title: Total Drama: Revenge of the Island

text: This line is used for the promotional trailer that Teletoon showed at the end of the "Total Drama World Tour" finale, which was first released on April 24, 2011 (Easter Sunday). On June 1, 2011 Christian Potenza released on his Facebook the new season logo along with more final designs of the characters. On June 3, 2011, there were snapshots of radioactive Camp Wawanakwa. On August 25, 2011 the "Total Drama: Revenge of the Island" flipbook was released. On August 28, 2011, the first official cast image was released to the public via Facebook. On September 24, 2011, the third official trailer was released which contained new scenes and confirmed the team assignments. On September 27, 2011, at the "Total Drama": Live Action Panel at Canada's 2011 Fan Expo, Cory Doran confirmed all of Mike's personalities, and Christian Potenza said that the team name for the Radioactive Rats was scrapped. On October 13, 2011, Christian Potenza revealed several things about the intro of the season and said that promos will be out soon. On November 28, 2011, the official Canadian air date for the season was announced to air on January 5, 2012. On December 14, 2011, an animated interview with Chris McLean and a fan was released and revealed that the new team name is now "Toxic Rats". On December 18, 2011, the official "Total Drama: Revenge of the Island" website was launched that includes interactive menus and biographies of all the new contestants, as well as videos and season descriptions. On December 21, 2011, France began to air the season's episodes, two weeks before Canada. On Christmas 2011, all of the auditions of the 13 new characters were released on Teletoon's official "Total Drama" site. Cartoon Network viewers were able to vote for Zoey before the finale due to them airing Episode 12 and 13 on the same day.


title: Total Drama: Revenge of the Island

text: "Total Drama: Revenge of the Island" has received mixed to positive reviews, attracting some criticism for its shorter length in comparison to the previous seasons. In the United States, over 3.3 million people have viewed the season, with over 51% growth over last year, making "Total Drama: Revenge of the Island" the #1 telecast of the year for Cartoon Network. Tom McGillis said in a 2012 interview that both "Total Drama: Revenge of the Island" and "Total Drama Island" are the highest rated seasons of the series. This season has been viewed in over 100 countries, worldwide. On Metacritic, the season currently holds a 7.3, which indicates "generally favorable reviews".


title: AVM Productions

text: AVM Productions is an Indian film production studio founded by A. V. Meiyappan. It is the final oldest survived studio in India. The filming studios are located in Vadapalani, Chennai. It has produced over 300 films in Tamil, Telugu, Kannada, Malayalam and Hindi cinema. AVM has introduced numerous actors in Southern industries, some of the prominent actors are Sivaji Ganesan, Rajkumar, S. S. Rajendran, Vyjayanthimala, Kamal Haasan and many more. The AVM Studios besides the shooting floors, has recording, dubbing and a preview theatre. The complex also houses facilities for production and post production processing.


title: AVM Productions

text: Born on 28 July 1907, Avichi Meiyappan hailed from a Nattukottai Nagarathar family of Karaikkudi in Tamil Nadu. Avichi Chettiar ran a mini-department store, named AV & Sons. It sold gramophone records. Meiyappan, who joined his father's shop even as a teenager, decided to produce gramophone records instead of merely marketing them. He came to Madras where along with friends K. S. Narayana Iyengar, Subbaiah Chettiar and others promoted Saraswathi Stores. He found excellent support in K. P. Varadachari, its manager, and his lawyer friend Thoothukudi Govindachari Raghavachari he produced many records.


title: AVM Productions

text: AVM Productions ventured into OTT space by announcing Tamil Stalkers, a crime thriller inspired from a real-life film piracy group, directed by Arivazhagan.


title: AVM Productions

text: The dawn of the talkie era (1931) inspired Meiyappan to start Saraswathi Sound Productions and he launched his maiden movie venture, Alli Arjuna a Hindu mythology based movie. The film was shot in Calcutta and proved a total flop as the one that followed named Ratnavali. At this point, an aspiring amateur actor and college graduate A. T. Krishnaswamy joined the unit as assistant director who was associated with Meiyappan for nearly a decade and wrote and directed the early AVM productions.


title: AVM Productions

text: The reverses forced Meiyappan to lie low but only for a while. In association with Jayanthilal, a cinema house owner based in Bangalore, he promoted a new company Pragati Pictures Limited. Grabbing an opportunity that came his way, AVM made Nandakumar, Tamil version of a Marathi film launching it as a Pragati production. The highlight of this movie was the debut of T. R. Mahalingam in to Tamil movie Industry.


title: AVM Productions

text: The film was a landmark because for the first time playback singing was tried, with Lalitha Venkatraman singing for the actor who played Devaki. Meiyappan took the sprawling Club House off Mount Road on lease and shot scenes without going to studios and erecting sets. Soon after he shifted his unit to another sprawling edifice known as Admiralty House in Adyar.


title: AVM Productions

text: In 1940, Meiyappan produced "Bhoo Kailas", a Telugu mythological film which created history. Its lead players were from Telugu and Kannada cinema, and was directed by Sundar Rao Nadkarni, a Mangalorean who had his training in Bombay. The film turned out to be a big hit and also won critical acclaim! Meiyappan struck gold with comedy next. "Sabapathy" (1941) with the saucer-eyed T. R. Ramachandran in the lead, along with Kali N. Rathnam and K. Sarangapani was a runaway success. Then came a series of hits like En Manaivi, Harishchandra (1943), Sri Valli.


title: AVM Productions

text: The city of Madras began to feel the strains of the World War II raging on elsewhere and to reduce the exposure from Japanese bombing, Meiyappan moved his facilities to Karaikudi. In the outskirts of the town karaikudi he occupied a drama auditorium with a large open area around it on long lease and erected a studio. Thus was born AVM Productions with AVM Studios.


title: AVM Productions

text: Hits like Nam Iruvar (1947), Parasakthi (1952), Andha Naal (1954), and the National Award-winning Hum Panchhi Ek Daal Ke (1957) followed.


title: AVM Productions

text: In 2007, AVM produced "Sivaji" which at 95 crore, was said to be the most expensive film ever made in the history of the Indian film industry at that time.


title: AVM Productions

text: In 2013, AVM led by Aruna Guhan and Aparna Guhan started producing films to be screened only on the net. Their first such production was a 55-minute-film, titled "Idhuvum Kadandhu Pogum".


title: Seo Ji-hoon (actor)

text: Seo Ji-hoon (born April 25, 1997) is a South Korean actor. Since his acting debut with "Signal" in 2016, Seo has appeared in dramas such as "Solomon's Perjury" (2016-2017), "School 2017" (2017), and "My First Love" (2018).


title: Seo Ji-hoon (actor)

text: Seo debuted as an actor in 2016, which he took a minor role in the drama "Signal" as the main culprit in the Inju female student case. Then he was cast in the web-drama "Matching! Boys Archery".


title: Seo Ji-hoon (actor)

text: Seo received his first KBS Drama Award nomination for his role in the one-act drama "The Legendary Shuttle". Later that year he starred in the mystery teen drama "Solomon's Perjury".


title: Seo Ji-hoon (actor)

text: In 2017, Seo starred in the teen drama "School 2017", and featured in the black comedy series "Prison Playbook".


title: Seo Ji-hoon (actor)

text: In 2018, Seo was cast in the fantasy romance drama "My First Love" as the younger version of Lee Jung-shin's character. He then played the younger version of the character Im Tae-kyung in the mystery drama "Misty". In May, Seo was cast in the fantasy romance drama "Tale of Fairy" alongside Moon Chae-won and Yoon Hyun-min.


title: Seo Ji-hoon (actor)

text: In 2019, Seo was cast in his first leading role in the tvN one-act drama "Drama Stage: Crumbling Friendship". The same year, Seo was cast in the youth historical drama "".


title: Seo Ji-hoon (actor)

text: In 2020, Seo starred in the romantic comedy "Men Are Men" alongside Hwang Jung-eum and Yoon Hyun-min. He won a KBS Drama Award for his roles in "Welcome" and "Men Are Men".


title: Ilaiyaraaja discography

text: Hindustan Coca-Cola collaborated with Ilaiyaraaja for its ad jingles. This is the first time Raja has done a commercial jingle.


title: Ilaiyaraaja discography

text: Many of Ilaiyaraaja's songs were used as jingles in advertisements. Filmmaker R Balki has used them in many of his ads either as a musical note or a jingle composed to the same tune.


title: Möbius transformation

text: In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form


title: Möbius transformation

text: of one complex variable "z"; here the coefficients "a", "b", "c", "d" are complex numbers satisfying "ad" − "bc" ≠ 0.


title: Möbius transformation

text: Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane.


title: Möbius transformation

text: These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.


title: Möbius transformation

text: The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics.


title: Möbius transformation

text: Möbius transformations are named in honor of August Ferdinand Möbius; they are also variously named homographies, homographic transformations, linear fractional transformations, bilinear transformations, fractional linear transformations, and spin transformations (in relativity theory).


title: Möbius transformation

text: Möbius transformations are defined on the extended complex plane formula_2 (i.e., the complex plane augmented by the point at infinity).


