Exceptional sets related to the largest digits in Lüroth expansions

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

Piecewise linear iterated function systems on the line of overlapping construction

In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

Penal Slavery in Early Modern Scandinavia

In Scandinavia, a penal institution known as “slavery” existed from the sixteenth to the nineteenth centuries. Penal slaves laboured in the creation and maintenance of military infrastructure. They were chained and often stigmatized, sometimes by branding. Their punishment was likened and, on a few occasions, linked to Atlantic slavery. Still, in reality, it was a wholly distinct form of enslavement that produced different experiences of coercion than those of the Atlantic. Such forms of penal slavery sit uneasily in historiographies of punishment but also offers a challenge for the dominant models of global labour history and its attempts to create comparative frameworks for coerced labour. This article argues for the need for contextual approaches to what such coercion meant to both coercers and coerced. Therefore, it offers an analysis of the meaning of early modern penal slavery based on an exceptional set of sources from 1723. In these sources, the status of the punished was negotiated and practiced by guards and slaves themselves. Court appearances by slaves were usually brief—typically revolving around escapes as authorities attempted to identify security breaches. The documents explored in this article are different: They present multiple voices speaking at length, negotiating their very status as voices. From that negotiation and its failures emerge a set of practiced meanings of penal “slavery” in eighteenth-century Copenhagen tied to competing yet intertwined notions of dishonour.

Normalizing Extreme Imprisonment

Normalizing Extreme Imprisonment offers a new explanation for how penal reforms and those driving them can end up normalizing, in the sense of making the public view as acceptable, incredibly severe punitive practices. Since its introduction in 1978 as an alternative to the death penalty, there has been a dramatic increase and expansion of life without parole (LWOP) in the United States, including beyond the scope of capital crimes for which it was originally conceived. Despite this growth, limited attention has been given to this punishment and very few attempts made to narrow its scope or curtail its proliferation. Emerging scholarship suggests the punishment has been ‘normalized’, in part because of how some death penalty abolitionists have framed and used LWOP. Drawing upon a range of evidence and using the development of LWOP in the Californian death penalty context over 40 years as an example, this book significantly deepens and extends this claim to offer a new explanation for how extreme forms of imprisonment become normalized. To discuss the extent to which some opponents to the death penalty may have facilitated, participated in, or perhaps even animated the three main normalizing mechanisms (visibility, denial, and routinization), this book focuses on three sites where death penalty abolitionists have lobbied, campaigned, pled and settled, for LWOP, namely Congress, the broader political sphere, and courtrooms. The book then contrasts these representations of LWOP’s severity with prisoners’ lived experiences detailed in an exceptional set of 299 letters.

On prime powers in linear recurrence sequences

In this paper we consider the Diophantine equation $$U_n=p^x$$ U n = p x where $$U_n$$ U n is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on $$U_n$$ U n , we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.

Exceptional Set in Waring–Goldbach Problem Involving Squares, Cubes and Sixth Powers

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+s) exceptions, all even positive integers up to N can be represented in the form where p1, p2, p3, p4, p5, p6 are prime numbers.

Conservation of a 2nd century Roman barge from Lyon (France)

The conservation of a 2nd-century AD shipwreck represents a challenge in many ways: the amount of material, different ones such as waterlogged wood, iron, lead, textiles, pitch, among others. A global project has to be planned, meaning that scale adaptations had to be thought of taking into account the exceptional size and diversity matters. On top of it, a common thread with pyrite has to be taken into account to avoid acidification by little footsteps at each conservation care stage. To do this, a complete dismantling has occurred, allowing the removal of 26 meters of waterproofing material which happened to be pitched textiles. To explore the data embedded inside, a new unfolding protocol had to be elaborated; that respected their fragility. New issues arose from this work, allowing to link the textiles to the barge, and also opened new exploration fields of an exceptional set of roman textiles.