text:
Rowan

The rowans ( or  ) or mountain-ashes are shrubs or trees in the genus Sorbus of the rose family, Rosaceae. They are native throughout the cool temperate regions of the Northern Hemisphere, with the highest species diversity in the Himalaya, southern Tibet and parts of western China, where numerous apomictic microspecies occur. The name rowan was originally applied to the species Sorbus aucuparia and is also used for other species in the genus Sorbus.

Natural hybrids, often including S. aucuparia and the whitebeam, Aria edulis (syn. Sorbus aria), give rise to many endemic variants in the UK.

Names
The Latin name sorbus was borrowed into Old English as syrfe. The Latin name sorbus is from a root for "red, reddish-brown" (PIE *sor-/*ser-); English sorb is attested from the 1520s in the sense "fruit of the service tree", adopted via French sorbe from Latin sorbum "service-berry". Sorbus domestica is also known as "whitty pear", the adjective whitty meaning "pinnate". The name "mountain-ash" for Sorbus domestica is due to a superficial similarity of the rowan leaves to those of the ash, not to be confused with Fraxinus ornus, a true ash that is also known as "mountain ash". Sorbus torminalis is also known as "chequer tree"; its fruits, formerly used to flavour beer, are called "chequers", perhaps from the spotted pattern of the fruit.

The traditional name rowan was applied to the species Sorbus aucuparia. The name "rowan" is recorded from 1804, detached from an earlier rowan-tree, rountree, attested from the 1540s in northern dialects of English and Scots. It is often thought to be from a North Germanic source, perhaps related to Old Norse reynir (c.f. Norwegian rogn, Danish røn, Swedish rönn), ultimately from the Germanic verb *raud-inan "to redden", in reference to the berries (as is the Latin name sorbus). Various dialectal variants of rowan are found in English, including ran, roan, rodan, royan, royne, round, and rune.

The Old English name of the rowan is cwic-beám, which survives in the name quickbeam (also quicken, quicken-tree, and variants). This name by the 19th century was reinterpreted as connected to the word witch, from a dialectal variant wick for quick and names such as wicken-tree, wich-tree, wicky, and wiggan-tree, giving rise to names such as witch-hazel and witch-tree.

The tree has two names in Welsh, cerdinen and criafol. Criafol may be translated as "The Lamenting Fruit", likely derived from the Welsh tradition that the Cross of Christ was carved from the wood of this tree, and the subsequent association of the Rowan's red fruit with the blood of Christ.

The Old Irish name is cairtheand, reflected in Modern Irish caorthann. The "arboreal" Bríatharogam in the Book of Ballymote associates the rowan with the letter luis, with the gloss "delightful to the eye (li sula) is luis, i.e. rowan (caertheand), owing to the beauty of its berries". Due to this, "delight of the eye" (vel sim.) has been reported as a "name of the rowan" by some commentators.

The most common Scots Gaelic name is caorann, which appears in numerous Highland place names such as Beinn a' Chaorainn and Loch a’ Chaorainn. Rowan was also the clan badge of the Malcolms and McLachlans. There were strong taboos in the Highlands against the use of any parts of the tree save the berries, except for ritual purposes. For example, a Gaelic threshing tool made of rowan and called a buaitean was used on grain meant for rituals and celebrations.

In the Canadian provinces of Newfoundland and Labrador and Nova Scotia, this species is commonly referred to as a "dogberry" tree. In German, Sorbus aucuparia is known as the Vogelbeerbaum ("bird-berry tree") or as Eberesche. The latter is a compound of the name of the ash tree (Esche) with what is contemporarily the name of the boar (Eber), but in fact the continuation of a Gaulish name, eburo- (also the name for a dark reddish-brown colour, cognate with Greek orphnos, Old Norse iarpr "brown"); like sorbus, eburo- seems to have referred to the colour of the berries; it is also recorded as a Gaulish name for the yew (which also has red berries), see also Eburodunum (disambiguation).

Botany

Rowans are mostly small deciduous trees 10–20 m tall, though a few are shrubs. Rowans are unrelated to the true ash trees of the genus Fraxinus, family Oleaceae. Though their leaves are superficially similar, those of Sorbus are alternate, while those of Fraxinus are opposite. Rowan leaves are arranged alternately, and are pinnate, with (7–)11–35 leaflets. A terminal leaflet is always present. The flowers are borne in dense corymbs; each flower is creamy white, and 5–10 mm across with five petals. The fruit is a small pome 4–8 mm diameter, bright orange or red in most species, but pink, yellow or white in some Asian species. The fruit are soft and juicy, which makes them a very good food for birds, particularly waxwings and thrushes, which then distribute the rowan seeds in their droppings. Due to their small size the fruits are often referred to as berries, but a true berry is a simple fruit produced from a single ovary, whereas a pome is an accessory fruit.

Rowan is used as a food plant by the larvae of some Lepidoptera species; see Lepidoptera that feed on Sorbus.

The best-known species is the European rowan Sorbus aucuparia, a small tree typically 4–12 m tall growing in a variety of habitats throughout northern Europe and in mountains in southern Europe and southwest Asia. Its berries are a favourite food for many birds and are a traditional wild-collected food in Britain and Scandinavia. It is one of the hardiest European trees, occurring to 71° north in Vardø Municipality in the far northern part of Arctic Norway, and has also become widely naturalised in northern North America.

