A fast Fourier transform is an algorithm that computes the discrete Fourier transform of a sequence, or its inverse. Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa; the DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is too slow to be practical. An FFT computes such transformations by factorizing the DFT matrix into a product of sparse factors; as a result, it manages to reduce the complexity of computing the DFT from O, which arises if one applies the definition of DFT, to O, where N is the data size. The difference in speed can be enormous for long data sets where N may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.

Fast Fourier transforms are used for applications in engineering, music and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805. In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime", it was included in Top 10 Algorithms of 20th Century by the IEEE magazine Computing in Science & Engineering; the best-known FFT algorithms depend upon the factorization of N, but there are FFTs with O complexity for all N for prime N. Many FFT algorithms only depend on the fact that e − 2 π i / N is an N-th primitive root of unity, thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms. Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/N factor, any FFT algorithm can be adapted for it; the development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished work in 1805 when he needed it to interpolate the orbit of asteroids Pallas and Juno from sample observations.

His method was similar to the one published in 1965 by James Cooley and John Tukey, who are credited for the invention of the modern generic FFT algorithm. While Gauss's work predated Joseph Fourier's results in 1822, he did not analyze the computation time and used other methods to achieve his goal. Between 1805 and 1965, some versions of FFT were published by other authors. Frank Yates in 1932 published his version called interaction algorithm, which provided efficient computation of Hadamard and Walsh transforms. Yates' algorithm is still used in the field of statistical analysis of experiments. In 1942, G. C. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallography, a field where calculation of Fourier transforms presented a formidable bottleneck. While many methods in the past had focused on reducing the constant factor for O computation by taking advantage of "symmetries", Danielson and Lanczos realized that one could use the "periodicity" and apply a "doubling trick" to get O runtime.

James Cooley and John Tukey published a more general version of FFT in 1965, applicable when N is composite and not a power of 2. Tukey came up with the idea during a meeting of President Kennedy's Science Advisory Committee where a discussion topic involved detecting nuclear tests by the Soviet Union by setting up sensors to surround the country from outside. To analyze the output of these sensors, an FFT algorithm would be needed. In discussion with Tukey, Richard Garwin recognized the general applicability of the algorithm not just to national security problems, but to a wide range of problems including one of immediate interest to him, determining the periodicities of the spin orientations in a 3-D crystal of Helium-3. Garwin gave Tukey's idea to Cooley for implementation. Cooley and Tukey published the paper in a short time of six months; as Tukey did not work at IBM, the patentability of the idea was doubted and the algorithm went into the public domain, through the computing revolution of the next decade, made FFT one of the indispensable algorithms in digital signal processing.

Let x0, …, xN−1 be complex numbers. The DFT is defined by the formula X k = ∑ n = 0 N − 1 x n e − i 2 π k n / N k = 0, …, N − 1, where e i 2 π / N is a primitive Nth root of 1. Evaluating this definition directly requires O {\displayst

Sir Henry Howard was a British diplomat, the first formal British envoy to the Vatican for over 300 years. Henry Howard was the elder son of Sir Henry Francis Howard a British diplomat and ambassador, through whom he was a descendant of Lord William Howard, younger son of Thomas Howard, 4th Duke of Norfolk, he was a member of the Howard family, a Roman Catholic, was educated at Downside School. He joined the Diplomatic Service as an attaché to the Legation in Washington, D. C. in 1865. He was promoted to Third Secretary in 1869 and to Second Secretary in 1873. While in Washington he was Her Majesty's Agent for British claims under the Treaty of Washington. For this service, he was invested as a Companion of the Order of the Bath in 1874, after he had left Washington, he served in The Hague, and, in early 1876, was in London as secretary to a Royal Commission on Fugitive Slaves. After this he was posted back to Washington to Guatemala in 1883. In 1885, Howard was promoted again to be Secretary to the legation at Athens, subsequently held the same post at the legations at Copenhagen, Peking and St Petersburg.

In 1894, he was appointed Secretary to the embassy at Paris. In 1896, he was appointed minister to the Netherlands and to Luxembourg. While at The Hague, Howard was knighted KCMG, in January 1899, and, a few months he was named as British co-representative at the Hague Convention of 1899. Sir Henry was promoted to Knight Commander of the Order of the Bath in the King's Birthday Honours of 1907. In October 1908 he left The Hague after presenting his letters of recall to Queen Wilhelmina, who conferred on him the Order of Orange-Nassau; some of his furniture and effects were shipped on the Great Eastern Railway Company's ship Yarmouth which sank with all hands on its way from Hook of Holland to Harwich on 27 October 1908. In December 1914, after the outbreak of the First World War, Sir Henry was appointed "His Majesty's Envoy Extraordinary and Minister Plenipotentiary on a Special Mission to His Holiness the Pope". Sir Henry was accompanied by a member of Foreign Office staff to be Secretary of the mission.

