title: We Have Always Lived in the Castle

text: Six years prior to the story, Constance and Merricat's parents John and Ellen, their aunt Dorothy, and their younger brother Thomas died after being poisoned with arsenic, which was mixed into the family's sugar bowl and sprinkled onto blackberries at dinner. Julian was also poisoned, but survived; Merricat was not present at the time as she had been sent to bed without dinner as punishment. Constance, the only member of the family who didn't put sugar on her berries, was arrested and charged with murder, but ultimately the verdict proved her not guilty. The people of the village believe that Constance got away with murder, leading them to exclude the family.


title: We Have Always Lived in the Castle

text: The three remaining Blackwoods have since grown accustomed to their isolation, leading a quiet, happy existence. Merricat is the family's sole contact with the outside world, walking into the village twice a week and carrying home groceries and library books; on these trips, she is faced directly with the hostility of the villagers, and often taunted by groups of children with an accusing rhyme about the poisoned sugar. Merricat is protective of her sister and practices sympathetic magic that maintains borders around the house.


title: We Have Always Lived in the Castle

text: Merricat feels that a dangerous change is approaching, but before she can warn Constance, their estranged cousin Charles appears for a visit and is welcomed into the home. Charles quickly begins to have a close relationship with Constance and gains her confidence. Charles is aware of Merricat's hostility and is increasingly rude to her and impatient of Julian's weaknesses. He makes many references to the money the sisters keep locked in their father's safe, and gradually forms something of an alliance with Constance, encouraging her to leave her home. Merricat perceives Charles as a threat and tries various magical and otherwise disruptive means to drive him from the house. Uncle Julian is increasingly disgusted by Charles and suspects that Charles came there for the Blackwoods' fortune.


title: We Have Always Lived in the Castle

text: One night before dinner, when Constance sends Merricat upstairs to wash her hands, Merricat, in a fit of anger, pushes Charles' smoking pipe into a wastebasket filled with newspapers. This soon causes a massive fire that consumes the family home. The villagers arrive and help put out the fire, but then finally unleash their long-repressed hostility toward the Blackwoods by vandalizing and ransacking the house. Driven outdoors, Merricat and Constance flee into the woods after being threatened by the villagers, while Julian dies of an apparent heart failure during the fire and Charles attempts to take the family safe. While Merricat and Constance shelter for the night under a tree Merricat has made into a hideaway, Constance confesses that she always knew that Merricat was the one who poisoned the family. Merricat readily admits to the deed, saying that she put the poison in the sugar bowl because she knew that Constance would not take sugar.


title: We Have Always Lived in the Castle

text: Upon returning to their ruined home, Constance and Merricat proceed to salvage what is left of their belongings, close off the rooms too damaged to use, and start their lives anew in the little space left to them. The house, now without a roof, resembles a castle "turreted and open to the sky." Constance and Merricat spend much of their time watching the outside world through peepholes hidden by vines that grow to cover the house. The villagers, feeling remorse at their actions, begin to leave food on their doorstep, while developing stories about the house akin to folklore. Charles returns once to try to renew his acquaintance with Constance, but she ignores him. The sisters choose to remain alone and unseen by the rest of the world.


title: We Have Always Lived in the Castle

text: The theme of persecution of people who exhibit "otherness" or become outsiders in small-town New England, by small-minded villagers, is at the forefront of "We Have Always Lived in the Castle" and is a repeated theme in Jackson's work. In her novels "The Haunting of Hill House" and, to a lesser extent, "The Sundial", this theme is also central to the psychology of the story. In all these works, the main characters live in a house that stands alone on many acres, and is entirely separate physically, socially, as well as ideologically, from the main inhabitants of the town. In his 2006 introduction of the Penguin Classics edition, Jonathan Lethem stated that the recurring town is "pretty well recognizable as North Bennington, Vermont", where Jackson and her husband, Bennington professor Stanley Edgar Hyman, encountered strong "reflexive anti-Semitism and anti-intellectualism".


title: We Have Always Lived in the Castle

text: All of Jackson's work creates an atmosphere of strangeness and contact with what Lethem calls "a vast intimacy with everyday evil..." and how that intimacy affects "a village, a family, a self". Only in "We Have Always Lived in the Castle", though, is there also a deep exploration of love and devotion despite the pervasive unease and perversity of character that runs through the story. Constance's complete absence of judgment of her sister and her crime is treated as absolutely normal and unremarkable, and it is clear throughout the story that Merricat loves and cares deeply for her sister.


title: We Have Always Lived in the Castle

text: The novel was described by Jackson's biographer, Judy Oppenheimer, as "a paean to agoraphobia", with the author's own agoraphobia and nervous conditions having greatly informed its psychology. Jackson freely admitted that the two young women in the story were liberally fictionalized versions of her own daughters, and Oppenheimer noted that Merricat and Constance were the "yin and yang of Shirley's own inner self". Written in deceptively simple language, by an entirely unreliable narrator, the novel implies that the two heroines may choose to live forever in the remaining three rooms of their house, since they prefer each other's company to that of any outsiders. Lethem calls this reversion to their pre-Charles stasis Merricat's "triumph".


title: We Have Always Lived in the Castle

text: "We Have Always Lived in the Castle" was named by "Time" magazine as one of the "Ten Best Novels" of 1962.


title: We Have Always Lived in the Castle

text: In March 2002, "Book" magazine named Mary Katherine Blackwood the seventy-first "best character in fiction since 1900". On Goodreads, the novel ranks #2 on the list of "Most Popular Books Published in 1962", as voted for by the website's users.


title: We Have Always Lived in the Castle

text: In 1966, the novel was adapted into a stage play by Hugh Wheeler, starring Heather Menzies, Shirley Knight, and Alan Webb. The play premiered at the Ethel Barrymore Theatre on Broadway on October 19, and closed after 11 performances on October 26. In 2010, Adam Bock and Todd Almond staged a musical version at the Yale Repertory Theatre in New Haven, Connecticut, which ran from September 23 to October 9.


title: We Have Always Lived in the Castle

text: In August 2009, the novel was optioned for the screen by Michael Douglas' production company Further Films, from a script written by Mark Kruger, with the support of Jackson's son Laurence Hyman. The film adaptation, from Further Films and Albyn Media, was filmed in Bray and Dublin, Ireland from August to September 2016. Directed by Stacie Passon, it stars Sebastian Stan as Charles, Taissa Farmiga as Merricat, Alexandra Daddario as Constance, and Crispin Glover as Uncle Julian. The film premiered at the LA Film Festival on September 22, 2018, and on May 17, 2019, was released by Brainstorm Media.


title: Nefertari

text: Nefertari, also known as Nefertari Meritmut, was an Egyptian queen and the first of the Great Royal Wives (or principal wives) of Ramesses the Great. She is one of the best known Egyptian queens, among such women as Cleopatra, Nefertiti, and Hatshepsut, and one of the most prominent not known or thought to have reigned in her own right. She was highly educated and able to both read and write hieroglyphs, a very rare skill at the time. She used these skills in her diplomatic work, corresponding with other prominent royals of the time. Her lavishly decorated tomb, QV66, is one of the largest and most spectacular in the Valley of the Queens. Ramesses also constructed a temple for her at Abu Simbel next to his colossal monument there.


title: Nefertari

text: There are different interpretations of the meaning of the name Nefertari. Nefertari means 'beautiful companion' and Meritmut means 'Beloved of the goddess Mut'. Some sources consider a more accurate translation for Nefertari as "the most beautiful one", "the most beautiful of them", "the most beautiful one of them all" "the most beautiful (one) among them", "the very best", or "the most beautiful of the women".


title: Nefertari

text: Nefertari held many titles, including: Great of Praises (wrt-hzwt), Sweet of Love (bnrt-mrwt), Lady of Grace (nbt-im3t), Great King's Wife (hmt-niswt-wrt), his beloved (hmt-niswt-wrt meryt.f), Lady of The Two Lands (nbt-t3wy), Lady of all Lands (hnwt-t3w-nbw), Wife of the Strong Bull (hmt-k3-nxt), god's Wife (hmt-ntr), Mistress of Upper and Lower Egypt (hnwt-Shm’w-mhw). Ramesses II also named her 'The one for whom the sun shines'.


title: Nefertari

text: Although Nefertari's family background is unknown, the discovery in her tomb of a knob inscribed with the cartouche of Pharaoh Ay has led people to speculate she was related to him. The time between the reign of Ay and Ramesses II means that Nefertari could not be a daughter of Ay and if any relation exists at all, she would be a great-granddaughter. There is no conclusive evidence linking Nefertari to the royal family of the 18th Dynasty, however.


title: Nefertari

text: Nefertari married Ramesses II before he ascended the throne. Nefertari had at least four sons and two daughters. Amun-her-khepeshef, the eldest, was Crown Prince and Commander of the Troops, and Pareherwenemef would later serve in Ramesses II's army. Prince Meryatum was elevated to the position of High Priest of Re in Heliopolis. Inscriptions mention he was a son of Nefertari. Prince Meryre is a fourth son mentioned on the façade of the small temple at Abu Simbel and is thought to be another son of Nefertari. Meritamen and Henuttawy are two royal daughters depicted on the façade of the small temple at Abu Simbel and are thought to be daughters of Nefertari.


title: Nefertari

text: Princesses named Bak(et)mut, Nefertari, and Nebettawy are sometimes suggested as further daughters of Nefertari based on their presence in Abu Simbel, but there is no concrete evidence for this supposed family relation.


title: Nefertari

text: Nefertari first appears as the wife of Ramesses II in official scenes during the first year of Ramesses II. In the tomb of Nebwenenef, Nefertari is depicted behind her husband as he elevates Nebwenenef to the position of High Priests of Amun during a visit to Abydos. Nefertari also appears in a scene next to a year 1 stela. She is depicted shaking two sistra before Taweret, Thoth, and Nut.