The greatest diversity of form as well as the largest number of rowan species is in Asia, with very distinctive species such as Sargent's rowan Sorbus sargentiana with large leaves 20–35 cm long and 15–20 cm broad and very large corymbs with 200–500 flowers, and at the other extreme, small-leaf rowan Sorbus microphylla with leaves 8–12 cm long and 2.5–3 cm broad. While most are trees, the dwarf rowan Sorbus reducta is a low shrub to 50 cm tall. Several of the Asian species are widely cultivated as ornamental trees.

North American native species in the genus Sorbus include the American mountain-ash Sorbus americana and Showy mountain-ash Sorbus decora in the east and Sitka mountain-ash Sorbus sitchensis in the west.

Numerous hybrids, mostly behaving as true species reproducing by apomixis, occur between rowans and whitebeams; these are variably intermediate between their parents but generally more resemble whitebeams and are usually grouped with them (q.v.).

Uses
Rowans are excellent small ornamental trees for parks, gardens and wildlife areas. Several of the Asian species, such as white-fruited rowan (Sorbus oligodonta) are popular for their unusual fruit colour, and Sargent's rowan (Sorbus sargentiana) for its exceptionally large clusters of fruit. Numerous cultivars have also been selected for garden use, several of them, such as the yellow-fruited Sorbus 'Joseph Rock', of hybrid origin. They are very attractive to fruit-eating birds, which is reflected in the old name "bird catcher".

The wood is dense and used for carving and turning and for tool handles and walking sticks. Rowan fruit are a traditional source of tannins for mordanting vegetable dyes. In Finland, it has been a traditional wood of choice for horse sled shafts and rake spikes.

The fruit of European rowan (Sorbus aucuparia) can be made into a slightly bitter jelly which in Britain is traditionally eaten as an accompaniment to game, and into jams and other preserves either on their own or with other fruit. The fruit can also be a substitute for coffee beans, and has many uses in alcoholic beverages: to flavour liqueurs and cordials, to produce country wine, and to flavour ale. In Austria a clear rowan schnapps is distilled which is called by its German name Vogelbeerschnaps, Czechs also make a rowan liquor called jeřabinka, the Polish Jarzębiak is rowan-flavoured vodka, and the Welsh used to make a rowan wine called diodgriafel.

Rowan cultivars with superior fruit for human food use are available but not common; mostly the fruits are gathered from wild trees growing on public lands.

Rowan fruit contains sorbic acid, and when raw also contains parasorbic acid (about 0.4%–0.7% in the European rowan ), which causes indigestion and can lead to kidney damage, but heat treatment (cooking, heat-drying etc.) and, to a lesser extent, freezing, renders it nontoxic by changing it to the benign sorbic acid. They are also usually too astringent to be palatable when raw. Collecting them after first frost (or putting in the freezer) cuts down on the bitter taste as well.

Mythology
In Sami mythology, the goddess Ravdna is the consort of the thunder-god Horagalles. Red berries of rowan were holy to Ravdna, and the name Ravdna resembles North Germanic words for the tree (for example, Old Norse reynir).

In Norse mythology, the goddess Sif is the wife of the thunder god Thor, who has been linked with Ravdna. According to Skáldskaparmál the rowan is called "the salvation of Thor" because Thor once saved himself by clinging to it. It has been hypothesized that Sif was once conceived in the form of a rowan to which Thor clung.

In the Fianna Cycle of Irish mythology, The Pursuit of Diarmuid and Grainne sees the couple eloping, trying to escape the vengeance of the legendary leader Fionn Mac Cumhaill, whom Grainne had spurned. The pair came to a forest guarded by the giant Searbhán. Searbhán allowed the pair to rest and hunt in his forest, as long as they did not eat the berries of his magical rowan tree. The pregnant Grainne desired the berries, and Diarmuid was compelled to kill Searbhán to obtain them. His mortal weapons being powerless against Searbhán, he used the giant's own iron club to kill him. The pair climbed high into the rowan tree to eat the sweetest berries, then rested in the tree afterwards. This was in violation of the advice of Aengus, the god of love, who had warned the couple that they should "not sleep in a cave with one opening, or a house with one door, or a tree with one branch, and that they would never be able to eat where they cooked, or sleep where they ate." Fionn Mac Cuimhaill tracked the couple to the rowan tree and tricked Diarmuid into revealing himself through a game of chess. Aengus spirited Grainne away and Diarmuid leapt to safety, and the pursuit continued.

Folk magic
The European rowan (Sorbus aucuparia) has a long tradition in European mythology and folklore. It was thought to be a magical tree and give protection against malevolent beings. The tree was also called "wayfarer's tree" or "traveller's tree" because it supposedly prevents those on a journey from getting lost. It was said in England that this was the tree on which the Devil hanged his mother.

British folklorists of the Victorian era reported the folk belief in apotropaic powers of the rowan-tree, in particular in the warding off of witches. Such a report is given by Edwin Lees (1856) for the Wyre Forest in the English West Midlands. Sir James Frazer (1890) reported such a tradition in Scotland, where the tree was often planted near a gate or front door.

According to Frazer, birds' droppings often contain rowan seeds, and if such droppings land in a fork or hole where old leaves have accumulated on a larger tree, such as an oak or a maple, they may result in a rowan growing as an epiphyte on the larger tree. Such a rowan is called a "flying rowan" and was thought of as especially potent against witches and black magic, and as a counter-charm against sorcery. In 1891, Charles Godfrey Leland also reported traditions of rowan's apotropaic powers against witches in English folklore, citing the Denham Tracts (collected between 1846 and 1859). Rowan also serves as protection against fairies. For example, according to Thomas Keightley mortals could safely witness fairy rades (mounted processions held by the fairies each year at the onset of summer) by placing a rowan branch over their doors.

