title: Strangling

text: Ligature strangulation (also known as "garroting") is strangling with some form of cord such as rope, wire, chain, or shoelaces, either partially or fully circumferencing the neck. Even though the mechanism of strangulation is similar, it is usually distinguished from hanging by the strangling force being something other than the person's own body weight. Incomplete occlusion of the carotid arteries is expected and, in cases of homicide, the victim may struggle for a period of time, with unconsciousness typically occurring in 10 to 15 seconds. Cases of ligature strangulation generally involve homicides of women, children, and the elderly. Compared to hanging, the ligature mark will most likely be located lower on the neck of the victim.


title: Strangling

text: During the Spanish Inquisition, victims who admitted their alleged sins and recanted were killed via ligature strangulation (i.e. the garrote) before their bodies were burnt during the "auto-da-fé". Throughout much of the 20th and 21st centuries, the American Mafia used ligature strangulation as a means of murdering their victims. Confessed American serial killer Altemio Sanchez used ligature strangulation in the rapes and/or murders of his victims, as did the Green River Killer, Gary Ridgway, and British serial killer Dennis Nilsen.


title: Factorization

text: In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several "factors", usually smaller or simpler objects of the same kind. For example, is a factorization of the integer , and is a factorization of the polynomial .


title: Factorization

text: Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any formula_1 can be trivially written as formula_2 whenever formula_3 is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator.


title: Factorization

text: Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography.


title: Factorization

text: Polynomial factorization has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up to ordering) factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials).


title: Factorization

text: A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals.


title: Factorization

text: "Factorization" may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix with all diagonal entries equal to one, an upper triangular matrix , and a permutation matrix ; this is a matrix formulation of Gaussian elimination.


title: Factorization

text: By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the order of the factors) factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one.


title: Factorization

text: For computing the factorization of an integer , one needs an algorithm for finding a divisor of or deciding that is prime. When such a divisor is found, the repeated application of this algorithm to the factors and gives eventually the complete factorization of .


title: Factorization

text: For finding a divisor of , if any, it suffices to test all values of such that and . In fact, if is a divisor of such that , then is a divisor of such that .


title: Factorization

text: If one tests the values of in increasing order, the first divisor that is found is necessarily a prime number, and the "cofactor" cannot have any divisor smaller than . For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of that is not smaller than and not greater than .


title: Factorization

text: There is no need to test all values of for applying the method. In principle, it suffices to test only prime divisors. This needs to have a table of prime numbers that may be generated for example with the sieve of Eratosthenes. As the method of factorization does essentially the same work as the sieve of Eratosthenes, it is generally more efficient to test for a divisor only those numbers for which it is not immediately clear whether they are prime or not. Typically, one may proceed by testing 2, 3, 5, and the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3.


title: Factorization

text: This method works well for factoring small integers, but is inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number


title: Factorization

text: is not a prime number. In fact, applying the above method would require more than , for a number that has 10 decimal digits.


title: Factorization

text: There are more efficient factoring algorithms. However they remain relatively inefficient, as, with the present state of the art, one cannot factorize, even with the more powerful computers, a number of 500 decimal digits that is the product of two randomly chosen prime numbers. This ensures the security of the RSA cryptosystem, which is widely used for secure internet communication.


title: Factorization

text: Manipulating expressions is the basis of algebra. Factorization is one of the most important methods for expression manipulation for several reasons. If one can put an equation in a factored form , then the problem of solving the equation splits into two independent (and generally easier) problems and . When an expression can be factored, the factors are often much simpler, and may thus offer some insight on the problem. For example,


title: Factorization

text: having 16 multiplications, 4 subtractions and 3 additions, may be factored into the much simpler expression


title: Factorization

text: with only two multiplications and three subtractions. Moreover, the factored form immediately gives roots "x" = "a","b","c" as the roots of the polynomial.


title: Factorization

text: On the other hand, factorization is not always possible, and when it is possible, the factors are not always simpler. For example, formula_7 can be factored into two irreducible factors formula_8 and formula_9.


title: Factorization

text: Solving algebraic equations may be viewed as a problem of polynomial factorization. In fact, the fundamental theorem of algebra can be stated as follows: every polynomial in of degree with complex coefficients may be factorized into linear factors formula_10 for , where the s are the roots of the polynomial. Even though the structure of the factorization is known in these cases, the


title: Factorization

text: s generally cannot be computed in terms of radicals ("n" roots), by the Abel–Ruffini theorem. In most cases, the best that can be done is computing approximate values of the roots with a root-finding algorithm.


title: Factorization

text: The systematic use of algebraic manipulations for simplifying expressions (more specifically equations)) may be dated to 9th century, with al-Khwarizmi's book "The Compendious Book on Calculation by Completion and Balancing", which is titled with two such types of manipulation.


title: Factorization

text: However, even for solving quadratic equations, the factoring method was not used before Harriot's work published in 1631, ten years after his death. In his book "Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas", Harriot drew tables for addition, subtraction, multiplication and division of monomials, binomials, and trinomials. Then, in a second section, he set up the equation , and showed that this matches the form of multiplication he had previously provided, giving the factorization .


title: Factorization

text: The following methods apply to any expression that is a sum, or that may be transformed into a sum. Therefore, they are most often applied to polynomials, though they also may be applied when the terms of the sum are not monomials, that is, the terms of the sum are a product of variables and constants.


title: Factorization

text: It may occur that all terms of a sum are products and that some factors are common to all terms. In this case, the distributive law allows factoring out this common factor. If there are several such common factors, it is preferable to divide out the greatest such common factor. Also, if there are integer coefficients, one may factor out the greatest common divisor of these coefficients.


title: Factorization

text: one may remark that the first two terms have a common factor , and the last two terms have the common factor . Thus


title: Factorization

text: In general, this works for sums of 4 terms that have been obtained as the product of two binomials. Although not frequently, this may work also for more complicated examples.


title: Factorization

text: Sometimes, some term grouping reveals part of a recognizable pattern. It is then useful to add and subtract terms to complete the pattern.


title: Factorization

text: Another example is the factorization of formula_16 If one introduces the non-real square root of –1, commonly denoted , then one has a difference of squares


title: Factorization

text: However, one may also want a factorization with real number coefficients. By adding and subtracting formula_18 and grouping three terms together, one may recognize the square of a binomial:


title: Factorization

text: These factorizations work not only over the complex numbers, but also over any field, where either –1, 2 or –2 is a square. In a finite field, the product of two non-squares is a square; this implies that the polynomial formula_22 which is irreducible over the integers, is reducible modulo every prime number. For example,


title: Factorization

text: Many identities provide an equality between a sum and a product. The above methods may be used for letting the sum side of some identity appear in an expression, which may therefore be replaced by a product.


title: Factorization

text: Below are identities whose left-hand sides are commonly used as patterns (this means that the variables and that appear in these identities may represent any subexpression of the expression that has to be factorized).


title: Factorization

text: The th roots of unity are the complex numbers each of which is a root of the polynomial formula_46 They are thus the numbers


title: Factorization

text: If and are real expressions, and one wants real factors, one has to replace every pair of complex conjugate factors by its product. As the complex conjugate of formula_52 is formula_53 and


title: Factorization

text: one has the following real factorizations (one passes from one to the other by changing into or , and applying the usual trigonometric formulas:


title: Factorization

text: The cosines that appear in these factorizations are algebraic numbers, and may be expressed in terms of radicals (this is possible because their Galois group is cyclic); however, these radical expressions are too complicated to be used, except for low values of . For example,


title: Factorization

text: Often one wants a factorization with rational coefficients. Such a factorization involves cyclotomic polynomials. To express rational factorizations of sums and differences or powers, we need a notation for the homogenization of a polynomial: if formula_60 its "homogenization" is the bivariate polynomial formula_61 Then, one has


title: Factorization

text: where the products are taken over all divisors of , or all divisors of that do not divide , and formula_64 is the th cyclotomic polynomial.


title: Factorization

text: For polynomials, factorization is strongly related with the problem of solving algebraic equations. An algebraic equation has the form


title: Factorization

text: If formula_70 is a factorization of as a product of two polynomials, then the roots of are the union of the roots of and the roots of . Thus solving is reduced to the simpler problems of solving and .


title: Factorization

text: If the coefficients of are real or complex numbers, the fundamental theorem of algebra asserts that has a real or complex root. Using the factor theorem recursively, it results that


title: Factorization

text: where formula_73 are the real or complex roots of , with some of them possibly repeated. This complete factorization is unique up to the order of the factors.


title: Factorization

text: If the coefficients of are real, one generally wants a factorization where factors have real coefficients. In this case, the complete factorization may have some quadratic (degree two) factors. This factorization may easily be deduced from the above complete factorization. In fact, if is a non-real root of , then its complex conjugate is also a root of . So, the product


title: Factorization

text: is a factor of with real coefficients. Repeating this for all non-real factors gives a factorization with linear or quadratic real factors.


title: Factorization

text: For computing these real or complex factorizations, one needs the roots of the polynomial, which may not be computed exactly, and only approximated using root-finding algorithms.