This appointment established full diplomatic relations with the Holy See for the first time since 1558. Sir Henry's instructions, in a letter to him from the Foreign Secretary, Sir Edward Grey, were published in a parliamentary paper:You will... in presenting your letters of credence to his Holiness, offering him the cordial congratulations of his Majesty the King on the occasion of his election, intimate to him that his Majesty's Government are anxious to put themselves into direct communication with him for the purpose of demonstrating the motives which have governed their attitude since the first moment that the normal relations between the Great Powers of Europe began to be disturbed and of establishing that his Majesty's Government used every effort to maintain the peace of Europe which his Holiness' venerated predecessor had so much at heart. In August 1916 Sir Henry retired from the Diplomatic Service and was appointed GCMG "in recognition of his long and eminent services, on the occasion of his retirement."

He died in Rome on 4 May 1921. On 2 October 1867 Henry Howard married daughter of George Washington Riggs, they had two sons. Lady Howard died on 3 December 1907. HOWARD, Sir Henry, Who Was Who, A & C Black, 1920–2008.

Marie Charlotte de La Tour d'Auvergne was a French noblewoman and member of the House of La Tour d'Auvergne. Married into the House of Beauvau, a powerful family originating in Anjou, she had a daughter at the age twenty, died of smallpox at the age of thirty three; the present Duke of Mouchy branch of the Noailles family are descended from her. Born at the Hôtel de Bouillon in Paris to Emmanuel Théodose de La Tour d'Auvergne, sovereign Duke of Bouillon, his last wife Louise Henriette Françoise of Lorraine, her mother was a daughter of Joseph of Count of Harcourt. Her father was a son of Godefroy Maurice de La Tour d'Auvergne and Marie Anne Mancini, the latter was a niece of Cardinal Mazarin and an infamous hostess in her day. Marie Charlotte was styled as Mademoiselle de Château-Thierry from birth; when her older half sister Anne Marie Louise, Mademoiselle de Bouillon was married to Charles de Rohan, Prince of Soubise in 1734, as the most senior unmarried princess of the La Tour d'Auvergne family, she was styled Mademoiselle de Bouillon until her marriage.

A first cousin included Antoine de Vignerot du Plessis, son of her aunt Élisabeth Sophie of Lorraine and the famous womaniser Armand de Vignerot du Plessis, duc de Richelieu. Her father died in 1730, her mother died in 1737. As such she became the ward of Count of Évreux. In 1741, her maternal uncle Louis de Lorraine-Harcourt was a proposed candidate for the hand of the 12-year-old, but the marriage never materialised and he died childless in 1747, she married Charles Juste de Beauvau, a member of the wealthy Beauvau family of the Duchy of Lorraine, on 3 April 1745. Her sister in law was the famous marquise de Boufflers, her daughter married Philippe Louis de Noailles, son of Philippe de Noailles and Anne d'Arpajon, lady-in-waiting to Marie Antoinette. She died of smallpox at her husband's town house in Lunéville, Lorraine, she and her daughter were heading to Paris from Lorraine. Notwithstanding the utmost care, she succumbed to the illness. Lauzun and Louise never married and were both affected by the death of Marie Charlotte – the two were in love – and with the death of Marie Charlotte, Lauzun lost his most valuable ally, regarding a union with Louise and himself.

Her daughter was placed in the Abbey of Port Royal in Paris where she remained till her marriage in 1767. After her death, her husband married again in 1764 to Marie Charlotte Sylvie de Rohan-Chabot, a cousin of Charles de Rohan, Prince of Soubise. Anne Louise Marie de Beauvau, Mademoiselle de Beauvau married Philippe Louis de Noailles, prince de Poix Duke of Mouchy and had issue. 20 December 1729 – 29 December 1734 Her Highness Mademoiselle de Château-Thierry 29 December 1734 – 3 April 1745 Her Highness Mademoiselle de Bouillon 3 April 1745 – 6 September 1763 Her Highness the Princess of Beauvau

Lapa Terra Ronca or Caverna Terra Ronca I, is a dolomitic limestone cave inside the area of the Terra Ronca State Park, which houses the largest collection of caves and grottoes in the midwest with a number of about 200 being that only Terra Ronca and Angelica are open for the tourists. It is located about 400 kilometers from Brasília, midway between the municipalities of São Domingos, Posse e Guarani de Goiás, in the State of Goiás, Brazil. Thousands of years ago, a landslide caused its division into two parts; the first can be visited but the second part requires a tour guide. It's called Terra Ronca II or Malhada of gigantic proportions, it has two skylights, one of them called Araras and a large gallery called Lovers' Hall decorated with rich stalactites and stalagmites and huge sand dunes crossed by the underground Lapa river. Indeed, geologists working for the state park say that the whole area where the cave is located was sea bottom about 600 million years, considering the many sea shells found around the area.