title: Möbius transformation

text: Stereographic projection identifies formula_3 with a sphere, which is then called the Riemann sphere; alternatively, formula_3 can be thought of as the complex projective line formula_5. The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of formula_5 as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, and is sometimes denoted formula_7.


title: Möbius transformation

text: The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds.


title: Möbius transformation

text: In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory.


title: Möbius transformation

text: Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations.


title: Möbius transformation

text: Möbius transformations can be more generally defined in spaces of dimension "n" > 2 as the bijective conformal orientation-preserving maps from the "n"-sphere to the "n"-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions.


title: Möbius transformation

text: where "a", "b", "c", "d" are any complex numbers satisfying . If , the rational function defined above is a constant since


title: Möbius transformation

text: Thus a Möbius transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere.


title: Möbius transformation

text: The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted formula_7 as it is the automorphism group of the Riemann sphere.


title: Möbius transformation

text: Every non-identity Möbius transformation has two fixed points formula_13 on the Riemann sphere. Note that the fixed points are counted here with multiplicity; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity.


title: Möbius transformation

text: are obtained by solving the fixed point equation "f"("γ") = "γ". For "c" ≠ 0, this has two roots obtained by expanding this equation to


title: Möbius transformation

text: Parabolic transforms have coincidental fixed points due to zero discriminant. For "c" nonzero and nonzero discriminant the transform is elliptic or hyperbolic.


title: Möbius transformation

text: When "c" = 0, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When "a" ≠ "d" the second fixed point is finite and is given by


title: Möbius transformation

text: In this case the transformation will be a simple transformation composed of translations, rotations, and dilations:


title: Möbius transformation

text: If "c" = 0 and "a" = "d", then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:


title: Möbius transformation

text: Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2:


title: Möbius transformation

text: Firstly, the projective linear group PGL(2,"K") is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially dimension counting, as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity.


title: Möbius transformation

text: Next, one can see by identifying the Möbius group with formula_22 that any Möbius function is homotopic to the identity. Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.


title: Möbius transformation

text: By contrast, the projective linear group of the real projective line, PGL(2,R) need not fix any points – for example formula_23 has no (real) fixed points: as a complex transformation it fixes ±"i" – while the map 2"x" fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.


title: Möbius transformation

text: Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form. We first treat the non-parabolic case, for which there are two distinct fixed points.


title: Möbius transformation

text: Every non-parabolic transformation is conjugate to a dilation/rotation, i.e., a transformation of the form


title: Möbius transformation

text: which sends the points ("γ", "γ") to (0, ∞). Here we assume that γ and γ are distinct and finite. If one of them is already at infinity then "g" can be modified so as to fix infinity and send the other point to 0.


title: Möbius transformation

text: If "f" has distinct fixed points ("γ", "γ") then the transformation formula_26 has fixed points at 0 and ∞ and is therefore a dilation: formula_27. The fixed point equation for the transformation "f" can then be written


title: Möbius transformation

text: Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers ("k") of "f" as the characteristic constant of "f". Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:


title: Möbius transformation

text: For loxodromic transformations, whenever |"k"| > 1, one says that γ is the repulsive fixed point, and γ is the attractive fixed point. For |"k"| < 1, the roles are reversed.


title: Möbius transformation

text: In the parabolic case there is only one fixed point "γ". The transformation sending that point to ∞ is


title: Möbius transformation

text: or the identity if "γ" is already at infinity. The transformation formula_26 fixes infinity and is therefore a translation:


title: Möbius transformation

text: Here, β is called the translation length. The fixed point formula for a parabolic transformation is then


title: Möbius transformation

text: Note that "β" is "not" the characteristic constant of "f", which is always 1 for a parabolic transformation. From the above expressions one can calculate:


title: Möbius transformation

text: The point formula_41 is called the pole of formula_42; it is that point which is transformed to the point at infinity under formula_42.


title: Möbius transformation

text: The inverse pole formula_44 is that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points:


title: Möbius transformation

text: These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation.


title: Möbius transformation

text: The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios formula_54 of the matrix


title: Möbius transformation

text: representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to


title: Möbius transformation

text: A Möbius transformation is equivalent to a sequence of simpler transformations. The composition makes many properties of the Möbius transformation obvious.


title: Möbius transformation

text: The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions "g", "g", "g", "g" such that each "g" is the inverse of "f". Then the composition


title: Möbius transformation

text: From this decomposition, we see that Möbius transformations carry over all non-trivial properties of circle inversion. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and isometries (translation, reflection, rotation), which trivially preserve angles.


title: Möbius transformation

text: Furthermore, Möbius transformations map generalized circles to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center.


title: Möbius transformation

text: Cross-ratios are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points formula_72 to four distinct points formula_73 respectively, then


title: Möbius transformation

text: If one of the points formula_72 is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of formula_76 is


title: Möbius transformation

text: The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles.


title: Möbius transformation

text: Two points "z" and "z" are conjugate with respect to a generalized circle "C", if, given a generalized circle "D" passing through "z" and "z" and cutting "C" in two points "a" and "b", are in harmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circle "D". This property is also sometimes referred to as being symmetric with respect to a line or circle.


title: Möbius transformation

text: Two points "z", "z" are conjugate with respect to a line, if they are symmetric with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle.


title: Möbius transformation

text: The point "z" conjugate to "z" when "L" is the line determined by the vector based "e" at the point "z" can be explicitly given as


title: Möbius transformation

text: The point "z" conjugate to "z" when "C" is the circle of radius "r" centered "z" can be explicitly given as


title: Möbius transformation

text: Since Möbius transformations preserve generalized circles and cross-ratios, they preserve also the conjugation.


title: Möbius transformation

text: The natural action of PGL(2,C) on the complex projective line CP is exactly the natural action of the Möbius group on the Riemann sphere, where the projective line CP and the Riemann sphere are identified as follows:


title: Möbius transformation

text: Here ["z":"z"] are homogeneous coordinates on CP; the point [1:0] corresponds to the point ∞ of the Riemann sphere. By using homogeneous coordinates, many concrete calculations involving Möbius transformations can be simplified, since no case distinctions dealing with ∞ are required.


title: Möbius transformation

text: The condition "ad" − "bc" ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero, i.e. that the matrix be invertible.


title: Möbius transformation

text: It is straightforward to check that then the product of two matrices will be associated with the composition of the two corresponding Möbius transformations. In other words, the map


title: Möbius transformation

text: from the general linear group GL(2,C) to the Möbius group, which sends the matrix formula_42 to the transformation "f", is a group homomorphism.


title: Möbius transformation

text: Note that any matrix obtained by multiplying formula_85 by a complex scalar λ determines the same transformation, so a Möbius transformation determines its matrix only up to scalar multiples. In other words: the kernel of consists of all scalar multiples of the identity matrix "I", and the first isomorphism theorem of group theory states that the quotient group GL(2,C) / ((C\{0})"I") is isomorphic to the Möbius group. This quotient group is known as the projective linear group and is usually denoted PGL(2,C).


title: Möbius transformation

text: The same identification of PGL(2,"K") with the group of fractional linear transformations and with the group of projective linear automorphisms of the projective line holds over any field "K", a fact of algebraic interest, particularly for finite fields, though the case of the complex numbers has the greatest geometric interest.


title: Möbius transformation

text: If one restricts formula_42 to matrices of determinant one, the map restricts to a surjective map from the special linear group SL(2,C) to the Möbius group; in the restricted setting the kernel is formed by plus and minus the identity, and the quotient group SL(2,C) / {±"I"}, denoted by PSL(2,C), is therefore also isomorphic to the Möbius group:


title: Möbius transformation

text: From this we see that the Möbius group is a 3-dimensional complex Lie group (or a 6-dimensional real Lie group). It is a semisimple non-compact Lie group.


title: Möbius transformation

text: Note that there are precisely two matrices with unit determinant which can be used to represent any given Möbius transformation. That is, SL(2,C) is a double cover of PSL(2,C). Since SL(2,C) is simply-connected it is the universal cover of the Möbius group. Therefore, the fundamental group of the Möbius group is Z.


title: Möbius transformation

text: Given a set of three distinct points formula_89 on the Riemann sphere and a second set of distinct points formula_90, there exists precisely one Möbius transformation formula_91 with formula_92 for formula_93. (In other words: the action of the Möbius group on the Riemann sphere is "sharply 3-transitive".) There are several ways to determine formula_91 from the given sets of points.


title: Möbius transformation

text: maps formula_97 to formula_98 respectively. If one of the "formula_99" is formula_100, then the proper formula for formula_101 is obtained from the above one by first dividing all entries by "formula_99" and then taking the limit "formula_103".


title: Möbius transformation

text: If formula_104 is similarly defined to map formula_90 to formula_98 then the matrix formula_42 which maps formula_108 to formula_109 becomes


title: Möbius transformation

text: in the formula_114-plane. The problem of constructing a Möbius transformation formula_115 mapping a triple formula_116 to another triple formula_117 is thus equivalent to finding the coefficients formula_118 of the hyperbola passing through the points formula_119. An explicit equation can be found by evaluating the determinant


title: Möbius transformation

text: for the coefficients formula_118 of the representing matrix formula_126. The constructed matrix formula_127 has determinant equal to formula_128 which does not vanish if the formula_99 resp. "formula_130" are pairwise different thus the Möbius transformation is well-defined. If one of the points "formula_99" or "formula_130" is formula_100, then we first divide all four determinants by this variable and then take the limit as the variable approaches formula_100.