title: Factorization

text: In practice, most algebraic equations of interest have integer or rational coefficients, and one may want a factorization with factors of the same kind. The fundamental theorem of arithmetic may be generalized to this case, stating that polynomials with integer or rational coefficients have the unique factorization property. More precisely, every polynomial with rational coefficients may be factorized in a product


title: Factorization

text: where is a rational number and formula_76 are non-constant polynomials with integer coefficients that are irreducible and primitive; this means that none of the formula_77 may be written as the product two polynomials (with integer coefficients) that are neither 1 nor –1 (integers are considered as polynomials of degree zero). Moreover, this factorization is unique up to the order of the factors and the signs of the factors.


title: Factorization

text: There are efficient algorithms for computing this factorization, which are implemented in most computer algebra systems. See Factorization of polynomials. Unfortunately, these algorithms are too complicated to use for paper-and-pencil computations. Besides the heuristics above, only a few methods are suitable for hand computations, which generally work only for polynomials of low degree, with few nonzero coefficients. The main such methods are described in next subsections.


title: Factorization

text: Every polynomial with rational coefficients, may be factorized, in a unique way, as the product of a rational number and a polynomial with integer coefficients, which is primitive (that is, the greatest common divisor of the coefficients is 1), and has a positive leading coefficient (coefficient of the term of the highest degree). For example:


title: Factorization

text: In this factorization, the rational number is called the content, and the primitive polynomial is the primitive part. The computation of this factorization may be done as follows: firstly, reduce all coefficients to a common denominator, for getting the quotient by an integer of a polynomial with integer coefficients. Then one divides out the greater common divisor of the coefficients of this polynomial for getting the primitive part, the content being formula_80 Finally, if needed, one changes the signs of and all coefficients of the primitive part.


title: Factorization

text: This factorization may produce a result that is larger than the original polynomial (typically when there are many coprime denominators), but, even when this is the case, the primitive part is generally easier to manipulate for further factorization.


title: Factorization

text: For example, for formula_86 one may easily see that the sum of its coefficients is 0, so is a root. As , and formula_87 one has


title: Factorization

text: For polynomials with rational number coefficients, one may search for roots which are rational numbers. Primitive part-content factorization (see above) reduces the problem of searching for rational roots to the case of polynomials with integer coefficients having no non-trivial common divisor.


title: Factorization

text: where both factors have integer coefficients (the fact that has integer coefficients results from the above formula for the quotient of by formula_92).


title: Factorization

text: Comparing the coefficients of degree and the constant coefficients in the above equality shows that, if formula_93 is a rational root in reduced form, then is a divisor of formula_94 and is a divisor of formula_95 Therefore, there is a finite number of possibilities for and , which can be systematically examined.


title: Factorization

text: has a rational root formula_93 with , then must divide 6; that is formula_98 and must divide 2, that is formula_99 Moreover, if , all terms of the polynomial are negative, and, therefore, a root cannot be negative. That is, one must have


title: Factorization

text: A direct computation shows that only formula_101 is a root, so there can be no other rational root. Applying the factor theorem leads finally to the factorization


title: Factorization

text: The above method may be adapted for quadratic polynomials, leading to the "ac method" of factorization.


title: Factorization

text: with integer coefficients. If it has a rational root, its denominator must divide evenly and it may be written as a possibly reducible fraction formula_104 By Vieta's formulas, the other root formula_105 is


title: Factorization

text: In summary, if formula_111 has rational roots there are integers and such formula_112 and formula_113 (a finite number of cases to test), and the roots are formula_114 and formula_115 In other words, one has the factorization


title: Factorization

text: If are all real, the factors are real if and only if the discriminant formula_124 is non-negative. Otherwise, the quadratic polynomial cannot be factorized into non-constant real factors.


title: Factorization

text: The quadratic formula is valid when the coefficients belong to any field of characteristic different from two, and, in particular, for coefficients in a finite field with an odd number of elements.


title: Factorization

text: There are also formulas for roots of cubic and quartic polynomials, which are, in general, too complicated for practical use. The Abel–Ruffini theorem shows that there are no general root formulas in terms of radicals for polynomials of degree five or higher.


title: Factorization

text: It may occur that one knows some relationship between the roots of a polynomial and its coefficients. Using this knowledge may help factoring the polynomial and finding its roots. Galois theory is based on a systematic study of the relations between roots and coefficients, that include Vieta's formulas.


title: Factorization

text: This implies that formula_125 is a common root of formula_130 and formula_131 It is therefore a root of the greatest common divisor of these two polynomials. It follows that this greatest common divisor is a non constant factor of formula_131 Euclidean algorithm for polynomials allows computing this greatest common factor.


title: Factorization

text: has two roots that sum to zero, one may apply Euclidean algorithm to formula_127 and formula_135 The first division step consists in adding formula_127 to formula_137 giving the remainder of


title: Factorization

text: Then, dividing formula_127 by formula_140 gives zero as a new remainder, and as a quotient, leading to the complete factorization


title: Factorization

text: The integers and the polynomials over a field share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a unit, ±1 in the case of integers) and a product of irreducible elements (prime numbers, in the case of integers), and this factorization is unique up to rearranging the factors and shifting units among the factors. Integral domains which share this property are called unique factorization domains (UFD).


title: Factorization

text: Greatest common divisors exist in UFDs, and conversely, every integral domain in which greatest common divisors exist is an UFD. Every principal ideal domain is an UFD.


title: Factorization

text: A Euclidean domain is an integral domain on which is defined a Euclidean division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD.


title: Factorization

text: In a Euclidean domain, Euclidean division allows defining a Euclidean algorithm for computing greatest common divisors. However this does not imply the existence of a factorization algorithm. There is an explicit example of a field such that there cannot exist any factorization algorithm in the Euclidean domain of the univariate polynomials over .


title: Factorization

text: In algebraic number theory, the study of Diophantine equations led mathematicians, during 19th century, to introduce generalizations of the integers called algebraic integers. The first ring of algebraic integers that have been considered were Gaussian integers and Eisenstein integers, which share with usual integers the property of being principal ideal domains, and have thus the unique factorization property.


title: Factorization

text: Unfortunately, it soon appeared that most rings of algebraic integers are not principal and do not have unique factorization. The simplest example is formula_142 in which


title: Factorization

text: This lack of unique factorization is a major difficulty for solving Diophantine equations. For example, many wrong proofs of Fermat's Last Theorem (probably including Fermat's ""truly marvelous proof of this, which this margin is too narrow to contain"") were based on the implicit supposition of unique factorization.


title: Factorization

text: This difficulty was resolved by Dedekind, who proved that the rings of algebraic integers have unique factorization of ideals: in these rings, every ideal is a product of prime ideals, and this factorization is unique up the order of the factors. The integral domains that have this unique factorization property are now called Dedekind domains. They have many nice properties that make them fundamental in algebraic number theory.


title: Factorization

text: Matrix rings are non-commutative and have no unique factorization: there are, in general, many ways of writing a matrix as a product of matrices. Thus, the factorization problem consists of finding factors of specified types. For example, the LU decomposition gives a matrix as the product of a lower triangular matrix by an upper triangular matrix. As this is not always possible, one generally considers the "LUP decomposition" having a permutation matrix as its third factor.


title: Factorization

text: A logical matrix represents a binary relation, and matrix multiplication corresponds to composition of relations. Decomposition of a relation through factorization serves to profile the nature of the relation, such as a difunctional relation.


title: Canon (fiction)

text: In fiction, canon is the material accepted as officially part of the story in an individual universe of that story by its fan base. It is often contrasted with, or used as the basis for, works of fan fiction. The alternative terms mythology, timeline, universe and continuity are often used, with the first of these being used especially to refer to a richly detailed fictional canon requiring a large degree of suspension of disbelief (e.g. an entire imaginary world and history), while the latter two typically refer to a single arc where all events are directly connected chronologically. Other times, the word can mean "to be acknowledged by the creator(s)".


title: Canon (fiction)

text: The use of the word "canon" originated in reference to a set of texts derived from Biblical canon, the set of books regarded as scripture, as contrasted with non-canonical Apocrypha. The term was first used by analogy in the context of fiction to refer to the Sherlock Holmes stories and novels, written by Sir Arthur Conan Doyle, as contrasted with numerous Holmes adventures added later by other writers. This usage was afterwards extended to the writings of various other authors.


title: Canon (fiction)

text: When there are multiple "official" works or original media, what material is canonical can be unclear. This is resolved either by explicitly excluding certain media from the status of canon (as in the case of "Star Trek" and "Star Wars"), by assigning different levels of canonicity to different media (as was in the case of "Star Wars" before its ownership by Disney), by considering different but licensed media treatments official and equally canonical to the series timeline within their own continuities' universe, but not across them, or not resolved at all. There is also no consensus regarding who has the "authority to decide" what is or isn't canon, with copyright holders usually declaring themselves the authorities when they want to erase or retcon materials that were approved by the setting's original creator (with "Star Wars" again being an example). The use of canon is of particular importance with regard to reboots or re-imaginings of established franchises, such as the "Star Trek" remake (2009), because of the ways in which it influences the viewer experience.