Pagan revivalism
In Neo-Druidism, the rowan is known as the "portal tree". It is considered the threshold, between this world and otherworld, or between here and wherever you may be going, for example, it was placed at the gate to a property, signifying the crossing of the threshold between the path or street and the property of someone. According to Elen Sentier, "Threshold is a place of both ingress (the way in) and egress (the way out). Rowan is a portal, threshold tree offering you the chance of 'going somewhere ... and leaving somewhere."

Weather-lore
In Newfoundland, popular folklore maintains that a heavy crop of fruit means a hard or difficult winter. Similarly, in Finland and Sweden, the number of fruit on the trees was used as a predictor of the snow cover during winter, but here the belief was that the rowan "will not bear a heavy load of fruit and a heavy load of snow in the same year", that is, a heavy fruit crop predicted a winter with little snow.

However, as fruit production for a given summer is related to weather conditions the previous summer, with warm, dry summers increasing the amount of stored sugars available for subsequent flower and fruit production, it has no predictive relationship to the weather of the next winter.

In Malax, Finland, the reverse was thought. If the rowan flowers were plentiful then the rye harvest would also be plentiful. Similarly, if the rowan flowered twice in a year there would be many potatoes and many weddings that autumn. And in Sipoo people are noted as having said that winter had begun when the waxwings (Bombycilla garrulus) had eaten the last of the rowan fruit.

In Sweden, it was also thought that if the rowan trees grew pale and lost colour, the autumn and winter would bring much illness.

Popular culture
References to the rowan fruit's red color and the flowers' beauty are common in Celtic music. For example, the song "Marie's Wedding" contains the verse

Red her cheeks as rowans are, bright her eyes as any star, fairest of them all by far, is our darling Marie.

J. R. R. Tolkien's novel The Two Towers employs rowans as the signature tree for the Ent, Quickbeam. The forest of Fangorn, where Quickbeam and other Ents live, is populated with numerous rowans that were said to have been planted by male Ents to please the female Entwives. Quickbeam declares his fondness for the tree by saying that no other "people of the Rose ... are so beautiful to me," a reference to the rowan's membership in the family Rosaceae.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Rowan


text:
William Bell Scott

William Bell Scott (1811–1890) was a Scottish artist in oils and watercolour and occasionally printmaking. He was also a poet and art teacher, and his posthumously published reminiscences give a chatty and often vivid picture of life in the circle of the Pre-Raphaelites; he was especially close to Dante Gabriel Rossetti. After growing up in Edinburgh, he moved to London, and from 1843 to 1864 was principal of the government School of Art in Newcastle upon Tyne, where he added industrial subjects to his repertoire of landscapes and history painting. He was one of the first British artists to extensively depict the processes of the Industrial Revolution. He returned to London, working for the Science and Art Department until 1885.

He painted a cycle of historical subjects mixed with scenes from modern industry for Wallington Hall in Northumberland (now National Trust), his best known works, and a purely historical cycle for Penkill Castle in Ayrshire in Scotland. He did not paint many portraits, but his striking portrait of his friend Algernon Charles Swinburne is the iconic image of the poet. His etchings were mostly designed to illustrate his books.

Life
The son of Robert Scott (1777–1841), the engraver, and brother of David Scott, the painter, he was born in Edinburgh. While a young man he studied art and assisted his father, and he published verses in the Scottish magazines.

In 1837, Scott went to London. There he became sufficiently well known as an artist to be appointed in 1844 master of the government school of design at Newcastle-on-Tyne. He held the post for twenty years, and did work in organizing art-teaching and examining under the Science and Art Department. In Newcastle, Scott was visited by all the Rossetti family, and Dante Gabriel Rossetti painted Maria Leathart's portrait at Scott's house 14 St Thomas' Crescent (plaque erected 2005). Algernon Charles Swinburne, who wrote two poems to Scott, spent much time with him in Newcastle after being sent down from Oxford.

After 1870 Scott was much in London, where he bought a house in Chelsea, and he was an intimate friend of Dante Gabriel Rossetti and in high repute as an artist and an author. He was, however, at daggers drawn with John Ruskin.

He resigned his appointment under the Science and Art Department in 1885, and from then until his death he was mainly occupied in writing his reminiscences, Autobiographical Notes, which were published posthumously in 1892, with a memoir by Professor Minto. It is for his connection with Rossetti's circle that Bell Scott will be chiefly remembered.

Artwork

In addition to oil paintings, Bell Scott did much decorative work, notably at Wallington Hall. There Pauline, Lady Trevelyan commissioned him, after coming across his 1850 Memoir of his brother David Scott. He produced eight large pictures illustrating Northumbrian history, with life-size figures, supplemented by eighteen pictures on The Ballad of Chevy Chase in the spandrels of the arches of the hall. For Penkill Castle he executed a similar series, illustrating James I's poem The Kingis Quair.

Writings
Scott's poetry, which he published at intervals (notably Poems, 1875, illustrated by etchings by himself and Alma-Tadema), recalled William Blake and Percy Bysshe Shelley, and was considerably influenced by Rossetti. He also wrote artistic and literary criticism, and edited John Keats, Letitia Elizabeth Landon, Lord Byron, Samuel Taylor Coleridge, Shelley, William Shakespeare and Sir Walter Scott.