title: Nefertari

text: Nefertari is an important presence in the scenes from Luxor and Karnak. In a scene from Luxor, Nefertari appears leading the royal children. Another scene shows Nefertari at the Festival of the Mast of Amun-Min-Kamephis. The king and the queen are said to worship in the new temple and are shown overseeing the Erection of the Mast before Amen-Re attended by standard bearers. Nefertari's speech during this ceremony is recorded:


title: Nefertari

text: Your beloved son, the Lord of Both Lands, Usermaatre Setepenre, has come to see you in your beautiful manifestation. He has erected for you the mast of the (pavilion)-framework. May you grant him eternity as King, and victory over those rebellious (against) His Majesty, L.P.H.


title: Nefertari

text: Nefertari appears as Ramesses II's consort on many statues in both Luxor and Karnak. In Western Thebes, Nefertari is mentioned on a statuary group from Deir el-Bahari, a stela and blocks from Deir el-Medina.


title: Nefertari

text: The greatest honor was bestowed on Nefertari however in Abu Simbel. Nefertari is depicted in statue form at the great temple, but the small temple is dedicated to Nefertari and the goddess Hathor. The building project was started earlier in the reign of Ramesses II, and seems to have been inaugurated by ca year 25 of his reign (but not completed until ten years later).


title: Nefertari

text: Nefertari's prominence at court is further supported by cuneiform tablets from the Hittite city of Hattusas (today Boghazkoy, Turkey), containing Nefertari's correspondence with the king Hattusili III and his wife Puduhepa. She is mentioned in the letters as "Naptera." Nefertari is known to have sent gifts to Puduhepa:


title: Nefertari

text: The great Queen Naptera of the land of Egypt speaks thus: Speak to my sister Puduhepa, the Great Queen of the Hatti land. I, your sister, (also) be well!! May your country be well. Now, I have learned that you, my sister, have written to me asking after my health. ... You have written to me because of the good friendship and brotherly relationship between your brother, the king of Egypt, The Great and the Storm god will bring about peace, and he will make the brotherly relationship between the Egptian king, the Great King, and his brother, the Hatti King, the Great King, last for ever... See, I have sent you a gift, in order to greet you, my sister... for your neck (a necklace) of pure gold, composed of 12 bands and weighing 88 shekels, coloured linen maklalu-material, for one royal dress for the king... A total of 12 linen garments.


title: Nefertari

text: Nefertari is shown at the inaugural festivities at Abu Simbel in year 24. Her daughter Meritamen is depicted taking part in place of her mother in some of the scenes. Nefertari may well have been in failing health at this point. After her death she was buried in tomb QV66 in the Valley of the Queens.


title: Nefertari

text: Nefertari appears twice as one of the royal women represented beside the colossal statues of Ramesses II that stand before the temple. To the left of the doorway, Nefertari, Queen-Mother Tuya and the king's son Amun-her-khepeshef (still called Amunhirwenemef here) flank the colossal statue of the king. To the right of the doorway Nefertari, Baketmut and the king's son Ramesses are shown with the Pharaoh.


title: Nefertari

text: Inside the temple Nefertari is depicted on one of the pillars in the great pillared hall worshipping Hathor of Ibshek.


title: Nefertari

text: On the wall of the inner pillared hall Nefertari appears behind Ramesses II. They stand before the barque of Amun, and Nefertari is shown playing the sistra. Elsewhere Nefertari and Ramesses II are shown before a barque dedicated to a deified Ramesses II. Nefertari is shown twice accompanying her husband in Triumph scenes.


title: Nefertari

text: The small temple at Abu Simbel was dedicated to Nefertari and Hathor of Ibshek. The dedication text on one of the buttresses states:


title: Nefertari

text: The two colossal standing statues of Nefertari in front of the small temple are equal in size to those of Ramesses II. Nefertari is shown holding a sistrum. She wears a long sheet dress and she is depicted with a long wig, Hathoric cow horns, the solar disk, and tall feathers mounted on a modius.


title: Nefertari

text: In the interior of the temple, Nefertari appears in a variety of scenes. She is shown for instance offering to a cow (Hathor) in a papyrus thicket, offering before Khnum, Satis, and Anuket, the triad of Elephantine, and offering to Mut and Hathor.


title: Nefertari

text: The tomb of Nefertari, QV66 is one of the largest in the Valley of the Queens. It is 520 square meters, and covered with pictures of Nefertari. Her husband, the pharaoh, is not represented in any of the pictures. Nefertari can be seen wearing Greek silver earrings with a labrys design in one of the portraits. These would have been sent to her as a gift for diplomatic reasons. The tomb was robbed in antiquity. In 1904 it was rediscovered and excavated by Ernesto Schiaparelli. Several items from the tomb, including parts of gold bracelets, shabti figures and a small piece of an earring or pendant are now in the Boston Museum of Fine Arts. Additional shabti figures are in the Egyptian Museum in Cairo.


title: Nefertari

text: It was reported that a pair of mummified legs found in QV66 and now at the Museo Egizio of Turin may indeed be Nefertari's based on the bone structure and the age of the person, which fits the profile of Nefertari.


title: Bra–ket notation

text: In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".


title: Bra–ket notation

text: A ket is of the form formula_1. Mathematically it denotes a vector, formula_2, in an abstract (complex) vector space formula_3, and physically it represents a state of some quantum system.


title: Bra–ket notation

text: A bra is of the form formula_4. Mathematically it denotes a linear form formula_5, i.e. a linear map that maps each vector in formula_3 to a number in the complex plane formula_7. Letting the linear functional formula_8 act on a vector formula_9 is written as formula_10.


title: Bra–ket notation

text: Assume that on formula_3 there exists an inner product formula_12 with antilinear first argument, which makes formula_3 an inner product space. Then with this inner product each vector formula_14 can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: formula_15. The correspondence between these notations is then formula_16. The linear form formula_17 is a covector to formula_18, and the set of all covectors form a subspace of the dual vector space formula_19, to the initial vector space formula_3. The purpose of this linear form formula_17 can now be understood in terms of making projections on the state formula_22, to find how linearly dependent two states are, etc.


title: Bra–ket notation

text: For the vector space formula_23, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication. If formula_23 has the standard Hermitian inner product formula_25, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted formula_26).


title: Bra–ket notation

text: It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator formula_27 on a two dimensional space formula_28 of spinors, has eigenvalues formula_29 with eigenspinors formula_30. In bra–ket notation, this is typically denoted as formula_31, and formula_32. As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors.


title: Bra–ket notation

text: Bra–ket notation was effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of formula_33 for inner products nearly 100 years earlier.


title: Bra–ket notation

text: Bra–ket notation is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread. Many phenomena that are explained using quantum mechanics are explained using bra–ket notation.


title: Bra–ket notation

text: In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" is much more specific: "vector" refers almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime. Such vectors are typically denoted with over arrows (formula_34), boldface (formula_35) or indices (formula_36).


title: Bra–ket notation

text: In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. Since the term "vector" is already used for something else (see previous paragraph), and physicists tend to prefer conventional notation to stating what space something is an element of, it is common and useful to denote an element formula_37 of an abstract complex vector space as a ket formula_18 using vertical bars and angular brackets and refer to them as "kets" rather than as vectors and pronounced "ket-formula_37" or "ket-A" for .


title: Bra–ket notation

text: Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the formula_40 making clear that the label indicates a vector in vector space. In other words, the symbol "" has a specific and universal mathematical meaning, while just the "" by itself does not. For example, is not necessarily equal to . Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as formula_41, formula_42, formula_43, etc.


title: Bra–ket notation

text: Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:


title: Bra–ket notation

text: Since the ket is an element of a vector space, a bra formula_45 is an element of its dual space, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts.


title: Bra–ket notation

text: A bra formula_17 and a ket formula_47 (i.e. a functional and a vector), can be combined to an operator formula_48 of rank one with outer product


title: Bra–ket notation

text: The bra–ket notation is particularly useful in Hilbert spaces which have an inner product that allows Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem). The inner product on Hilbert space formula_50 (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in the bra ket notation: for a vector ket formula_51 define a functional (i.e. bra) formula_52 by


title: Bra–ket notation

text: In the simple case where we consider the vector space formula_23, a ket can be identified with a column vector, and a bra as a row vector. If moreover we use the standard Hermitian inner product on formula_23, the bra corresponding to a ket, in particular a bra and a ket with the same label are conjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply matrix multiplication. In particular the outer product formula_56 of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix).


title: Bra–ket notation

text: For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:


title: Bra–ket notation

text: The conjugate transpose (also called "Hermitian conjugate") of a bra is the corresponding ket and vice versa:


title: Bra–ket notation

text: Writing elements of a finite dimensional (or mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a basis. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "" and "".


title: Bra–ket notation

text: In quantum mechanics, it is common practice to write down kets which have infinite norm, i.e. non-normalizable wavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves. These do not, technically, belong to the Hilbert space itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). The bra–ket notation continues to work in an analogous way in this broader context.


title: Bra–ket notation

text: Banach spaces are a different generalization of Hilbert spaces. In a Banach space , the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.


title: Bra–ket notation

text: Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow:


title: Bra–ket notation

text: The Hilbert space of a spin-0 point particle is spanned by a "position basis" , where the label extends over the set of all points in position space. This label is the eigenvalue of the position operator acting on such a basis state, formula_62. Since there are an uncountably infinite number of vector components in the basis, this is an uncountably infinite-dimensional Hilbert space. The dimensions of the Hilbert space (usually infinite) and position space (usually 1, 2 or 3) are not to be conflated.


title: Bra–ket notation

text: Starting from any ket in this Hilbert space, one may "define" a complex scalar function of , known as a wavefunction,


title: Bra–ket notation

text: On the left-hand side, is a function mapping any point in space to a complex number; on the right-hand side, is a ket consisting of a superposition of kets with relative coefficients specified by that function.


