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Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x: e i x = cos ⁡ x + i sin ⁡ x, where e is the base of the natural logarithm, i is the imaginary unit, cos and sin are the trigonometric functions cosine and sine with the argument x given in radians; this complex exponential function is sometimes denoted cis x. The formula is still valid if x is a complex number, so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous in mathematics and engineering; the physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". When x = π, Euler's formula evaluates to e i π + 1 = 0, known as Euler's identity. Johann Bernoulli noted that 1 1 + x 2 = 1 2, and since ∫ d x 1 + a x = 1 a ln ⁡ + C, the above equation tells us something about complex logarithms by relating natural logarithms to imaginary numbers.

Bernoulli, did not evaluate the integral. Bernoulli's correspondence with Euler shows that Bernoulli did not understand complex logarithms. Euler suggested that the complex logarithms can have infinitely many values. Meanwhile, Roger Cotes in 1714 discovered that i x = ln ⁡. Cotes missed the fact that a complex logarithm can have infinitely many values, differing by multiples of 2iπ, due to the periodicity of the trigonometric functions. Around 1740 Euler turned his attention to the exponential function instead of logarithms and obtained the formula used today, named after him, it was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions. The view of complex numbers as points in the complex plane was described about 50 years by Caspar Wessel. Interpretation of the formulaThis formula can be interpreted as saying that the function eiφ is a unit complex number, i.e. it traces out the unit circle in the complex plane as φ ranges through the real numbers.

Here φ is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians. The original proof is based on the Taylor series expansions of the exponential function ez and of sin x and cos x for real numbers x. In fact, the same proof shows that Euler's formula is valid for all complex numbers x. A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates; the polar form simplifies the mathematics. Any complex number z = x + iy, its complex conjugate, z = x − iy, can be written as z = x + i y = | z | = r e i φ, z ¯ = x − i y = | z | = r e − i φ, where x = Re z is the real part, y = Im z is the imaginary part, r = |z| = √x2 + y2 is the magnitude of z φ = arg z = atan2.φ is the argument of z, i.e. the angle between the x axis and the vector z measured counterclockwise in radians, defined up to addition of 2π.

Many texts write φ = tan−1 y/x instead of φ = atan2, but the first equation needs adjustment when x ≤ 0. This is because for any real x and y


Catalepsy is a nervous condition characterized by muscular rigidity and fixity of posture regardless of external stimuli, as well as decreased sensitivity to pain. Symptoms include: rigid body, rigid limbs, limbs staying in same position when moved, no response, loss of muscle control, slowing down of bodily functions, such as breathing. Catalepsy is a symptom of certain nervous disorders or conditions such as Parkinson's disease and epilepsy, it is a characteristic symptom of cocaine withdrawal, as well as one of the features of catatonia. It can be caused by schizophrenia treatment with anti-psychotics, such as haloperidol, by the anesthetic ketamine. Protein kinase A has been suggested as a mediator of cataleptic behavior. Unsuggested waxy catalepsy, sometimes accompanied by spontaneous anesthesia, is seen as an indicator of hypnotic trance. Suggested or induced rigid catalepsy, of extended limbs or the entire body, sometimes tested with heavy weights, has been a staple of stage hypnosis shows and academic demonstrations of hypnotism since the late 18th Century, as proof of extraordinary physical abilities possible in trance states.

Such demonstrations have been performed by Asian martial artists to prove the presence of "ki" or "chi" power, a kind of psychological or spiritual resource. St. Teresa of Avila experienced a prolonged bout of catalepsy that began in 1539; this episode was precipitated by the stress she was suffering at the Carmelite Convent of the Incarnation. Her legs became rigid. Teresa endured intermittent attacks of catalepsy from on. In the arts, catalepsy is used for dramatic effect, sometimes as a plot device. In Alexandre Dumas, père's novel The Count of Monte Cristo, the Abbé Faria has fits of catalepsy from time to time, before dying from one. In Eugène Sue's The Mysteries of Paris, the villain Jacques Ferrand experiences a fit described as cataleptic in his final confrontation with Rodolphe, blinded by lamplight and hallucinating with visions of his fantasized Cecily. In George Eliot's Silas Marner, the main character Silas Marner has cataleptic fits and seizures, it is not mentioned. In Arthur Conan Doyle's "The Adventure of the Resident Patient", a man feigns catalepsy to gain access to a neurologist's rooms.