Family
In 1839 Scott married Letitia Margery Norquoy (1813–1898); the marriage was childless. In Newcastle he had as a pupil Alice Boyd, the sister of the then-laird of Penkill Castle in Scotland. In 1860, Bell Scott visited Penkill Castle and began a liaison with Alice that would last until his death at Penkill in 1890. Although unhappily married, Bell Scott refused to cause a scandal by leaving his wife, and a workable ménage à trois was established: Alice spent winters with William and Letitia in London, while they came to Penkill in the summers.

title: William Bell Scott


text:
Nauru Pacific Line

The Nauru Pacific Line (NPL) is the national shipping line of the Republic of Nauru, founded in 1969. Its fleet consists of five owned and two leased ships, which travel throughout all of Oceania. It is loss-making and gets subventions from the state.

It is a member of the Pacific Forum Line, a joint venture between several national shipping lines established in 1977 after a dispute between the NPL and the Maritime Union of New Zealand.

title: Nauru Pacific Line


text:
Pacific Forum Line

Pacific Forum Line (PFL) is a regional shipping line in Polynesia. Established in 1976 by the Pacific Islands Forum to ensure a regional shipping service, it was purchased in 2012 by the government of Samoa. It is currently operated as a joint venture with Neptune Pacific Line.

History
The Pacific Forum Line was born from concern over the deterioration of traditional island tramp services due to containerisation. At the South Pacific Forum in Nauru in 1976 pacific nations agreed to establish a shipping line to ensure regular shipping services for the islands and encourage development. The initial memorandum of understanding required the line to be incorporated in Samoa, operate a viable shipping service and attempt to make a profit, with any profit allocated between founding nations. The initial owners - the governments of the Cook Islands, Fiji, Kiribati, Nauru, New Zealand, Papua New Guinea, Samoa, Solomon Islands, and Tonga - all had an equal say in governance of the line, though each had a different financial stake. Tuvalu, the Marshall Islands and Niue later became shareholders.

The first services began in May 1978, with a ship serving Australia, Fiji, Samoa, Tonga, and New Zealand. The company was immediately in financial trouble, and by 1981 was in danger of collapse. Losses in the first three years of operations were more than US$16 million, and it continued to be supported by member nations. The line finally became profitable in the late 1980's. By 1998, twenty years after being established, it was seen as an example of successful regionalism and had finally begun paying dividends.

In 2007 the Forum received an apparently unsolicited bid from Pacific International Lines to buy the company. This was followed by bids from Swire Shipping and Fiji Water-Neptune Pacific Line. An extraordinary shareholders' meeting in March 2008 rejected any proposal to sell the company. Following the meeting, leaked emails showed that CEO John MacLennan had actively promoted the sale.

Purchase by Samoa
Heavy losses in 2011 caused several of the owner countries to consider selling their stake in 2012. Following a bid by Singaporean company Sofrana, the Samoan government bought out the other countries in October 2012. In September 2013 it sold a 50 percent stake in the company to Neptune Pacific Line, a subsidiary of Roll Global. The deal required PFL to continue to serve every Forum country except Tuvalu, Niue and the Marshall Islands.

In 2016 the company merged with Polynesian Shipping Line.

Operations
Pacific Forum Line Limited the parent, is registered in Apia, Samoa and the operating arm – Pacific Forum Line (N.Z) Limited is located in Auckland, New Zealand. Pacific Forum Line has also set up agency divisions in many of the ports they service.

Pacific Forum Line vessels operate on five routes that connect the Pacific Islands to New Zealand and Australia and provide an important South Pacific Inter-island link. PFL carries containerised and break-bulk cargo on the New Zealand and Australian trade vessels. In addition, to and from Australia bulk liquids are also carried in purpose built tanks.

Current fleet
PFL's current vessels include:
 * Capitaine Tasman
 * Capitaine Dampier
 * Southern Moana
 * Kokopo Chief
 * Southern Trader

Accidents
In April 2004 the captain of the Forum Fiji II died after being washed overboard 320 km south of Tonga.

In August 2009 the Forum Samoa II ran aground on the reef at Apia harbour after suffering engine trouble. It was freed in September.

title: Pacific Forum Line


text:
Polydivisible number

In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... that has the following properties:

 * 1) Its first digit a is not 0.
 * 2) The number formed by its first two digits ab is a multiple of 2.
 * 3) The number formed by its first three digits abc is a multiple of 3.
 * 4) The number formed by its first four digits abcd is a multiple of 4.
 * 5) etc.

Definition
Let $$n$$ be a positive integer, and let $$k = \lfloor \log_{b}{n} \rfloor + 1$$ be the number of digits in n written in base b. The number n is a polydivisible number if for all $$1 \leq i \leq k$$,
 * $$\left\lfloor\frac{n}{b^{k - i}}\right\rfloor \equiv 0 \pmod i$$.