title: Bra–ket notation

text: It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by


title: Bra–ket notation

text: One occasionally even encounters a expressions such as formula_67, though this is something of an abuse of notation. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, formula_68


title: Bra–ket notation

text: even though, in the momentum basis, this operator amounts to a mere multiplication operator (by ). That is, to say,


title: Bra–ket notation

text: In quantum mechanics the expression is typically interpreted as the probability amplitude for the state to collapse into the state . Mathematically, this means the coefficient for the projection of onto . It is also described as the projection of state onto state .


title: Bra–ket notation

text: where is the state with a definite value of the spin operator equal to + and is the state with a definite value of the spin operator equal to −.


title: Bra–ket notation

text: Since these are a basis, "any" quantum state of the particle can be expressed as a linear combination (i.e., quantum superposition) of these two states:


title: Bra–ket notation

text: depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.


title: Bra–ket notation

text: There is a mathematical relationship between formula_76, formula_77, formula_78 and formula_79; see change of basis.


title: Bra–ket notation

text: There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.


title: Bra–ket notation

text: A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as formula_80, and formula_12 for the inner product. Consider the following dual space bra-vector in the basis formula_82:


title: Bra–ket notation

text: It has to be determined by convention if the complex numbers formula_84 are inside or outside of the inner product, and each convention gives different results.


title: Bra–ket notation

text: It is common to use the same symbol for "labels" and "constants". For example, formula_87, where the symbol formula_88 is used simultaneously as the "name of the operator" formula_89, its "eigenvector" formula_90 and the associated "eigenvalue" formula_88. Sometimes the "hat" is also dropped for operators, and one can see notation such as formula_92


title: Bra–ket notation

text: It is common to see the usage formula_93, where the dagger (formula_94) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, formula_95, represents a vector in a complex Hilbert-space formula_96, and the bra, formula_97, is a linear functional on vectors in formula_96. In other words, formula_95 is just a vector, while formula_97 is the combination of a vector and an inner product.


title: Bra–ket notation

text: This is done for a fast notation of scaling vectors. For instance, if the vector formula_101 is scaled by formula_102, it may be denoted formula_103. This can be ambiguous since formula_88 is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. formula_105.


title: Bra–ket notation

text: A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have certain properties.) In other words, if formula_106 is a linear operator and formula_95 is a ket-vector, then formula_108 is another ket-vector.


title: Bra–ket notation

text: In an formula_109-dimensional Hilbert space, we can impose a basis on the space and represent formula_95 in terms of its coordinates as a formula_111 column vector. Using the same basis for formula_106, it is represented by an formula_113 complex matrix. The ket-vector formula_108 can now be computed by matrix multiplication.


title: Bra–ket notation

text: Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.


title: Bra–ket notation

text: Operators can also be viewed as acting on bras "from the right hand side". Specifically, if is a linear operator and is a bra, then is another bra defined by the rule


title: Bra–ket notation

text: In an -dimensional Hilbert space, can be written as a row vector, and (as in the previous section) is an matrix. Then the bra can be computed by normal matrix multiplication.


title: Bra–ket notation

text: then this expression gives the expectation value, or mean or average value, of the observable represented by operator for the physical system in the state .


title: Bra–ket notation

text: A convenient way to define linear operators on a Hilbert space is given by the outer product: if is a bra and is a ket, the outer product


title: Bra–ket notation

text: For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:


title: Bra–ket notation

text: One of the uses of the outer product is to construct projection operators. Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is


title: Bra–ket notation

text: Just as kets and bras can be transformed into each other (making into ), the element from the dual space corresponding to is , where denotes the Hermitian conjugate (or adjoint) of the operator . In other words,


title: Bra–ket notation

text: Self-adjoint operators, where , play an important role in quantum mechanics; for example, an observable is always described by a self-adjoint operator. If is a self-adjoint operator, then is always a real number (not complex). This implies that expectation values of observables are real.


title: Bra–ket notation

text: Bra–ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, and denote arbitrary complex numbers, denotes the complex conjugate of , and denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.


title: Bra–ket notation

text: Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the associative property holds). For example:


title: Bra–ket notation

text: and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously "because" of the equalities on the left. Note that the associative property does "not" hold for expressions that include nonlinear operators, such as the antilinear time reversal operator in physics.


title: Bra–ket notation

text: Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called "dagger", and denoted ) of expressions. The formal rules are:


title: Bra–ket notation

text: These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows:


title: Bra–ket notation

text: Two Hilbert spaces and may form a third space by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in and respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.)


title: Bra–ket notation

text: If is a ket in and is a ket in , the direct product of the two kets is a ket in . This is written in various notations:


title: Bra–ket notation

text: In quantum mechanics, it often occurs that little or no information about the inner product of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients and of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more.


title: Bra–ket notation

text: In his book (1958), Ch. III.20, Dirac defines the "standard ket" which, up to a normalization, is the translationally invariant momentum eigenstate formula_141 in the momentum representation, i.e., formula_142. Consequently, the corresponding wavefunction is a constant, formula_143, and


title: Bra–ket notation

text: Typically, when all matrix elements of an operator such as formula_146 are available, this resolution serves to reconstitute the full operator,


title: Bra–ket notation

text: The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space).


title: Bra–ket notation

text: Let formula_148 be a Hilbert space and a vector in . What physicists would denote by is the vector itself. That is,


title: Bra–ket notation

text: Let be the dual space of . This is the space of linear functionals on . The embedding formula_150 is defined by formula_151, where for every the linear functional formula_152 satisfies for every the functional equation formula_153.


title: Bra–ket notation

text: Notational confusion arises when identifying and with and respectively. This is because of literal symbolic substitutions. Let formula_154 and let . This gives


title: Bra–ket notation

text: Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an asterisk but an overline (which the physicists reserve for averages and the Dirac spinor adjoint) to denote complex conjugate numbers; i.e., for scalar products mathematicians usually write


title: Murder of Lee Rigby

text: On the afternoon of 22 May 2013, a British Army soldier, Fusilier Lee Rigby of the Royal Regiment of Fusiliers, was attacked and killed by Michael Adebolajo and Michael Adebowale near the Royal Artillery Barracks in Woolwich, southeast London.


title: Murder of Lee Rigby

text: Rigby was off duty and walking along Wellington Street when he was attacked. Adebolajo and Adebowale ran him down with a car, then used knives and a cleaver to stab and hack him to death. The men dragged Rigby's body into the road and remained at the scene until police arrived, informing passers-by that they had murdered Rigby to avenge Muslims killed by the British military. Unarmed police arrived at the scene nine minutes after an emergency call was received and set up a cordon. Armed police officers arrived five minutes later. The assailants, armed with a cleaver and brandishing a gun, charged at the police, who fired shots that wounded them both. They were apprehended and taken to separate hospitals. Adebolajo and Adebowale are British of Nigerian descent, were raised as Christians, and converted to Islam.


title: Murder of Lee Rigby

text: On 19 December 2013, both of the attackers were found guilty of Rigby's murder. On 26 February 2014, they were sentenced to life imprisonment, with Adebolajo given a whole life order and Adebowale ordered to serve at least 45 years. The attack was condemned by political and Muslim leaders in the United Kingdom and in the international press.


title: Murder of Lee Rigby

text: The soldier killed in the attack was 25-year-old Lee James Rigby, a drummer and machine-gunner in the 2nd Battalion of the Royal Regiment of Fusiliers. Rigby, from Middleton, Greater Manchester, was born in 1987 and had served in Cyprus, Germany, and Afghanistan before becoming a recruiter and assisting with duties in the Tower of London. He was attacked when he was returning to barracks from working at the Tower. Rigby married in 2007 and had a two-year-old son, but had separated from his wife. He was engaged to a new fiancée at the time of his death. A post-mortem examination showed that Rigby died from "multiple incised wounds".


title: Murder of Lee Rigby

text: Rigby supported British Armed Forces charity Help for Heroes and was wearing a hoodie supporting the charity when he was attacked. In the five days after his death the charity received more than £600,000 in donations.


title: Murder of Lee Rigby

text: Rigby was given a military funeral at Bury Parish Church on 12 July 2013. The service was attended by several thousand people, including present and former soldiers, the prime minister David Cameron, and Mayor of London Boris Johnson. A private burial service was then held at nearby Middleton Cemetery. The first permanent memorial to him was installed in February 2014 at The Valley, a football stadium less than from the site of his murder.


title: Murder of Lee Rigby

text: The attack took place shortly before 14:20 in Wellington Street, and near its junction with John Wilson Street, part of the South Circular Road (A205) in Woolwich, near the perimeter of the Royal Artillery Barracks where Rigby was stationed. Rigby had arrived at Woolwich Arsenal station at 14:10 and was walking down Wellington Street towards the Barracks.


title: Murder of Lee Rigby

text: While Rigby was crossing the road to get to a shop, two men, who were later identified as Michael Adebolajo and Michael Adebowale, drove a Vauxhall Tigra at him at , knocking him to the ground. They attacked Rigby with knives and a cleaver, and attempted to behead him.


title: Murder of Lee Rigby

text: Immediately after the attack, several passers-by stood over Rigby's body to protect him from further injury. Ingrid Loyau-Kennett, a cub scout leader from Cornwall, disembarked from a passing bus with the intention of rendering first aid, after she saw what she thought was a road accident. On discovering that the victim was dead she engaged one of the assailants in conversation. The man said he was responsible for killing the man on the ground – a British soldier who the attacker claimed had "killed Muslims in Iraq and in Afghanistan".


title: Murder of Lee Rigby

text: In a video shot by a bystander, Adebolajo said "The only reason we have killed this man today is because Muslims are dying daily by British soldiers". Adebolajo also gave a bystander at the scene a handwritten two-page note which set out his justification for his actions. The assailants remained at the scene and asked bystanders to call the police. The Metropolitan Police received the first 999 call about an assault at 14:20 and unarmed police were deployed. Subsequent 999 calls said the attackers had a firearm, and armed police were ordered to the scene at 14:24. Unarmed police arrived at 14:29, set up a cordon, and remained behind it. Authorised firearms officers arrived at 14:34. The two men, one brandishing a cleaver and the other a revolver, charged at the police. Armed police fired eight times and both men were wounded. They were arrested and taken to separate hospitals. A revolver, knives, and a cleaver were seized at the scene. The victim, Rigby, was pronounced dead and formally identified. The revolver was later determined to be a non-functioning 90-year-old Dutch KNIL 9.4mm. Adebowale pointed the gun at responding armed police officers, who opened fire and shot off one of his thumbs.


