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An examination of the practice of petitioning at the grassroots level shows how it stimulated a vibrant popular politics. Revisionist scholarship emphasising the supposed taming or disciplining of political culture has ignored the lively local culture of petitioning. The chapter first outlines the process and practice of petitioning: the drafting, signing, and presentation and reception of petitions. Of all these different stages in the process of petitioning, it was the practice of signing petitions that was most important to nineteenth-century popular politics. Not only did it underpin other forms of political activity, such as public meetings, but opened up new informal spaces for political activity and engendered new forms of political behaviour. The practice of petitioning stimulated a never-ending cycle of claim and counter-claim about the forging of signatures, the undue influence of landlords or employers, and outright misrepresentation. This endless contestation was intrinsic to the practice and process of petitioning and one of the most important ways in which it energised popular politics at the local level.
This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results.
The signature of a path can be described as its full non-commutative exponential. Following T. Lyons, we regard its expectation, the expected signature, as a path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions with motivations ranging from financial mathematics to statistical physics. From an affine semimartingale perspective, the functional relation may be interpreted as a type of generalized Riccati equation.
Chapter 2 explores the material properties of royal correspondence, focussing on evidence that correlates with the scribal/holograph provenance of the texts. Five features are examined in a corpus of over 100 royal letters issued by the Tudor monarchs: material provenance markers, handwriting, page orientation, signature placement, and signature style. The chapter finds that royal scribal letters have distinctive material features that make their royal source explicit, with these characteristics used very consistently throughout the Tudor period. Holograph royal letters show a reduced propensity to follow these material codes, and instead show a greater individuality more typical of non-royal letter-writing in the period. The differences are proposed to arise from the different production processes of the letter types, affecting the degree of institutionalised power presented to the letter's recipient. Elizabeth's correspondence shows a wider variation in material choices than that of her predecessors, potentially indicative of shifts in how correspondence was utilised, and the values placed on holograph writing by the end of the sixteenth century.