 * Example

For example, 10801 is a seven-digit polydivisible number in base 4, as
 * $$\left\lfloor\frac{10801}{4^{7 - 1}}\right\rfloor = \left\lfloor\frac{10801}{4096}\right\rfloor = 2 \equiv 0 \pmod 1,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 2}}\right\rfloor = \left\lfloor\frac{10801}{1024}\right\rfloor = 10 \equiv 0 \pmod 2,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 3}}\right\rfloor = \left\lfloor\frac{10801}{256}\right\rfloor = 42 \equiv 0 \pmod 3,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 4}}\right\rfloor = \left\lfloor\frac{10801}{64}\right\rfloor = 168 \equiv 0 \pmod 4,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 5}}\right\rfloor = \left\lfloor\frac{10801}{16}\right\rfloor = 675 \equiv 0 \pmod 5,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 6}}\right\rfloor = \left\lfloor\frac{10801}{4}\right\rfloor = 2700 \equiv 0 \pmod 6,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 7}}\right\rfloor = \left\lfloor\frac{10801}{1}\right\rfloor = 10801 \equiv 0 \pmod 7.$$

Enumeration
For any given base $$b$$, there are only a finite number of polydivisible numbers.

Maximum polydivisible number
The following table lists maximum polydivisible numbers for some bases b, where $A−Z$ represent digit values 10 to 35.

Estimate for Fb(n) and Σ(b)

Let $$n$$ be the number of digits. The function $$F_b(n)$$ determines the number of polydivisible numbers that has $$n$$ digits in base $$b$$, and the function $$\Sigma(b)$$ is the total number of polydivisible numbers in base $$b$$.

If $$k$$ is a polydivisible number in base $$b$$ with $$n - 1$$ digits, then it can be extended to create a polydivisible number with $$n$$ digits if there is a number between $$bk$$ and $$b(k + 1) - 1$$ that is divisible by $$n$$. If $$n$$ is less or equal to $$b$$, then it is always possible to extend an $$n - 1$$ digit polydivisible number to an $$n$$-digit polydivisible number in this way, and indeed there may be more than one possible extension. If $$n$$ is greater than $$b$$, it is not always possible to extend a polydivisible number in this way, and as $$n$$ becomes larger, the chances of being able to extend a given polydivisible number become smaller. On average, each polydivisible number with $$n - 1$$ digits can be extended to a polydivisible number with $$n$$ digits in $$\frac{b}{n}$$ different ways. This leads to the following estimate for $$F_{b}(n)$$:

 * $$F_b(n) \approx (b - 1)\frac{b^{n-1}}{n!}.$$

Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately

 * $$\Sigma(b) \approx \frac{b - 1}{b}(e^{b}-1)$$

Specific bases
All numbers are represented in base $$b$$, using A−Z to represent digit values 10 to 35.

Base 5
The polydivisible numbers in base 5 are
 * 1, 2, 3, 4, 11, 13, 20, 22, 24, 31, 33, 40, 42, 44, 110, 113, 132, 201, 204, 220, 223, 242, 311, 314, 330, 333, 402, 421, 424, 440, 443, 1102, 1133, 1322, 2011, 2042, 2200, 2204, 2231, 2420, 2424, 3113, 3140, 3144, 3302, 3333, 4022, 4211, 4242, 4400, 4404, 4431, 11020, 11330, 13220, 20110, 20420, 22000, 22040, 22310, 24200, 24240, 31130, 31400, 31440, 33020, 33330, 40220, 42110, 42420, 44000, 44040, 44310, 110204, 113300, 132204, 201102, 204204, 220000, 220402, 223102, 242000, 242402, 311300, 314000, 314402, 330204, 333300, 402204, 421102, 424204, 440000, 440402, 443102, 1133000, 1322043, 2011021, 2042040, 2204020, 2420003, 2424024, 3113002, 3140000, 3144021, 4022042, 4211020, 4431024, 11330000, 13220431, 20110211, 20420404, 24200031, 31400004, 31440211, 40220422, 42110202, 44310242, 132204314, 201102110, 242000311, 314000044, 402204220, 443102421, 1322043140, 2011021100, 3140000440, 4022042200

The smallest base 5 polydivisible numbers with n digits are
 * 1, 11, 110, 1102, 11020, 110204, 1133000, 11330000, 132204314, 1322043140, none...

The largest base 5 polydivisible numbers with n digits are
 * 4, 44, 443, 4431, 44310, 443102, 4431024, 44310242, 443102421, 4022042200, none...

The number of base 5 polydivisible numbers with n digits are
 * 4, 10, 17, 21, 21, 21, 13, 10, 6, 4, 0, 0, 0...

Base 10
The polydivisible numbers in base 10 are
 * 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 105, 108, 120, 123, 126, 129, 141, 144, 147, 162, 165, 168, 180, 183, 186, 189, 201, 204, 207, 222, 225, 228, 243, 246, 249, 261, 264, 267, 282, 285, 288...

The smallest base 10 polydivisible numbers with n digits are
 * 1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640, 10200056405, 102006162060, 1020061620604, 10200616206046, 102006162060465, 1020061620604656, 10200616206046568, 108054801036000018, 1080548010360000180, 10805480103600001800, ...

The largest base 10 polydivisible numbers with n digits are
 * 9, 98, 987, 9876, 98765, 987654, 9876545, 98765456, 987654564, 9876545640, 98765456405, 987606963096, 9876069630960, 98760696309604, 987606963096045, 9876062430364208, 98485872309636009, 984450645096105672, 9812523240364656789, 96685896604836004260, ...

The number of base 10 polydivisible numbers with n digits are
 * 9, 45, 150, 375, 750, 1200, 1713, 2227, 2492, 2492, 2225, 2041, 1575, 1132, 770, 571, 335, 180, 90, 44, 18, 12, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

Programming example
The example below searches for polydivisible numbers in Python.