In Ford Madox Ford's The Good Soldier, the protagonist Dowell experiences catalepsy following the death of his wife. In Robert A. Heinlein's Stranger in a Strange Land, the main character Valentine Michael Smith is believed to have catalepsy when he is returned to Earth. In Edgar Allan Poe's "The Premature Burial", the narrator develops catalepsy, he fears being mistakenly declared dead and buried alive, goes to great lengths to prevent this. In another of Poe's short stories, "The Fall of the House of Usher", Madeline Usher has catalepsy, is buried alive by her unstable brother Roderick. Catalepsy is depicted in "Berenice", thus becoming one of the recurrent themes in Poe's fiction. In Poppy Z. Brite's Exquisite Corpse, the main character—Comptom, a serial killer facing a lifetime sentence—uses shamanistic techniques to induce catalepsy, convincingly appearing deceased, is able to escape prison. In Émile Zola's short story "La Mort d'Olivier Becaille", the title character is buried alive and notes that "I must have fallen into one of those cataleptic states that I had read of".

In Sax Rohmer's Fu Manchu novels, Dr. Fu-Manchu has a serum that induces a state of catalepsy so extreme as to be indistinguishable from death. In Charles Dickens's novel Bleak House, Mrs. Snagsby has violent spasms before becoming cataleptic and being carried upstairs like a grand piano. In Hegel's Lectures on the History of Philosophy: Greek Philosophy to Plato, Hegel describes Socrates as having catalepsy caused by magnetic somnambulism when in deep meditation. In Charles Williams's novel Many Dimensions, Sir Giles Tumulty says to Lord Arglay, the Chief Justice of England: "You are a louse-brained catalept, Arglay." In Philip K. Dick's novel Now Wait for Last Year, Kathy Sweetscent becomes immobilized by withdrawal from JJ-180, an alien drug. "My God, Kathy thought. I can't free myself; this is catalepsy!" In the second chapter of Álvares de Azevedo's Noite na Taverna, character Solfieri rescues a woman who has catalepsy from inside a coffin. In the Ted Hughes poem titled "Conjuring in Heaven" from Crow, the eponymous character is left in a state of catalepsy.

In the old time radio show Suspense theater, episode titled "Dead Ernest." Episodes number 205, 1946, number 244, 1947. Catatonia Mood disorder#Depressive disorders

Margery Palmer McCulloch

Margery Palmer McCulloch was a Scottish literary scholar, author and co-editor of the Scottish Literary Review. Dr. Margery Palmer McCulloch was educated at the former Hamilton Academy, she was an elected member of Council of the Association for Scottish Literary Studies and served as Convener of the Association's Publications Board, as well as being co-editor of the Scottish Literary Review. A past Convener of the Society’s Glasgow Branch, McCulloch was a former Honorary Secretary of the Saltire Society, she contributed to literary and cultural programmes on BBC Radio and articles to newspapers and journals, including The Herald, The Scotsman and The Times Literary Supplement. In 1999, Dr McCulloch discovered a poem by Hugh MacDiarmid, lost for 60 years. News of the poem's discovery was welcomed in the literary world. Dr McCulloch made the discovery whilst researching the archives of Catherine Carswell, a novelist and critic. McCulloch died in Glasgow on 29 October 2019, aged 84. McCulloch has an extensive list of published books and articles on Scottish literature and writers, her published works including: The Novels of Neil M. Gunn: a critical study Edwin Muir: poet and novelist The Man Who Came Back: Essays and Short Stories by Neil M. Gunn Liz Lochhead’s Mary Queen of Scots Got Her Head Chopped Off A Flame in the Mearns: Lewis Grassic Gibbon: a centenary celebration Modernism and Nationalism: Literature and Society in Scotland, 1918-1939 Scottish Modernism and its Contexts 1918–1959: Literature, National Identity and Cultural Exchange