title: Möbius transformation

text: If we require the coefficients formula_118 of a Möbius transformation to be real numbers with formula_136, we obtain a subgroup of the Möbius group denoted as PSL(2,R). This is the group of those Möbius transformations that map the upper half-plane to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps . If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane "H", the Poincaré half-plane model, and PSL(2,R) is the group of all orientation-preserving isometries of "H" in this model.


title: Möbius transformation

text: The subgroup of all Möbius transformations that map the open disk to itself consists of all transformations of the form


title: Möbius transformation

text: with formula_138 ∈ R, "b" ∈ C and |"b"| < 1. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps . By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the Poincaré disk model, and this group is the group of all orientation-preserving isometries of "H" in this model.


title: Möbius transformation

text: Since both of the above subgroups serve as isometry groups of "H", they are isomorphic. A concrete isomorphism is given by conjugation with the transformation


title: Möbius transformation

text: Alternatively, consider an open disk with radius "r", centered at "r""i". The Poincaré disk model in this disk becomes identical to the upper-half-plane model as "r" approaches ∞.


title: Möbius transformation

text: and corresponds under the isomorphism formula_142 to the projective special unitary group PSU(2,C) which is isomorphic to the special orthogonal group SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the point groups in three dimensions.


title: Möbius transformation

text: Icosahedral groups of Möbius transformations were used by Felix Klein to give an analytic solution to the quintic equation in ; a modern exposition is given in .


title: Möbius transformation

text: If we require the coefficients "a", "b", "c", "d" of a Möbius transformation to be integers with "ad" − "bc" = 1, we obtain the modular group PSL(2,Z), a discrete subgroup of PSL(2,R) important in the study of lattices in the complex plane, elliptic functions and elliptic curves. The discrete subgroups of PSL(2,R) are known as Fuchsian groups; they are important in the study of Riemann surfaces.


title: Möbius transformation

text: In the following discussion we will always assume that the representing matrix formula_143 is normalized such that formula_144.


title: Möbius transformation

text: Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, with the hyperbolic ones being a subclass of the loxodromic ones. The classification has both algebraic and geometric significance. Geometrically, the different types result in different transformations of the complex plane, as the figures below illustrate.


title: Möbius transformation

text: The four types can be distinguished by looking at the trace formula_145. Note that the trace is invariant under conjugation, that is,


title: Möbius transformation

text: and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix formula_42 has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations formula_148 (both not equal to the identity transform) with formula_149 are conjugate if and only if formula_150


title: Möbius transformation

text: A non-identity Möbius transformation defined by a matrix formula_42 of determinant one is said to be "parabolic" if


title: Möbius transformation

text: (so the trace is plus or minus 2; either can occur for a given transformation since formula_42 is determined only up to sign). In fact one of the choices for formula_42 has the same characteristic polynomial "X"−2"X"+1 as the identity matrix, and is therefore unipotent. A Möbius transform is parabolic if and only if it has exactly one fixed point in the extended complex plane formula_155, which happens if and only if it can be defined by a matrix conjugate to


title: Möbius transformation

text: The set of all parabolic Möbius transformations with a "given" fixed point in formula_3, together with the identity, forms a subgroup isomorphic to the group of matrices


title: Möbius transformation

text: this is an example of the unipotent radical of a Borel subgroup (of the Möbius group, or of SL(2,C) for the matrix group; the notion is defined for any reductive Lie group).


title: Möbius transformation

text: with the complex number "λ" not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number "k" = λ, called the characteristic constant or multiplier of the transformation.


title: Möbius transformation

text: The transformation is said to be "elliptic" if it can be represented by a matrix formula_85 whose trace is real with


title: Möbius transformation

text: A transform is elliptic if and only if |"λ"| = 1 and "λ" ≠ ±1. Writing formula_162, an elliptic transform is conjugate to


title: Möbius transformation

text: Note that for "any" formula_42 with characteristic constant "k", the characteristic constant of formula_165 is "k". Thus, all Möbius transformations of finite order are elliptic transformations, namely exactly those where λ is a root of unity, or, equivalently, where α is a rational multiple of . The simplest possibility of a fractional multiple means "α" = /2, which is also the unique case of formula_166, is also denoted as a ; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as:


title: Möbius transformation

text: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: formula_168 which fixes 1 and swaps 0 with "∞" (rotation by 180° about the points 1 and −1), formula_169, which fixes "∞" and swaps 0 with 1 (rotation by 180° about the points 1/2 and "∞"), and formula_170 which fixes 0 and swaps 1 with "∞" (rotation by 180° about the points 0 and 2).


title: Möbius transformation

text: The transform is said to be "hyperbolic" if it can be represented by a matrix formula_85 whose trace is real with


title: Möbius transformation

text: The transform is said to be "loxodromic" if formula_173 is not in [0,4]. A transformation is loxodromic if and only if formula_174.


title: Möbius transformation

text: Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below.


title: Möbius transformation

text: Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real conics. The terminology is due to considering half the absolute value of the trace, |tr|/2, as the eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/"n" is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. Alternatively one may use half the trace "squared" as a proxy for the eccentricity squared, as was done above; these classifications (but not the exact eccentricity values, since squaring and absolute values are different) agree for real traces but not complex traces. The same terminology is used for the classification of elements of SL(2, R) (the 2-fold cover), and analogous classifications are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.


title: Möbius transformation

text: The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case:


title: Möbius transformation

text: When expressed in this way, the real number ρ becomes an expansion factor. It indicates how repulsive the fixed point "γ" is, and how attractive "γ" is. The real number "α" is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about "γ" and clockwise about "γ".


title: Möbius transformation

text: If "ρ" = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be "elliptic". These transformations tend to move all points in circles around the two fixed points. If one of the fixed points is at infinity, this is equivalent to doing an affine rotation around a point.


title: Möbius transformation

text: If we take the one-parameter subgroup generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the "same" two points. All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere. In general, the two fixed points can be any two distinct points.


title: Möbius transformation

text: Imagine that some observer rotates with constant angular velocity about some axis. Then we can take the two fixed points to be the North and South poles of the celestial sphere. The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, ∞, and with the number α corresponding to the constant angular velocity of our observer.


title: Möbius transformation

text: Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):


title: Möbius transformation

text: These pictures illustrate the effect of a single Möbius transformation. The one-parameter subgroup which it generates "continuously" moves points along the family of circular arcs suggested by the pictures.


title: Möbius transformation

text: If "α" is zero (or a multiple of 2), then the transformation is said to be "hyperbolic". These transformations tend to move points along circular paths from one fixed point toward the other.


title: Möbius transformation

text: If we take the one-parameter subgroup generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the "same" two points. All other points flow along a certain family of circular arcs "away" from the first fixed point and "toward" the second fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.


title: Möbius transformation

text: This too has an important physical interpretation. Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points 0, ∞, with the real number ρ corresponding to the magnitude of his acceleration vector. The stars seem to move along longitudes, away from the South pole toward the North pole. (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane.)


title: Möbius transformation

text: Here are some figures illustrating the effect of a hyperbolic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):


title: Möbius transformation

text: These pictures resemble the field lines of a positive and a negative electrical charge located at the fixed points, because the circular flow lines subtend a constant angle between the two fixed points.


title: Möbius transformation

text: If both "ρ" and "α" are nonzero, then the transformation is said to be "loxodromic". These transformations tend to move all points in S-shaped paths from one fixed point to the other.


title: Möbius transformation

text: The word "loxodrome" is from the Greek: "λοξος (loxos), "slanting" + δρόμος (dromos), "course"". When sailing on a constant bearing – if you maintain a heading of (say) north-east, you will eventually wind up sailing around the north pole in a logarithmic spiral. On the mercator projection such a course is a straight line, as the north and south poles project to infinity. The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of "k". Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes.


title: Möbius transformation

text: If we take the one-parameter subgroup generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the "same" two points. All other points flow along a certain family of curves, "away" from the first fixed point and "toward" the second fixed point. Unlike the hyperbolic case, these curves are not circular arcs, but certain curves which under stereographic projection from the sphere to the plane appear as spiral curves which twist counterclockwise infinitely often around one fixed point and twist clockwise infinitely often around the other fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.


title: Möbius transformation

text: You can probably guess the physical interpretation in the case when the two fixed points are 0, ∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the "same" axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ρ, α determined respectively by the magnitude of the actual linear and angular velocities.


title: Möbius transformation

text: These images show Möbius transformations stereographically projected onto the Riemann sphere. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location.


title: Möbius transformation

text: If a transformation formula_42 has fixed points γ, γ, and characteristic constant "k", then formula_177 will have formula_178.


title: Möbius transformation

text: These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants.


title: Möbius transformation

text: And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. Note that in the elliptical and loxodromic images, the α value is 1/10 .