title: Murder of Lee Rigby

text: The two men who carried out the attack, Michael Olumide Adebolajo, 28, and Michael Oluwatobi Adebowale, 22, are British of Nigerian descent. Both men were known to British security services.


title: Murder of Lee Rigby

text: On 23 May, a man aged 29 and two women aged 31 and 29 were arrested on suspicion of conspiracy to murder. The Metropolitan Police arrested three people aged between 21 and 28 in south-east London, at two separate locations on the evening of 25 May. On 26 May, a 22-year-old male was arrested in Highbury. On 27 May, a 50-year-old male was arrested in Welling. Of the eight people arrested, six were freed on bail, and two released without charge.


title: Murder of Lee Rigby

text: Michael Olumide Adebolajo, born in Lambeth to a Christian family, went to Marshalls Park School and Havering Sixth Form College and then went to study sociology at the University of Greenwich. He has a history of involvement in radical Islamist activities and had been arrested at a violent protest and later released. According to Anjem Choudary, a radical Muslim cleric, Adebolajo converted to Islam in 2003 and was linked to the outlawed Islamist group al-Muhajiroun. In 2006, Adebolajo was arrested outside the Old Bailey during a protest about the trial of Mizanur Rahman.


title: Murder of Lee Rigby

text: In 2009, Adebolajo spoke at a demonstration against the English Defence League and Stop Islamisation of Europe organised by Unite Against Fascism at Harrow Central Mosque. He was recorded saying: "Don't be scared of them, do not be scared of the police or the cameras. You are here only to please Allah. You're not here for any other reason, if you are here just for a fight, please leave our ranks. We only want those who are sincere to Allah. Purify your intention."


title: Murder of Lee Rigby

text: In 2010, Adebolajo was arrested in Kenya with five others. He travelled using a British passport in the name Michael Olemendis Ndemolajo. Boniface Mwaniki, head of Kenya's anti-terrorism unit, said he believed Adebolajo was planning to train with Al-Shabaab, a militant group linked to al-Qaeda. He was released to British authorities in Kenya and deported. The British Foreign Office confirmed "a British national was arrested in Kenya in 2010" and was given consular assistance. No charges were filed against Adebolajo.


title: Murder of Lee Rigby

text: Abu Nusaybah, a friend of Adebolajo, said on BBC's "Newsnight" on 25 May that Adebolajo had complained of persistent questioning by the British Security Service (MI5) specifically concerning his knowledge of "certain individuals". He said Adebolajo alleged that MI5 had asked him to work with them and he had refused. He also said Adebolajo claimed he had been tortured and sexually assaulted by Kenyan troops after his arrest.


title: Murder of Lee Rigby

text: Adebolajo was released from hospital on 31 May and taken into police custody. The following day he was charged with Rigby's murder, two charges of attempting to murder police officers, and possession of a firearm. At a court appearance on 3 June, he asked to be known as Mujahid Abu Hamza. On 17 July, Adebolajo lost two of his front teeth while being restrained by five prison officers at Belmarsh Prison.


title: Murder of Lee Rigby

text: Michael Oluwatobi Adebowale attended Kidbrooke School, later studying at the University of Greenwich with Michael Adebolajo. Adebowale's mother is a probation officer and his father a member of staff at the Nigerian High Commission. His parents separated when he was young and he converted to Islam in 2009, adopting the name Ismael. Adebowale was convicted of drug dealing offences in 2009 and was sentenced to 15 months' imprisonment.


title: Murder of Lee Rigby

text: On 28 May, Adebowale was released from hospital and taken to a police station in south London. Police charged him with the murder of Rigby and possession of a firearm.


title: Murder of Lee Rigby

text: Investigators searched four houses in Greenwich, south London; one in Romford, east London; another in north London; and a property in Saxilby, Lincolnshire.


title: Murder of Lee Rigby

text: Sir Malcolm Rifkind, the chairman of the Intelligence and Security Committee, said the committee would use new powers to retrieve documents from intelligence agencies. A written report is to be provided by Andrew Parker, the Director General of MI5.


title: Murder of Lee Rigby

text: An Independent Police Complaints Commission investigation into the use of firearms by Metropolitan Police Officers, which was published on 19 December 2013 once a verdict had been reached in the defendants' trial, concluded that the officers who had used force on 22 May 2013 had "acted entirely appropriately" and had shown "skill and professionalism".


title: Murder of Lee Rigby

text: The Home Secretary, Theresa May, chaired a meeting of the Cabinet Office Briefing Room committee (COBRA) attended by Defence Secretary Philip Hammond, the Mayor of London Boris Johnson, the Metropolitan Police Commissioner Sir Bernard Hogan-Howe, the Metropolitan Police Assistant Commissioner Cressida Dick and other unnamed members of the intelligence agencies. The Prime Minister David Cameron cut short a visit to Paris to chair a second COBRA meeting.


title: Murder of Lee Rigby

text: On 31 May, the inquest into Rigby's death was opened and adjourned at Southwark Coroner's Court. The inquest heard that Rigby had been identified by his dental records.


title: Murder of Lee Rigby

text: On 27 September 2013, the two accused men appeared via videolink in court at the Old Bailey, where they both pleaded not guilty to the murder of Lee Rigby, and to other charges relating to the incident. The trial began at the Old Bailey on 29 November 2013. Adebolajo asked to be known as Mujaahid Abu Hamza in court with Adebowale wishing to be known as Ismail Ibn Abdullah.


title: Murder of Lee Rigby

text: On 19 December 2013, Michael Adebolajo and Michael Adebowale were found guilty of the murder of Lee Rigby. The judge, Mr Justice Sweeney, said that he would pass sentence after a key appeal court ruling on the use of whole life terms. On 26 February 2014, both men were sentenced to life imprisonment. Adebolajo was given a whole life order excluding the possibility of parole, and Adebowale, the younger of the two, was given a minimum term of 45 years in prison.


title: Murder of Lee Rigby

text: During the sentencing, Mr Justice Sweeney said that the extremist views of the attackers were a "betrayal of Islam", prompting Adebowale to shout "That’s a lie", while Adebolajo shouted ""Allahu Akbar"". Following a scuffle with security guards in the dock, both men were removed from the court and the sentencing continued in their absence.


title: Murder of Lee Rigby

text: On 8 April 2014, Adebolajo launched an appeal against his whole life term. On 29 July, he was refused permission to appeal, and the case was heard by a panel of Court of Appeal judges.


title: Murder of Lee Rigby

text: In July 2014, a freedom of information request filed by "The Sun" showed that Adebolajo and Adebowale had received a combined £212,613.32 in legal aid.


title: Murder of Lee Rigby

text: On 3 December 2014, Rigby's killers lost legal challenges to their sentences. Michael Adebolajo had attempted to have his conviction overturned and whole-life sentence reduced, while Michael Adebowale attempted a reduction in his minimum sentence of 45 years. Both requests were rejected at the Court of Appeal.


title: Murder of Lee Rigby

text: The Ministry of Defence investigated the incident. Immediately after the death, British service members were advised not to wear military uniforms in public, although the advice was later relaxed.


title: Murder of Lee Rigby

text: In the immediate aftermath, Julie Siddiqi of the Islamic Society of Britain expressed concern that the killing would be used to create ethnic and community divisions. Sir Bernard Hogan-Howe condemned the attack and called for a calm and "measured response", adding "we have met with community representatives, and extra officers remain on duty there tonight. Across London our officers are in contact with their communities too." Commander Simon Letchford later noted community concerns following the incident and assured that an investigation was under way. He also appealed for calm and avoidance of speculation. An additional 1,200 police officers were deployed across London to prevent revenge attacks on Muslim communities.


title: Murder of Lee Rigby

text: The British National Party (BNP) leader, Nick Griffin, posted a series of Twitter messages blaming "mass immigration" for the attack and called for a protest rally in Woolwich. After the English Defence League called on its supporters to mobilise, some members staged a protest at Woolwich Arsenal station in which bottles were thrown at police. The BNP scheduled their protest for 1 June, but Scotland Yard refused to permit them to march from Woolwich Barracks; the demonstration instead took place at Whitehall in central London. Unite Against Fascism mounted a counter-protest. Police arrested 58 people, for breaches of the Public Order Act.


title: Murder of Lee Rigby

text: On 7 June 2013, a 21-year-old woman from Harrow was ordered to complete 250 hours of unpaid work after tweeting that people in Help for Heroes T-shirts "deserve to be beheaded". On 14 March 2014, a married couple from London – who pleaded guilty to disseminating a terrorist publication – were jailed for posting videos on YouTube which condoned the death of Lee Rigby, with one video describing it as a "brilliant day".


title: Murder of Lee Rigby

text: In the aftermath of the attack, an anti-Muslim backlash occurred across the United Kingdom. A representative of Hope not Hate said the number of phone calls to its helpline concerning anti-Muslim incidents greatly increased after the murder. Hope not Hate reported 193 Islamophobic incidents, including attacks on 10 mosques, as of 27 May. On 1 June, Tell MAMA, a government-funded project, reported 212 anti-Muslim incidents, including 125 online incidents, 17 incidents involving physical attack, and 11 attacks on mosques. , although the decision had been made before Rigby's death.


title: Murder of Lee Rigby

text: Incidents ranged from verbal abuse to physical assaults in which women's headscarves were pulled off. Graffiti was scrawled over mosques and Muslim-owned businesses. Hope not Hate claimed that online activity suggested some of the attacks on Muslims were co-ordinated. At least seven people were arrested for a range of social media-related issues.


