Robert Burns known familiarly as Rabbie Burns, the National Bard, Bard of Ayrshire and the Ploughman Poet and various other names and epithets, was a Scottish poet and lyricist. He is regarded as the national poet of Scotland and is celebrated worldwide, he is the best known of the poets who have written in the Scots language, although much of his writing is in English and a light Scots dialect, accessible to an audience beyond Scotland. He wrote in standard English, in these writings his political or civil commentary is at its bluntest, he is regarded as a pioneer of the Romantic movement, after his death he became a great source of inspiration to the founders of both liberalism and socialism, a cultural icon in Scotland and among the Scottish diaspora around the world. Celebration of his life and work became a national charismatic cult during the 19th and 20th centuries, his influence has long been strong on Scottish literature. In 2009 he was chosen as the greatest Scot by the Scottish public in a vote run by Scottish television channel STV.

As well as making original compositions, Burns collected folk songs from across Scotland revising or adapting them. His poem "Auld Lang Syne" is sung at Hogmanay, "Scots Wha Hae" served for a long time as an unofficial national anthem of the country. Other poems and songs of Burns that remain well known across the world today include "A Red, Red Rose", "A Man's a Man for A' That", "To a Louse", "To a Mouse", "The Battle of Sherramuir", "Tam o' Shanter" and "Ae Fond Kiss". Burns was born two miles south of Ayr, in Alloway, the eldest of the seven children of William Burnes, a self-educated tenant farmer from Dunnottar in the Mearns, Agnes Broun, the daughter of a Kirkoswald tenant farmer, he was born in a house built by his father, where he lived until Easter 1766, when he was seven years old. William Burnes sold the house and took the tenancy of the 70-acre Mount Oliphant farm, southeast of Alloway. Here Burns grew up in poverty and hardship, the severe manual labour of the farm left its traces in a premature stoop and a weakened constitution.

He had little regular schooling and got much of his education from his father, who taught his children reading, arithmetic and history and wrote for them A Manual of Christian Belief. He was taught by John Murdoch, who opened an "adventure school" in Alloway in 1763 and taught Latin and mathematics to both Robert and his brother Gilbert from 1765 to 1768 until Murdoch left the parish. After a few years of home education, Burns was sent to Dalrymple Parish School in mid-1772 before returning at harvest time to full-time farm labouring until 1773, when he was sent to lodge with Murdoch for three weeks to study grammar and Latin. By the age of 15, Burns was the principal labourer at Mount Oliphant. During the harvest of 1774, he was assisted by Nelly Kilpatrick, who inspired his first attempt at poetry, "O, Once I Lov'd A Bonnie Lass". In 1775, he was sent to finish his education with a tutor at Kirkoswald, where he met Peggy Thompson, to whom he wrote two songs, "Now Westlin' Winds" and "I Dream'd I Lay".

Despite his ability and character, William Burnes was unfortunate, migrated with his large family from farm to farm without being able to improve his circumstances. At Whitsun, 1777, he removed his large family from the unfavourable conditions of Mount Oliphant to the 130-acre farm at Lochlea, near Tarbolton, where they stayed until William Burnes's death in 1784. Subsequently, the family became integrated into the community of Tarbolton. To his father's disapproval, Robert joined a country dancing school in 1779 and, with Gilbert, formed the Tarbolton Bachelors' Club the following year, his earliest existing letters date from this time, when he began making romantic overtures to Alison Begbie. In spite of four songs written for her and a suggestion that he was willing to marry her, she rejected him. Robert Burns was initiated into the Masonic lodge St David, Tarbolton, on 4 July 1781, when he was 22. In December 1781, Burns moved temporarily to Irvine to learn to become a flax-dresser, but during the workers' celebrations for New Year 1781/1782 the flax shop caught fire and was burnt to the ground.