title: Canon (fiction)

text: The official "Star Trek" website describes "Star Trek" canon as "the events that take place within the episodes and movies" referring to the live-action television series and films, with "" having long existed in a . Events, characters and storylines from tie-in novels, comic books, and video games are explicitly excluded from the "Star Trek" canon, but the site notes that elements from these sources have been subsequently introduced into the television series, and says that "canon is not something set in stone." Some non-canonical elements that later became canonical in the Star Trek universe are Uhura's first name Nyota, introduced in the novels and made canonical in the 2009 film "Star Trek", and James T. Kirk's middle name Tiberius, introduced in the "Star Trek" animated series and made canonical in "".


title: Canon (fiction)

text: During George Lucas' time with the franchise, "Star Wars" canon was divided into discrete tiers that incorporated the Expanded Universe (EU), with continuity tracked by Lucasfilm creative executive Leland Chee. Higher-tier and newer material abrogated lower-tier and older material in case of contradiction. The live-action theatrical films, the 2008 and its , and statements by Lucas himself were at the top of this hierarchy; such works invariably superseded EU material in case of contradiction. The EU itself was further divided into several descending levels of continuity. After Disney's acquisition of the franchise, Lucasfilm designated all Expanded Universe material published prior to 25 April 2014 (other than the first six theatrical films and the 2008 "The Clone Wars" film and TV series) as the non-canonical "Legends" continuity. Material released since this announcement is a separate canonical timeline from the original George Lucas Canon, with all narrative development overseen by the Lucasfilm Story Group.


title: Canon (fiction)

text: The makers of "Doctor Who" have generally avoided making pronouncements about canonicity, with Russell T Davies explaining that he does not think about the concept for the "Doctor Who" television series or its spin-offs.


title: Canon (fiction)

text: In literature, the term "canon" is used to distinguish between the original works of a writer who created certain characters and/or settings, and the later works of other writers who took up the same characters or setting. For example, the canon of Sherlock Holmes consists of the 56 short stories and four novels written by Sir Arthur Conan Doyle that feature the detective Sherlock Holmes. The subsequent works by other authors who also used the character Sherlock Holmes are considered "non-canonical".


title: Canon (fiction)

text: Some works by the original writer such as "The Field Bazaar" but not the same publisher may be debated as forming part of canon. This is because copyright used to be exercised by the publisher of the work of literature rather than the author. Campaigning by Victor Hugo led to the Berne Convention which introduced author's rights.


title: Canon (fiction)

text: However, sometimes in literature, original writers have not approved works as canon, but original publishers or literary estates of original writers posthumously approve subsequent works as canon, such as "The Royal Book of Oz" (1921) (by original publisher), "Porto Bello Gold" (1924) (by estate), and "Heidi Grows Up" (1938) (by estate).


title: Canon (fiction)

text: In film and television this is common that the original writer does not decide canon. In literature, the estate of H. G. Wells authorised sequels by Stephen Baxter, "The Massacre of Mankind" (2017) and "The Time Ships" (1995). "Scarlett" was a 1991 sequel to "Gone with the Wind" authorised by the estate.


title: Canon (fiction)

text: Sequels to the stories by P G Wodehouse about the butler Jeeves were sanctioned by Wodehouse's estate for "Jeeves and the Wedding Bells" (2013) by Sebastian Faulks and "Jeeves and the King of Clubs" (2018) by Ben Schott. "The Monogram Murders" (2014) by Sophie Hannah is a sequel to Hercule Poirot novels authorised by the Agatha Christie estate.


title: Canon (fiction)

text: Fan fiction is almost never regarded as canonical. However, certain ideas may become influential or widely accepted within fan communities, who refer to such ideas as "fanon", a blend of "fan" and "canon". Similarly, the jargon "headcanon" is used to describe a fan's personal interpretation of a fictional universe.


title: Ark: The Animated Series

text: Ark: The Animated Series is an upcoming animated fantasy television series based on the video game "".


title: Ark: The Animated Series

text: The series will consist of 14 episodes set in various points of history that flesh out the world of "" on The Island. There is no confirmation on whether or not this series will continue through the Story DLC such as Scorched Earth, Aberration, and Extinction.


title: Ark: The Animated Series

text: The series was announced with a trailer during the 2020 Video Game Awards, along with the cast that included Gerard Butler, David Tennant, Michelle Yeoh, Jeffrey Wright, Zahn McClarnon, Elliot Page, Karl Urban, Malcolm McDowell, Alan Tudyk, Russell Crowe and Vin Diesel, who will also serve as an executive producer.


title: Ark: The Animated Series

text: The series was created by the game's creators Jeremy Stieglitz and Jesse Rapczak and written by Marguerite Bennett and Kendall Deacon Davis. Jay Oliva will direct the series, which will consist of 14 thirty-minute episodes that develop the world of the game, in addition to two seasons being put into production. In June 2022, the series entered post-production, with Monica Bellucci and Dee Bradley Baker added to the voice cast.


title: Ishmael in Islam

text: Ismail () is regarded as a prophet and messenger and the ancestor to the Ishmaelites in Islam. He is the son of Ibrahim (Abraham), born to Hajar (Hagar). Ismail is also associated with Mecca and the construction of the Kaaba"." Ismail is considered the ancestor to Muhammad.


title: Ishmael in Islam

text: Ismail is the figure known as Ishmael in Judaism and Christianity. These sources include the Quran, Quranic commentary (tafsir), "hadith", historiographic collections like that of Muhammad ibn Jarir al-Tabari, and "Isra'iliyat" (Islamic texts about Biblical or ancient Israelite figures that originate from Jewish or Christian sources).


title: Ishmael in Islam

text: Ishmael was the first son of Abraham; his mother was Hagar. There are many versions of the story, some of which include a prophecy about Ishmael's birth. One such example is from Ibn Kathir (d.1373) whose account states that an angel tells the pregnant Hagar to name her child Ishmael and prophesies, "His hand would be over everyone, and the hand of everyone would be against him. His brethren would rule over all the lands." Ibn Kathir comments that this foretells of Muhammad's leadership.


title: Ishmael in Islam

text: Ishmael and Hagar being taken to Mecca by Abraham in Islamic texts is an important part in the story of Ishmael, as it brings the focus to Mecca and is the beginning of Mecca's sanctification as a holy area. Islamic tradition says Abraham was ordered by God to take Hagar and Ishmael to Mecca, and later Abraham returned to Mecca to build the Kaaba. In many of these accounts, the "Sakina" (something like a wind or spirit sent by God), or the angel Gabriel (Jibreel) guides them to the location of the Kaaba, at which point Abraham builds it and afterwards, leaves the other two there (other versions discussed below say the construction of the Kaaba occurred later and that Ishmael took part in it). Generally, it is said that Hagar asks Abraham who he is entrusting herself and Ishmael to as he leaves them. He answers that he is entrusting them to God, to which Hagar then makes a reply that shows her faith, stating that she believes God will guide them. Hagar and Ishmael then run out of water and Ishmael becomes extremely thirsty. Hagar is distressed and searches for water, running back and forth seven times between the hills of Al-Safa and Al-Marwah. Hagar is later remembered by Muslims for this act during the "Hajj", or pilgrimage, in which Muslims run between these same hills as part of the "Sa'yee". When she returns to Ishmael, she finds either him or an angel scratching the ground with their heel or finger, whereupon water begins flowing and Hagar collects some or dams it up. This spring or well is known as Zamzam. At some point, a passing tribe known as the Jurhum sees birds circling the water and investigates. They ask Hagar if they can settle there, which she allows, and many versions say as Ishmael grew up he learned various things from the tribe. There are numerous versions of this story, each differing in various ways. The versions used in this summary, as well as others, can be found in al-Tabari's history and are recounted in Reuven Firestone's "Journeys in Holy Lands".


title: Ishmael in Islam

text: While the Tawrat (the Arabic-language name for the Torah within its context as an Islamic holy book), mentions that it was Isaac who was binded, some Muslims believe that Abraham was told to sacrifice his other son, Ishmael, though the Qur'an does not mention the name of the son. The multiple versions suggest that the "dhabih" was originally an oral story that had been circulating before being written as it is in the Qur'an and in additional commentaries. Norman Calder explains, "oral narrative is marked by instability of form and detail from version to version, and by an appropriate creative flexibility which makes of every rendering a unique work of art." Each version is indeed a "unique work of art," differing from another in various ways to present certain ideas, such as the importance of Ishmael over Isaac because he was the first child.


title: Ishmael in Islam

text: The general narrative pertaining to Ishmael describes the sacrifice either as a test or as part of a vow. Some versions tell of the devil trying to stop God's command from being obeyed by visiting Hagar, Ishmael, and Abraham. Every time the devil says Abraham is going to sacrifice Ishmael, each person answers that if God commanded it, they should obey. Eventually, Abraham tells Ishmael about the order and Ishmael is willing to be sacrificed and encourages Abraham to listen to God. Often, Ishmael is portrayed as telling Abraham some combination of instructions to bring his shirt back to Hagar, bind him tightly, sharpen the knife, and place him face down, all so that there will be no wavering in the resolve to obey God.