title: Möbius transformation

text: In higher dimensions, a Möbius transformation is a homeomorphism of formula_179, the one-point compactification of formula_180, which is a finite composition of inversions in spheres and reflections in hyperplanes. Liouville's theorem in conformal geometry states that in dimension at least three, all conformal transformations are Möbius transformations. Every Möbius transformation can be put in the form


title: Möbius transformation

text: In summary, the action of the restricted Lorentz group SO(1,3) agrees with that of the Möbius group PSL(2,C). This motivates the following definition. In dimension "n" ≥ 2, the Möbius group Möb("n") is the group of all orientation-preserving conformal isometries of the round sphere "S" to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R, there is an isomorphism of Möb("n") with the restricted Lorentz group SO(1,"n"+1) of Lorentz transformations with positive determinant, preserving the direction of time.


title: Möbius transformation

text: He identified the Lorentz group with transformations for which {"x" : Q("x") = -1} is stable. Then he interpreted the x's as homogeneous coordinates and {"x" : Q("x") = 0}, the null cone, as the Cayley absolute for a hyperbolic space of points {"x" : Q("x") < 0}. Next, Coxeter introduced the variables


title: Möbius transformation

text: so that the Lorentz-invariant quadric corresponds to the sphere formula_185 Coxeter notes that Felix Klein also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane formula_186 Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes.


title: Möbius transformation

text: As seen above, the Möbius group PSL(2,C) acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. Restricting to the points where "Q"=1 in the positive light cone, which form a model of hyperbolic 3-space "H", we see that the Möbius group acts on "H" as a group of orientation-preserving isometries. In fact, the Möbius group is equal to the group of orientation-preserving isometries of hyperbolic 3-space.


title: Möbius transformation

text: If we use the Poincaré ball model, identifying the unit ball in R with "H", then we can think of the Riemann sphere as the "conformal boundary" of "H". Every orientation-preserving isometry of "H" gives rise to a Möbius transformation on the Riemann sphere and vice versa; this is the very first observation leading to the AdS/CFT correspondence conjectures in physics.


title: Cytokinesis

text: Cytokinesis () is the part of the cell division process during which the cytoplasm of a single eukaryotic cell divides into two daughter cells. Cytoplasmic division begins during or after the late stages of nuclear division in mitosis and meiosis. During cytokinesis the spindle apparatus partitions and transports duplicated chromatids into the cytoplasm of the separating daughter cells. It thereby ensures that chromosome number and complement are maintained from one generation to the next and that, except in special cases, the daughter cells will be functional copies of the parent cell. After the completion of the telophase and cytokinesis, each daughter cell enters the interphase of the cell cycle.


title: Cytokinesis

text: Particular functions demand various deviations from the process of symmetrical cytokinesis; for example in oogenesis in animals the ovum takes almost all the cytoplasm and organelles. This leaves very little for the resulting polar bodies, which in most species die without function, though they do take on various special functions in other species.


title: Cytokinesis

text: Another form of mitosis occurs in tissues such as liver and skeletal muscle; it omits cytokinesis, thereby yielding multinucleate cells.


title: Cytokinesis

text: Plant cytokinesis differs from animal cytokinesis, partly because of the rigidity of plant cell walls. Instead of plant cells forming a cleavage furrow such as develops between animal daughter cells, a dividing structure known as the cell plate forms in the cytoplasm and grows into a new, doubled cell wall between plant daughter cells. It divides the cell into two daughter cells.


title: Cytokinesis

text: Cytokinesis largely resembles the prokaryotic process of binary fission, but because of differences between prokaryotic and eukaryotic cell structures and functions, the mechanisms differ. For instance, a bacterial cell has a Circular chromosome (a single chromosome in the form of a closed loop), in contrast to the linear, usually multiple, chromosomes of eukaryote. Accordingly, bacteria construct no mitotic spindle in cell division. Also, duplication of prokaryotic DNA takes place during the actual separation of chromosomes; in mitosis, duplication takes place during the interphase before mitosis begins, though the daughter chromatids do not separate completely before the anaphase.


title: Cytokinesis

text: The word "cytokinesis" () uses combining forms of "cyto-" + "kine-" + "-sis", New Latin from Classical Latin and Ancient Greek, reflecting "cell" and "kinesis" ("motion, movement"). It was coined by Charles Otis Whitman in 1887.


title: Cytokinesis

text: Origin of this term is from Greek (', a hollow), Latin derivative ' (cellular), Greek ("", movement).


title: Cytokinesis

text: Animal cell cytokinesis begins shortly after the onset of sister chromatid separation in the anaphase of mitosis. The process can be divided to the following distinct steps: anaphase spindle reorganization, division plane specification, actin-myosin ring assembly and contraction, and abscission. Faithful partitioning of the genome to emerging daughter cells is ensured through the tight temporal coordination of the above individual events by molecular signaling pathways.


title: Cytokinesis

text: Animal cell cytokinesis starts with the stabilization of microtubules and reorganization of the mitotic spindle to form the central spindle. The central spindle (or "spindle midzone") forms when non-kinetochore microtubule fibers are bundled between the spindle poles. A number of different species including "H. sapiens", "D. melanogaster" and "C. elegans" require the central spindle in order to efficiently undergo cytokinesis, although the specific phenotype associated with its absence varies from one species to the next (for example, certain Drosophila cell types are incapable of forming a cleavage furrow without the central spindle, whereas in both "C. elegans" embryos and human tissue culture cells a cleavage furrow is observed to form and ingress, but then regress before cytokinesis is complete).


title: Cytokinesis

text: The process of mitotic spindle reorganization and central spindle formation is caused by the decline of CDK1 activity during anaphase. The decline of CDK1 activity at the metaphase-anaphase transition leads to dephosphorylating of inhibitory sites on multiple central spindle components. First of all, the removal of a CDK1 phosphorylation from a subunit of the CPC (the chromosomal passenger complex) allows its translocalization to the central spindle from the centromeres, where it is located during metaphase. Besides being a structural component of the central spindle itself, CPC also plays a role in the phosphoregulation of other central spindle components, including PRC1 (microtubule-bundling protein required for cytokinesis 1) and MKLP1 (a kinesin motor protein). Originally inhibited by CDK1-mediated phosphorylation, PRC1 is now able to form a homodimer that selectively binds to the interface between antiparallel microtubules, facilitating spatial organization of the microtubules of the central spindle. MKLP1, together with the Rho-family GTPase activating protein CYK-4 (also termed MgcRacGAP), forms the centralspindlin complex. Centralspindlin binds to the central spindle as higher-order clusters. The centralspindlin cluster formation is promoted by phosphorylation of MLKP1 by Aurora B, a component of CPC. In short, the self-assembly of central spindle is initiated through the phosphoregulation of multiple central spindle components by the decline of CDK1 activity, either directly or indirectly, at the metaphase-anaphase transition. The central spindle may have multiple functions in cytokinesis including the control of cleavage furrow positioning, the delivery of membrane vesicles to the cleavage furrow, and the formation of the midbody structure that is required for the final steps of division.


title: Cytokinesis

text: The second step of animal cell cytokinesis involves division plane specification and cytokinetic furrow formation. Precise positioning of the division plane between the two masses of segregated chromosomes is essential to prevent chromosome loss. Meanwhile, the mechanism by which the spindle determines the division plane in animal cells is perhaps the most enduring mystery in cytokinesis and a matter of intense debate. There exist three hypotheses of furrow induction. The first is the astral stimulation hypothesis, which postulates that astral microtubules from the spindle poles carry a furrow-inducing signal to the cell cortex, where signals from two poles are somehow focused into a ring at the spindle. A second possibility, called the central spindle hypothesis, is that the cleavage furrow is induced by a positive stimulus that originates in the central spindle equator. The central spindle may contribute to the specification of the division plane by promoting concentration and activation of the small GTPase RhoA at the equatorial cortex. A third hypothesis is the astral relaxation hypothesis. It postulates that active actin-myosin bundles are distributed throughout the cell cortex, and inhibition of their contraction near the spindle poles results in a gradient of contractile activity that is highest at the midpoint between poles. In other words, astral microtubules generate a negative signal that increases cortical relaxation close to the poles. Genetic and laser-micromanipulation studies in C. elegans embryos have shown that the spindle sends two redundant signals to the cell cortex, one originating from the central spindle, and a second signal deriving from the spindle aster, suggesting the involvement of multiple mechanisms combined in the positioning of the cleavage furrow. The predominance of one particular signal varies between cell types and organisms. And the multitude and partial redundancy of signals may be required to make the system robust and to increase spatial precision.


