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The tilde ("tIldeI,-di,-d@,_"tIld) or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin titulus, meaning "title" or "superscription". Its primary use is as a diacritic (accent) in combination with a base letter; but for historical reasons, it is also used in standalone form within a variety of contexts. The tilde was originally written over an omitted letter or several letters as a scribal abbreviation, or "mark of suspension" and "mark of contraction", shown as a straight line when used with capitals. Thus, the commonly used words Anno Domini were frequently abbreviated to Ao Dñi, with an elevated terminal with a suspension mark placed over the "n". Such a mark could denote the omission of one letter or several letters. This saved on the expense of the scribe's labor and the cost of vellum and ink. Medieval European charters written in Latin are largely made up of such abbreviated words with suspension marks and other abbreviations; only uncommon words were given in full. The text of the Domesday Book of 1086, relating for example, to the manor of Molland in Devon (see adjacent picture), is highly abbreviated as indicated by numerous tildes. The text with abbreviations expanded is as follows: Mollande tempore regis Edwardi geldabat pro quattuor hidis et uno ferling. Terra est quadraginta carucae. In dominio sunt tres carucae et decem servi et triginta villani et viginti bordarii cum sedecim carucis. Ibi duodecim acrae prati et quindecim acrae silvae. Pastura tres leugae in longitudine et latitudine. Reddit quattuor et viginti libras ad pensam. Huic manerio est adjuncta Blachepole. Elwardus tenebat tempore regis Edwardi pro manerio et geldabat pro dimidia hida. Terra est duae carucae. Ibi sunt quinque villani cum uno servo. Valet viginti solidos ad pensam et arsuram. Eidem manerio est injuste adjuncta Nimete et valet quindecim solidos. Ipsi manerio pertinet tercius denarius de Hundredis Nortmoltone et Badentone et Brantone et tercium animal pasturae morarum.

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Tilde

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Towards the K(2)-local homotopy groups of Z

Philip Egger

Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class (Z) over tilde of 2-local finite spectra and showed that all spectra Z is an element of (Z) over tilde admit a v(2)-self-map of periodicity 1. The aim here is to compute the K(2)-local homotopy groups pi(*)L(K(2))Z of all spectra Z is an element of (Z) over tilde using a homotopy fixed point spectral sequence, and we give an almost complete answer. The incompleteness lies in the fact that we are unable to eliminate one family of d(3)-differentials and a few potential hidden 2-extensions, though we conjecture that all these differentials and hidden extensions are trivial.

GEOMETRY & TOPOLOGY PUBLICATIONS2020