Prunus cerasus is a species of Prunus in the subgenus Cerasus, native to much of Europe and southwest Asia. It is related to the sweet cherry, but has a fruit, more acidic, its sour pulp is edible. The tree is smaller than the sweet cherry, has twiggy branches, its crimson-to-near-black cherries are borne upon shorter stalks. There are two main varieties of the sour cherry: the dark-red morello cherry and the lighter-red amarelle cherry. Prunus cerasus, a tetraploid with 2n=32 chromosomes, is thought to have originated as a natural hybrid between Prunus avium and Prunus fruticosa in the Iranian Plateau or Eastern Europe where the two species come into contact. Prunus fruticosa is believed to have provided its smaller size and sour tasting fruit; the hybrids stabilised and interbred to form a new, distinct species. Cultivated sour cherries were selected from wild specimens of Prunus cerasus and the doubtfully distinct P. acida from around the Caspian and Black Seas, were known to the Greeks in 300 BC.

They were extremely popular with Persians and the Romans who introduced them into Britain long before the 1st century AD. The fruit remains popular in modern-day Iran. In England, their cultivation was popularised in the 16th century in the time of Henry VIII, they became a popular crop amongst Kentish growers, by 1640 over two dozen named cultivars were recorded. In the Americas, by 1704 the Vestry of New Kent County, Virginia recorded "The DePriest of Kent" planted 354 acres of Prunus cerasus along the Pamunkey River as the'Kent' variety, that spawned other Virginia colonists throughout Richmond to plant sour cherry trees,'Early Richmond' variety or'Kentish Red', when they arrived. Before the Second World War there were more than fifty cultivars of sour cherry in cultivation in England; this is a late-flowering variety, thus misses more frosts than its sweet counterpart and is therefore a more reliable cropper. The Morello cherry ripens in mid toward the end of August in southern England, it is self-fertile, would be a good pollenizer for other varieties if it did not flower so late in the season.

Sour cherries require similar cultivation conditions to pears, that is, they prefer a rich, well-drained, moist soil, although they demand more nitrogen and water than sweet cherries. Trees will do badly if waterlogged, but have greater tolerance of poor drainage than sweet varieties; as with sweet cherries, Morellos are traditionally cultivated by budding onto strong growing rootstocks, which produce trees too large for most gardens, although newer dwarfing rootstocks such as Colt and Gisella are now available. During spring, flowers should be protected, trees weeded and sprayed with natural seaweed solution; this is the time when any required pruning should be carried out. Morello cherry trees fruit on younger wood than sweet varieties, thus can be pruned harder, they are grown as standards, but can be fan trained, cropping well on cold walls, or grown as low bushes. Sour cherries suffer fewer pests and diseases than sweet cherries, although they are prone to heavy fruit losses from birds. In summer, fruit should be protected with netting.

When harvesting fruit, they should be cut from the tree rather than risking damage by pulling the stalks. Unlike most sweet cherry varieties, sour cherries are self pollenizing. Two implications of this are that seeds run true to the cultivar, that much smaller pollinator populations are needed because pollen only has to be moved within individual flowers. In areas where pollinators are scarce, growers find that stocking beehives in orchards improves yields; some cultivars of sour cherry trees, such as Montmorency and North Star, have been documented to perform better than other cherry trees in Colorado's Front Range region. Dried sour cherries are used in cooking including soups, pork dishes, cakes and pies. Sour cherries or sour cherry syrup are used in drinks, such as the portuguese ginjinha. In Iran, Turkey and Cyprus, sour cherries are prized for making spoon sweets by boiling pitted sour cherries and sugar. A particular use of sour cherries is in the production of kriek lambic, a cherry-flavored variety of a fermented beer made in Belgium.

Fruit trees Fruit tree forms Fruit tree propagation Ginjinha, a Portuguese liqueur made from Sour Cherry Griotte de Kleparow Kirsch Kriek, a traditional Belgian beer made with sour cherries Marasca cherry Amarena cherry North Star cherry, a dwarf variety Pruning fruit trees Sour cherry soup Syzygium corynanthum, an Australian rainforest tree known as the Sour cherry Vişinată, a Romanian liqueur made with sour cherries Vişne, Vişne Receli is a sour cherry preserve and Vişne Suyu is a popular fruit juice in Turkey

In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function f: D → C from the unit disk D onto a subset of the complex plane contains the disk whose center is f and whose radius is |f′|/4; the theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1916; the example of the Koebe function shows. A related result is the Schwarz lemma, a notion related to both is conformal radius. Suppose that g = z + b 1 z − 1 + b 2 z − 2 + ⋯ is univalent in |z| > 1. ∑ n ≥ 1 n | b n | 2 ≤ 1. In fact, if r > 1, the complement of the image of the disk |z| > r is a bounded domain X. Its area is given by ∫ X d x d y = 1 2 i ∫ ∂ X z ¯ d z = 1 2 i ∫ | z | = r g ¯ d g = π r 2 − π ∑ n | b n | 2 r − 2 n. Since the area is positive, the result follows by letting r decrease to 1; the above proof shows equality holds if and only if the complement of the image of g has zero area, i.e. Lebesgue measure zero.