title: Murder of Lee Rigby

text: During the night after Rigby's death, two mosques were attacked. In Braintree, Essex, a man entered a mosque with two knives, threatened the congregation, and threw an explosive device, which witnesses said was a grenade or gas canister. In Gillingham, Kent, a man ran into a mosque and smashed windows and bookcases, specifically targeting those containing copies of the Quran; two men were arrested in connection with the attacks. On 26 May, several petrol bombs were thrown into a mosque in Grimsby, but no one was injured and the fires were rapidly extinguished. Two former soldiers were arrested in connection with the attack.


title: Murder of Lee Rigby

text: On 5 June, the Al-Rahma Islamic Centre in Muswell Hill – which was used by children after school – was destroyed by a fire, and the building had been sprayed with graffiti making reference to the English Defence League. The fire investigation was conducted by Scotland Yard's counter-terrorism command, because of a possible link to domestic extremism. On 8 June, a fire at Darul Uloom School, an Islamic boarding school in southeast London, forced the evacuation of 128 students and teachers, with police suspecting that the incident may have been a revenge attack. On 10 June, a senior Metropolitan Police officer confirmed there had been an eight-fold increase in the number of Islamophobic incidents since Rigby's death, and that the real figure may be higher due to under-reporting.


title: Murder of Lee Rigby

text: In the London Borough of Hackney the Stamford Hill Shomrim, a Jewish volunteer Neighbourhood Patrol Group, made an offer of help to the local Muslim community which was welcomed and subsequently commended by Hackney Police Borough Commander Chief Superintendent Matthew Horne.


title: Murder of Lee Rigby

text: Video footage of one of the perpetrators justifying the killing of Lee Rigby was obtained by "The Sun" and ITN. ITN's video, which was edited before it was broadcast, aired during the 18:30 "ITV News" bulletin before the 21:00 watershed, and again in its 22:00 bulletin. After being posted on the ITN website in the afternoon, the high level of visits caused the site to crash and go offline for around half an hour. Total traffic on the site, which averages 860,000 unique users per week, reached 1.2 million for the day of the attack.


title: Murder of Lee Rigby

text: Managing editor of "The Sun", Richard Caseby, said the newspaper had faced "a very difficult decision". Both media outlets argued they had released the video "in the public interest". BBC News showed some parts of the video. Sky News decided not to follow suit, as senior editors were of the opinion that the graphic images were "unnecessarily distressing". Both ITV and the BBC ran warnings before showing the footage. Most of Britain's national daily newspapers grabbed still images from the video footage for their front pages the next morning. A BBC executive said that the news organisation edited the footage before broadcasting, and "dealt with the material as carefully as we could." The spokesman said they "thought very carefully about the pictures...and gave great consideration to how we used the footage". They argued that the footage was an important element of the story and shed light on the perpetrators and the possible motives for the attack."


title: Murder of Lee Rigby

text: "The Guardian" reported there were "around 800 complaints from distressed viewers". Most complaints were targeted at the television coverage, with ITV receiving 400 complaints in the 24 hours following the broadcast. Sky News, which showed a still image of one of the suspected attackers with bloodied hands, received "a handful of complaints".


title: Murder of Lee Rigby

text: On 17 June, the broadcasting standards watchdog Ofcom launched an investigation into broadcast of footage from the attack after receiving about 700 complaints. Ofcom published its findings on 6 January 2014, ruling that the news footage had not breached broadcasting regulations. Ofcom issued new guidelines to news outlets on giving appropriate warnings before airing distressing content.


title: Murder of Lee Rigby

text: The UK government established a task force to look at ways of stemming the growth of Islamic extremism in Britain, focusing on the radicalisation of worshippers in mosques, university students and prisoners. The task force – chaired by David Cameron – had its inaugural meeting at 10 Downing Street on 3 June 2013, and includes Cabinet Ministers, and representatives from the police and intelligence services. Later that day Cameron made a House of Commons statement on the Woolwich attack, saying that lessons must be learned. "When young men born and bred in this country are radicalised and turned into killers, we have to ask some tough questions about what is happening in our country. It is as if that for some young people there is a conveyor belt to radicalisation that has poisoned their minds with sick and perverted ideas. We need to dismantle this process at every stage – in schools, colleges, universities, on the internet, in our prisons, wherever it is taking place."


title: Murder of Lee Rigby

text: On 25 November 2014, the findings of a British parliamentary inquiry into Rigby's murder was published. The report found that his death could not have been prevented, although his killers had appeared in seven intelligence investigations. In December 2012, Michael Adebowale had discussed killing a soldier on Facebook with a foreign-based extremist known as "Foxtrot". The UK authorities did not have access to the details of the conversation until June 2013, when they were disclosed to GCHQ. The Intelligence and Security Committee stated "Had MI5 had access to this exchange, their investigation into Adebowale would have become a top priority." Facebook said that it did not comment on individual cases, but responded that "Facebook’s policies are clear, we do not allow terrorist content on the site and take steps to prevent people from using our service for these purposes." In an interview with BBC News on 26 November 2014, Richard Barrett, the former Director of Global Counter-terrorism at MI6, said that it was unfair to expect companies to monitor websites for all potentially extremist content. Facebook had blocked seven of Adebowale's accounts prior to the killing, five of which had been flagged for links with extremism. The accounts had been flagged by an automated process, and no person at Facebook had manually checked the accounts.


title: Murder of Lee Rigby

text: On 3 June 2018, Rigby's murderer Michael Adebolajo said he regretted the act and apologised for the first time, according to prison sources. Adebolajo added that he "misinterpreted" the Koran to justify his actions and that he was brainwashed. Lee's mother Lyn Rigby said she does not accept his apology and will "never" forgive him.


title: Murder of Lee Rigby

text: Queen Elizabeth II, political leaders and religious leaders variously expressed concern and distress over the incident, and called for calm. The Prime Minister, David Cameron, made the following statement:This country will be absolutely resolute in its stand against extremism and terror. This action was a betrayal of Islam and the Muslim communities that give so much to our country. We will defeat violent extremism by standing together. We will not rest until we know every detail. [The attackers told Ingrid Loyau-Kennett that] they wanted to start a war in London and she replied, "You are going to lose, it is you against many." She speaks for all of us.


title: Murder of Lee Rigby

text: Many Muslim leaders denounced the attack. The Prime Minister's statement was echoed by Shaykh Ibrahim Mogra, with the Archbishop of Canterbury, Justin Welby, the co-chair of the Christian Muslim Forum, in a joint statement. The Muslim Council of Britain said the attack "has no basis in Islam and we condemn this unreservedly". The head of the Ramadhan Foundation, Mohammed Shafiq, also condemned the attack. The director of Faith Matters and co-ordinator of the government-backed anti-Islamophobic project Tell MAMA stated: "We, as the Muslim community, will work against anyone who promotes such hatred."


title: Murder of Lee Rigby

text: Anjem Choudary refused to condemn the attack. He said, "I'm not in the business of condemnation or condoning. I think if anyone needs to be condemned it is the British government and their foreign policy. It's so clear that that is the cause." On BBC's "Newsnight", when Choudary was questioned about his role in the radicalisation of Michael Adebolajo, he denied any responsibility, and talked about such radicalisation as a means to an end. He stated that he believed that not many Muslims would disagree with what Adebolajo had said in his videoed statement.


title: Murder of Lee Rigby

text: Asghar Bukhari of the UK Muslim Public Affairs Committee said that both the British Government and the Muslim community were at fault in dealing with "extremism". He criticised the British Government for being involved in wars in Iraq and Afghanistan while "completely denying that it has anything to do with the political situation around the Muslim world", and said that Muslim organisations "have failed their own community by not teaching these young, angry men how to get a democratic change to this policy that's ruining so many lives". He described Muslim leaders as unwilling to bring about change, focussing on points of theology, rather than the practical education of young people in ways to achieve political change.


title: Murder of Lee Rigby

text: Baroness Neville-Jones, a former security minister and chairman of the British Joint Intelligence Committee, and Colonel Richard Kemp, a former Army commander, suggested blame could be put on internet hate preaching. Neville-Jones told the BBC Radio 4<nowiki>'</nowiki>s "Today" programme that "the inspiration that comes from internet hate preaching and jihadist rhetoric...is a very, very serious problem now."


title: Murder of Lee Rigby

text: George Galloway, then an MP, said that the attack on Lee Rigby was "indefensible". He criticised British support for the Syrian rebels, stating that similar attacks are likely to occur "as long as we are, as a country, involved in spreading murder and mayhem across the Muslim world."


title: Murder of Lee Rigby

text: Former UK Prime Minister Tony Blair saw the attack not as an isolated expression of two crazed individuals but part of the broader "trouble within Islam".


title: Murder of Lee Rigby

text: In foreign press reports there was widespread outrage and condemnation of the killing. Yusif al-Shihab, in Kuwait's "Al-Abas", stated that the assailants have "deformed the image of Islam" while Batir Mohammad Wardum in the Jordanian daily "Al-Dustur", and other Middle Eastern newspapers, stressed that their actions have endangered the lives of thousands of Muslims.


title: Murder of Lee Rigby

text: In a statement issued on 28 May, Adebolajo's relatives condemned terrorism and violence in the name of religion, and expressed their horror at Rigby's death.


title: Murder of Lee Rigby

text: In October 2013 British anti-terrorist police warned several Muslims who had spoken out against Islamist extremism, some of them explicitly against the murder of Rigby, that they had been targeted in a video created by Al-Shabaab, the group responsible for the attack on the Westgate shopping mall in Kenya.