Related problems
Polydivisible numbers represent a generalization of the following well-known problem in recreational mathematics:

 * Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.

The solution to the problem is a nine-digit polydivisible number with the additional condition that it contains the digits 1 to 9 exactly once each. There are 2,492 nine-digit polydivisible numbers, but the only one that satisfies the additional condition is

 * 381 654 729

Other problems involving polydivisible numbers include:

 * Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible number that only uses even digits is

 * 48 000 688 208 466 084 040

 * Finding palindromic polydivisible numbers - for example, the longest palindromic polydivisible number is

 * 30 000 600 003

 * A common, trivial extension of the aforementioned example is to arrange the digits 0 to 9 to make a 10 digit number in the same way, the result is 3816547290. This is a pandigital polydivisible number.

title: Polydivisible number


text:
Giulio Douhet

General Giulio Douhet (30 May 1869 – 15 February 1930) was an Italian general and air power theorist. He was a key proponent of strategic bombing in aerial warfare. He was a contemporary of the air warfare advocates Walther Wever, Billy Mitchell, and Hugh Trenchard.

Biography
Born in Caserta, Campania, Italy, from a family of Savoyard exiles who had migrated there after the cession of Savoy to France he attended the Military Academy of Modena and was commissioned into the artillery of the Italian Army in 1882. Later he attended the Polytechnic Institute in Turin where he studied science and engineering.

Assigned to the General Staff, after the beginning of the new century, Douhet published lectures on military mechanization. With the arrival of dirigibles and then fixed-wing aircraft in Italy, he quickly recognized the military potential of the new technology. Douhet saw the pitfalls of allowing air power to be fettered by ground commanders and began to advocate the creation of a separate air arm commanded by airmen. He teamed up with the young aircraft engineer Gianni Caproni to extol the virtues of air power in the years ahead.

In 1911, Italy went to war against the Ottoman Empire for control of Libya. During the war, aircraft operated for the first time in reconnaissance, transport, artillery spotting and even limited bombing roles. Douhet wrote a report on the aviation lessons learned in which he suggested high altitude bombing should be the primary role of aircraft. In 1912, Douhet assumed command of the Italian aviation battalion at Turin and wrote a set of Rules for the Use of Airplanes in War (Regole per l'uso degli aeroplani in guerra) — one of the first doctrine manuals of its kind. However, Douhet's preaching on air power marked him as a radical. After an incident in which he ordered construction of Caproni bombers without authorization, he was exiled to the infantry.

When World War I began, Douhet began to call for Italy to launch a massive military build-up, particularly in aircraft. "To gain command of the air," he said, was to render an enemy "harmless". When Italy entered the war in 1915 Douhet was shocked by the army's incompetence and unpreparedness. He proposed a force of 500 bombers that could drop 125 tons of bombs daily to break the bloody stalemate with Austria, but he was ignored. He corresponded with his superiors and government officials, criticising the conduct of the war and advocating an air power solution. Douhet was court-martialed and was imprisoned for one year for criticizing Italian military leaders in a memorandum to the cabinet.

Douhet continued to write about air power from his cell, finishing a novel on air power and proposing a massive Allied fleet of aircraft in communications to ministers. He was released and returned to duty shortly after the disastrous Battle of Caporetto in 1917. Douhet was recalled to service in 1918 to serve as head of the Italian Central Aeronautic Bureau.

He was exonerated in 1920 and promoted to general officer in 1921. The same year, he completed a hugely-influential treatise on strategic bombing, The Command of the Air (Il dominio dell'aria), and retired from military service soon after. Except for a few months as the head of aviation in Benito Mussolini's government in 1922, Douhet spent much of the rest of his life theorizing about the impact of military air power.

Douhet died of a heart attack in 1930 in Rome, Italy.

Aerial strategy
In his book Douhet argued that air power was revolutionary because it operated in the third dimension. Aircraft could fly over surface forces, relegating them to secondary importance. The vastness of the sky made defense almost impossible, so the essence of air power was the offensive. The only defense was a good offense. The air force that could achieve command of the air by bombing the enemy air arm into extinction would doom its enemy to perpetual bombardment. Command of the air meant victory.

Douhet believed in the morale effects of bombing. Air power could break a people's will by destroying a country's "vital centers". Armies became superfluous because aircraft could overfly them and attack these centers of the government, military and industry with impunity, a principle later called "The bomber will always get through". Targeting was central to this strategy and he believed that air commanders would prove themselves by their choice of targets. These would vary from situation to situation, but Douhet identified the five basic target types as: industry, transport infrastructure, communications, government and "the will of the people".

The last category was particularly important to Douhet, who believed in the principle of total war.

The chief strategy laid out in his writings, the Douhet model, is pivotal in debates regarding the use of air power and bombing campaigns. The Douhet model rests on the belief that in a conflict, the infliction of high costs from aerial bombing can shatter civilian morale. This would unravel the social basis of resistance, and pressure citizens into asking their governments to surrender. The logic of this model is that exposing large portions of civilian populations to the terror of destruction or the shortage of consumer goods would damage civilian morale into submission. By smothering the enemy's civilian centers with bombs, Douhet argued the war would become so terrible that the common people would rise against their government, overthrow it with revolution, then sue for peace.