This result was proved in 1914 by the Swedish mathematician Thomas Hakon Grönwall. The Koebe function is defined by f = z 2 = ∑ n = 1 ∞ n z n Application of the theorem to this function shows that the constant 1/4 in the theorem cannot be improved, as the image domain f does not contain the point z = −1/4 and so cannot contain any disk centred at 0 with radius larger than 1/4; the rotated Koebe function is f α = z 2 = ∑ n = 1 ∞ n α n − 1 z n with α a complex number of absolute value 1. The Koebe function and its rotations are schlicht: that is, univalent and satisfying f = 0 and f′ = 1. Let g = z + a 2 z 2 + a 3 z 3 + ⋯ be univalent in |z| < 1. | a 2 | ≤ 2. This follows by applying Gronwall's area theorem to the odd univalent function g − 1 / 2 = z − 1 2 a 2 z − 1 + ⋯. Equality holds; this result was proved by Ludwig Bieberbach in 1916 and provided the basis for his celebrated conjecture that |an| ≤ n, proved in 1985 by Louis de Branges. Applying an affine map, it can be assumed that f = 0, f ′ = 1, so that f = z + a 2 z 2 + ⋯.

If w is not in f h = w f w − f = z + z 2 + ⋯ is univalent in |z| < 1. Applying the coefficient inequality to f and h gives | w | − 1 ≤ | a 2 |

Major General Harold Augustus Freeman-Attwood, was a British Army officer who fought in both World Wars. Born Harold Freeman on 30 December 1897, he was the eldest son of Edward Freeman, a British Army officer, Katherine Margaret. Freeman was educated at Summer Fields School, Marlborough College and, during the First World War, attended the Royal Military College, where he graduated on 13 July 1915 and was subsequently commissioned as a second lieutenant into his father's regiment, the Royal Welch Fusiliers, he served with the 1st Battalion, Royal Welch Fusiliers, part of the 22nd Brigade of the 7th Division, a Regular Army unit, on the Western Front, where he was awarded the Military Cross during the Battle of Passchendaele in August 1917, with his battalion, was sent to the Italian Front in the year, where it remained until the end of the war. Remaining in the army between the wars, Freeman married Jessie Job on 10 September 1921 and together they had three children, he served with his regiment throughout the interwar period with the 1st Battalion, in operations in Wazaristan in the early 1920s before returning to the United Kingdom where he became adjutant to a Territorial Army battalion of his regiment from 1924 to 1928.

He attended the Staff College, Camberley from 1928 to 1929 and served in Cyprus, where he was appointed an Officer of the Order of the British Empire for suppressing a Greek Cypriot rebellion in 1931–1932. He returned again to the United Kingdom where he served at the staff of the War Office from 1932 to 1934. At the outbreak of the Second World War in September 1939, Freeman-Attwood by now a lieutenant colonel, was serving as a General Staff Officer with the 50th Motor Division, a TA formation, he was sent with the division to France in January 1940 where it became part of the British Expeditionary Force. He served with the division throughout the Battle of France in May 1940 and took part in the Dunkirk evacuation, and, in late July, was promoted to brigadier and assumed command of the 5th London Brigade, another TA unit, part of the 2nd London Division. In November 1941 he was promoted to major general and became General Officer Commanding of the 46th Infantry Division, another TA unit, in succession to Major General Miles Dempsey.

The division, recruiting from the North Midlands and the West Riding of Yorkshire, was composed of the 137th, 138th and 139th Infantry Brigades, along with supporting divisional troops. In January 1943, Freeman-Attwood led the division overseas to French North Africa, upon its arrival in Tunisia, came under command of Lieutenant General Charles Allfrey's V Corps, itself part of Lieutenant General Kenneth Anderson's British First Army; the division fought in the Tunisian Campaign, most notably in the final stages of the Battle of Kasserine Pass and in Operation Ochsenkopf, until the campaign came to an end in May 1943, with Freeman-Attwood being awarded the Distinguished Service Order for his division's actions at Djebel Abiod. In August, as the division was preparing to take part in the Allied invasion of Italy, Freeman-Attwood was relieved of his command and retired from the army in October, after being court-martialled for writing home in a letter to his wife expressing a wish to be drinking champagne in Italy on their wedding anniversary and disclosing details of future military operations.

Returning to civilian life, he joined the Imperial Chemical Industries and, by 1949, was Staff Manager. Freeman-Attwood divorced in 1945, remarried to Marion Louise the following year, he retired to Nottinghamshire where he was involved in politics and active for the Conservative Party. Smart, Nick. Biographical Dictionary of British Generals of the Second World War. Barnesley: Pen & Sword. ISBN 1844150496. Generals of World War II