title: Murder of Lee Rigby

text: On 19 February 2015, 19-year-old Brusthom Ziamani was found guilty of preparing a terrorist act. He was arrested in London in August 2014 while carrying a 12-inch knife, hammer and black jihadist flag. Ziamani had said that he intended to attack and kill soldiers, and had described Adebolajo as a "legend". On 20 March Ziamani was sentenced to 22 years in prison. On 29 April 2015, 18-year-old Kazi Islam, who was inspired by the murder, was convicted by a jury at the Old Bailey of grooming a vulnerable friend to kill two soldiers, and buying ingredients for a pipe bomb. On 29 May, he was sentenced to eight years in a young offenders' institution. On 14 January 2015, 26-year-old white supremacist Zack Davies of Mold, Flintshire attacked a Sikh dentist in a Tesco supermarket with a machete and a hammer. He claimed in court that the attack was revenge for the murder of Rigby. Davies was sentenced to life imprisonment on 11 September 2015.


title: Murder of Lee Rigby

text: On 1 September 2014, Rigby was honoured at a ceremony in Staffordshire, with his name added to the Armed Forces Memorial at the National Memorial Arboretum.


title: Murder of Lee Rigby

text: A memorial to Rigby in his home town of Middleton, Greater Manchester, consisting of a bronze drum and a plaque, was unveiled on 29 March 2015.


title: Murder of Lee Rigby

text: Plans for a memorial to Rigby in Woolwich initially ran into opposition from local MP Nick Raynsford, who expressed concerns that it would generate "undesirable interest from extremists" or become a target for vandals. Greenwich Council said that it had not received a request from the Army to erect a memorial at the site. Meanwhile, the site of the murder on Wellington Street developed into an unofficial memorial site. Following a campaign for a memorial supported by Boris Johnson and a petition with 25,000 signatures, plans for a memorial near the site of the attack were announced on 11 June 2014. The memorial was revealed on 11 November 2015 after considerable delays, as "the council had had to balance different opinions about how Lee Rigby should be commemorated". Lee Rigby's name appears on a plaque on the south wall of the memorial garden inside the ruined St George's Garrison Church in Woolwich, opposite the Royal Artillery Barracks. The memorial consists of a white marble plaque marking Woolwich's history as a barracks town, and two bronze plaques with the names of 11 men who served or lived in Woolwich and gave their lives in the service of their country, including Rigby and the victims of the 1974 King's Arms bombing nearby. In April 2016 the family stated, in relation to calls for a memorial: "There is a permanent memorial to Lee at St George's Chapel in Woolwich, which is what we wanted".


title: Murder of Lee Rigby

text: On 29 February 2020, a memorial to Rigby was unveiled at Millwall F.C.'s stadium, The New Den, with members of Rigby's regiment in attendance. Fans of the club raised the funds to pay for the plaque, which was donated free of charge by local funeral directors with the money going to charity.


title: Murder of Lee Rigby

text: The Lee Rigby Foundation was established by Lyn and Ian Rigby after their son's murder. The organisation was set up as a support network for bereaved military families, establishing a series of static caravans that families can use for a holiday break. Lee Rigby House in Staffordshire was also established for this purpose, with the premises donated by former professional wrestler Peter Thornley.


title: Thomas Wayne

text: Thomas Alan Wayne, M.D. is a fictional character appearing in American comic books published by DC Comics. He is the father of Bruce Wayne (Batman), and husband of Martha Wayne as well as the paternal grandfather of Damian Wayne. Wayne was introduced in "Detective Comics" #33 (November 1939), the first exposition of Batman's origin story. A gifted surgeon and philanthropist to Gotham City, Wayne inherited the Wayne family fortune after Patrick Wayne. When Wayne and his wife are murdered in a street mugging, Bruce is inspired to fight crime in Gotham as the vigilante Batman.


title: Thomas Wayne

text: Wayne was revived in Geoff Johns' alternate timeline comic "Flashpoint" (2011), in which he plays a major role as a hardened version of Batman, whose son was killed instead of his wife and himself, and dies again by the end of the storyline. Wayne returned to the main DC Universe in DC Rebirth, as a revived amalgamation of his original self killed by Joe Chill and his "Flashpoint" Batman self killed in "The Button".


title: Thomas Wayne

text: As a key figure in the origin of Batman, Thomas Wayne has appeared in multiple forms of media. Notable portrayals of the character in live-action include Linus Roache in "Batman Begins", Jeffrey Dean Morgan in "", Brett Cullen in "Joker," and Luke Roberts in "The Batman". Ben Aldridge also portrays him in the television series "Pennyworth".


title: Thomas Wayne

text: Thomas Wayne, M.D. is seldom shown outside of Bruce Wayne's and Alfred Pennyworth's memories of him, and Bruce's dreams and nightmares. He is frequently depicted as looking very much like his son, but with a mustache.


title: Thomas Wayne

text: A notable occurrence in Thomas’ biography was when Bruce falls through a fissure on the Wayne property, into what would one day become the Batcave (sometimes the fissure is replaced with an abandoned well). Thomas eventually rescues his terrified son from the cave.


title: Thomas Wayne

text: Dr. Wayne's role in his son's future vigilante career is expanded upon in "The First Batman", a Silver Age tale from "Detective Comics" #235, which reveals that Dr. Wayne attacks and defeats hoodlums while dressed like a "Bat-Man" for a masquerade ball. According to the story, Dr. Wayne's actions result in crime boss Lew Moxon being imprisoned; ten years later, Moxon orders Joe Chill to murder Dr. Wayne. Realizing Moxon ordered his parents killed, Batman confronts Moxon, now suffering from amnesia and thus has no memory of Dr. Wayne. When his costume is torn, Batman wears his father's in order to frighten Moxon. Sure enough, the costume restores Moxon's memory; the former crime boss panics, believing that Thomas’ ghost is attacking, and flees into the streets and is struck and killed by a truck. These events were retold in the 1980 miniseries "The Untold Legend of the Batman". Following the Crisis on Infinite Earths, Thomas as the "First Batman" was retconned - he instead attends the masquerade ball as Zorro. This was retconned once more in the pages of "Superman/Batman", where Superman, hoping to reverse some universe-altering change in the time streams, lands in a version of Gotham City in which Thomas never died, finding him giving out Halloween candy in the original Batman costume.


title: Thomas Wayne

text: In many of the modern interpretations of the character, such as those by Frank Miller and Jeph Loeb, Thomas Wayne is portrayed as having been a somewhat distant, stern father, bestowing more kindness and generosity on his patients than his own son.


title: Thomas Wayne

text: Dr. Wayne was once suspected to be the father of Bane. However, DNA testing proved this to be false, and Bane's real father was recently revealed to be King Snake.


title: Thomas Wayne

text: In "", a flashback reveals that Thomas Wayne saved the life of gangster Carmine Falcone. Falcone's father Vincent Falcone came to Wayne Manor and begged Thomas to save his dying son, who had been shot by rival gangster Luigi Maroni. Thomas wanted to take the younger Falcone to the hospital, but Vincent insisted that nobody know about the shooting; the surgery was thus performed in the dining room with Alfred assisting. After saving Carmine's life, he was offered a reward or favor, but refused to accept any form of payment. Unbeknownst to Thomas, young Bruce watched this all in silence from afar. Years later, Bruce contemplates whether Gotham would have been better off had his father let Falcone die; Alfred replies that Thomas would have helped anyone in need.


title: Thomas Wayne

text: In "Superman/Batman" #50, it is revealed that, while on a drive with a pregnant Martha Wayne, Thomas witnesses a strange object fall to Earth. As he inspects it, Thomas' consciousness is transported to Krypton, and presented in a holographic form. There, he encounters Jor-El, wishing to know what kind of a world Earth is, as it is one of many possible candidates for him to send his son Kal-El to. Thomas tells Jor-El that the people of Earth aren't perfect, but are essentially a good and kind race, who would raise the child right, convincing Jor-El to send Kal-El there. Upon returning to his body, Thomas uses the technology in the Kryptonian probe to revitalize a failing Wayne Enterprises. Years later, the alien technology would be the basis of much of Batman's crimefighting technology. Thomas recorded his encounter in a diary, which was discovered by Bruce in the present day.


title: Thomas Wayne

text: When exiting a movie theater (opera in some versions), Thomas and Martha Wayne are murdered by a mugger in front of their son, Bruce Wayne. This tragedy shocks Gotham and leads to Park Row (the street where it occurred) being labeled Crime Alley. Most importantly, it serves as the motivation for Bruce to become Batman.


title: Thomas Wayne

text: During "Batman R.I.P.", it is alleged that Thomas Wayne and Martha Wayne were leading a double life, secretly partaking in criminal endeavors, drug abuse and orgies while presenting a façade of respectability to the outside world. The alleged evidence is revealed to be doctored in the aftermath of the storyline, however.


title: Thomas Wayne

text: Doctor Simon Hurt, head of the Black Glove and the mastermind behind "Batman R.I.P.", actually claims to "be" Thomas Wayne to both Bruce Wayne and Alfred Pennyworth. Although both of them rebuke him without hesitation, Hurt never explicitly drops the claim.


title: Thomas Wayne

text: In the ongoing follow-up series, "Batman & Robin", it is suggested that some, if not all, of these allegations have begun to circulate around Gotham; Dick Grayson and Damian Wayne attend a high society function where a few party guests vaguely mention the existence of rumors surrounding the family, and Dick tries to tie Bruce's absence from the public eye with being occupied with clearing his family's reputation. Matters come to a head when Hurt returns to the city, claiming to be Thomas Wayne in order to take control of Wayne Manor and establish himself as the new Batman, but Grayson and Damian outsmart him.


title: Thomas Wayne

text: It is hinted at during the course of the "Batman and Robin" series that Simon Hurt's actual identity is Thomas Wayne, albeit one from the 17th century who was a 'black sheep' of the Wayne family and prolongs his life through occult rituals. The "Return of Bruce Wayne" miniseries and its fallout "" cements Hurt's status as the elder Thomas Wayne from the Puritan Ages, driven insane by his meeting with "Barbatos", the Hyper-Adapter sent through time along with Bruce Wayne to ensure the effectiveness of Darkseid's "Omega Sanction".