title: Cytokinesis

text: At the cytokinesis furrow, it is the actin-myosin contractile ring that drives the cleavage process, during which cell membrane and wall grow inward, which eventually pinches the mother cell in two. The key components of this ring are the filamentous protein actin and the motor protein myosin II. The contractile ring assembles equatorially (in the middle of the cell) at the cell cortex (adjacent to the cell membrane). Rho protein family (RhoA protein in mammalian cells) is a key regulator of contractile ring formation and contraction in animal cells. The RhoA pathway promotes assembly of the actin-myosin ring by two main effectors. First, RhoA stimulates nucleation of unbranched actin filaments by activation of Diaphanous-related formins. This local generation of new actin filaments is important for the contractile ring formation. This actin filament formation process also requires a protein called profilin, which binds to actin monomers and helps load them onto the filament end. Second, RhoA promotes myosin II activation by the kinase ROCK, which activates myosin II directly by phosphorylation of the myosin light chain and also inhibits myosin phosphatase by phosphorylation of the phosphatase-targeting subunit MYPT. Besides actin and myosin II, the contractile ring contains the scaffolding protein anillin. Anillin binds to actin, myosin, RhoA, and CYK-4, and thereby links the equatorial cortex with the signals from the central spindle. It also contributes to the linkage of the actin-myosin ring to the plasma membrane. Additionally, anillin generates contractile forces by rectifying thermal fluctuations. Another protein, septin, has also been speculated to serve as a structural scaffold on which the cytokinesis apparatus is organized. Following its assembly, contraction of the actin-myosin ring leads to ingression of the attached plasma membrane, which partitions the cytoplasm into two domains of emerging sister cells. The force for the contractile processes is generated by movements along actin by the motor protein myosin II. Myosin II uses the free energy released when ATP is hydrolyzed to move along these actin filaments, constricting the cell membrane to form a cleavage furrow. Continued hydrolysis causes this cleavage furrow to ingress (move inwards), a striking process that is clearly visible through a light microscope.


title: Cytokinesis

text: The cytokinetic furrow ingresses until a midbody structure (composed of electron-dense, proteinaceous material) is formed, where the actin-myosin ring has reached a diameter of about 1–2 μm. Most animal cell types remain connected by an intercellular cytokinetic bridge for up to several hours until they are split by an actin-independent process termed abscission, the last step of cytokinesis.


title: Cytokinesis

text: The process of abscission physically cleaves the midbody into two. Abscission proceeds by removal of cytoskeletal structures from the cytokinetic bridge, constriction of the cell cortex, and plasma membrane fission. The intercellular bridge is filled with dense bundles of antiparallel microtubules that derive from the central spindle. These microtubules overlap at the midbody, which is generally thought to be a targeting platform for the abscission machinery.


title: Cytokinesis

text: The microtubule severing protein spastin is largely responsible for the disassembly of microtubule bundles inside the intercellular bridge. Complete cortical constriction also requires removal of the underlying cytoskeletal structures. Actin filament disassembly during late cytokinesis depends on the PKCε–14-3-3 complex, which inactivates RhoA after furrow ingression. Actin disassembly is further controlled by the GTPase Rab35 and its effector, the phosphatidylinositol-4,5-bisphosphate 5-phosphatase OCRL. Understanding the mechanism by which the plasma membrane ultimately splits requires further investigation.


title: Cytokinesis

text: Cytokinesis must be temporally controlled to ensure that it occurs only after sister chromatids separate during the anaphase portion of normal proliferative cell divisions. To achieve this, many components of the cytokinesis machinery are highly regulated to ensure that they are able to perform a particular function at only a particular stage of the cell cycle. Cytokinesis happens only after APC binds with CDC20. This allows for the separation of chromosomes and myosin to work simultaneously.


title: Cytokinesis

text: After cytokinesis, non-kinetochore microtubules reorganize and disappear into a new cytoskeleton as the cell cycle returns to interphase (see also cell cycle).


title: Cytokinesis

text: Due to the presence of a cell wall, cytokinesis in plant cells is significantly different from that in animal cells, Rather than forming a contractile ring, plant cells construct a cell plate in the middle of the cell. The stages of cell plate formation include (1) creation of the phragmoplast, an array of microtubules that guides and supports the formation of the cell plate; (2) trafficking of vesicles to the division plane and their fusion to generate a tubular-vesicular network; (3) continued fusion of membrane tubules and their transformation into membrane sheets upon the deposition of callose, followed by deposition of cellulose and other cell wall components; (4) recycling of excess membrane and other material from the cell plate; and (5) fusion with the parental cell wall


title: Cytokinesis

text: The phragmoplast is assembled from the remnants of the mitotic spindle, and serves as a track for the trafficking of vesicles to the phragmoplast midzone. These vesicles contain lipids, proteins and carbohydrates needed for the formation of a new cell boundary. Electron tomographic studies have identified the Golgi apparatus as the source of these vesicles, but other studies have suggested that they contain endocytosed material as well.


title: Cytokinesis

text: These tubules then widen and fuse laterally with each other, eventually forming a planar, fenestrated sheet . As the cell plate matures, large amounts of membrane material are removed via clathrin-mediated endocytosis Eventually, the edges of the cell plate fuse with the parental plasma membrane, often in an asymmetrical fashion, thus completing cytokinesis. The remaining fenestrae contain strands of endoplasmic reticulum passing through them, and are thought to be the precursors of plasmodesmata .


title: Cytokinesis

text: The construction of the new cell wall begins within the lumen of the narrow tubules of the young cell plate. The order in which different cell wall components are deposited has been determined largely by immuno-electron microscopy. The first components to arrive are pectins, hemicelluloses, and arabinogalactan proteins carried by the secretory vesicles that fuse to form the cell plate. The next component to be added is callose, which is polymerized directly at the cell plate by callose synthases. As the cell plate continues to mature and fuses with the parental plasma membrane, the callose is slowly replaced with cellulose, the primary component of a mature cell wall . The middle lamella (a glue-like layer containing pectin) develops from the cell plate, serving to bind the cell walls of adjoining cells together.


title: Cytokinesis

text: Cytokinetic furrow ingression is powered by Type II Myosin ATPase. Since Myosins are recruited to the medial region, the contractile forces acting on the cortex resemble a 'purse string' constriction pulling inwards. This leads to the inward constriction. The plasma membrane by virtue of its close association with the cortex via crosslinker proteins To the constriction of the cleavage furrow, the total surface area should be increased by supplying the plasma membrane via exocytosis </ref>.


title: Cytokinesis

text: CEP55 is a mitotic phosphoprotein that plays a key role in cytokinesis, the final stage of cell division.


title: NCIS (season 8)

text: The eighth season of the police procedural drama "NCIS" premiered on September 21, 2010 in the same time slot as the previous season.


title: NCIS (season 8)

text: The season story arc involves Ziva's largely-unseen boyfriend, Ray, and the CIA continuing to meddle in NCIS's day-to-day workings. Notable events include the terrorism and internal affairs threat during the "Enemies" two-parter, and the arrival of another Major Case Response Team from Rota, Spain, the team that Tony was offered to be lead of in the beginning of season four. The season ends on a five-part story arc involving the Port-to-Port killer that menaced both teams. The finale aired on May 17, 2011.


title: Trouble (rapper)

text: Mariel Semonte Orr (November 4, 1987 – June 5, 2022), known professionally as Trouble, was an American rapper.


title: Trouble (rapper)

text: Mariel Semonte Orr was born on November 4, 1987, in Atlanta, Georgia. He started rapping at the age of 14.


title: Trouble (rapper)

text: In April 2011, Trouble released his debut mixtape "December 17th", which featured on "Complex" "25 Best Mixtapes of 2011". The mixtape included the song "Bussin'", which featured Yo Gotti, Waka Flocka Flame, and Trae tha Truth on the remix. In August 2011, "Complex" included Trouble in its "15 New Rappers To Watch Out For" list. In 2013, he was included in "XXL" magazine's "15 Atlanta Rappers You Should Know" list.


title: Trouble (rapper)

text: In 2015, Trouble appeared on Lupe Fiasco's album "Tetsuo & Youth", along with Glasses Malone, Trae tha Truth, Billy Blue, Buk, and Fam-Lay in the track "Chopper".


title: Trouble (rapper)

text: In 2016, Trouble appeared on rapper YFN Lucci's single "Key to the Streets", which peaked at number 70 on the "Billboard" Hot 100.


title: Trouble (rapper)

text: In March 2016, Trouble and labelmate Veli Sosa filmed a music video in North Carolina titled "Straight Out" from his 2016 mixtape "Skoobzilla" and "Like A" which featured ReeseDaGreat, who is also signed to DTE. The videos were filmed by director LookImHD and guest starred Big Bank Black as a cameo in the videos, a snippet of which was later posted on Instagram.


title: Trouble (rapper)

text: In 2017, Trouble signed a record deal with Mike Will Made It's record label, Ear Drummer Records and Interscope Records.


title: Trouble (rapper)

text: On June 5, 2022, Orr was shot during a home invasion at the Rockdale County apartment of a female companion. He was transported to a hospital, where he died at age 34. The suspect, Jamichael Jones, did not know Orr and was involved in a domestic dispute with the woman.


title: Murder, She Baked

text: Murder, She Baked is an American/Canadian television film series, based on the cozy mystery novels written by Joanne Fluke. The television films centered around small-town baker Hannah Swensen, portrayed by Alison Sweeney, and Detective Mike Kingston, played by Cameron Mathison. Five films for the Hallmark Movies & Mysteries channel were created based on Fluke's book series, with the fifth movie airing on March 26, 2017. The films took a four year hiatus before returning as the Hannah Swensen Mysteries. The films are set in the fictional town of Eden Lake in Minnesota, reverting to Lake Eden like in the novels in the sixth film.