It was formed by the continuous action of rain water and the Lapa river that comes down from the Serra Geral. Its name comes from the roaring sound produced by this river running through its 5.8 km long route the two craters on the surface arisen from seismic activity millions of years ago, many halls of stalactites and stalagmites of impressive dimensions of up to 150,000 m² and columns with more than 590 meters high. Terra Ronca has a cave mouth of 96 meters high and 120 meters wide, with an altar measuring 760 meters long and 100 meters high where the religious ceremony of Bom Jesus da Lapa da Terra Ronca takes place at the beginning of August; the Salao dos Namorados, besides the ceiling of the cave shaped in the form of a clock, the Oco das Araras, a fantastic place where many parrots live, Clarabóia, another output of Terra Ronca for those who do not want to stay longer inside the cave are other highlights found here. There are several other output options like the Pesqueiro 1 and 2.

List of caves in Brazil Base de Dados do Ministerio do Meio Hambiente Governo Federal - ICMBIO Official Website

Randy Spears, sometimes credited as Greg Ory, is a retired American pornographic actor and director who has appeared in numerous adult feature films. Spears is noted among porn actors for his comedic acting talent. Spears left Kankakee for California during the early 1980s, he landed small roles in the mainstream films Trading Blow Out. Spears was seen in three mainstream films under the name Gregory Patrick. In the 1988 comedy Sexpot he played the role of Damon. A year Spears starred, along with Georgina Spelvin and Linda Blair, in Bad Blood, a gothic thriller known as A Woman Obsessed, he acted in commercials and in a one-week stint on Ryan's Hope in 1989 where he played a bartender. Spears recalls this time as being "fun". In 2009, Spears provided the voice of a character called John Q. Mind in the episode "In Country... Club", the fifth-season premiere of American Dad! Spears was introduced to the industry in 1987 by Ona Zee. In a 2016 interview, he said he was an aspiring actor who, after a few years in show business, fell on hard times due to the 1988 Writers Guild of America strike.

He said he turned to modeling when a make-up artist handed him her card and told him to dial the number if he felt like doing nude work. With nothing more than a can of beans and a sack of potatoes in his cupboard, he called a month despite his concern that working in the adult entertainment industry could hurt his professional career on. Spears said he felt conflicted after his first shoot, for which he earned $400. Still, he fielded and accepted more offers, which came in, he said, once producers noticed he had some acting experience, he kept his mother uninformed about his new profession. Spears' first billing was as detective Hank Smith in The Case of the Sensuous Sinners, released in 1988. In 1990, Spears and Tom Byron traveled to Australia and made the films Singles Holiday, Lost in Paradise, Dial a Sailor, Sexual Healer and Outback Assignment; the same year, he won the Adult Video News Best Actor Award for his performance in The Masseuse with Hyapatia Lee. Spears played Captain Jim Quirk in the "Sex Trek" series, a sex-parody of Star Trek.

He has since been inducted into the AVN Hall of Fame. In 2011, he shot his last film in the pornography film industry, having broken down in tears as he left the set, he publicly announced in 2015. He was best friends with Jon Dough. Spears expressed. Following Dough's suicide in 2006, Spears and Dough's widow set up a memorial fund on MySpace to help pay for the funeral costs. In a 2015 interview, Spears said that he had severe drug problems during his time in the adult business and that a vicious circle kept him in the business all those years: "I had to go to work, to do the porn, so that I could buy the drugs, to bury the pain of doing the porn. So I’d go to work, do the porn, so I could buy the drugs, to bury the pain, and around and around it went." Spears claimed that the business changed his view of women, perceiving them as sexual objects, adding "I lost the ability to have a loving and caring relationship. I thought. I was fooled." AVN Awards: 1991 Best Actor, Film 1994 Best Supporting Actor, Video 2000 Best Actor, Video 2002 Hall of Fame inductee 2003 Best Supporting Actor, Video 2004 Best Actor, Film 2004 Best Supporting Actor, Video 2004 Best Oral Sex Scene, Film with Sunrise Adams 2006 Best Supporting Actor, Film 2006 Best Group Sex Scene, Film 2007 Best Group Sex Scene, Film 2007 Best Actor, Film 2008 Best Supporting Actor, Film 2010 Best Group Sex Scene with Special Guest Jamie Wright & Jessica StoneXRCO Awards: 1989 Stud Of The Year 1990 Male Performer Of The Year 1991 Best Actor All That Sex 1991 Male Performer Of The Year 1999 Best Actor Double Feature 2002 Hall of Fame inductee 2003 Best Actor Space Nuts 2004 Best Actor Misty Beethoven - The Musical 2005 Best Actor Eternity 2006 Best Actor Curse Eternal 2007 Best Actor Black WidowOther Awards: 1993 F.

