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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.

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

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

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 (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.

A sharp upper bound is given for the degree of nonconvexity of the partition range of a finite-dimensional vector measure, in terms of the maximum (one-dimensional) mass of the atoms of that measure. This upper bound improves on a bound of Hill and Tong (1989, Anal. Stat.17, 1325–1334) by an order of magnitude n. Its proof uses several ideas from graph theory, combinatorics, and convex geometry. Applications are given to optimal-partitioning and fair division problems.

Minimax risk inequalities are obtained for the location-parameter classification problem. For the classical single observation case with continuous distributions, best possible bounds are given in terms of their Lévy concentration, establishing a conjecture of Hill and Tong (1989). In addition, sharp bounds for the minimax risk are derived for the multiple (i.i.d.) observations case, based on the tail concentration and the Lévy concentration. Some fairly sharp bounds for discontinuous distributions are also obtained.

The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk is constant for every fixed k.

This paper examines level sets of two families of continuous, nowhere differentiable functions (one a subfamily of the other) defined in terms of the "tent map". The well-known Takagi function is a special case. Sharp upper bounds are given for the Hausdorff dimension of the level sets of functions in these two families. Furthermore, the case where a function f is chosen at random from either family is considered, and results are given for the Hausdorff dimension of the zero set and the set of maximum points of f., 34 pages, 5 figures. The statement of Theorem 1.1 was expanded and various improvements to the presentation were made

The purpose of this note is to correct an error in an earlier paper by the author about the level sets of the Takagi function [Monatsh. Math. 167 (2012), 311-331 and arXiv:1102.1616], and to prove a stronger form of one of the main results of that paper about the propensity of level sets containing uncountably many local level sets., Comment: 6 pages, correction of arXiv:1102.1616. A small mistake in the proof of Theorem 8 was corrected

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as signed Takagi functions. The case when r_n=1 for all n is the classical Takagi function, a well-known example of a continuous but nowhere differentiable function. For f of the above form, the maximum and minimum values of f are expressed in terms of the sequence {r_n}. It is then shown that almost all level sets of f are finite (with respect to Lebesgue measure on the range of f), but the set of ordinates y with an uncountably large level set is residual in the range of f. The concept of a local level set of the Takagi function, due to Lagarias and Maddock, is extended to arbitrary signed Takagi functions. It is shown that the average number of local level sets contained in a level set of f is the reciprocal of the height of the graph of f, and consequently, this average lies between 3/2 and 2., Comment: This is a stand-alone version of Section 5 in arXiv:1102.1616, with more proof details. 15 pages, 2 figures. An error in the proof of Theorem 1.3 was corrected, and the theorem now has a slightly stronger statement

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1. Several inequalities are derived for S_b(N). Some of the inequalities can be interpreted as comparing the average value of s_b(n) over integer intervals of certain lengths to the average value of a beginning subinterval. Two of the main results are applied to derive a pair of "approximate convexity" inequalities for a sequence of Takagi-like functions. One of these inequalities was discovered recently via a different method by V. Lev; the other is new., 15 pages

This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the tent map, are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.

This paper considers a class of continuous functions constructed as a series of iterates of the “tent map” multiplied by variable signs. This class includes Takagi’s nowhere-differentiable function, and contains the functions studied by Hata and Yamaguti [Japan J. Appl. Math., 1 (1984), 183-199] and Kono [Acta Math. Hungar. 49 (1987), 315324] as a proper subclass. A complete description is given of the differentiability properties of the functions in this class, and several statements are proved concerning their uniform and local moduli of continuity. The results are applied to generation of random functions. AMS 2000 subject classification: 26A27 (primary); 26A15, 26A30, 60G50 (secondary)

Let X_t, 0, 20 pages; added a few specific examples and additional references

Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang" type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\tau^*\equiv T$ is the drift is positive. This result generalizes recent findings by S. Yam, S. Yung and W. Zhou [{\em J. Appl. Probab.} {\bf 46} (2009), 651--668] and J. Du Toit and G. Peskir [{\em Ann. Appl. Probab.} {\bf 19} (2009), 983--1014], and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good ones as long as possible., Comment: 13 pages

Levy's dragon curve (P. L´, Les courbes planes ou gauches et les surfaces composees de parties semblables au tout, J. Ecole Polytechn., 227-247, 249-291 (1938)) is a well-known self-similar planar curve with non-empty interior. We derive an arithmetic expression for the coordi- nate functions of Levy's dragon curve, and show that the 3 -dimensional Hausdorff measure of the graph of each coordinate function is strictly pos- itive and finite. This complements known dimensional results concerning the coordinate functions of space-filling curves of Peano and Hilbert. The proof is based on deriving suitable uniform upper bounds for the sizes of the graphs' level sets.

Supplementary methods. Figure S1. DAS44 over time. Table S1. Baseline characteristics of patients who achieved DMARD-free remission, stratified by treatment arm. Table S2. Univariable Cox-regression analyses studying the association between baseline characteristics and late flares in patients at risk for a late flare. (DOCX 79 kb)

Supplementary methods. Figure S1. DAS44 over time. Table S1. Baseline characteristics of patients who achieved DMARD-free remission, stratified by treatment arm. Table S2. Univariable Cox-regression analyses studying the association between baseline characteristics and late flares in patients at risk for a late flare. (DOCX 79 kb)