title: Orange River

text: The Orange River (from Afrikaans/Dutch: Oranjerivier) is a river in Southern Africa. It is the longest river in South Africa. With a total length of , the Orange River Basin extends from Lesotho into South Africa and Namibia to the north. It rises in the Drakensberg mountains in Lesotho, flowing westwards through South Africa to the Atlantic Ocean. The river forms part of the international borders between South Africa and Lesotho and between South Africa and Namibia, as well as several provincial borders within South Africa. Except for Upington, it does not pass through any major cities. The Orange River plays an important role in the South African economy by providing water for irrigation and hydroelectric power. The river was named the Orange River in honour of the Dutch ruling family, the House of Orange, by the Dutch explorer Robert Jacob Gordon. Other names include simply the word for river, in Khoekhoegowab orthography written as !Garib, which is rendered in Afrikaans as Gariep River with the intrusion of a velar fricative in place of the alveolar click, Groote River (derived from Kai !Garib) or Senqu River (used in Lesotho), derived from ǂNū "Black". It is known in isiZulu as isAngqu


title: Orange River

text: The Orange rises in the Drakensberg mountains along the border between South Africa and Lesotho, about west of the Indian Ocean and at an altitude of over . The extremity of the Orange River inside Lesotho is known as the Senqu. Parts of the Senqu River freeze in winter because of the high altitude. This creates droughts downstream, which mainly affect goat and cattle production.


title: Orange River

text: The Orange River then runs westward through South Africa, forming the south-western boundary of the Free State province. In this section, the river flows first into the Gariep Dam and later into the Vanderkloof Dam. From the border of Lesotho to below the Vanderkloof Dam, the river bed is deeply incised. Further downstream, the land is flatter, and the river is used extensively for irrigation.


title: Orange River

text: At the western point of the Free State, southwest of Kimberley, the Orange meets with its main tributary, the Vaal River, which forms much of the northern border of the province. From here, the river flows further westward through the arid wilderness of the southern Kalahari region and Namaqualand in the Northern Cape province to meet with Namibia at 20°E longitude. From here, it flows westward for , forming the international border between the province and Namibia's ǁKaras Region. On the border, the river passes the town of Vioolsdrif, the main border post between South Africa and Namibia.


title: Orange River

text: In the last of its course, the Orange receives many intermittent streams, and several large wadis lead into it. In this section, the Namib Desert terminates on the north bank of the river, so under normal circumstances, the volume of water added by these tributaries is negligible. Here, the bed of the river is once again deeply incised. The Augrabies Falls are located on this section of the Orange, where the river descends in a course of .


title: Orange River

text: The Orange empties into the Atlantic Ocean between the small towns of Oranjemund (meaning "Orange mouth") in Namibia and Alexander Bay in South Africa, about equidistant between Walvis Bay and Cape Town. Some from its mouth, it is obstructed by rapids and sand bars and is generally not navigable for long stretches.


title: Orange River

text: In the dry season, the volume of the water in the river is considerably reduced because of the rapid run-off and evaporation. At the source of the Orange, the rainfall is about per annum, but precipitation decreases as the river flows westward; at its mouth, the rainfall is less than per year. The factors that support evaporation, though, tend to increase in a westerly direction. In the wet season (summer), the Orange river becomes a brown colored torrent. The huge mass of sediment carried constitutes a long-term threat to engineering projects on the river.


title: Orange River

text: The total catchment of the Orange River (including the Vaal) extends over , i.e. equivalent to about 77% of the land area of South Africa (). Around (38%), however, are situated outside the country in Lesotho, Botswana, and Namibia.


title: Orange River

text: Some of the earliest precolonial inhabitants called the river "ǂNūǃarib", referring to its black colour, or sometimes just "Kai !Arib" ("Great River"), from which is derived the Afrikaans version "Gariep", and translation "Groote Rivier". The early Dutch name for the river was just that translation, Groote Rivier, meaning "Great River". The river was named the Orange River by Colonel Robert Gordon, commander of the United East India Company (VOC) garrison at Cape Town, on a trip to the interior in 1779. Gordon named the river in honor of William V of Orange. A popular but incorrect belief is that the river was named after the supposedly orange color of its water, as opposed to the color of its tributary, the Vaal River, which name is derived from the name ǀHaiǃarib "pale river" ("vaal" being Afrikaans for pale or grey). Since the end of apartheid, the name "Gariep" has had greater favour in official correspondence in South Africa, although the name "Orange" has greater international recognition. In Lesotho, where the river rises, it is known as the Senqu River, derived from the original Khoemana name.


title: Orange River

text: The Eastern Cape Geographical Names Committee has advertised its intention to consider a name change from the colonial name, for that portion of the river that forms the border between the Eastern Cape and the Free State, with suggestions being IGqili or Senqu. The advertisement placed in the "Aliwal Weekblad" newspaper states that the "present name is perceived to have a strong association with the history of colonial subjugation and has therefore no place under the current democratic dispensation."


title: Orange River

text: In South African folklore, the Orange River is often associated with the Grootslang, a mythical being resembling a giant serpent, which is often connected to the river's alluvial diamonds. The Grootslang is described as living in a gem-filled cave connected to the Orange River by a natural pipe through which the diamonds gradually enter the river. Other sites said to be lairs of the creature include the pool beneath the King George Cataract at Aughrabies Falls, which is also said to be a source of diamonds, and a large rock in the middle of the river itself. In this version of the legend, the Grootslang is also said to prey on cattle from the river's banks.


title: Orange River

text: As the collection point for the majority of South Africa's water, the Orange River plays a major role in supporting agriculture, industry, and mining. To assist in this, two large water schemes have been created, the Orange River Project and the Lesotho Highlands Water Project. Historically, the river played an important role in the South African diamond rush, with the first diamonds in the country being discovered in alluvial deposits on the Orange. Today, several commercial diamond mines operate along the final stretch of the Orange River and around its mouth.


title: Orange River

text: Because of the lack of dangerous animals and high water levels during summer, the river is used for recreational canoeing and rafting. Orange River rafting has become very popular with many companies using their camps along the river from which to operate. The most popular trips are four-day and six-day river trips that take place either along the gorge below the Augrabies Falls or along the Richtersveld area.


title: Orange River

text: The Orange River Project (ORP) was one of the largest and most imaginative projects of its kind in South Africa. It was constructed by Hendrik Verwoerd's government at the height of the apartheid era. The ORP was built to exploit the waters of the Orange River—which, without the Vaal River, represents some 14.1% of the total runoff in South Africa—and in the process, to satisfy an increasing demand for water. The main objectives of the project were:


title: Orange River

text: The Gariep Dam near Colesberg is the main storage structure within the Orange River. From here, the water is supplied in two directions, westward along the Orange River (via hydroelectric power generators) to the Vanderkloof Dam and southward through the Orange-Fish Tunnel to the Eastern Cape.


title: Orange River

text: Eskom operates hydroelectric power stations at both the Gariep Dam and the Vanderkloof Dam. The hydroelectric power station at the Vanderkloof Dam was the first power-generation station in South Africa situated entirely underground. The towns Oviston and Oranjekrag were established to facilitate the construction and operation of the new infrastructure.


title: Orange River

text: Irrigation in the vast area downstream of the Vanderkloof Dam, which has turned thousands of hectares of arid veld into highly productive agricultural land, was made possible by the construction of the Gariep and Vanderkloof Dams. Old established irrigation schemes such as those at Buchuberg, Upington, Kakamas, and Vioolsdrif have also benefitted because regulation of the flow is now possible. On the Namibian side of the river, Aussenkehr produces grapes with the help of water from the Orange.


title: Orange River

text: In recent years, the wine-producing areas along the Orange River have grown in importance. Irrigation in the Eastern Cape has also received a tremendous boost, not only from the additional water being made available, but also owing to improvement in water quality. Without this improvement, the citrus farmers along the Lower Sundays River would almost certainly have continued to suffer losses of productivity.


title: Orange River

text: The Lesotho Highlands Water Project was conceived to supplement the water supply in the Vaal River System. Water is delivered to South Africa by means of the delivery tunnel which passes under the Lesotho South Africa border at the Caledon River, and then under the Little Caledon River south of Clarens in the Free State, and discharges into the Ash River about further to the north. The scheme became viable when water demands in Gauteng reached levels that could no longer be supported economically by alternative schemes such as the Tugela River-Vaal River pumped storage scheme, which used the Sterkfontein Dam, located near Harrismith in the Free State.


title: Orange River

text: In 1867, the first diamond discovered in South Africa, the Eureka Diamond, was found near Hopetown on the Orange River. Two years later, a much larger diamond known as the Star of South Africa was found in the same area, causing a diamond rush. This was soon eclipsed by the diamond rush to mine diamonds directly from kimberlite at Kimberley in 1871, although alluvial diamonds continued to be found in the Orange. Today, several commercial diamond mines operate on the last stretch of the river, as well as the beaches around its mouth. Diamond mines also operate on the middle stretch of the river.


title: Orange River

text: During the temperate months of March and April, given good rains and the sluices of the dams being open, a canoeist (or rafter) can easily travel per day. The lower reaches of the river are most popular, because of the spectacular topography. Commercial tours are available, and these expeditions depart from the border town of Vioolsdrif.