title: Ishmael in Islam

text: As Abraham attempts to slay Ishmael, either the knife is turned over in his hand or copper appears on Ishmael to prevent the death and God tells Abraham that he has fulfilled the command. Unlike in the Bible, there is no mention in the Qur'an of an animal (ram) replacing the boy; rather he is replaced with a 'great sacrifice' ("dhibḥin ʿaẓīm"). Since the sacrifice of a ram cannot be greater than that of Abraham's son (and a prophet in Islam at that), this replacement seems to point to either the religious institutionalisation of sacrifice itself, or to the future self-sacrifices of the Islamic prophet Muhammad and his companions (who were destined to emerge from the progeny of Ishmael) in the cause of their faith. Every Eid al-Adha once a year Muslims around the world slaughter an animal to commemorate Abraham's sacrifice and to remind themselves of self-abnegation in the way of God. Later historiographic literature incorporates the Biblical narrative in which a ram is provided which is slaughtered instead of Ishmael.


title: Ishmael in Islam

text: The actions of Ishmael in this narrative have led him to become a prominent model of hospitality and obedience. This story in the Quran is unique when compared to that in the Bible because Abraham talks with his son, whichever it is believed to be, and the son is thus aware of the plan to become a sacrifice and approves of it. As noted above, in some versions, Ishmael makes sure in different ways that neither he nor his father hesitate in their obedience to God. In this way, Ishmael is a model of surrendering one's will to God, an essential characteristic in Islam.


title: Ishmael in Islam

text: Though it is generally believed by modern Muslims that Ishmael was the son who was almost sacrificed, among scholars and historiographers of early Islam, there is much debate. There are such persuasive arguments for both, in fact, it is estimated that 131 traditions say Isaac was the son, while 133 say Ishmael. Such dispute over which son suggests that the story, and where and to whom it happens, is extremely important. It is argued that the story originated from rabbinic texts and was adapted to Islam over time in order to give Mecca religious importance and connect the story with the pilgrimage.Arguments by early Muslim scholars for Ishmael as the intended sacrifice include that Jews claim it is Isaac only because they are jealous that it was actually the ancestor of Arabs, Ishmael, and that the horns of the ram that was sacrificed instead hung in the Kaaba at one time. In looking solely at the text of the Quran to determine which son was to be sacrificed, there still are various views. The strongest case for Ishmael in the Quran is that directly after the sacrifice narrative, Abraham is told of the coming of Isaac's birth, therefore, it must be Ishmael who was about to be sacrificed. Authentic hadiths are said to not contradict each other because that negates the definition of the hadith.


title: Ishmael in Islam

text: At some point, often believed to be after Hagar's death, Ishmael married a woman from the Jurhum, the tribe who settled in the area around Zamzam. Abraham visited Ishmael in Mecca and when he arrived at his home, Ishmael was not there. Instead Ishmael's wife greets Abraham, but she was not welcoming or generous to him. Abraham instructed her to tell Ishmael some version of the statement that he was not pleased with or to change "the threshold of his door." When Ishmael returns home and his wife told him that, he knows it is from his father and taking the advice, divorced the woman. He then married another woman from Jurhum. Abraham once again visited and was met by Ishmael's second wife, as Ishmael was out. This wife was very kind and provided food for him. Abraham instructed her to tell Ishmael some version of the statement that he was pleased with "the threshold of his door." When Ishmael arrived and his wife repeated Abraham's statement, Ishmael knew it was from his father and kept his wife.


title: Ishmael in Islam

text: There are many versions of the construction of the Kaaba that differ in fairly significant ways, although all have Abraham build or cleanse the Kaaba and then immediately after, or at an unknown time, God called Abraham to establish the Hajj, or pilgrimage. These narratives differ in when these events occurred, if and how there was supernatural involvement, the inclusion or omission of the Black Stone, and whether Ishmael assisted his father. Of those that say Ishmael took part in the construction, most describe Abraham visited Ishmael a third time in Mecca, during which they raised the Kaaba. Some say Ishmael looked for a final stone, but Abraham did not accept the one he brought back. Instead an angel had brought the Black Stone, which Abraham put into place. Ishmael was left at the Kaaba, in charge of its care and to teach others about the Hajj. The starting of the Hajj has many versions, and some scholars believe this reflects the late association of Abraham with the Hajj after Islam had developed to help remove its connection to early pagan rituals.


title: Ishmael in Islam

text: Ishmael is considered a prophet in Islam and is listed in the Quran with other prophets in many instances. In other verses, such as 21:85 and 38:48, Ishmael is praised for being patient, good, and righteous. A particular example which describes Ishmael individually is 19:54-55 – "And call to mind, through this divine writ, Ishmael. Behold, he was always true to his promise, and was an apostle [of God], a prophet, who used to enjoin upon his people prayer and charity, and found favour in his Sustainer's sight." As a descendant of Ishmael, Muhammad is considered justified as the Prophet and continues the line of prophets from pre-Islamic times.


title: Ishmael in Islam

text: Ishmael's place as the "founder of the Arabs" was first stated by Josephus. As Islam became established, the figure Ishmael and those descended from him, the Ishmaelites, became connected, and often equated, with the term "Arab" in early Jewish and Christian literature. Before Islam developed as a religion, Ishmael was depicted in many ways, but after its establishment, Ishmael was almost always seen in a negative light in Jewish and Christian texts, as he becomes the symbol for the "other" in these religions. As the Islamic community became more powerful, some Jewish midrash about Ishmael was modified so that he was portrayed more negatively in order to challenge the Islamic view that Ishmael, and thus the Muslims, were the favoured descendants of Abraham. This became the genealogy according to Jewish sources and the Bible, in contrast with the genealogy of Arabs according to Muslims. The development of Islam created pressure for Islam to be somehow different from Judaism and Christianity, and accordingly, Ishmael's lineage to Arabs was stressed.


title: Ishmael in Islam

text: Today, some Christians believe that God fulfills his promises to Ishmael today by blessing the Arab nations with oil and political strength. In pre-Islamic times, there were three distinct groups of Arabs- the "Ba'ida", "Ariba", and "Musta'riba". The Ba'ida were the "legendary Arabs of the past," while the Ariba were the "Southern Arabs." Ishmael's descendants became the Northern Arabs known as the Musta'riba or the "Arabized Arabs." The Musta'riba were described as Arabized since it is believed Ishmael learned Arabic when he moved to Mecca and married into the Arabic tribe of Jurhum. Ishmael's line is then traced from his son Kedar, then down through to Adnan, then to the Musta'riba, to the Quraysh. In this manner, Muhammad's ancestry leads back to Ishmael, joining "original biblical ancestry of Abraham with a distinctively Arab afinal stock," and connecting Muhammad with Mecca and the Kaaba.


title: Turbine

text: A turbine ( or ) (from the Greek , "tyrbē", or Latin "turbo", meaning vortex) is a rotary mechanical device that extracts energy from a fluid flow and converts it into useful work. The work produced by a turbine can be used for generating electrical power when combined with a generator. A turbine is a turbomachine with at least one moving part called a rotor assembly, which is a shaft or drum with blades attached. Moving fluid acts on the blades so that they move and impart rotational energy to the rotor. Early turbine examples are windmills and waterwheels.


title: Turbine

text: Gas, steam, and water turbines have a casing around the blades that contains and controls the working fluid. Credit for invention of the steam turbine is given both to Anglo-Irish engineer Sir Charles Parsons (1854–1931) for invention of the reaction turbine, and to Swedish engineer Gustaf de Laval (1845–1913) for invention of the impulse turbine. Modern steam turbines frequently employ both reaction and impulse in the same unit, typically varying the degree of reaction and impulse from the blade root to its periphery. Hero of Alexandria demonstrated the turbine principle in an aeolipile in the first century AD and Vitruvius mentioned them around 70 BC.


title: Turbine

text: The word "turbine" was coined in 1822 by the French mining engineer Claude Burdin from the Greek , "tyrbē", meaning "vortex" or "whirling", in a memo, "Des turbines hydrauliques ou machines rotatoires à grande vitesse", which he submitted to the Académie royale des sciences in Paris. Benoit Fourneyron, a former student of Claude Burdin, built the first practical water turbine.


title: Turbine

text: A working fluid contains potential energy (pressure head) and kinetic energy (velocity head). The fluid may be compressible or incompressible. Several physical principles are employed by turbines to collect this energy:


title: Turbine

text: Impulse turbines change the direction of flow of a high velocity fluid or gas jet. The resulting impulse spins the turbine and leaves the fluid flow with diminished kinetic energy. There is no pressure change of the fluid or gas in the turbine blades (the moving blades), as in the case of a steam or gas turbine, all the pressure drop takes place in the stationary blades (the nozzles). Before reaching the turbine, the fluid's "pressure head" is changed to "velocity head" by accelerating the fluid with a nozzle. Pelton wheels and de Laval turbines use this process exclusively. Impulse turbines do not require a pressure casement around the rotor since the fluid jet is created by the nozzle prior to reaching the blades on the rotor. Newton's second law describes the transfer of energy for impulse turbines. Impulse turbines are most efficient for use in cases where the flow is low and the inlet pressure is high.