title: Thomas Wayne

text: In "", Doctor Bruce Wayne discovers the preserved brain of his father deep below the research university where he works. He steals this and other body parts in a desperate attempt to revive his beloved father from death. This doesn't work out well due to various factors outside his control, but the part of the creature that still recognizes and loves his son does its best to help Bruce escape his enemies and expose the person responsible for the Waynes' murders; Thomas had discovered that one of his colleagues was killing people to perfect the preservation fluid used to store organs (Including Thomas's own brain, claimed by his killer after his death).


title: Thomas Wayne

text: In "", Thomas Wayne and Martha Wayne are saved from death when Valentin Sinclair- really Vandal Savage, a man who has a long-standing interest and admiration for the Wayne family despite the fact that they often end up opposing him when they learn about his plans- scares off Joe Chill. Sinclair then becomes a partner in Wayne Enterprises, only to have the Waynes killed when they threaten to expose his plan to divert the meteor that gave him his powers back to Earth in order to study it. Their deaths come at the hands of Sinclair's fear-inducing henchman Scarecrone who causes them to remember the mugging, which drives them to flee Chill by running off their balcony. This prompts Bruce to become Batman to investigate.


title: Thomas Wayne

text: In "", depicting an alternate timeline where Oliver Cromwell established a theocratic government across the globe, Thomas Wayne is the chief physician of the Privy Council, but when it is discovered that he is treating various 'enemies of the state' such as Jews or homosexuals, the Star Chamber votes by secret ballot to have him and Martha executed and make it look like a random mugging.


title: Thomas Wayne

text: In the graphic novel written by Geoff Johns and with art by Gary Frank, "", Thomas Wayne was a physician who had run for mayor against Oswald Cobblepot. Cobblepot had attempted to arrange his opponent's murder during the latter's outing to a movie with his family, but a mugger got to them first and killed Thomas and his wife. It is also implied that both he and Alfred are keeping a traumatic secret.


title: Thomas Wayne

text: In "" by Grant Morrison and Frank Quitely, the Thomas Wayne of the antimatter universe is the father of the supervillain Owlman. Like his original counterpart, he married Martha Kane and fathered Bruce Wayne. However, this version also had a second child named Thomas Wayne Jr. After Martha and Bruce are gunned down by a crooked police officer when Thomas Sr. refused to accompany him for questioning as Thomas Wayne had performed an "illegal medical operation," Thomas Jr. becomes Owlman to get revenge on the justice system. It is later revealed that Thomas Wayne Sr. is currently acting as commissioner of the Gotham City Police Department, seeking to bring his son to justice with the help of a cadre of idealistic officers. Owlman finds it his goal to "punish" his father for letting his mother and brother die.


title: Thomas Wayne

text: In Mark Millar's "", Batman's (unnamed) parents are anti-communist protesters in the Soviet Union. They are executed in their home by NKVD Commissar Pyotr Roslov for printing and distributing anti-communist pamphlets. Their son witnesses the murders and attempts to overthrow the Communist Party of the Soviet Union when he is an adult.


title: Thomas Wayne

text: The fifth issue of the comic book continuation of the television series "Smallville", written by executive story editor Bryan Q. Miller, reveals that Lionel Luthor invited Thomas to join the secret society Veritas with Virgil Swann, which Thomas declined.


title: Thomas Wayne

text: In the revised Earth-Two alternate universe of "The New 52", Thomas Wayne is revealed to be the second version of Batman, having succeeded his son as the incumbent through use of the Miraclo pill of Hourman which enhances his strength and agility. He states that he is 65 years old. In contrast to his depiction on the Prime Earth, he and Martha Wayne are targeted for assassination due to Thomas's Mafia connections and subsequent efforts to "straighten out" when Bruce was born. Thomas hides the fact that he survived for many years in order to keep Bruce safe. Eventually, when the first version of Batman tracks Thomas down, he learns the truth and rejects Thomas's reason for being "dead" for most of Bruce's life, thus discouraging him from ever having any future relationship with his son, and by extension Bruce and Selina Kyle Wayne's family. However, he watched them from afar and tried to be as involved as he could, particularly with their daughter Helena. After Bruce's death saving Earth 2 from an Apokoliptan invasion, Thomas honors him by taking up the "mantle of the Bat" and makes use of Miraclo to help him fight crime.


title: Thomas Wayne

text: After Earth-2's destruction in the war with Apokolips as seen in the "Convergence" storyline, Thomas Wayne is one of the few survivors transferred to Telos's world, along with Dick Grayson and other heroes. While investigating this world, Thomas travels with Dick into a variation of the pre-"Flashpoint" Gotham City where he has an unknown conversation with Bruce before he departs with Dick in a flying Batmobile. When they are cornered by the Club of Villains, who pursued them out of the city, Thomas sacrifices himself in a suicide bomb blast, destroying most of the Club of Villains in the process, his last words being to inform the Club of Villains that they will never hurt another Batman.


title: Thomas Wayne

text: In the revised Earth-Three alternate universe of "The New 52", Thomas Wayne Sr. is featured in Owlman's origin story. In stark contrast to Batman's father who was a highly skilled surgeon and philanthropist who had saved countless lives during his time, Earth 3's Thomas Wayne is a cowardly and sociopathic doctor who often kills his patients (Martha claims that this is due to a "surgical fetish") and spends his money on lawyers to cover up his patients' deaths as accidents. Earth 3's Alfred kills him along with Martha and Bruce Wayne as per Thomas Wayne Jr.'s orders. Owlman later states that his father was a weak man and wonders why Batman would devote his life to the memory of Thomas Wayne of Prime Earth.


title: Thomas Wayne

text: When Earth 3 was rebooted during the "Infinite Frontier" storyline, Thomas Wayne Sr. and Martha Wayne were depicted as criminals. After they caused the death of Jimmy Gordon Jr., Boss Gordon sent Harvey Bullock to kill them where Bruce was also killed and Thomas Wayne Jr. was left alive. Years later, Thomas Wayne Jr. in the identity of Owlman would learn this fact when he interrogates Bullock.


title: Thomas Wayne

text: In the opening of the first issue of the comics "DC Comics Bombshells", set in an alternate history 1940, Thomas and Martha Wayne's lives are saved by an already-existing Batwoman, implying that Bruce Wayne will never grow up to become Batman. However, in the final issue, set in 1960, the now grown up Bruce Wayne takes up training and becomes Batman, to honor the person who saved him and his parents' lives.


title: Thomas Wayne

text: In this film, Thomas Wayne is portrayed much less sympathetically than in other incarnations. He is a successful businessman who is running to become Mayor of Gotham City, but he has little sympathy for the lower classes, dismissing the poor as "clowns" who have no one but themselves to blame for their misfortune. During the course of the movie, Arthur Fleck (Joaquin Phoenix) discovers from a letter his mother Penny (Frances Conroy) is writing that he may be Wayne's son, the product of Wayne's affair with Penny while she worked for him as a maid. Arthur sneaks backstage at a campaign event and confronts Wayne, who denies that Arthur is his son or that he and Penny ever had an affair; he says that Penny is delusional and made up the whole thing. Thomas then punches Arthur in the face for earlier speaking to his son Bruce, and threatens to kill him if he ever comes near Bruce again. Considering later revelations that some parts of Arthur's life were all a delusion, it is unclear how much of his meetings with Thomas should be taken literally or if any of these events happened at all.


title: Thomas Wayne

text: At the end of the film, Thomas, Martha (Carrie Louise Putrello) and Bruce attempt to flee a riot that Arthur - now calling himself "Joker" - indirectly caused, only for one of the masked rioters to follow them and kill him and Martha in front of Bruce, starting him on the path to becoming Batman.


title: Thomas Wayne

text: Luke Roberts portrayed Thomas in the 2022 film "The Batman". As in the comics, Wayne saved Carmine Falcone's (John Turturro) life. While running for mayor, he was met by a reporter who threatened to reveal his wife Martha's family history of mental illness. Desperate, Wayne turned to Falcone to have the latter intimidate the reporter, only for Falcone to kill him so he could have something on the Wayne family. Wayne, wracked with guilt, threatened to reveal everything to the police; he and his wife to be killed were murdered a week later. It is implied they may have been assassinated on the orders of either Falcone or Salvatore Maroni, but it is left ambiguous.


title: Coccyx

text: The coccyx (: coccyges or coccyxes), commonly referred to as the tailbone, is the final segment of the vertebral column in all apes, and analogous structures in certain other mammals such as horses. In tailless primates (e.g. humans and other great apes) since "Nacholapithecus" (a Miocene hominoid), the coccyx is the remnant of a vestigial tail. In animals with bony tails, it is known as "tailhead" or "dock", in bird anatomy as "tailfan". It comprises three to five separate or fused coccygeal vertebrae below the sacrum, attached to the sacrum by a fibrocartilaginous joint, the sacrococcygeal symphysis, which permits limited movement between the sacrum and the coccyx.


title: Coccyx

text: The coccyx is formed of three, four or five rudimentary vertebrae. It articulates superiorly with the sacrum. In each of the first three segments may be traced a rudimentary body and articular and transverse processes; the last piece (sometimes the third) is a mere nodule of bone. The transverse processes are most prominent and noticeable on the first coccygeal segment. All the segments lack pedicles, laminae and spinous processes. The first segment is the largest; it resembles the lowest sacral vertebra, and often exists as a separate piece; the remaining ones diminish in size rostrally.


title: Coccyx

text: Most anatomy books incorrectly state that the coccyx is normally fused in adults. It has been shown that the coccyx may, in some people, consist of up to five separate bony segments, the most common configuration being two or three segments.


