For the first iteration the current intersection will be the starting point and the distance to it ( the intersection's label ) will be zero.

For the Julia set is the unit circle and on this the iteration is given by doubling of angles ( an operation that is chaotic on the non-rational points ).

For the Julia set is the line segment between-2 and 2, and the iteration corresponds to in the unit interval.

For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes.

For the more general iteration, it has been proved that if the Julia set is connected ( that is, if c belongs to the ( usual ) Mandelbrot set ), then there exist a biholomorphic map between the outer Fatou domain and the outer of the unit circle such that.

For example, a collection of walkers in a Markov chain Monte Carlo iteration is called an ensemble in some literature.

For example, a first iteration prototype may be called an " Alpha " prototype.

For the production of human ovum cells, the colony's founding fathers brought and maintained a set of ovarian tissue cultures which are abstractly referred to with a code indicating the original donor ’ s initials and the iteration of the culture ; e. g., CJB-9 is the 9th production of the tissue culture donated by Dr. Cynthia Jane Baruch.

For example, in binary search, the size of the problem to be solved is halved with each iteration, and therefore roughly lb n iterations are needed to obtain a problem of size 1, which is solved easily in constant time.

For example, study the geometrical basis of graffiti, the transformational geometry and iteration in cornrow hairstyles, taxation and the creativity of children with decorative arts can all encompass and rely on mathematical concepts.

For example, in the case of variables whose conditional distribution is not easy to sample from, a single iteration of slice sampling or the Metropolis-Hastings algorithm can be used to sample from the variables in question.

For an ( n, k ) LDPC code of rate k / n, a full iteration occurs when n variable and n − k constraint nodes have been updated, no matter the order in which they were updated.

For general matrices, algorithms are iterative, producing better approximate solutions with each iteration.

For example, for power iteration,.

For example, hito-bito, " people ", is usually written 人々, using the kanji for 人 with an iteration mark, 々, rather than 人人, using the same kanji twice, though this latter is allowed, and in this simple case might be used because it is easier to write.

For example, tokorodokoro could be written horizontally as ところ〲 ; the voiced iteration mark only applies to the first sound と.

* For the first iteration the two piece and one piece estimates are used in the formula ( 4 X ( more accurate )-( less accurate ))/ 3 The same formula is then used to compare the four piece and the two piece estimate, and likewise for the higher estimates

* For the second iteration the values of the first iteration are used in the formula ( 16 ( more accurate )- less accurate )/ 15

For example, the Barnes – Hut simulation-based method FADE can improve running time to n * log ( n ) per iteration.

For non-linear systems similar reasoning shows that the normal equations for an iteration cycle can be written as

For example, in the case of finding a line which fits the data set illustrated in the above figure, the RANSAC algorithm typically chooses 2 points in each iteration and computes as the line between the points and it is then critical that the two points are distinct.

For example one could assign a 2, 000 max iteration image to the red channel, a 200 max iteration image to the green channel, and a 20 max iteration image to the blue channel of an image in an RGB color space.

For example, No. 56 printed the patent giving the Electoral Prince the title of Duke of Cambridge.

For want of funds, Alexander was unable to resist the Grand Master of the Teutonic Knights or prevent Grand Duke of Muscovy Ivan III from ravaging Grand Duchy of Lithuania with the Tatars.

For the rest of the two spacecrafts ' pass over the near side of the Moon, Mattingly prepared to shift Casper to a circular orbit while Young and Duke prepared Orion for the descent to the lunar surface.

For the following year the Duke proposed a campaign along the valley of the River Moselle to carry the war deep into France.

For his part, Davis was fully aware of the importance of the project, which he pursued to the point of turning down a job with Duke Ellington's orchestra.

Informally, he may have been known as " Dickon ", according to a sixteenth-century legend of a note, warning of treachery, that was sent to the Duke of Norfolk on the eve of Bosworth: " Jack of Norffolke be not to bolde ,/ For Dyckon thy maister is bought and solde ".

For the 1735 campaign the allied forces in northern Italy came under the command of the Duke de Noailles, elevated to Marshal after his successful contributions to the Rhine campaign.

For the Duke thought it unseemly to receive money for such merchandise, and equally he considered it wrong that Harold should be buried as his mother wished, since so many men lay unburied because of his avarice.

For example, during the Warring States Period, Duke Xiang of Song was at war with the state of Chu, and had an opportunity to attack the enemy force, commanded by Zhu, while they were crossing a river.

For example, in June 2008, a Duke University study was published assessing the neuropsychological effects and attitudes in patients after ECT.

For most of the rest of that year, George and May were styled Their Royal Highnesses The Duke and Duchess of Cornwall and York.

For example, Prince William and his wife Catherine Middleton, a commoner, became the Duke and Duchess of Cambridge upon their marriage on 29 April 2011.

For example, Emperor Lizong of Song granted the posthumous title Duke of Hui ( 徽国公 ) to the Neo-Confucian thinker Zhu Xi.

For example, a similar tale is recorded in Arabian Nights where Harun al-Rashid plays the same trick on a man he finds sleeping in an alley, and in De Rebus Burgundicis by the Dutch historian Pontus de Heuiter, where the trick is performed by Philip the Good, i. e. Philip III, Duke of Burgundy.

: For the operetta by Gilbert and Sullivan, see The Grand Duke.

For the Grand Duke butterfly, see Bassarona iva.

For the première, La Fenice had cast Felice Varesi as Rigoletto, the young tenor Raffaele Mirate as the Duke, and Teresa Brambilla as Gilda ( although Verdi would have preferred Teresa De Giuli Borsi ).

For example, the double marriage in 1385 at Cambrai of his son, John the Fearless, and his daughter, Marguerite, to Margaret of Bavaria and William of Bavaria, son and daughter of Albert, Count of the neighbouring Hainault and Holland, prepared the later union of Hainault and Holland with Burgundy and Flanders, as carried out by Philip's grandson, Philip the Good ; the marriages also inserted the new Valois Burgundy dynasty into the Wittelsbach network of alliances: the other daughters of Count Albert had married William I, Duke of Guelders and Wenceslaus, King of Bohemia ; their cousin, Isabeau of Bavaria, had married Charles VI of France, and become Queen of France.

For her own part, Margaret's family were far more powerful and secure than they had been in 1454: her father had been killed at the Battle of Wakefield on 30 December 1460, but her brother was now Edward IV, opposed ineffectively only by Margaret of Anjou and her son, Edward of Westminster ; this made Margaret a far more valuable bride than she had been as the mere daughter of a Duke.

For details, see Duke of Denver.

For his support of Richard III's claim to the throne, John Howard, the son of Thomas Mowbray's elder daughter Margaret, was created 1st Duke of Norfolk in 1483, in the title's third creation.

For example, the Duke of Norfolk is also the Earl of Arundel and the Lord Maltravers.

For instance, the eldest son of the Duke of Buccleuch and Queensberry is styled Earl of Dalkeith, even though the Duke is also the Marquess of Dumfriesshire, a title which outranks the Earldom.

For instance, the Duke of Westminster is also the Marquess of Westminster and the Earl Grosvenor ( amongst other titles ).