O. X. E Male Fan Favorite 1999 Nightmoves Award for Best Actor/Male Performer 2001 F. O. X. E Male Fan Favorite 2002 Nightmoves Award for Best Actor/Male Performer 2005 Nightmoves Award for Best Actor/Male Performer 2006 F. O. X. E Male Fan Favorite 2006 Ninfa Award - Best Supporting Actor 2007 F. A. M. E. Award – Favorite Male Star 2008 Eroticline Award - Best International Actor What about Boob? Sex Trek: The Next Penetration The Plumber's Revenge 2 XXX Training Jane Millionaire Snatch Adams Rawhide Babes Behind Bars Ethnic City Full Exposure Pandora's Box Prisoner Two Naughty Bookworms Batman XXX: A Porn Parody The Alibi Heat Randy Spears on IMDb Randy Spears at the Internet Adult Film Database Randy Spears at the Adult Film Database Randy Spears on Facebook Profile of Randy Spears from Violet Blue 2008 FAME Awards hosting bio Randy Spears Interview podcast

A puer mingēns is a figure in a work of art depicted as a prepubescent boy in the act of urinating, either actual or simulated. The puer mingens could represent anything from whimsy and boyish innocence to erotic symbols of virility and masculine bravado; the term puer mingens come from the Latin puer, meaning "boy", from the Latin mingens, the present participle of the verb mingere which means "to urinate". In Latin, verbs for urinating like mingere were employed in the sense of "to ejaculate"; this connotation was preserved in various descendents of Latin, including Italian with such words as pisciare. On account of this, the urine emitted from the penis of the puer mingens can be interpreted symbolically as semen. Lorenzo Lotto's Venus and Cupid is an example. In several languages, such as Italian and English, "to make water" was a euphemism for urinating. In allusion to this, one can find depictions of a puer mingens "making water" in works such as Michelangelo's Children's Bacchanal, or in church lavabos whose waterspouts are positioned in front of naked boys' groins.

Pueri mingentes were incorporated as functioning statues whose pipes shot forth streams of water out of the statues' penises. One of the most famous examples of this is the Mannekin Pis in Brussels. Pueri mingentes are a classical motif found in antiquity. Ancient Roman examples of pueri mingentes occurred on children's sarcophagi; the puer mingens was revived during the Renaissance. Donatello, who paved the way in the reinvention of the larger motif of the putti in sculpture, depicted one of the earliest Renaissance examples of a puer mingens on the base of his Judith and Holofernes statue. From its revival in 1400s Florence, the artistic motif of urinating boys spread throughout the rest of Europe, reaching its height of popularity during the late Renaissance in the sixteenth and seventeenth centuries before receding in popularity. In Roman times the puer mingens was found in depictions of Bacchic rites on children's sarcophagi. From the Renaissance onward, the puer mingens can be found in both secular and religious art and across a range of media, from illuminated manuscripts, functional fountains, frescoes, to apotropaic amulets.

Owing to the abovementioned associations with fertility, pueri mingentes are found on deschi da parto – trays given to pregnant women and those who had given birth in order to betoken and celebrate the healthy birth of male offspring. Paintings intended as wedding gifts, such as Lorenzo Lotto's Venus and Cupid, might feature urinating boys; the puer mingens was prominently incorporated into fountains that would shoot water out of the statue's penis. Although this artistic motif is Roman in origin, there is scant attestation of working fountains incorporating pueri mingentes in Roman times. In addition to public spaces, such as the Mannekin Pis's location in central Brussels, functional fountains graced many private sixteenth- and seventeenth-century gardens across Europe. Putti – Artistic depictions of naked boys Manneken Pis – A bronze statue in Brussels of a naked little boy urinating Media related to Pueri mingentes at Wikimedia Commons