title: Turbine

text: Reaction turbines develop torque by reacting to the gas or fluid's pressure or mass. The pressure of the gas or fluid changes as it passes through the turbine rotor blades. A pressure casement is needed to contain the working fluid as it acts on the turbine stage(s) or the turbine must be fully immersed in the fluid flow (such as with wind turbines). The casing contains and directs the working fluid and, for water turbines, maintains the suction imparted by the draft tube. Francis turbines and most steam turbines use this concept. For compressible working fluids, multiple turbine stages are usually used to harness the expanding gas efficiently. Newton's third law describes the transfer of energy for reaction turbines. Reaction turbines are better suited to higher flow velocities or applications where the fluid head (upstream pressure) is low.


title: Turbine

text: In the case of steam turbines, such as would be used for marine applications or for land-based electricity generation, a Parsons-type reaction turbine would require approximately double the number of blade rows as a de Laval-type impulse turbine, for the same degree of thermal energy conversion. Whilst this makes the Parsons turbine much longer and heavier, the overall efficiency of a reaction turbine is slightly higher than the equivalent impulse turbine for the same thermal energy conversion.


title: Turbine

text: In practice, modern turbine designs use both reaction and impulse concepts to varying degrees whenever possible. Wind turbines use an airfoil to generate a reaction lift from the moving fluid and impart it to the rotor. Wind turbines also gain some energy from the impulse of the wind, by deflecting it at an angle. Turbines with multiple stages may use either reaction or impulse blading at high pressure. Steam turbines were traditionally more impulse but continue to move towards reaction designs similar to those used in gas turbines. At low pressure the operating fluid medium expands in volume for small reductions in pressure. Under these conditions, blading becomes strictly a reaction type design with the base of the blade solely impulse. The reason is due to the effect of the rotation speed for each blade. As the volume increases, the blade height increases, and the base of the blade spins at a slower speed relative to the tip. This change in speed forces a designer to change from impulse at the base, to a high reaction-style tip.


title: Turbine

text: Classical turbine design methods were developed in the mid 19th century. Vector analysis related the fluid flow with turbine shape and rotation. Graphical calculation methods were used at first. Formulae for the basic dimensions of turbine parts are well documented and a highly efficient machine can be reliably designed for any fluid flow condition. Some of the calculations are empirical or 'rule of thumb' formulae, and others are based on classical mechanics. As with most engineering calculations, simplifying assumptions were made.


title: Turbine

text: Velocity triangles can be used to calculate the basic performance of a turbine stage. Gas exits the stationary turbine nozzle guide vanes at absolute velocity "V". The rotor rotates at velocity "U". Relative to the rotor, the velocity of the gas as it impinges on the rotor entrance is "V". The gas is turned by the rotor and exits, relative to the rotor, at velocity "V". However, in absolute terms the rotor exit velocity is "V". The velocity triangles are constructed using these various velocity vectors. Velocity triangles can be constructed at any section through the blading (for example: hub, tip, midsection and so on) but are usually shown at the mean stage radius. Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:


title: Turbine

text: Modern turbine design carries the calculations further. Computational fluid dynamics dispenses with many of the simplifying assumptions used to derive classical formulas and computer software facilitates optimization. These tools have led to steady improvements in turbine design over the last forty years.


title: Turbine

text: The primary numerical classification of a turbine is its specific speed. This number describes the speed of the turbine at its maximum efficiency with respect to the power and flow rate. The specific speed is derived to be independent of turbine size. Given the fluid flow conditions and the desired shaft output speed, the specific speed can be calculated and an appropriate turbine design selected.


title: Turbine

text: The specific speed, along with some fundamental formulas can be used to reliably scale an existing design of known performance to a new size with corresponding performance.


title: Turbine

text: The number of blades in the rotor and the number of vanes in the stator are often two different prime numbers in order to reduce the harmonics and maximize the blade-passing frequency.


title: Turbine

text: Gas turbines have very high power densities (i.e. the ratio of power to mass, or power to volume) because they run at very high speeds. The Space Shuttle main engines used turbopumps (machines consisting of a pump driven by a turbine engine) to feed the propellants (liquid oxygen and liquid hydrogen) into the engine's combustion chamber. The liquid hydrogen turbopump is slightly larger than an automobile engine (weighing approximately 700 lb) with the turbine producing nearly 70,000 hp (52.2 MW).


title: Pikmin

text: The "Pikmin" series features five main entries, as well as a spin-off. The first two, "Pikmin" (2001) and "Pikmin 2" (2004), were released for the GameCube and later ported to the Wii as "New Play Control!" titles in 2008 and 2009. "Pikmin 2" was re-released in the "Nintendo Selects" series in 2012. A third installment, "Pikmin 3", was released for the Wii U in 2013. The series' first handheld game, "Hey! Pikmin", was released for the Nintendo 3DS in 2017. "Pikmin 4" was announced to be in development by Miyamoto in September 2015, and was officially confirmed for release in 2023 during a Nintendo Direct in September 2022. "Pikmin Bloom", the series' mobile spin-off, was released for iOS and Android in 2021.


title: Pikmin

text: The series' gameplay combines elements of action platformer with puzzle and strategy gameplay. The "Pikmin" games focus on exploring an Earth-like planet named PNF-404, and controlling a crowd of the titular Pikmin. Pikmin are intelligent, loyal, multicolored, plant-animal hybrids that follow the orders of their leader. There have been multiple leaders over the course of the series, with the first and most notable being Captain Olimar, a tiny alien from the planet Hocotate introduced in the original "Pikmin". "Pikmin 2" features Louie and The President, Olimar's co-worker and boss, respectively, who are also playable leaders. "Pikmin 3" features leaders from a new alien species. The leaders are Alph, Brittany, and Charlie, who come from the planet Koppai and are similar to the Hocotatians.


title: Pikmin

text: Pikmin are directed to perform a number of tasks, with the primary one being the retrieval of items integral to the mission of each game. The player must direct the Pikmin to overcome obstacles and PNF-404's fauna. Because individual Pikmin are small and weak against most predators, and some are best suited for certain tasks, it is the responsibility of the player to direct the Pikmin appropriately in order to ensure successful retrievals of any items, avoid any hazards, and quickly tend to any obstacles and predators. Game time is divided into individual days, each of which consist of approximately 13 minutes, in which the player is encouraged to accomplish as much as possible before sunset.


title: Pikmin

text: Pikmin appear in a variety of colors, which signify the Pikmin's abilities and resistance to environmental hazards. The number of Pikmin and their abilities have shifted over the course of the franchise, with the first "Pikmin" game featuring three colors: Red, Yellow, and Blue. Red Pikmin can resist fire, Blue Pikmin can move underwater thanks to their gills, and Yellow Pikmin can be thrown higher than other Pikmin, and are able to carry bomb rocks. In "Pikmin 2", Yellow Pikmin have lost their ability to carry bomb rocks in exchange for being immune to electrical hazards, and White and Purple Pikmin are introduced. White Pikmin are the fastest, resistant to poisonous gases, capable of poisoning enemies, and capable of spotting objects buried underground. In comparison, Purple Pikmin are significantly stronger and heavier than the other types, which makes them excellent at inflicting damage and carrying heavy items.


title: Pikmin

text: "Pikmin 3" introduced two more types: the gray-colored Rock Pikmin, and the pink-colored Winged Pikmin. Rock Pikmin can break crystalline materials, cannot be crushed or impaled, and inflict the most damage out of all Pikmin types when thrown. In comparison, Winged Pikmin can fly freely, which enables them to recover elevated items and attack airborne enemies with ease. In addition to making each type capable of carrying bomb rocks, "Pikmin 3" also adjusted the abilities of Yellow, Blue, White, and Purple Pikmin. Yellow Pikmin are tied with Rock Pikmin for the fastest digging speed; Blue Pikmin are capable of swimming; White Pikmin are tied with Winged Pikmin for the lowest attack strength, and slowest digging speed; Purple Pikmin are tied with Red Pikmin for second highest attack strength, and second fastest digging speed.


title: Pikmin

text: Outside of coloration, all types of Pikmin share the same method of indicating their strength and swiftness: the stalk atop their heads will sport either a leaf, bud or flower, which develops when Pikmin are fed nectar or left planted in the ground for a certain amount of time. By extension, all types of Pikmin are stored in their respective motherships, referred to as "Onions", for safety after sunset, as they are vulnerable to nocturnal predators. The Onions play a vital role in Pikmin reproduction: when any food, such as prey or pellets, is delivered to an Onion, it propagates seeds, which grow and are plucked from the ground as fully developed Pikmin. An Onion serves as an incubator for all Pikmin of its respective color, housing the Pikmin. The Onions travel alongside Olimar as he flies to different locations on the planet.


title: Pikmin

text: "Pikmin" is the first game in the series, released in 2001. The game's plot focuses on Captain Olimar's predicament of having crash-landed on an unknown planet and befriending plant-animal creatures that he dubs as Pikmin. Olimar has to gather the missing pieces of his broken spaceship in order to escape, before his air supply completely runs out and he succumbs to the planet's oxygen-rich atmosphere, which is poisonous to his species.


