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Fourier series

In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle of the summation can be made to approximate an arbitrary function in that interval; as such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series; the process of deriving the weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform; the Fourier series is named in honour of Jean-Baptiste Joseph Fourier, who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides, publishing his Théorie analytique de la chaleur in 1822.

The Mémoire introduced Fourier analysis Fourier series. Through Fourier's research the fact was established that an arbitrary function can be represented by a trigonometric series; the first announcement of this great discovery was made by Fourier in 1807, before the French Academy. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles; the heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave; these simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves, to write the solution as a superposition of the corresponding eigensolutions.

This superposition or linear combination is called the Fourier series. From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality. Although the original motivation was to solve the heat equation, it became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids; the Fourier series has many such applications in electrical engineering, vibration analysis, optics, signal processing, image processing, quantum mechanics, thin-walled shell theory, etc. Consider a real-valued function, s, integrable on an interval of length P, which will be the period of the Fourier series. Common examples of analysis intervals are: x ∈, P = 1.

X ∈, P = 2 π. The analysis process determines the weights, indexed by integer n, the number of cycles of the n th harmonic in the analysis interval. Therefore, the length of a cycle, in the units of x, is P / n, and the corresponding harmonic frequency is n / P. The n t h harmonics are sin ⁡ and cos ⁡, their amplitudes are found by integration over the interval of length P: If s is P -periodic any interval of that length is sufficient. A 0 and b 0 can be reduced to a 0 = 2 P ∫ P s d x and b 0 = 0. Many texts choose P = 2 π to simplify the argument of the sinusoid functions; the synthesis process is: In general, integer N is theoretically infinite. So, the series might not converge or equate to s ( x

Lady Lumley's School

Lady Lumley's School is a coeducational secondary school and sixth form located in Pickering, North Yorkshire, England. It was founded in Thornton le Dale in 1670 and named after Viscountess Lumley, a family title of the Earl of Scarborough, it has school links worldwide within Tanzania, Morocco and France. The school has been awarded Sportsmark 2008, an iNET qualification, Specialist Schools and Academies Trust, a British Schools Orienteering award and was classified as a Healthy School. In 2010 Ofsted Inspection Report rated Lady Lumley's School as overall grade 2. Ashley Hicklin, singer/songwriter Craig and Chris Short, professional footballers Duncan Dowson: Emeritus Professor John Healey, Labour Party MP for Wentworth and Dearne Richard Buck, athlete

Bundanoon, New South Wales

Bundanoon is a town in the Southern Highlands of New South Wales, Australia, in Wingecarribee Shire. At the 2016 census, Bundanoon had a population of 2,729, it is an Aboriginal name meaning "place of deep gullies" and was known as Jordan's Crossing. Bundanoon is colloquially known as Bundy/Bundi. Bundanoon, like its fellow Southern Villages of the Southern Highlands, has had a boom-and-bust economic cycle; the town became a well-known tourist destination early in the 20th century. By the 1950s, changes in lifestyle the affordability of the motor car, gave city dwellers more options and Bundanoon declined; the Sydney real estate boom of the early 21st century made Bundanoon an affordable haven within commuting distance of the city. Property values increased several-fold, houses in Bundanoon were selling for over a million dollars by 2007. Several houses in the town were destroyed by a bushfire in January 2020; the 2016 census recorded 2,729 people living in Bundanoon. The town's population was older than the general population: their median age was 56 years, 18 years older than the national median age of 38.

Children aged under 15 made up 13.8% of the population and people aged 65 years and over made up 36.0% of the population. This is reflected in the low workforce participation, with only 1,078 of the 2,729 people reporting themselves as being in the labour force. Of these, 49.6% were employed full-time, 41.7% were employed part-time and 4.2% were unemployed.76.6% of people living in Bundanoon were born in Australia. 92.5% of people only spoke English at home. The most common responses for religion were No Religion 29.4%, Anglican 27.4% and Catholic 18.7%. Brigadoon Highland Gathering - every April - attracts Scottish participants and tourists from around the world. Garden Ramble - every October Bundanoon hosts the Sydney Gilbert & Sullivan Opera every spring over 1 x weekend. Performances are a much loved annual event. Anglican: Part of the Sutton Forest parish. In 1879, Holy Trinity Anglican church was built on its present site. Catholic: Part of the parish of St Paul's in Moss Vale. St Brigid's Catholic Church was built in 1895.

Prior to that, Roman Catholic services were conducted at Sutton Forest. Uniting: Part of the Moss Vale–Bundanoon–Robertson parish. By 1870 the Primitive Methodist Church was established on the corner where the Soldiers’ Memorial Hall now stands; the Primitive Methodists moved to a new building in 1885 and leased the original site to shop owners. Bundanoon Public School was established in 1871 and had an enrolment of 191 students as at 2016; the Primitive Methodist Church hall served with Mrs Dinah Osborne as teacher. A new school of two rooms is still used as the school library. Bundanoon railway station is located on the Southern Highlands line. Most services terminate at Moss Vale, meaning Bundanoon only receives limited services. Sydney to Canberra services stop at the station. Local bus services are provided by Berrima Bus Lines. On the day of the 2011 census, 4.1% of employed people travelled to work on public transport and 65.3% by car. Rugby League – Bundanoon has a rugby league team playing in the Second Division competition of Group 6.