title: Orange River

text: The Orange River has no large animals. It lies outside the range of the Nile crocodile, and although hippopotami were once abundant, they were hunted to extermination in the 19th century. The Orange River has a relative paucity of species diversity. A 2011 survey of 13,762 fish found only 16 species of fish present. Three of these, the common carp, the Mozambique tilapia, and the western mosquitofish are not indigenous. Another exotic species, rainbow trout, is found in the river headwaters in Lesotho.


title: Three Days of Happiness

text: Unable to make ends meet in Japan with his part-time job, 20 year old Kusonoki prepares to sell the last of his possessions. After visiting a library to sell books, the clerk tells Kusonoki of a place where he can sell his lifespan for actual money. Everything between how impactful someone is to the world and how many people you have affected is taken into consideration for your lifespan. After a complicated path towards this fabled place, he enters the store and meets a mysterious girl named Miyagi. After a couple hours for his evaluation, Kusonoki is informed his lifespan is only worth ¥300,000 (about $2000), abysmally lower than the universal average. Realizing that his future is worthless, he decides to sell it keeping only 3 months so he may enjoy the money. The next day he awakens to Miyagi knocking on his door. As a policy of the company, she was sent to watch over Kusonoki for the next 3 months until his death. What follows is an introspective journey pondering themes such as potential, true happiness, and what makes time valuable.


title: Three Days of Happiness

text: Written by Sugaru Miaki and illustrated by Shōichi Taguchi, "Three Days of Happiness" was published in a single volume by ASCII Media Works under their Media Works Bunko imprint on December 25, 2013.


title: Three Days of Happiness

text: The novel was licensed for English release in North America by Yen Press; it was released on October 20, 2020.


title: Three Days of Happiness

text: A manga adaptation of the novel by Shōichi Taguchi, was serialized on Shueisha's Shueisha's website and smartphone app "Shōnen Jump+" from August 10, 2016, to October 25, 2017. Shueisha collected its chapters in three "tankōbon" volumes, released from July 4 to December 4, 2017.


title: Skull Island

text: Skull Island is the name most often used to describe a fictional island that first appeared in the 1933 film "King Kong" and later appearing in its sequels, the three remakes, and any other King Kong-based media. It is the home of the eponymous King Kong and several other species of creatures, mostly prehistoric and in some cases species that should have been extinct long before the rise of mammalian creatures, along with a primitive society of humans.


title: Skull Island

text: In the 1962 film "King Kong vs. Godzilla" and the 1967 film "King Kong Escapes", the comparable islands are called "Farou Island" and "Mondo Island", respectively. Kong plays a similar role in these islands as the god-like being of the land, a role he plays in all versions of the King Kong story. Skull Island's origins are unknown, but Kong appears to be the only giant gorilla known to exist on the island. However, the 2005 remake shows other skeletons of Kong-sized gorillas, indicating that there was once a group of such creatures of an unknown number living on the island. Additionally, 2017's "" depicts the skeletons of Kong's parents.


title: Skull Island

text: In "King Kong", the island is never mentioned by name and is located at approximately  – 2000 miles west from the coast of Periuk, Indonesia. There is a distinctive rocky knoll in the center of the island that is shaped like a human skull, which is referred to as "Skull Mountain".


title: Skull Island

text: According to the first film, the captain of a Norwegian barque discovered a canoe blown off course with only one native left alive. Before the native died, the captain of the barque was able to get a rough location of the island and some details on it, including its most distinctive feature - a huge wall built by the ancestors of the natives back when they had high civilization. When the protagonists of the picture arrive at the island and examine it they find,


title: Skull Island

text: in addition to the expected superstitious natives, prehistoric creatures of all sorts and one extremely large ape, known by those on the island as "Kong".


title: Skull Island

text: Other than being the descendants of a high civilization, the ancestry of the natives is never explained.


title: Skull Island

text: In the sequel film "The Son of Kong", Carl Denham returns to Skull Island when there was a mention of some treasure that was hidden there. He also encounters a large white gorilla who is the son of King Kong. Skull Island sinks into the sea during a powerful earthquake. Kong's son drowns while holding Carl Denham above the water. Denham survives unscathed, while the treasure is claimed by him and the other three survivors.


title: Skull Island

text: The term "Skull Island" is never used in the original films. In "King Kong", only "Skull Mountain" is named, while in the sequel "Son of Kong", its simply referred to as "Kong's island". In the novelization of "King Kong" by Delos Lovelace, it is called "Skull Mountain Island", but RKO referred to it as "Skull Island" in some of their publicity materials.


title: Skull Island

text: In the 1976's "King Kong", Skull Island is mentioned as being "the beach of the skull". It is located somewhere in Indian Ocean south of Java, covered by a permanent cloud bank. Various people are said to have visited the island through the centuries but returned with no clear proof. By the 1970s, it is an urban legend whose only evidence is in government secret files. It is also revealed to have a huge deposit of oil, which led a greedy oil company executive to go in search of it. Besides featuring a primitive native tribe and the giant, bipedal gorilla Kong, there is also a giant snake who appeared in Kong's lair and wanted to kill him and Dwan, but it was killed by the former.


title: Skull Island

text: In 1986 sequel film "King Kong Lives", Skull Island does not appear but it was mentioned by adventurer Hank Mitchell (mistakenly calling it Kong Island) who believed that the islands Borneo and Skull Island were once a part of the same landmass in the past. The fate of this incarnation is unknown.


title: Skull Island

text: In "", the island was named Kong Island. Unlike previous incarnations, Kong Island was situated in the infamous Bermuda Triangle, not the Indian Ocean. Although various prehistoric creatures are seen living there, Kong Island also contains some ruins where one of them serves as the prison of the demon Chiros.


title: Skull Island

text: It is also where Jason Jenkins and his grandmother Dr. Lorna Jenkins also live along with Jason's friend Tan. Another human inhabitant is Lua, the sole survivor of the native people of the island and a female shaman.


title: Skull Island

text: A 2004 illustrated novel that serves as both prequel and sequel of the original "King Kong" story, conceived by Merian C. Cooper and novelized by Delos Lovelace in 1932, and authorized by the Cooper family.


title: Skull Island

text: Created and Illustrated by Joe DeVito [and novelized by Brad Strickland and DeVito, with John Michlig], "Kong: King of Skull Island" depicts a Skull Island far larger than originally thought. It is either the last vestige of a volatile volcanic series of islands or the remnant of a larger landmass. Skull Island is located in the Indian Ocean, west of Sumatra, and has several much smaller islands in various locations around its perimeter, with the most prevalent of these off a small peninsula on its southeastern corner.


title: Skull Island

text: Skull Island contains two main mountains, the larger being shaped in the visage of a human skull. Skull Mountain is riddled with caves and passageways carved out by natural erosion, but also man-made. In earlier times, these were often used by islanders to avoid the monsters on the surface, but proved to have a multitude of dangerous denizens of their own. These include previously unknown species such as snake-like amphibians with enormous heads capable of swallowing a human whole; foot-long creatures that combine squid and crustacean characteristics with bio-luminescent sails tipped with poisonous stingers; giant spiders with 7-foot bodies and 8-foot-long legs ending in hand-like appendages.


title: Skull Island

text: The human civilization that inhabits Skull Island at least through the 20th century is the last remnant of a previously unknown super-race called the Tagatu (a combination of two formerly separate tribes: the Tagu and the Atu). The original culture is believed to have Asian origins that bled into an island group west of Sumatra that no longer exists. As a result of a natural disaster, they were forced to migrate to Skull Island untold millennia ago. Drawn to its spectacular makeup by their insatiable curiosity, the Tagatu believed their mastery of biological and organic sciences could overcome Skull Island's dangers. It was they who originally brought the gigantic simian Kongs there for protection and who, with the help of the Kongs, built the iconic wall across the Skull Island's peninsula for protections against Skull Island's prehistoric denizens.


title: Skull Island

text: Among the many mysteries revealed in the work is the reality that on Skull Island the dinosaurs never died out, but continued to evolve over the intervening 65 million years. This has resulted in strange variations on previously known species, as well as many new ones. Chief among the latter is a race of sentient dinosaurs, called Deathrunners. Bipedal, extremely aggressive and 6 to 9 feet tall, they once ruled the island and were at war with the Tagatu and the Kongs. Their race is propagated every few generations by one queen that grows to gigantic size. It is one of these, called “Gaw”, that ruled Skull Island when King Kong was born and who Kong had to defeat in order to become a king.


title: Skull Island

text: In the story, Carl Denham's son, Vincent (now a paleontologist), and an older Jack Driscoll return to Skull Island in 1957 to discover pieces of the earlier civilization's history and relationship to the island through an enigmatic Tagatu elder simply called the “Storyteller”, as well as in the form of archaeological discoveries such as specially pigmented paintings in underground passageways that move when illuminated by torchlight to reveal past events on Skull Island. Primary among these is the discovery of remnants of “the Old City”, which was established by the Tagatu at the zenith of their civilization in the center of the island in an ideal valley between the two mountains. The Storyteller's tale, along with Vincent and Driscoll's findings, suggest that the eerie skull visage that gave the island its name may have been the work of human hands. These and other discoveries hint that the true extent of Skull Island's secrets have yet to be revealed.