This emphasis on the strategic offensive would blind Douhet to the possibilities of air defense or tactical support of armies. In his second edition of The Command of the Air he maintained such aviation was "useless, superfluous and harmful". He proposed an independent air force composed primarily of long-range load-carrying bombers. He believed interception of these bombers was unlikely, but allowed for a force of escort aircraft to ward off interceptors. Attacks would not require great accuracy. On a tactical level he advocated using three types of bombs in quick succession; explosives to destroy the target, incendiaries to ignite the damaged structures, and poison gas to keep firefighters and rescue crews away.

The entire population was in the front line of an air war and they could be terrorized with urban bombing. In his book The War of 19-- he described a fictional war between Germany and a Franco-Belgian alliance in which the Germans launched massive terror bombing raids on the populace, reducing their cities to ashes before their armies could mobilize. Because bombing would be so terrible, Douhet believed that wars would be short. As soon as one side lost command of the air it would capitulate rather than face the terrors of air attack. In other words, the enemy air force was the primary target. A decisive victory here would hasten the end of the war.

Critical reception
Though the initial response to The Command of the Air was muted, the second edition generated virulent attacks from his military peers, particularly those in the navy and army. Douhet's was an apocalyptic vision that gripped the popular imagination. his theories were unproven and unchallengeable for another 20 years. In many cases, he had hugely exaggerated the effects of bombing. His calculations for the amount of bombs and poison gas required to destroy a city were ludicrously optimistic. In "Rivista Aeronautica" in July 1928, he wrote that he believed that 300 tons of bombs over the most important cities would end a war in less than a month.

title: Giulio Douhet


text:
User talk:Gp3sharon

Hi Sharon :) I hope you like the place and choose to stay.

Some links that may be of use:
 * Welcome, newcomers
 * How to edit a page
 * Village pump - ask questions you may have here, or leave a message on my talk page

It may be useful to write the content before you make a new page. You can do this offline in your user space such as User:Gp3sharon/Men slimming first and then moving it to Men slimming or just write the text offline and then copy your own text here. Otherwise it might get deleted. Thanks :) Dysprosia 13:52, 29 Nov 2003 (UTC)

Please don't remove the votes for deletion notice. Also - the article you are writing is not in a suitable style for an encyclopedia. Secretlondon 11:09, Dec 4, 2003 (UTC)

title: User talk:Gp3sharon


text:
Talk:Polyphase system

Random posts

 * does any one know if Japan has a 230v single phase 50hz supply avaible commercialy —Preceding unsigned comment added by 204.191.51.253 (talk) 18:26, 10 November 2007 (UTC)

From the article:
 * A polyphase coil takes advantage of the three phase's energy is made up of: induction, neutralization, and capacitance.

Ewwww. -- The Anome 13:55, 29 Nov 2003 (UTC)

What is LCR?

This could really use some graphs or diagrams or...some sort of visualization. --Atario 19:11, 8 Jul 2004 (UTC)

"Three phase power is commonly found in industrial applications and LCR circuits."

What is an LCR circuit? Did you mean RLC circuit? if so, what does it have to do with three-phase power? - Omegatron 13:39, Jul 16, 2004 (UTC)

Polyphase_coil says something about it.

there is no mention of other voltages availbile eg, in Canada\US

277/480, 347/600, 240/416.. as well as the difference in polyphase systems such as delta connections, wye connections etc.

Volates in north america polyphase are not limited to 120/208
 * delta is a live-live connection and wye is live-neutral connection. So this is covered in the article. Also some of the differences you mention might result from measuring either peek or mean(sp?) voltage.

Not all polyphase motors are self-starting (nor simple in construction). Ie. synchronius (sp? synchronic?) motor won't start and reach synchronic speed without either running in asynchronious mode (this means rotor has to be capable of both modes) or controlled frequency changes (this practically means computerised dedicated power supply).

Why a page for "poly"-phase, without a list of applicable poly-phases?
Generally this article seems to be useless because it seems to claim a huge variety of polyphase power systems exist. But there are only two types of phased power systems in common use today, known as single-phase or three-phase. Split-phase can be a combination of either of these two. Two-phase is considered an antique relic and is not used anymore.

Common use:
 * Single-phase electric power - Used by most high amperage American/European residential devices, 230v Amercian or 560v European
 * Three-phase electric power - Used primarily for high horsepower motors, but can still be found in residential settings if a building is a large condominium or commercial skyscraper with apartments. Direct voltage is 480-volt American.

Combination use:
 * Split-phase electric power - For single-phase power, it is used by most low amperage American/European residential devives, 115v American or 230v European. For three-phase power, the voltages can be 208-volt American if 2 of the 3 phases are used for single-phase loads, and can be used as three American 120v split-phase circuits for residential/commercial/industrial applications.

Considered obsolete:
 * Two-phase electric power
 * 25-hertz systems (at one time trolley cars still used it)

So what are these other phase wiring designs that this article tries to suggest may exist? It's either single phase or three phase. Nobody used four-phase or five-phase, etc. Polyphase suggests that other systems than three-phase power are in use in the world today.... but if not, then why not just call it three-phase electrical power? DMahalko (talk) 00:00, 16 May 2008 (UTC)

 * US (and I believe Canadian) homes use 240V center-tapped, allowing for 120V or 240V devices. As I understand it, (not living there) Europe normally distributes three-phase wye with 230V (+/- 10) phase to neutral, for (about) 400V phase-to-phase. I suspect you never find 560V in European residences or 480 in American residences. Three-phase wye 120/208 occurs in larger US residences, such as college dormitories and condominiums. Three phase 277/480 is common for US industrial lighting, where it allows for more efficient fluorescent lighting systems, and also more efficient large three-phase motors. Gah4 (talk) 21:07, 20 July 2015 (UTC)

