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Bodiam Castle

Bodiam Castle is a 14th-century moated castle near Robertsbridge in East Sussex, England. It was built in 1385 by Sir Edward Dalyngrigge, a former knight of Edward III, with the permission of Richard II, ostensibly to defend the area against French invasion during the Hundred Years' War. Of quadrangular plan, Bodiam Castle has no keep, having its various chambers built around the outer defensive walls and inner courts, its corners and entrance are marked by towers, topped by crenellations. Its structure and situation in an artificial watery landscape indicate that display was an important aspect of the castle's design as well as defence, it was the centre of the manor of Bodiam. Possession of Bodiam Castle passed through several generations of Dalyngrigges, until their line became extinct, when the castle passed by marriage to the Lewknor family. During the Wars of the Roses, Sir Thomas Lewknor supported the House of Lancaster, when Richard III of the House of York became king in 1483, a force was despatched to besiege Bodiam Castle.

It is unrecorded whether the siege went ahead, but it is thought that Bodiam was surrendered without much resistance. The castle was confiscated, but returned to the Lewknors when Henry VII of the House of Lancaster became king in 1485. Descendants of the Lewknors owned the castle until at least the 16th century. By the start of the English Civil War in 1641, Bodiam Castle was in the possession of Lord Thanet, he supported the Royalist cause, sold the castle to help pay fines levied against him by Parliament. The castle was subsequently dismantled, was left as a picturesque ruin until its purchase by John Fuller in 1829. Under his auspices, the castle was restored before being sold to George Cubitt, 1st Baron Ashcombe, to Lord Curzon, both of whom undertook further restoration work; the castle is protected as Scheduled Monument. It has been owned by The National Trust since 1925, donated by Lord Curzon on his death, is open to the public. Edward Dalyngrigge was a younger son and thus deprived of his father's estates through the practice of primogeniture, hence he had to make his own fortunes.

By 1378, he owned the manor of Bodiam by marrying into a land-owning family. From 1379 to 1388, Dalyngrigge was a Knight of the Shire for Sussex and one of the most influential people in the county. By the time he applied to the king for a licence to crenellate, the Hundred Years' War had been fought between England and France for nearly 50 years. Edward III of England pressed his claim for the French throne and secured the territories of Aquitaine and Calais. Dalyngrigge was one of many Englishmen who travelled to France to seek their fortune as members of Free Companies – groups of mercenaries who fought for the highest bidder, he left for France in 1367 and journeyed with Lionel, Duke of Clarence and son of Edward III. After fighting under the Earl of Arundel, Dalyngrigge joined the company of Sir Robert Knolles, a notorious commander, reputed to have made 100,000 gold crowns as a mercenary from pillage and plunder, it was as a member of the Free Companies. The Treaty of Bruges ensured peace for two years, but after it expired, fighting resumed between England and France.

In 1377 Edward III was succeeded by Richard II. During the war and France struggled for control of the English Channel, with raids on both coasts. With the renewed hostilities, Parliament voted that money should be spent on defending and fortifying England's south coast, defences were erected in Kent in anticipation of a French invasion. There was internal unrest as well as external threats, Dalyngrigge was involved in suppressing the Peasants' Revolt of 1381; the manor of Bodiam was granted a charter in 1383 permitting a weekly market and an annual fair to be held. In 1385, a fleet of 1,200 ships – variously cogs and galleys – gathered across the English Channel at Sluys, Flanders. In the year, Edward Dalyngrigge was granted a licence to fortify his manor house. Know that of our special grace we have granted and given licence on behalf of ourselves and our heirs, so far as in us lies, to our beloved and faithful Edward Dalyngrigge Knight, that he may strengthen with a wall of stone and lime, crenellate and may construct and make into a Castle his manor house of Bodiam fat man, near the sea, in the County of Sussex, for the defence of the adjacent country, the resistance to our enemies...

In witness of which etc. The King at Westminster 20 October. Dalyngrigge's licence from Richard II permitted him to refortify his existing manor house, but instead he chose a fresh site to build a castle on. Construction was completed in one phase, most of the castle is in the same architectural style. Archaeologist David Thackray has deduced from this that Bodiam Castle was built probably because of the threat from the French. Stone castles were time-consuming and expensive to build costing thousands of pounds. Dalyngrigge was Captain of the port of Brest in France from 1386 to 1387, as a result was absent for the first years of the castle's construction, it replaced the old manor house as Dalyngrigge's main residence and the administrative centre of the manor. It is not recorded when Bodiam Castle was completed, but Thackray suggests that it was before 1392. Danlyngrigge's estates, including the castle, were inherited by John Dalyngrigge. Like his father, John enjoyed the favour of the king and was described as the "King's K

Orophus

Orophus is a small genus of katydids native to Mexico, Central America, South America. Katydids in this genus have an elongated head with ovoid eyes; the ovipositor is medium-sized crenulated, curving upwards, one fifth of the length of the posterior femur. They are found in the understory rather than in the canopy in contrast with other members of the subfamily Phaneropterinae; the group was named in 1859 by Swiss entomologist Henri Louis Frédéric de Saussure as a subgenus of Phylloptera. It was erected as a separate genus in 1869 by British entomologist Francis Walker, it is in the tribe Amblycoryphini within the subfamily Phaneropterinae. The type species is Orophus mexicanus. Other genera with species placed in Orophus include Eurycorypha and Microcentrum; as of 2019, the genus includes seven species split into three species groups: Species group Orophus mexicanusSpecies in this group are light green to yellowish in coloration. Orophus amazonicus Cadena-Castañeda, 2014 – Colombia Orophus guatemalae – Central America Orophus mexicanus – Mexico, Central AmericaSpecies group Orophus ovatusThis species group has variable coloration.

Orophus ovatus – Costa RicaSpecies group Orophus tessellatusSpecies in this group are quite variable in their coloration and the density of the spots on the forewings, ranging from light to dark green, pink, or brown. Orophus andinus Cadena-Castañeda, 2014 – Colombia Orophus conspersus – Central America and northern South America Orophus tessellatus – Mexico, Central America, South America

Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions; this article provides links to functions of both classes. Many of the functions mentioned in this article have expansions as series involving these sums. An arithmetic function a is additive if a = a + a for all natural numbers m and n. An arithmetic function a is additive if a = a + a for all coprime natural numbers m and n. ∑ p f and ∏ p f mean that the sum or product is over all prime numbers: ∑ p f = f + f + f + ⋯ ∏ p f = f f f ⋯.

∑ p k f and ∏ p k f mean that the sum or product is over all prime powers with positive exponent: ∑ p k f = ∑ p ∑ k > 0 f = f + f + f + f + f + f + f + ⋯ ∑ d ∣ n f and ∏ d ∣ n f mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if n = 12, ∏ d ∣ 12 f = f f f f f f; the notations can be combined: ∑ p ∣ n f and ∏ p ∣ n f mean that the sum or product is over all prime divisors of n. For example, if n = 18, ∑ p ∣ 18 f = f + f, ∑ p k ∣ n f and ∏ p k ∣ n f mean that the sum or product is over all prime powers dividing n. For example, if n = 24, ∏ p k ∣ 24 f = f f f f; the fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: n = p 1 a 1 ⋯ p k a k {\displaystyle n=p_^{a_{