = V(Y 1,…,Y n ), y = E(max 1≤j≤n Y j ) for some sequence Y 1,…,Y n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ n, β(x)}, where Ψ n, β(x) = [(1 − β)n + 2β]x − β−(n−2) x 2 if x ≤ β n−1, and Ψ n, β(x) = min j≥1{(1 − β)jx + β j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.

Let (Sn)n≥0 be a correlated random walk on the integers, let M0 ≥ S0 be an arbitrary integer, and let Mn = max{M0, S1,…, Sn}. An optimal stopping rule is derived for the sequence Mn - nc, where c > 0 is a fixed cost. The optimal rule is shown to be of threshold type: stop at the first time that Mn - Sn ≥ Δ, where Δ is a certain nonnegative integer. An explicit expression for this optimal threshold is given.

Let X 1,X 2 , … be any sequence of nonnegative integrable random variables, and let N∈{1,2 , …} be a random variable with known distribution, independent of X 1,X 2 , …. The optimal stopping value sup t E(Xt I(N≥ t)) is considered for two players: one who has advance knowledge of the value of N, and another who does not. Sharp ratio and difference inequalities relating the two players' optimal values are given in a number of settings. The key to the proofs is an application of a prophet region for arbitrarily dependent random variables by Hill and Kertz [T.P. Hill and R.P. Kertz (1983). Stop rule inequalities for uniformly bounded sequences of random variables. Trans. Amer. Math. Soc., 278, 197–207].

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

This note considers the average-optimal expected return of two players observing independent random variables X 1, … , X n , whose distributions are generated at random. One player, the pseudo prophet, knows the distributions prior to observing the random variables. The other player, the gambler, has no such foresight. Sharp difference and ratio comparisons of the two players' optimal expected returns are given. The key step in the proof is a reduction to a classical prophet inequality for i.i.d. random variables proved by Hill and Kertz (Hill, T.P.; Kertz, R.P. Comparisons of stop rule and supremum expectations of i.i.d. random variables. Ann. Probab. 1982, 10 (2), 336–345).

Let X, X1, ..., Xk be i.i.d. random variables, and for k∈ N let Dk(X) = E(X1 V ... V Xk+1) −EX be the kth centralized maximal moment. A sharp lower bound is given for D1(X) in terms of the Levy concentration Ql(X) = supx∈ R P(X∈[x, x + l]). This inequality, which is analogous to P. Levy's concentration-variance inequality, illustrates the fact that maximal moments are a gauge of how much spread out the underlying distribution is. It is also shown that the centralized maximal moments are increased under convolution.