title: Skull Island

text: In Peter Jackson's remake, Skull Island's position west of Sumatra remains the same, in a region afflicted by magnetic anomalies and violent sea storms. According to the book "", Skull Island was geologically unstable and has been slowly sinking into the sea for the past thousand years. By 1933, the island was on the verge of destruction. Fifteen years after its discovery to the modern world, Skull Island finally sank into the ocean.


title: Skull Island

text: In its prehistory, Skull Island was a refuge for a variety of prehistoric creatures. Over time, more and more species arrived either by swimming, flying, rafting, or migrating through temporary land bridges. As the island slowly receded into the sea, life was forced to adapt, resulting in an ecosystem of bizarre and nightmarish creatures.


title: Skull Island

text: Three thousand years before, an advanced civilization from Southeast Asia migrated to Skull Island, bringing with them domesticated animals such as Gaur and the giant ancestors of Kong. This culture eventually died off, leaving behind only gigantic eroding ruins scattered around the island (such as the enormous wall) and a small society of primitive people that became the Skull Island natives.


title: Skull Island

text: To coincide with the 80th anniversary of both characters, Altus Press announced on January 29, 2013, that King Kong would meet pulp hero Doc Savage in a new, officially sanctioned book written by Will Murray and artist Joe DeVito, who will also do the cover artwork. Set in 1920, shortly after returning from military service during World War I, Doc Savage searches for his long-lost grandfather (the legendary mariner Stormalong Savage) with his father, the explorer Clark Savage, Sr., that ultimately leads father and son to the mysterious Skull Island and its prehistoric denizens, including King Kong. "Doc Savage: Skull Island" was released in March 2013. In his review for the "New York Journal of Books", playwright-author Mark Squirek concluded:


title: Skull Island

text: Altus Press has announced an authorized crossover novel, "King Kong vs. Tarzan", for release in the summer of 2016. Written by Will Murray, it tells the previously untold story of the transportation by cargo ship of Kong from Skull Island to America, and King Kong's inevitable encounter with Tarzan of the Apes.


title: Skull Island

text: Skull Island is the main setting of "", which is set in the same universe of Gareth Edwards' 2014 film "Godzilla". Kong is tall in the film and there is evidence that more of Kong's species once existed on the island. The island is located in the South Pacific and sits in the eye of a massive swirling storm system that conceals it from the outside world. This version of the island resembles a human skull when seen from the air. The island is situated atop an entrance to the Hollow Earth, which is home to large, bipedal, predators known as "Skullcrawlers" that are referred to as the island's "devils". The Skullcrawlers wiped out King Kong's family, making him the last of his kind.


title: Skull Island

text: There is also evidence of dinosaurs living, or having lived, on the island, in the form of a "Triceratops" skull found in the "graveyard" of the island. A number of other species are referred to as "florafauna" for displaying physiological traits of plants. Similar to previous incarnations, there is a human native tribe present (characterized as Iwis) who are much less hostile than in previous versions, mainly because a stranded air pilot from World War II named Hank Marlow makes peace between them and modern-day humans. The film sees an expedition in 1973 landing upon the island after Skull Island is detected by Landsat.


title: Skull Island

text: During the credits of "", a newspaper clipping states that the awakened Titans are converging on Skull Island, which is becoming unstable. It is also revealed that Monarch has an outpost on Skull Island. Between the events of the film and "Godzilla vs. Kong", the island's climate deteriorated, with a remnant of King Ghidorah's global superstorm merging with Skull Island's pre-existing storm system causing it to move inwards and make it uninhabitable, save for a domed habitat created to contain Kong.


title: Skull Island

text: Besides King Kong and the natives, much of the Wildlife on Skull Island consist of many kinds of prehistoric animals from each different era such as Dinosaurs. However some of them are fictional descendants of real animals.


title: Skull Island

text: All the creatures in Peter Jackson's remake of "King Kong" are established in the companion book "The World of Kong: A Natural History of Skull Island" to be fictional descendants of real animals.


title: Skull Island

text: The MonsterVerse's version of Skull Island does not appear to be inhabited by any living dinosaurs, though an oversize Triceratops skull can be seen in the Valley of Fallen Giants, which indicates that there were dinosaurs living on the island. Much of the island's ecosystem in the film and supplementary material is populated by naturally-evolved hybrids of plant and animal, referred to as "Florofauna" by Monarch scientists.


title: Skull Island

text: All versions of "King Kong" present an island inhabited by giant animals, both invertebrate and vertebrate, but even though biophysical analysis of Kong and other creatures concludes that some are biophysically viable, the ecosystem of the island could not support them.


title: Nothing

text: Nothing, the complete absence of anything, has been a matter of philosophical debate since at least the 5th century BC. Early Greek philosophers argued that it was impossible for "nothing" to exist. The atomists allowed "nothing" but only in the spaces between the invisibly small atoms. For them, all space was filled with atoms. Aristotle took the view that there exists matter and there exists space, a receptacle into which matter objects can be placed. This became the paradigm for classical scientists of the modern age like Newton. Nevertheless, some philosophers, like Descartes, continued to argue against the existence of empty space until the scientific discovery of a physical vacuum.


title: Nothing

text: Existentialists like Sartre and Heidegger (as interpreted by Sartre) have associated "nothing" with consciousness. Some writers have made connections between Heidegger's concept of "nothing" and the nirvana of Eastern religions.


title: Nothing

text: Modern science does not equate "vacuum" with "nothing". Indeed, the vacuum in quantum field theory is filled with virtual particles. The quantum vacuum is often viewed as a modern version of an aether theory.


title: Nothing

text: Some would consider the study of "nothing" to be foolish. A typical response of this type is voiced by Giacomo Casanova (1725–1798) in conversation with his landlord, one Dr. Gozzi, who also happens to be a priest:


title: Nothing

text: However, "nothingness" has been treated as a serious subject for a very long time. In philosophy, to avoid linguistic traps over the meaning of "nothing", a phrase such as "not-being" is often employed to make clear what is being discussed.


title: Nothing

text: One of the earliest Western philosophers to consider nothing as a concept was Parmenides (5th century BC), who was a Greek philosopher of the monist school. He argued that "nothing" cannot exist by the following line of reasoning: To speak of a thing, one has to speak of a thing that exists. Since we can speak of a thing in the past, this thing must still exist (in some sense) now, and from this he concludes that there is no such thing as change. As a corollary, there can be no such things as "coming-into-being", "passing-out-of-being", or "not-being".


title: Nothing

text: Parmenides was taken seriously by other philosophers, influencing, for instance, Socrates and Plato. Aristotle gives Parmenides serious consideration but concludes, "Although these opinions seem to follow logically in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts."


title: Nothing

text: In modern times, Albert Einstein's concept of spacetime has led many scientists, including Einstein himself, to adopt a position remarkably similar to Parmenides. On the death of his friend Michele Besso, Einstein consoled his widow with the words, "Now he has departed from this strange world a little ahead of me. That signifies nothing. For those of us that believe in physics, the distinction between past, present and future is only a stubbornly persistent illusion."


title: Nothing

text: Leucippus (early 5th century BC), one of the atomists, along with other philosophers of his time, made attempts to reconcile this monism with the everyday observation of motion and change. He accepted the monist position that there could be no motion without a void. The void is the opposite of being. It is "not-being". On the other hand, there exists something known as an absolute plenum, a space filled with matter, and there can be no motion in a plenum because it is completely full. But, there is not just one monolithic plenum, for existence consists of a multiplicity of plenums. These are the invisibly small "atoms" of Greek atomist theory, later expanded by Democritus (c. 460–370 BC), which allows the void to "exist" between them. In this scenario, macroscopic objects can "come-into-being", move through space, and pass into "not-being" by means of the coming together and moving apart of their constituent atoms. The void must exist to allow this to happen, or else the "frozen world" of Parmenides must be accepted.


title: Nothing

text: Bertrand Russell points out that this does not exactly defeat the argument of Parmenides but, rather, ignores it by taking the rather modern scientific position of starting with the observed data (motion, etc.) and constructing a theory based on the data, as opposed to Parmenides' attempts to work from pure logic. Russell also observes that both sides were mistaken in believing that there can be no motion in a plenum, but arguably motion cannot "start" in a plenum. Cyril Bailey notes that Leucippus is the first to say that a "thing" (the void) might be real without being a body and points out the irony that this comes from a materialistic atomist. Leucippus is therefore the first to say that "nothing" has a reality attached to it.


title: Nothing

text: Aristotle (384–322 BC) provided the classic escape from the logical problem posed by Parmenides by distinguishing things that are "matter" and things that are "space". In this scenario, space is not "nothing" but, rather, a receptacle in which objects of matter can be placed. The true void (as "nothing") is different from "space" and is removed from consideration. This characterization of space reached its pinnacle with Isaac Newton who asserted the existence of absolute space. René Descartes, on the other hand, returned to a Parmenides-like argument of denying the existence of space. For Descartes, there was matter, and there was extension of matter leaving no room for the existence of "nothing".