Bundanoon has a number of heritage-listed sites, including: Main Southern railway: Bundanoon railway station In July 2009 "Bundy on Tap", a community initiative in Bundanoon, declared itself opposed to the sale of bottled drinking water on environmental grounds. The issue of bottled water was to protest against companies Norlex and Coca-Cola extracting water from the town's groundwater. In January 2020 the town was affected by a bushfire during the 2019–20 Australian bushfire season. Several homes were destroyed. Bundanoon Village Website Brigadoon Highland Gathering at Bundanoon Morton National Park at Bundanoon "The Southern Highlands News".. Archived from the original on 21 May 2008. Sunnataram Forest Monastery Bundanoon Community Web Bundanoon Garden Ramble Bundy on Tap

Caladenia whiteheadii

Caladenia whiteheadii is a plant in the orchid family Orchidaceae and is endemic to New South Wales. It is a ground orchid with a single hairy leaf and a single pale yellow flower with thick reddish tips on the sepals and petals, it is only known from a single hill near Eugowra. Caladenia whiteheadii is a terrestrial, deciduous, herb with an underground tuber and a single, dull green, lance-shaped leaf, 50–140 mm long and 7–13 mm wide with red to purple blotches near its base; the leaf and the flowering stem are densely covered with erect transparent or whitish hairs up to 4 mm long. A single pale yellow flower 40–60 mm wide is borne on a wiry flowering stem 150–300 mm tall; the dorsal sepal is 35–40 mm long, 2–2.5 mm wide, oblong to elliptic near the base tapers to a reddish glandular tip. The lateral sepals are lance-shaped to egg-shaped near their bases, 40–45 mm long, 3–4 mm wide and taper to a reddish glandular tip; the petals are 30–35 mm long, 2.5–3 mm wide, narrow lance-shaped near the base gradually taper to a thin glandular tip.

The labellum is egg-shaped to lance-shaped, 11–14 mm long, 8–10 mm wide and has up to eight pairs of red, linear teeth up to 1 mm long on the edges. The tip of the labellum curls downward and there are four rows of red linear calli along the mid-line of the labellum. Flowering occurs in October. Caladenia whiteheadii was first formally described in 2006 by David Jones who gave it the name Arachnorchis whiteheadii; the specimen was collected near Eugowra and the description was published in Australian Orchid Research. In 2010 Gary Backhouse changed the name to Caladenia whiteheadii and published the change in The Victorian Naturalist; the specific epithet honours Brian Whitehead. This spider orchid is only known from a single hill near Eugowra where it grows in shrubby forest

2002–03 Golden State Warriors season

The 2002–03 NBA season was the Warriors' 57th season in the National Basketball Association, 41st in the San Francisco Bay Area. Under new head coach Eric Musselman, the Warriors began to show signs of life after a slow start losing 11 of their first 15 games. For the first time in nearly a decade, the Warriors reached the.500 mark late in the season with a 30–30 record as of March 4. However, they would win just eight of their final 22 games to finish sixth in the Pacific Division with a 38–44 record. Second-year guard Gilbert Arenas was named Most Improved Player of The Year, second-year star Jason Richardson won the Slam Dunk Contest in Atlanta for the second year in a row. Following the season, Arenas signed as a free agent with the Washington Wizards, Antawn Jamison and Danny Fortson were both traded to the Dallas Mavericks, Bob Sura was dealt to the Detroit Pistons and Chris Mills retired. Z - clinched division title y - clinched division title x - clinched playoff spot 2002-03 NBA season

Acoustic (John Lennon album)

Acoustic is a compilation album of John Lennon demos and live performances that feature his acoustic guitar work and was released in 2004. Although it failed to chart in the United Kingdom, Acoustic reached number 31 in the United States with sales of 27,858 copies, becoming John Lennon's best charting posthumous US release since 1988's Imagine: John Lennon soundtrack, it spent eight weeks on the chart. In keeping with the overall theme of the album, the booklet contains the lyrics and guitar chords for each track, as well as a chord index on the last page. All tracks written except where noted. "Working Class Hero" – 3:58 "Love" – 2:30 "Well Well Well" – 1:14 "Look at Me" – 2:49 "God" – 2:38 "My Mummy's Dead" – 1:13 "Cold Turkey" – 3:26 "The Luck of the Irish" – 3:41 "John Sinclair" – 3:22 Tracks 8-9 recorded live on 10 December 1971 in Ann Arbor, Michigan "Woman Is the Nigger of the World" – 0:39 "What You Got" – 2:24 "Watching the Wheels" – 3:04 "Dear Yoko" – 4:05 "Real Love" – 4:00 "Imagine" – 3:08 Recorded live on 17 December 1971 at the Apollo Theater, New York City, New York "It's Real" – 1:04 The album received much criticism from both critics and fans, as nine of the sixteen tracks had been released on the 1998 John Lennon Anthology box set, while the remaining seven recordings had been available on several different bootleg CDs for years.

Tracks 1-2, 4, 8-10, 12 & 15-16 first released on John Lennon Anthology "Reviews: Acoustic". Bagism. 13 March 2011. Retrieved 18 February 2015. Tracks 1-6 taken from 8-track John Lennon/Plastic Ono Band sessions "John Lennon – Acoustic". Uncut. 13 December 2004. Retrieved 18 February 2015. Track 7 taken from home demo September 1969 Nat. "Beatles Rarity Of The Week – "Cold Turkey"". The Beatles Rarity. Archived from the original on 18 February 2015. Retrieved 18 February 2015. Tracks 8 & 9 taken from John Sinclair benefit concert "Solo Gigs". Norwegian Wood. 10 February 2009. Retrieved 18 February 2015. Track 10 taken from home recording Track 11 taken from June 1974 work-outs "John Lennon – Acoustic". Uncut. 13 December 2004. Retrieved 18 February 2015. Tracks 12-14 taken from home recordings "John Lennon – Acoustic". Uncut. 13 December 2004. Retrieved 18 February 2015. Track 15 taken from live performance at the Apollo "Solo Gigs". Norwegian Wood. 10 February 2009. Retrieved 18 February 2015. Track 16 taken from home recordings