title: Pikmin

text: "Pikmin" features three endings, which are dependent on the number of spaceship parts reclaimed. Reclaiming all thirty parts by the final day results in the best ending: Olimar waves goodbye to the Pikmin, who proceed to defend themselves on their own at nighttime, while Onions of various colors observe his departure off as he looks back on the unknown planet before returning home. Reclaiming at least the twenty-five required parts by the final day results in the normal ending: Olimar quickly departs, while one of each type of Pikmin run toward his ship to watch him depart. Lastly, failing to reclaim the twenty-five required parts by the final day results in the worst ending: Olimar crash-lands back onto the planet, succumbs to oxygen poisoning, and is taken by a number of Pikmin to be placed into an Onion, which converts him into a Pikmin-Hocotatian hybrid.


title: Pikmin

text: "Pikmin" was ported to the Wii in 2009 as a New Play Control! title, which was made available on the Wii U's Virtual Console service in North America in 2016.


title: Pikmin

text: "Pikmin 2", released in 2004, takes place immediately following the events of "Pikmin". When Olimar returns to his home planet Hocotate, he discovers that his employer, Hocotate Freight, has gone into severe debt. When the company's president discovers that the souvenir that Olimar has brought back with him is significantly valuable, he orders Olimar and Louie, another employee, to journey back to the Pikmin planet and gather treasure to pay off the debt.


title: Pikmin

text: The main objective of "Pikmin 2" is to collect human garbage that is referred to as treasure on Hocotate, including bottle caps and gadgets. The player controls both Olimar and Louie, alternating between the two characters in order to divide and accomplish more tasks during a single day. Unlike the first game, "Pikmin 2" has no deadline, so the player can spend as many days as desired to collect all the treasure.


title: Pikmin

text: Like its predecessor, "Pikmin 2" was ported to the Wii in 2009 (or 2012 in North America) as a New Play Control! title, which was made available on the Wii U's Virtual Console service in Europe in 2016.


title: Pikmin

text: Shigeru Miyamoto hinted about the possibilities of a new "Pikmin" game in a July 2007 interview with IGN, saying "I certainly don't think we've seen the last of "Pikmin". I definitely would like to do something with them, and I think the Wii interface in particular is very well suited to that franchise." A later CNET.com interview in April 2008 reported that "For now, Miyamoto looks ahead to other projects for the Wii, mentioning his desire to continue the "Pikmin" series."


title: Pikmin

text: A new "Pikmin" game was confirmed at E3 2008 during Nintendo's developer roundtable, in which Miyamoto stated that his team were working on a new entry in the series. Miyamoto confirmed that "Pikmin 3" would be released on the Wii. He also stated that the Wii's controls were "working well" with the game.


title: Pikmin

text: The announcement in October 2008 of a re-release of the first two "Pikmin" games on the Wii with updated motion controls, raised concerns about whether "Pikmin 3" was in fact in development. However, in a subsequent interview with IGN, it was made clear that the re-releases of "Pikmin" and "Pikmin 2" under the New Play Control! brand were separate from "Pikmin 3". The game was not shown at E3 2009, 2010, or 2011, but Miyamoto confirmed in June 2010 that the game was indeed still in development.


title: Pikmin

text: At Miyamoto's roundtable discussion at E3 2011, Miyamoto stated that "Pikmin 3" development was moved to Wii U, the Wii successor. He said that the HD graphics for the next system and its special screen-based controller would work better for it.


title: Pikmin

text: On April 15, 2012, Miyamoto said in a Spanish interview that a new "Pikmin" game and a new "New Super Mario Bros." game will both be showcased for the Wii U at E3 2012. He also added the comment "Anyone who played the original "Pikmin" games will enjoy playing it."


title: Pikmin

text: On June 5, 2012, at Nintendo's E3 press conference, "Pikmin 3" was announced and featured a trailer that revealed the inclusion of the new Rock Pikmin. Six days later, a second trailer was released that revealed the new Winged Pikmin and other aspects of the game. Although originally intended to be released within the Wii U's launch window, "Pikmin 3" was delayed and instead released in mid-2013.


title: Pikmin

text: On August 5, 2020, Nintendo announced "Pikmin 3 Deluxe", an enhanced port of "Pikmin 3" for the Nintendo Switch, for release on October 30, 2020 via Twitter.


title: Pikmin

text: A Nintendo 3DS entry in the series was announced in a Nintendo Direct in September 2016 as a side-scrolling action game due for release in 2017. In another Nintendo Direct in April 2017, the game's title was announced as "Hey! Pikmin" and its release date was announced to be July 28, 2017. The game was developed by Arzest.


title: Pikmin

text: On March 22, 2021, Niantic, known primarily for the mobile game "Pokémon Go", announced a Pikmin game for mobile devices. Little was known at the time of announcement, except that it will feature similar gameplay to "Pokémon Go", using AR technology. A playable prototype was released for select users in Singapore the same day the game was announced. On October 26, 2021, the game was revealed to be titled "Pikmin Bloom" and launched immediately in Australia and Singapore.


title: Pikmin

text: In September 2015, Miyamoto said that "Pikmin 4" was in development and "very close to completion". At E3 2017, he said that its development was still progressing. "Pikmin 4" was officially announced in a Nintendo Direct on September 13, 2022, with the game set for release in 2023 on the Nintendo Switch.


title: Pikmin

text: The main protagonist of the series is Captain Olimar, an employee of the Freight Company, who flies a ship known as the S.S. Dolphin, a reference to the GameCube's developmental code name. He has a wife, son and daughter. "Pikmin 2" introduces Louie, a co-worker of Olimar, who places the Hocotate Freight Company in debt after losing a shipment of gold pikpik carrots to a "ravenous space bunny". The names "Olimar" and "Louie" are based on Mario and Luigi, who were also created by Shigeru Miyamoto.


title: Pikmin

text: In "Pikmin 3", three new characters succeed Olimar as playable leaders: Alph the engineer, Brittany the botanist, and Captain Charlie.


title: Pikmin

text: All three characters come from Koppai, a planet whose inhabitants are similar to the Hocotations, and they interact with each other through in-game dialogue. Though not playable during the game's story, Olimar and Louie both appear later in the game's story and become integral to its plot, are playable in Bingo Battle Mode and five of the downloadable mission levels, and Olimar's expedition logs are later discovered by the crew and offer helpful advice regarding PNF-404.


title: Pikmin

text: After becoming stranded on the unknown planet in "Pikmin", Olimar becomes acquainted with the species of plant-like beings that he names "Pikmin" because they resemble the "Pikpik" brand of carrots on his home planet of Hocotate. They are small, brightly colored creatures who are each as tall as a US dime. They follow any command: from attacking enemies and defending Olimar to retrieving large artifacts. Pikmin are generally very weak, unless they are used in hordes, as the Pikmin are excellent at working together. Pikmin speak with strange, squeaking noises. Simple commands can, however, be communicated to them by the use of a whistle or any other device capable of producing high-pitched noises.


title: Pikmin

text: Pikmin come in several colors, each with their own strengths and weaknesses. In "Pikmin", there are three types of Pikmin: Red Pikmin, who are better in combat and can withstand extreme heat and fire; Yellow Pikmin, who can be thrown higher, can use bomb rocks ("Pikmin" only), are immune to electricity ("Pikmin 2", "Pikmin 3", and "Hey! Pikmin"), and can dig faster ("Pikmin 3" only); and Blue Pikmin, which have gills and can walk in water (or get splashed with water-based attacks) without drowning. There are also Mushroom Pikmin who attack the other types of Pikmin. They are created when regular Pikmin are infected by spores let out by the Puffstool.


title: Pikmin

text: In "Pikmin 2", three new species of Pikmin were introduced: Purple Pikmin, White Pikmin, and Bulbmin. Purple Pikmin move slower than normal Pikmin, but have ten times the strength and weight of other Pikmin, giving them the ability to pick up much heavier objects and attack and destroy obstacles and other enemies with fewer numbers. They also have the potential to stun enemies when they land near or on them. White Pikmin, when ingested by enemies, damage the enemy through being toxic. White Pikmin themselves are immune to poison, can locate buried treasure with their special eyes, and can run faster than other Pikmin. Bulbmin are a subterranean parasitic species of Pikmin that infect Bulborbs (the most common enemy found within the "Pikmin" series) and take control of them. These Pikmin are found exclusively within certain cave systems in the game, usually as a pack of up to 10 juvenile Bulbmin and a single adult Bulbmin. While the adult forms of these Pikmin are hostile towards Olimar, the juvenile Bulbmin will follow Olimar like regular Pikmin once the adult is killed. They are immune to most hazards (save explosions, being eaten, being crushed, and bottomless pits), and will not follow Olimar when he leaves the cave they are found in. The only way to "keep" them is if they are transformed into other Pikmin species by Candypop Buds.


title: Pikmin

text: "Pikmin 3" introduced two new species of Pikmin: Rock Pikmin and Winged Pikmin. The gray-colored Rock Pikmin look like small rocks and can smash through glass, crystals, ice, and the carapaces of large creatures. These Pikmin can also survive crushing attacks from boulders and enemies, and are invulnerable to piercing attacks as well. Rock Pikmin also cause more initial damage when thrown at an enemy, though they are unable to latch onto them. The pink-colored Winged Pikmin are the smallest species of Pikmin and hover in the air using their small wings. They can fly over hazards such as water and pits, and can carry items in the air.