WHTV, virtual channel 18, was a television station licensed to Jackson, United States, which served the Central Lower Peninsula of Michigan, including the capital city of Lansing. The station was owned by Venture Technologies Group. WHTV's studios were located on West Saint Joseph Street in downtown Lansing, its transmitter was located on M-52 in Lyndon Township, Washtenaw County. Through its history, WHTV was affiliated with UPN, followed by MyNetworkTV, had a secondary affiliation with Jewelry TV that became primary several times. WHTV sold its spectrum in the Federal Communications Commission's incentive auction and ceased operations at 11:59 p.m. on August 31, 2017. The station signed on August 20, 1999 with programming from Bloomberg Television and Jewelry Television, it aired an analog signal on UHF channel 18 from a transmitter near Onondaga on the Ingham and Jackson county line. It became a UPN affiliate on October 16, 2000. From 2002 until 2006, WHTV's internal operations were housed at the studios of ABC affiliate WLAJ, owned by Freedom Communications, on South Pennsylvania Avenue in Lansing.

It relocated to the facilities of CBS outlet WLNS-TV on East Saginaw Street after entering into a joint sales agreement with WLNS' then-owner Young Broadcasting. While managed by Young, WHTV carried CBS programming preempted by WLNS, including the CBS Sports feed of the US Open Tennis Championship while WLNS aired the Jerry Lewis MDA Telethon on Labor Day, as well as programs preempted by WLNS in the event of a local special or breaking news. WHTV occasionally carried Saturday NCAA football games from ABC when they conflicted with WLAJ's commitment to broadcast the ESPN Plus Big Ten football and basketball packages. WHTV would place its digital transmitter at Van Atta Road in Meridian Charter Township; the station had a construction permit to increase power and relocate its transmitter to a tower used by Detroit's Ion Television owned-and-operated station WPXD-TV in Lyndon Township. The new transmitter would be located in the Detroit market but the station would continue to serve as a Lansing–Jackson market television outlet.

The planned move date of its signal to the new antenna was scheduled to occur on November 1, 2012, but the parts delivery for the new transmitter was delayed until December 20. On December 4, 2012, WLAJ was sold from the Sinclair Broadcast Group to Shield Media, LLC. Shield entered into certain shared services and joint sales agreements with Young Broadcasting. At some point in February 2013, WLAJ moved from its studios into the WLNS facility. In effect, this move reunited WHTV's intellectual unit with WLAJ, due to WHTV's existing relationship with WLNS and its prior partnership with WLAJ. Young Broadcasting would merge with Media General on November 12, 2013; as a result of these changes, WHTV announced that it would not renew its operational outsourcing agreement with WLNS. In July 2014, it entered into a new local marketing agreement with WSYM and relocated its advertising sales operation to the Fox affiliate's studios. In April 2017, WHTV announced that it would shut down on April 30, 2017 revised to May 17, 2017 May 31, 2017, August 31, 2017, following the Federal Communications Commission's incentive auction.

The station sold its spectrum for $13,906,280. On May 1, 2017, WHTV dropped the MyNetworkTV affiliation and Scripps LMA with WSYM-TV and elected to air Jewelry Television in the interim discontinuing the 18.2 subchannel. After the final revision to their off-air date, the station signed-off for the last time several minutes before midnight on August 31. WHTV's license was cancelled at the station's request on September 11, 2017; the programming which aired on WHTV, including MyNetworkTV, remained in limbo for several months. On September 18, 2017, it was announced that WSYM would launch a fourth subchannel on October 9, containing both a MyNetworkTV affiliation and much of WHTV's previous programming. WHTV shut down its analog signal, over UHF channel 18, on December 1, 2008; the station's digital signal remained on its pre-transition UHF channel 34, using PSIP to display WHTV's virtual channel as 18 on digital television receivers. Syndicated programming on WHTV until May 1, 2017 included Two and a Half Men, The Middle, Jerry Springer, Divorce Court among others.