Efficiency
The article says "Polyphase systems are particularly useful for transmitting power to electric motors." It seems to me that three phase allows for efficient long distance power distribution and also works well for motors, not necessarily both at the same time. Gah4 (talk) 12:36, 15 November 2009 (UTC)

Counting phases
"A polyphase system must provide a defined direction of phase rotation, so mirror image voltages do not count towards the phase order. A 3-wire system with two phase conductors 180 degrees apart is still only single phase. Such systems are sometimes described as split-phase." I do understand not counting phases that are 180 degrees apart. It comes naturally from transformer winding physics. But why is this related to direction of rotation? Also, in describing rectifiers, the article mentions six phases. Seems like if you use the same phase counting method it should still be three, but rectifier configurations that uses six diodes. One is the three-phase extension of the usual center-tap full-wave rectifier, another the extension of the normal bridge rectifier. Is there a proper name for these rectifiers? Gah4 (talk) 20:56, 20 July 2015 (UTC)

 * Continuing the question, as new additions have been made regarding six-phase, it seems to me that it is common to use a six phase rectifier on three phase power. That is, a full wave rectifier has twice the number of phases of its input source. Does Brookhaven have a six phase input, and use a 12 phase rectifier? Gah4 (talk) 08:49, 26 December 2015 (UTC)

Describes polyphase better
I found this webpage http://www.thecoffeebrewers.com/electricity.html describes polyphase better than this wiki. Haimiyahya (talk) 16:23, 3 August 2015 (UTC)haimiyahyaHaimiyahya (talk) 16:23, 3 August 2015 (UTC)

 * Interesting, though most of what it says is close to right, but not quite right. For one, real generators never look like the diagram shown, which is also shown in many physics books. It is right in theory, but in reality you need an iron core, and that completely changes the way it works, though in the end the result is the same sinusoid. For another, generators never rotate at 60Hz (3600RPM).  The usual large power generators run at 72RPM. That is, they have 100 poles instead of two.  Next, it indicates that power distributed through a neighborhood isn't in the tens of kV. I do know that the distribution through the neighborhood where I grew up was changed from 2.3kV to 22kV in about 1970.  I believe 22kV is still commonly used for residential delivery.  Next, 50Hz appliances with transformers should not have any problem at 60Hz, but not all 60Hz transformers will run properly at 50Hz.  Most likely they will, though. Next, it indicates that the usual US/Canada home power is two phase. The way electricians count, it is still single phase. That is, you don't separately count phases that are opposite sides of a center tapped transformer.  If you tell the electrician that you have, or want, two phase power, he/she will be very confused. Also, circuit breakers should be able to run at their specified current rating. Not that you shouldn't leave some margin, but if it trips at exactly the specified current, it is defective. Since the target audience is interesting in heating appliances, much of the bottom half is appropriate for that case, but not quite right for other types of loads. Next, heaters, light bulbs, and wiring works on RMS current, not peak current. The heat capacity is large enough to average out the power, such that instantaneous power isn't important, but average power is.  (At least over a 50Hz or 60Hz cycle.) The important point is almost lost in the discussion: if you run at 240V heating appliance at 208V, as is commonly done for ovens, stoves, water heaters, and clothes dryers, the power output will be reduced by about (240/208) squared. Nameplates on appliances commonly give ratings for both common power sources. A small autotransformer will get you from 208V to 230V or 240V if you want one. Gah4 (talk) 09:37, 26 December 2015 (UTC)

Comments and inventors discrepancy
I don't like the introduction. The initial description/explanation is not good. It really shouldn't be so hard to come up with a clear statement.

The sentence “Induction motors using a rotating magnetic field were independently invented by Galileo Ferraris and Nikola Tesla and developed in a three-phase form by Mikhail Dolivo-Dobrovolsky in 1889.” is contradicting the sentence found in the Three-phase electric power article which says “The three-phase system was independently invented by Galileo Ferraris, Mikhail Dolivo-Dobrovolsky, Jonas Wenström and Nikola Tesla in the late 1880s.” The two should be consistent.

In my opinion the motors section should directly follow the introduction and state clearly what's the relationship between a polyphase system, AC/induction motors and AC electrical power generation.

ICE77 (talk) 03:33, 6 November 2016 (UTC)

Higher phase order
"Common practice for rectifier installations and in HVDC converters is to provide six phases, with 60 degree phase spacing, to reduce harmonic generation in the AC supply system and to provide smoother direct current."

B.S. The source is 3-phase, split into delta and wye components to provide unidirectional 6-pulse output of the rectifier. Sbalfour (talk) 23:06, 2 November 2017 (UTC)

 * As far as I can tell, when counting phases of AC systems, you don't count a 180 degree (other half of a transformer) phase, but you do when counting phases for a rectifier. A three phase input, into a full wave rectifier, is a six phase rectifier system. Gah4 (talk) 23:40, 2 November 2017 (UTC)

three phase power
From the article "Compared to a single-phase, two-wire system, a three-phase three-wire system transmits three times as much power for the same conductor size and voltage." Isn't the formula P=I*U*root(3) which makes it 1.73 times the power, not three times ? 176.11.34.234 (talk) —Preceding undated comment added 16:55, 8 June 2019 (UTC)

title: Talk:Polyphase system