title: Pikmin

text: Another main character in "Pikmin 2" is the President of Hocotate Freight, a large man that runs the company and keeps debt collectors away. The president is forced to sell the Dolphin, and it is replaced by an unnamed ship that is able to speak, name and value the treasure on the planet.


title: Pikmin

text: The "Pikmin" series has received generally positive reviews, and has sold over 6 million units, with the Nintendo Switch port of "Pikmin 3" being the best seller at 2.04 million units. The series has been praised for its strategic gameplay based on time and resource management, as well as its graphics; Philip Kollar of Polygon described "Pikmin 3" as "one of those rare pieces of evidence that a modern Nintendo can still produce something wholly unique."


title: Pikmin

text: Olimar was introduced as a playable character in the "Super Smash Bros." series beginning with 2008's "Super Smash Bros. Brawl", and has appeared in every subsequent entry. He commands the Pikmin in battle and, like in the "Pikmin" series, he is almost entirely dependent on them. Olimar is able to pluck new Pikmin from the ground at any time unless he has the maximum number of Pikmin allowed; he can command up to six in "Brawl", and three in subsequent games. The Pikmin, as in the "Pikmin" series, are fragile and can be defeated easily, but new ones can be plucked from the stage immediately after. Olimar's Final Smash, a signature one-use technique, shows him getting into his Hocotate Spaceship and flying off into the sky, while Red Bulborbs attack the other players on the ground. The ship then plummets to the ground and causes an explosion, dealing damage and knockback to opposing players as Olimar is ejected from it. "Super Smash Bros. Brawl" also features a playable stage called Distant Planet, based on the general setting of the Pikmin's home planet featured in the first three "Pikmin" games. The same stage returned in "Super Smash Bros. for Nintendo 3DS", while "Super Smash Bros. for Wii U" added a stage based on "Pikmin 3"; both stages returned in "Super Smash Bros. Ultimate". Alph from "Pikmin 3" appears as an alternate costume for Olimar beginning with "3DS" and "Wii U".


title: Pikmin

text: Pikmin appear in the Nintendo 3DS built-in application "AR Games". Pikmin also make a cameo during the process of transferring downloadable content from a Nintendo DSi to a 3DS, (or transferring from a 3DS to another 3DS or 2DS) with various types of Pikmin carrying the data over. A similar animation occurs when transferring save data and other information from a Wii to an SD card using the Wii U transfer application, and a second animation, picking up where the first one leaves off, appears when completing the transfer from the SD card to a Wii U.


title: Pikmin

text: "Nintendo Land" includes the series in a team attraction titled "Pikmin Adventure", where players' Miis fight robotic versions of familiar Pikmin foes. "Pikmin Adventure" utilizes the Wii U GamePad and Wii Remote; the former controls Olimar, whereas the latter allows other players to play as Pikmin. On the Game Boy Advance, the card-based Nintendo e-Reader peripheral included Pikmin minigames exclusive to Japan following the release of "Pikmin 2", titled "Pikmin 2-e". In the various minigames, players control either Olimar or the Pikmin to complete various puzzles.


title: Pikmin

text: Several red, yellow and blue Pikmin can be found across the land on ledges of Super Nintendo World at Universal Studios Japan, with some carrying objects like coins and a berry.


title: Pikmin

text: On November 5, 2014, three short animated "Pikmin" films were released on the Wii U and Nintendo 3DS eShop, after being shown at the Tokyo International Film Festival. The short films, "The Night Juicer", "Treasure in a Bottle", and "Occupational Hazards", each feature Captain Olimar and groups of Pikmin. To commemorate the films, puzzles related to them were made available in the Puzzle Swap game mode of "StreetPass Mii Plaza". The films were later uploaded to YouTube to promote "Pikmin 3 Deluxe".


title: Duke of Cornwall

text: Duke of Cornwall is a title in the Peerage of England, traditionally held by the eldest son of the reigning British monarch, previously the English monarch. The duchy of Cornwall was the first duchy created in England and was established by a royal charter in 1337. The present duke is Prince William. His wife, Catherine, is the current duchess of Cornwall.


title: Duke of Cornwall

text: Some folkloric histories of the British Isles, such as Geoffrey of Monmouth's "History of the Kings of Britain" (1136), claim that the first leader of Cornwall was Corineus, a Trojan warrior and ally of Brutus of Troy, portrayed as the original settler of the British Isles. From then through the Arthurian period, such legendary dukes of Cornwall stood apart from the high-king of Britain, while serving as his closest ally and, at times, as his protector (all per Monmouth's collected yarns). Notably in this tale, Gorlois, duke of Cornwall under King Uther Pendragon, rebelled when the king became obsessed with Gorlois' wife Igraine. Uther killed Gorlois and took Igraine: the son was King Arthur.


title: Duke of Cornwall

text: The historical record suggests that, following the Anglo-Saxon settlement of Britain, Cornwall formed part of a separate Kingdom of Dumnonia, which included Devon, although there is evidence that it may have had its own rulers at times. The Celtic southwest of Britain was gradually conquered by the emerging Germanic Kingdom of England, and after the Norman Conquest in 1066 the new rulers of England appointed their own men as earl of Cornwall, the first of whom was in fact a Breton of Cornouaille in Brittany.


title: Duke of Cornwall

text: Edward of Woodstock (widely known as 'The Black Prince'), the eldest son of Edward III, was made the first duke of Cornwall in 1337, after Edward III had lost the title of duke of Normandy. Cornwall was the first dukedom conferred within the Kingdom of England.


title: Duke of Cornwall

text: The charter that established the estate on 17 March 1337 set out the rule that the duke and possessor of the estate would be the eldest son and heir apparent of the monarch. There were some deviations from this rule until a legal case (the Prince's Case) in 1606, which held that the rule should be adhered to.


title: Duke of Cornwall

text: When the estate is without a duke, the possessor is the monarch, even if the former duke left surviving descendants. The monarch's grandson, even if he is the heir apparent, does not succeed to the dukedom. Similarly, no female may ever be duke of Cornwall, even if she is heir presumptive or heir apparent (this being a distinct and even likely possibility in the future after the passage of the Succession to the Crown Act 2013), although a queen regnant without a son would be the de facto duke. However, if a duke of Cornwall should die without descendants (and also no sister between the two brothers if the younger one was born after 28 October 2011), his next brother obtains the duchy, this brother being both the oldest living son and heir apparent.


title: Duke of Cornwall

text: It is possible for an individual to be prince of Wales and heir apparent without being duke of Cornwall. The title 'prince of Wales' is the traditional title of the heir apparent to the throne, granted at the discretion of the sovereign (not automatically) and is not restricted to the eldest son.


title: Duke of Cornwall

text: For example, after the death of Frederick, Prince of Wales, George II's heir apparent was his grandson George (Frederick's eldest son and the future George III). The young Prince George was created prince of Wales but did not become duke of Cornwall because he was the king's grandson, rather than his son. When the sovereign has no legitimate son, or when the heir apparent is not the sovereign's son, the estates of the duchy of Cornwall revert to the Crown until a legitimate son is born or until the accession of a new monarch who has a son.


title: Duke of Cornwall

text: James Francis Edward Stuart, son of James II, was born duke of Cornwall in 1688. Although his father lost the throne, James Francis Edward was not deprived of his own titles and honours as a result of his father's deposition. Instead, from the (prevailing) Hanoverian perspective, it was as a result of his claiming his father's lost thrones that James was attainted for treason on 2 March 1702, and his titles were thus forfeited under English law. However, from the (minority) Jacobite perspective, on his father's death in 1701 the duchy of Cornwall was merged with the Crown.


title: Duke of Cornwall

text: The duchy includes over 220 square miles (570 square kilometres) of land, more than half of which lies in Devon. The duke has some rights over the territory of Cornwall, and for this and other reasons there is debate as to the constitutional status of Cornwall. The High Sheriff of Cornwall is appointed by the duke, not the monarch, in contrast to the other counties of England and Wales. The duke has the right to the estates of all those who die without named heirs ("bona vacantia") in the whole of Cornwall. In 2013, the duchy had a revenue surplus of £19 million, a sum that was exempt from income tax, though the previous prince of Wales chose to pay the tax voluntarily.


title: Duke of Cornwall

text: Until 2011, if there was no duke of Cornwall at any time, then the income of the duchy went to the Crown. Since the passing into law of the Sovereign Grant Act 2011, revenues of the duchy of Cornwall pass to the heir to the throne, regardless of whether that heir is the duke of Cornwall. When the heir is a minor, 10% of revenues pass to the heir, with the balance passing to the Crown (and the Sovereign Grant is reduced by the same amount).


title: Duke of Cornwall

text: The coat of arms of the duke of Cornwall is blazoned as "sable, fifteen bezants", that is, a black field bearing fifteen golden discs. The arms are now used as a badge by the prince of Wales, and they appear below the shield in his coat of arms, along with his other badges.


title: Duke of Cornwall

text: The arms were adopted late in the 15th century, based on the arms of Richard, Earl of Cornwall. The bezants in Richard's arms were intended to represent peas, known in French as "pois", as a punning reference to the French region of Poitou, of which he was count.