The only remaining non-Jewelry Television content until August 31, 2017 was E/I programming on weekday mornings. Query the FCC's TV station database for WHTV

5 Broken Cameras

5 Broken Cameras is a 94-minute documentary film co-directed by Palestinian Emad Burnat and Israeli Guy Davidi. It was shown at film festivals in 2011 and placed in general release by Kino Lorber in 2012. 5 Broken Cameras is a first-hand account of protests in Bil'in, a West Bank village affected by the Israeli West Bank barrier. The documentary was shot entirely by Palestinian farmer Emad Burnat, who bought his first camera in 2005 to record the birth of his youngest son. In 2009 Israeli co-director Guy Davidi joined the project. Structured around the destruction of Burnat's cameras, the filmmakers' collaboration follows one family's evolution over five years of turmoil; the film won a 2012 Sundance Film Festival award, it won the Golden Apricot at the 2012 Yerevan International Film Festival, for Best Documentary Film, won the 2013 International Emmy Award, was nominated for a 2013 Academy Award. There are five cameras — each with its own story; when his fourth son, Gibreel, is born in 2005, self-taught cameraman Emad Burnat, a Palestinian villager, gets his first camera.

At the same time in his village of Bil’in, the Israelis begin bulldozing village olive groves to build a barrier to separate Bil'in from the Jewish Settlement Modi'in Illit. The barrier's route cuts off 60% of Bil'in farmland and the villagers resist this seizure of more of their land by the settlers. During the next year, Burnat films this struggle, led by two of his best friends including his brother Iyad, while at the same time recording the growth of his son. Soon, these events begin to affect his family and his own life. Emad films the Army and Police beating and arresting villagers and activists who come to support them. Settlers attack Burnat when he tries to film them; the Army raids the village in the middle of the night to arrest children. He, his friends, brothers are arrested or shot; each camera used to document these events is smashed. In 2009, Burnat approaches Guy Davidi – an Israeli filmmaker and together, from these five broken cameras and the stories that they represent, these two filmmakers create the film.

Israel began construction of an Israeli West Bank barrier in the West Bank village of Bil’in, Palestine in 2005. Discovering that the wall would cut through their agricultural land, confiscating half of it, the villagers initiated popular protests and were joined by Israeli and international activists. At that point Burnat received a camera to document the movement. In 2007 the Israeli Supreme Court ordered the barrier rerouted, four years after village access to some of the land was restored, the demonstrations were called off. A case against Canada, for failing to prevent Canadian corporations from being complicit in the building of the settlements, is pending before the UN Human Rights Committee; the first year, Burnat filmed to serve the purposes of activists. His footage was introduced as evidence in Israeli court and posted on YouTube to spread awareness of the growing movement; as media interest in Bil’in grew, Burnat's footage gained international recognition and was used by local and international news agencies.

He started working as a freelance photographer for Reuters and provided footage documenting the villagers' fight to professional filmmakers. This footage was used in such notable films as Shai Carmeli Pollac's Bil’in, My Love and Guy Davidi's and Alesandre Goetschmann Interrupted Streams. Burnat was approached in 2009 by Greenhouse, a Mediterranean film development project, to develop a documentary; the project focused on the non-violent movement and on Bassem Abu-Rahme, killed earlier that year at a demonstration in Bil’in. After some difficulties, Burnat approached Israeli filmmaker, Guy Davidi who had just finished editing “Interrupted Streams”, Davidi's first feature documentary, released in 2010 at the Jerusalem International Film Festival. Earlier, Davidi had been involved in the left-wing organizations Indymedia and Anarchists Against the Wall. “Until my twenties,” Davidi has said in an interview, “it was hard for me to work in Israel. I felt it was a destructive environment, a violent environment....

There is a lot of aggression expressed towards the arts in Israel. I connect it with the political situation.... So I left for Paris and I found time to reflect on my life.... I kind of found a freedom in Paris and I wanted to express it as well in Israel, and since my life was connected with the West Bank.”Davidi provided Burnat's film with a new concept: Burnat himself, the cameraman, would be the protagonist, the story would be told from his point of view. Davidi proposed that the film be structured around the history of the destruction of Burnat's cameras. Footage that Burnat shot of his family was incorporated into the film, thus enhancing the personal element. Beginning in 2009, adhering to the new concept for the film, focused more extensively on his family's reactions to events. A few important scenes shot by other cameramen were used to supplement the narrative, to introduce Burnat as a character. Starting in 2009, Davidi structuring the film. In 2011 French editor Véronique Lagoarde–Ségot joined the project to edit the final cut of the 90-minute film and to create the 52-minute television version.