title: Coccyx

text: The anterior surface is slightly concave and marked with three transverse grooves which indicate the junctions of the different segments. It gives attachment to the anterior sacrococcygeal ligament and the levatores ani and supports part of the rectum. The posterior surface is convex, marked by transverse grooves similar to those on the anterior surface, and presents on either side a linear row of tubercles – the undeveloped articular processes of the coccygeal vertebrae. Of these, the superior pair are the largest, and are called the coccygeal cornua they project caudally, and articulate with the cornua of the sacrum, and on either side complete the foramen for the transmission of the posterior division of the fifth sacral nerve.


title: Coccyx

text: The lateral borders are thin and exhibit a series of small bony protrusions, which represent the transverse processes of the coccygeal vertebrae. Of these, the first is the largest; it is flattened anteriorly, and often extends to join the lower part of the thin lateral edge of the sacrum, thus completing the foramen for the transmission of the anterior division of the fifth sacral nerve; the others diminish in size from caudally, and are often lacking. The borders of the coccyx are narrow, and give attachment on either side to the sacrotuberous and sacrospinous ligaments, to the coccygeus and levator ani in front of the ligaments, and to the gluteus maximus behind them.


title: Coccyx

text: The apex is rounded, and has attached to it the tendon of the external anal sphincter; it may be divided in two.


title: Coccyx

text: The coccygeal fossa is a shallow depression on the posterior surface between the sacrum and the perineum, located in the intergluteal cleft that runs from just below the sacrum to the perineum. It is not consistently present in all humans. The coccygeal fossa marks the deepest part of the pelvic floor, next to the coccyx. The levator ani has its origin here.


title: Coccyx

text: The extensor coccygis is a slender muscle fascicle, which is not always present. It extends over the caudal portion of the posterior surface of the sacrum and coccyx. It arises by tendinous fibers from the last segment of the sacrum, or first piece of the coccyx, and passes downward to be inserted into the lower part of the coccyx. It is an evolutionary relic of the extensor muscle of the caudal vertebrae of other animals, enabling limited coccygeal motion.


title: Coccyx

text: The joints are variable and may be: (1) synovial joints; (2) thin discs of fibrocartilage; (3) intermediate between these two; (4) ossified.


title: Coccyx

text: The anterior side of the coccyx has attachments to the levator ani muscle, coccygeus, iliococcygeus, and pubococcygeus, anococcygeal raphe. Attached to the posterior side is the gluteus maximus, which extends the thigh at the hip joint. The ligaments attached to the coccyx include the anterior and posterior sacrococcygeal ligaments which are the continuations of the anterior and posterior longitudinal ligaments that extend along the entire spine. The lateral sacrococcygeal ligaments complete the foramina for the last sacral nerve. Some fibers of the sacrospinous and sacrotuberous ligaments (arising from the spine of the ischium and the ischial tuberosity respectively) also attach to the coccyx. An extension of the pia mater, the filum terminale, extends from the apex of the conus, and inserts on the coccyx.


title: Coccyx

text: The coccyx is not entirely useless in humans, based on the fact that the coccyx has attachments to various muscles, tendons and ligaments. However, these muscles, tendons and ligaments are also attached at many other points, to stronger structures than the coccyx. It is doubtful that the coccyx attachments are important to the well-being of humans, given the large number of coccygectomy procedures performed annually to treat coccydynia. Reviews of studies covering more than 700 coccygectomies found the operation was successful in relieving pain in 84% of cases. 12% of the time, the only major complication faced was infection due to the proximity to the anus. One notable complication of coccygectomy is an increased risk of perineal hernia.


title: Coccyx

text: Injuring the coccyx can give rise to a painful condition called coccydynia and one or more of the bones or the connections thereof may be broken, fractured tailbone. A number of tumors are known to involve the coccyx; of these, the most common is sacrococcygeal teratoma. Both coccydynia and coccygeal tumors may require surgical removal of the coccyx (coccygectomy). One very rare complication of coccygectomy is a type of perineal hernia known as a coccygeal hernia.


title: Coccyx

text: The term "coccyx" is derived from the ancient Greek word "kokkyx" "cuckoo"; the latter is attested in the writings of the Greek physician Herophilus to denote the end of the vertebral column. This Greek name for the cuckoo was applied as the last three or four bones of the coccyx resemble the beak of this bird, when viewed from the side.


title: Coccyx

text: This established etymological explanation can also be found in the writings of the 16th century anatomist Andreas Vesalius who wrote: "os cuculi, a similitudine rostri cuculi avis" (the cuckoo bone shows a likeness to the beak of the cuckoo bird). Vesalius used the Latin expression "os cuculi", with "os", bone and "cuculus", the Latin name for the cuckoo. The 16th/17th century French anatomist Jean Riolan the Younger gives a rather hilarious etymological explanation, as he writes: "quia crepitus, qui per sedimentum exeunt, ad is os allisi, cuculi vocis similitudinem effingunt" (because the sound of the farts that leave the anus and dash against this bone, shows a likeness to the call of the cuckoo). The latter is not considered as potential candidate.


title: Coccyx

text: Besides "os cuculi", "os caudae", with "caudae", of the tail is attested. This Latin expression might be the source of the English, French language, German and Dutch terms "tailbone", "l'os de la queue", "Schwanzbein" and "staartbeen". In the current official anatomic Latin nomenclature, "Terminologia Anatomica", "coccyx" and "os coccygis" is used.


title: All Time Low

text: All Time Low is an American rock band from Towson, Maryland, formed in 2003. Consisting of lead vocalist/rhythm guitarist Alex Gaskarth, lead guitarist Jack Barakat, bassist/backing vocalist Zack Merrick, and drummer Rian Dawson, the band took its name from lyrics in the song "Head on Collision" by New Found Glory. The band has consistently done year-long tours, has headlined numerous tours, and has appeared at music festivals including Warped Tour, Reading and Leeds, and Soundwave.


title: All Time Low

text: Beginning as a band in high school, All Time Low released their debut EP, "The Three Words to Remember in Dealing with the End EP", in 2004 through local label Emerald Moon. Since then the band has released eight studio albums: "The Party Scene" (2005), "So Wrong, It's Right" (2007), "Nothing Personal" (2009), "Dirty Work" (2011), "Don't Panic" (2012), "Future Hearts" (2015), "Last Young Renegade" (2017), and "Wake Up, Sunshine" (2020). They released their first live album, "Straight to DVD", in 2010, and released their second live album, "Straight to DVD II: Past, Present and Future Hearts", on September 9, 2016.


title: All Time Low

text: Formed while still in high school in 2003, All Time Low began covering songs by pop punk bands such as Blink-182. The band's line-up included Alex Gaskarth on vocals, Jack Barakat on guitar, TJ Ihle on lead guitar and backing vocals, Chris Cortilello on bass, and Rian Dawson on drums. Cortilello and Ihle left the band, resulting in the band lying dormant until Zack Merrick joined on bass and Gaskarth picked up guitar. They released a four-song EP in November before signing to Emerald Moon Records in 2004. They released their second EP, titled "The Three Words to Remember in Dealing with the End EP" later that same year. The band released their debut studio album, "The Party Scene", in July 2005.


title: All Time Low

text: In December, it was announced that the band was no longer signed, but were attracting attention from a number of record labels. In late 2006, the band performed a showcase for John Janick the founder of record label Fueled by Ramen. They were not signed because Cute Is What We Aim For had recently been taken on by the label, which was not in a position to sign another band at the time. The band was brought to the attention of Hopeless Records by fellow touring band Amber Pacific; on March 28, 2006, it was announced that All Time Low had signed with Hopeless. The band said in an interview that they were starting to get serious about music while in their senior year of high school; following their graduation, the members focused on the group full-time, and released the "Put Up or Shut Up" EP in July. The EP entered the Independent Albums chart at No. 20 and the Top Heatseekers at No. 12.


title: All Time Low

text: All Time Low began a busy tour in support of the EP in late 2006. After the tour, the band began writing material for their second studio album.


title: All Time Low

text: In the summer of 2007, All Time Low played the Vans Warped Tour on the Smartpunk Stage. They made their live debut in the UK in late 2007 supporting Plain White T's.


title: All Time Low

text: All Time Low released their second studio album "So Wrong, It's Right" in September 2007. It peaked at No. 62 on the "Billboard" 200 and No. 6 on the Independent Albums chart. The second single from the album, "Dear Maria, Count Me In", which was written about a stripper, became the band's first single to reach the charts and peaked at No. 86 on the Pop 100. In 2011, the single was certified Gold for 500,000 shipments.


title: All Time Low

text: In early 2008 the band completed their first headlining tour, the Manwhores and Open Sores Tour with opening acts Every Avenue, Mayday Parade, and Just Surrender.


title: All Time Low

text: Following the release of "So Wrong, It's Right", All Time Low quickly gained popularity, eventually making their TRL debut on February 12, 2008. They have also been featured on MTV's "Discover and Download" and Music Choice's "Fresh Crops", and have been added to both MTV's "Big Ten" and MTV Hits playlists. On March 7, 2008, the band made their live television debut on Jimmy Kimmel Live! and then performed live at the mtvU Woodie Awards.


title: All Time Low

text: From March 2008 to May 2008, they co-headlined the AP Tour 2008 with The Rocket Summer; supported by acts such as The Matches, Sonny Moore, and Forever the Sickest Kids. In May 2008 they played at the Give It a Name Festival. Also in May 2008, they co-headlined a UK tour with Cobra Starship. In July 2008, the band headlined the Shortest Tour Ever with supporting acts Hit the Lights, Valencia, and There for Tomorrow. From mid-July to mid-August they played the 2008 Vans Warped Tour. They ended 2008 with their headlining tour, The Compromising of Integrity, Morality and Principles in Exchange for Money Tour with Mayday Parade, The Maine, and Every Avenue.


title: All Time Low

text: In December 2008, All Time Low was named "Band of the Year" by "Alternative Press" magazine and featured on the cover of their January 2009 issue.


title: All Time Low

text: In early 2009, All Time Low confirmed in an interview with UK magazine "Rock Sound" that they had begun writing new material for a third studio album and revealed they had collaborated with artists and producers to help co-write a number of songs.