This venture accordingly came to an end, Burns went home to Lochlea farm. During this time he befriended Captain Richard Brown who encouraged him to become a poet, he continued to write poems and songs and began a commonplace book in 1783, while his father fought a legal dispute with his landlord. The case went to the Court of Session, Burnes was upheld in January 1784, a fortnight before he died. Robert and Gilbert made an ineffectual struggle to keep on the farm, but after its failure they moved to the farm at Mossgiel, near Mauchline, in March, which they maintained with an uphill fight for the next four years. In mid-1784 Burns came to know a group of girls known collectively as The Belles of Mauchline, one of whom was Jean Armour, the daughter of a stonemason from Mauchline, his first child, Elizabeth "Bess" Burns, was born to his mother's servant, Elizabeth Paton, while he was embarking on a relationship with Jean Armour, who became pregnant with twins in March 1786. Burns signed a paper attesting his marriage to Jean, but her father "was in the greatest distress, fainted away".

To avoid disgrace, her parents sent her to live with her uncle in Paisley. Although Armour's father forbade it, they were married in 1788. Armour bore him nine chi

In mathematics, a p-adically closed field is a field that enjoys a closure property, a close analogue for p-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965. Let K be the field ℚ of rational numbers and v be its usual p-adic valuation. If F is a extension field of K, itself equipped with a valuation w, we say, formally p-adic when the following conditions are satisfied: w extends v, the residue field of w coincides with the residue field of v, the smallest positive value of w coincides with the smallest positive value of v: in other words, a uniformizer for K remains a uniformizer for F; the formally p-adic fields can be viewed as an analogue of the formally real fields. For example, the field ℚ of Gaussian rationals, if equipped with the valuation w given by w = 1 is formally 5-adic; the field of 5-adic numbers is formally 5-adic. On the other hand, the field of Gaussian rationals is not formally 3-adic for any valuation, because the only valuation w on it which extends the 3-adic valuation is given by w = 1 and its residue field has 9 elements.

When F is formally p-adic but that there does not exist any proper algebraic formally p-adic extension of F F is said to be p-adically closed. For example, the field of p-adic numbers is p-adically closed, so is the algebraic closure of the rationals inside it. If F is p-adically closed, then: there is a unique valuation w on F which makes F p-adically closed, F is Henselian with respect to this place, the valuation ring of F is the image of the Kochen operator, the value group of F is an extension by ℤ of a divisible group, with the lexicographical order; the first statement is an analogue of the fact that the order of a real-closed field is uniquely determined by the algebraic structure. The definitions given above can be copied to a more general context: if K is a field equipped with a valuation v such that the residue field of K is finite, the value group of v admits a smallest positive element, K has finite absolute ramification, i.e. v is finite we can speak of formally v-adic fields and v-adically complete fields.

If K is a field equipped with a valuation v satisfying the hypothesis and with the notations introduced in the previous paragraph, define the Kochen operator by: γ = 1 π z q − z 2 − 1. It is easy to check; the Kochen operator can be thought of as a p-adic analogue of the square function in the real case. An extension field F of K is formally v-adic if and only if 1 π does not belong to the subring generated over the value ring of K by the image of the Kochen operator on F; this is an analogue of the statement that a field is formally real when − 1 {\displa

Renia flavipunctalis, the yellow-dotted renia, yellow-spotted renia or even-lined renia, is a litter moth of the family Erebidae. The species was first described by Carl Geyer in 1832, it is found from southern Canada to Texas. The wingspan is 26–31 mm. Adults are on wing from June to August. There is one generation in the north-east; the larvae feed including dead leaves of deciduous trees. Wagner, David L.. Owlet Caterpillars of Eastern North America. Princeton University Press. ISBN 978-0691150420. "930536.00 – 8384.1 – Renia flavipunctalis – Yellow-spotted Renia Moth –". North American Moth Photographers Group. Mississippi State University. Retrieved January 31, 2020. McLeod, Robin. "Species Renia flavipunctalis - Yellow-spotted Renia - Hodges#8384.1". BugGuide. Retrieved January 31, 2020. Anweiler, G. G. & Robinson, E. "Species Page - Renia flavipunctalis". Entomology Collection. University of Alberta E. H. Strickland Entomological Museum. Retrieved January 31, 2020