The film takes the form of a diary, is divided into 5 sections, each of which recounts the story of one of the five cameras that Burnat used over the years. In a prologue, Burnat is shown with his 5 broken cameras laid out on a table; this scene is returned to at the end of the film. Title

Physics outreach

Physics outreach encompasses facets of science outreach and physics education, a variety of activities by schools, research institutes, universities and institutions such as science museums aimed at broadening the audience for and awareness and understanding of physics. While the general public may sometimes be the focus of such activities, physics outreach centers on developing and providing resources and making presentations to students, educators in other disciplines, in some cases researchers within different areas of physics. Ongoing efforts to expand the understanding of physics to a wider audience have been undertaken by individuals and institutions since the early 19th century. Historic works, such as the Dialogue Concerning the Two Chief World Systems, Two New Sciences by Galileo Galilei, sought to present revolutionary knowledge in astronomy, frames of reference, kinematics in a manner that a general audience could understand with great effect. In the mid 1800s, English physicist and chemist, Michael Faraday gave a series of nineteen lectures aimed towards young adults with the hopes of conveying scientific phenomena.

His intentions were to inspire them and generate revenue of the Royal Institution. This series became known as the Christmas lectures, still continues today. By the early 20th century, the public notoriety of physicists such as Albert Einstein and Marie Curie, inventions such as radio led to a growing interest in physics. In 1921, in the United States, the establishment of Sigma Pi Sigma physics honor society at universities was instrumental in the expanding number of physics presentations, led to the creation of physics clubs open to all students. Museums were an important form of outreach but most early science museums were focused on natural history; some specialized museums, such as the Cavendish Museum at University of Cambridge, housed many of the important pieces of apparatus that contributed to the major discoveries by Maxwell, Rutherford, etc. However, such venues provided little opportunity for hands-on learning or demonstrations. In August 1969, Frank Oppenheimer dedicated his new Exploratorium in San Francisco to interactive science exhibits that demonstrated principles in physics.

The Exploratorium published the details of their own exhibits in "Cookbooks" that served as an inspiration to many other museums around the world, since has diversified into many outreach programs. Oppenheimer had researched European science museums while on a Guggenheim Fellowship in 1965, he noted that three museums served as important influences on the Exploratorium: the Palais de la Découverte, which displayed models to teach scientific concepts and employed students as demonstrators, a practice that directly inspired the Exploratorium's much-lauded High School Explainer Program. In the ensuing years, physics outreach, science outreach more continued to expand and took on new popular forms, including successful television shows such as Cosmos: A Personal Voyage, first broadcast in 1980; as a form of outreach within the physics education community for teachers and students, in 1997 the US National Science Foundation and Department of Energy USDOE established QuarkNet, a professional teacher development program.

In 2012, the University of Notre Dame received a $6.1M, five-year grant to support a nationwide expansion of the Quarknet program. In 1997, the European Particle Physics Outreach Group, led by Christopher Llewellyn Smith, FRS, Director General of CERN, was formed to create a community of scientists, science educators, communication specialists in science education and public outreach for particle physics; this group became the International Particle Physics Outreach Group in 2011 after the start up of the LHC. Many contemporary initiatives in physics outreach have begun to shift focus, transcending traditional field boundaries, seeking to engage students and the public by integrating elements of aesthetic design and popular culture; the goal has been not only to push physics out of a science education framework but to draw in professionals and students from other fields to bring their perspectives on physical phenomena. Such work includes artists creating sculptures using ferrofluids, art photography using high speed and ultra high speed photography.

Other efforts, such as University of Cambridge's Physics at Work program have created annual events to demonstrate to secondary students uses of physics in everyday life and a Senior Physics Challenge. Seeing the importance these initiatives, Cambridge has established a full-time physics outreach organization, an Educational Outreach Office, aspirations for a Center of Physics and expanded industrial partnerships that "would include a well equipped core team of outreach officers dedicated to demonstrating the real life applications of physics, showing that physics is an accessible and relevant subject"; the French research group, La Physique Autrement, of the Laboratoire de Physique des Solides, works on research about new ways to present modern solid-state physics and to engage the general public. In 2013, Physics Today covered this group in an article entitled "Quantum Physics For Everyone" which discussed how with the help of designers and unconventional demonstrations, the project sought out and succeeded to engage people who never thought of themselves as interested in science.

The Science & Entertainment Exchange was developed by the United States National Academy of Sciences