title: Duke of Cornwall

text: On 21 June 1968 a royal warrant augmented the aforementioned arms with the heir apparent's coronet, which consists of four crosses patée and four fleurs-de-lises with one arch (used only by the prince of Wales). The supporters are two Cornish choughs, each supporting an ostrich feather. The motto used with the arms is "Houmout", meaning "High-spirited", the personal motto of the Black Prince.


title: Duke of Cornwall

text: All dukes of Cornwall who have been the eldest living son of the sovereign are generally considered to have held the same "creation" of the dukedom. The following is a table of these dukes of Cornwall, with the processes by which they became duke and by which they ceased to hold the title:


title: Duke of Cornwall

text: When his heir apparent, Edward of Woodstock, Prince of Wales and Duke of Cornwall, predeceased him, Edward III granted a new creation of the title 'duke of Cornwall' to his grandson, Richard of Bordeaux. When he acceded the throne as Richard II in 1377, this creation merged with the Crown.


title: Duke of Cornwall

text: When Richard Plantagenet, Duke of York pressed his claim to the throne, he was made heir apparent to Henry VI by the Act of Accord. On 31 October 1460, he was made prince of Wales and earl of Chester, duke of Cornwall and Lord Protector of England. Since he was not the eldest living son of the monarch, this creation was outside the terms of the 1337 warrant; York died in battle two months later, on 30 December 1460.


title: Duke of Cornwall

text: Charles Edward Stuart ('The Young Pretender'), eldest son and heir apparent of James Francis Edward Stuart ('The Old Pretender'), was born in Rome on 31 December 1720, and shortly after his birth, he was declared prince of Wales, duke of Cornwall and earl of Chester in the Jacobite succession. With the death of the Old Pretender on 1 January 1766, he acceded to his father's claim to be King of England, Scotland, France, and Ireland. He died on 31 January 1788.


title: I've Been Killing Slimes for 300 Years and Maxed Out My Level

text: After living a painful life as an office worker, Azusa ends up dying from overworking. She finds herself reincarnated as an undying, unaging witch in a new world, and now she vows to spend her days as pleasantly and stress-free as possible. She makes a living by hunting slimes, the easiest targets. But after centuries of doing this simple job, she has become extremely powerful and finds that she can no longer maintain her low-key lifestyle.


text: Originally, the series was published by Kisetsu Morita on the user-generated web novel site Shōsetsuka ni Narō in 2016. The publisher SB Creative acquired the series and then began publishing it on their GA Novel imprint as a light novel in January 2017. Twenty volumes have been released as of April 2022.


text: In the volumes 5 to 7, 20, and 21 of the light novel, a special story about the character Beelzebub was published at the end of those volumes. In September 2019, those stories were compiled into its own volume and the spin-off, called , published by SB Creative on the GA Novel imprint with Kisetsu Morita and Benio as the author and illustrator, respectively.


text: In the volumes 8 to 10 of the light novel, another spin-off light novel called , a special story about the character Halkara was published at the end of those volumes.


text: In the volumes 11 to 13, 16, and 18 to 19 of the light novel, another spin-off light novel called a special story about the character Laika was published at the end of those volumes.


text: In the volume 17 of the light novel, another spin-off light novel called , a special story about the character Wynona was published at the end of those volume.


text: In June 2017, Square Enix began publishing a manga adaptation of the series by Yūsuke Shiba in their online website and smartphone app "Gangan Online", with eleven "tankōbon" volumes released as of June 2022.


text: A manga adaptation of the spin-off light novel "I Was a Bottom-Tier Bureaucrat for 1,500 Years, and the Demon King Made Me a Minister" was launched on Square Enix's "Gangan Online" website and smartphone app, with Meishi Murakami reprising his work.


text: A manga adaptation of the spin-off light novel "Red Dragon Women Academy" was launched on Square Enix's "Manga UP!" smartphone app, with Hitsujibako reprising his work.


text: An anime television series adaptation was announced during a livestream for the "GA Fes 2019" event on October 19, 2019. The series is animated by Revoroot and directed by Nobukage Kimura, with Tatsuya Takahashi handling series' composition, Keisuke Goto designing the characters, and Keiji Inai composing the series' music. The series aired from April 10 to June 26, 2021, on AT-X, Tokyo MX, and BS11. Aoi Yūki performed the opening theme song , while Azumi Waki performed the ending theme song "Viewtiful Days!". Crunchyroll licensed the series outside of Asia. In Southeast Asia and South Asia, Muse Communication licensed the series and will stream it on iQIYI, Bilibili, CATCHPLAY+, and other platforms. The company also has licensed the anime to Animax Asia for TV broadcasts.


text: The light novel ranked ninth in 2019 in Takarajimasha's annual light novel guide book "Kono Light Novel ga Sugoi!" in the "tankōbon" category.


title: Wahoo

text: Wahoo (Acanthocybium solandri) is a scombrid fish found worldwide in tropical and subtropical seas. It is best known to sports fishermen, as its speed and high-quality flesh makes it a prized and valued game fish. In Hawaii, the wahoo is known as ono." The species is sometimes called hoo" in the United States.


title: Wahoo

text: Its body is elongated and the back is an iridescent blue, while the sides are silvery with a pattern of irregular vertical blue bars. These colors fade rapidly at death. The mouth is large, and the teeth of the wahoo are razor sharp. Both the upper and lower jaws have a somewhat sharper appearance than those of king or Spanish mackerel. Specimens have been recorded at up to in length, and weighing up to . Growth can be rapid. They are among the fastest fish in the sea.


title: Wahoo

text: The wahoo may be distinguished from the related Atlantic king mackerel and from the Indo-Pacific narrow-barred Spanish mackerel by a fold of skin that covers the mandible when its mouth is closed. In contrast, the mandible of the king mackerel is always visible, as is also the case for the smaller Spanish mackerel and Cero mackerel. The teeth of the wahoo are similar to those of king mackerel, but shorter and more closely set together.


title: Wahoo

text: The barracuda is sometimes confused with the mackerel and wahoo, but it is easy to distinguish from the latter two species. Barracuda have prominent scales and larger, dagger-like teeth, and lack the caudal keels and blade-like (forked) tail characteristic of the scombrids.


title: Wahoo

text: Wahoo have a circumtropical distribution and are found in Atlantic, Pacific, and Indian Oceans. Population genomic research using RAD sequencing indicates that two weakly differentiated fish stocks are in the Atlantic and Indo-Pacific Oceans, likely with a considerable degree of migration and gene flow between these populations.


title: Wahoo

text: The eggs of the species are buoyant and the larvae are pelagic. Wahoo tend to be solitary or occur in loose-knit groups of two or three fish. Where conditions are suitable, they can be found in schools around 100 or more.


title: Wahoo

text: Their diet is made up of other fish and squid. From a study surrounding the Western and Central Pacific Ocean (WCPO), an analysis of the stomach contents of wahoo indicated that their diet consists of 84.64% native fish, 14.26% cephalopods (e.g. cuttle fish), and 1.1% crustaceans. The gender ratio of wahoo favors females over males with ratios ranging from 1:0.9 (Puerto Rico) to 3.5:1 (North Carolina), which is common for most pelagic marine species.


title: Wahoo

text: Most wahoo taken from waters have a trematode parasite, the giant stomach worm "(Hirudinella ventricosa)," living in their stomachs, but it does not appear to harm the fish.


title: Wahoo

text: The flesh of the wahoo is white and/or grey, delicate to dense, and highly regarded by many cuisines. The taste has been said to be similar to mackerel. This has created some demand for the wahoo as a premium-priced commercial food fish. In many areas of its range, such as Hawaii, Bermuda, and many parts of the Caribbean, local demand for the wahoo is met by artisanal commercial fishermen who take them primarily by trolling.


title: Wahoo

text: Although local wahoo populations can be affected by heavy commercial and sport-fishing pressure, wahoo as a species is less susceptible to industrial commercial fishing than more tightly schooling and abundant species such as tuna. Wahoo are regularly taken as a bycatch in various commercial fisheries, including longline fisheries for tuna, billfish, and dolphinfish (mahi-mahi or dorado). It is also taken in tuna purse seine fisheries, especially in sets made around floating objects, which act as a focal point for a great deal of other marine life besides tuna. In 2003, the South Atlantic Fishery Management Council issued a Dolphin Wahoo Fishery Management Plan for the Atlantic. The species as a whole, though, is not considered overfished.


title: Wahoo

text: In most parts of its range, the wahoo is a highly prized sport-fishing catch. It reaches a good size, and is often available not too far from land; it is also a very good fighter on light to medium tackle. It is known in sport-fishing circles for the speed and strength of its first run. Recreational sports fishermen sometimes sell their catch.


title: Wahoo

text: Wahoo are successfully fished with live bait around deep-water oil and gas platforms in the Gulf of Mexico during the winter.


title: Eulerian path

text: In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this:


title: Eulerian path

text: Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem:


title: Eulerian path

text: The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs.


title: Eulerian path

text: For the existence of Eulerian trails it is necessary that zero or two vertices have an odd degree; this means the Königsberg graph is "not" Eulerian. If there are no vertices of odd degree, all Eulerian trails are circuits. If there are exactly two vertices of odd degree, all Eulerian trails start at one of them and end at the other. A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian.


title: Eulerian path

text: An Eulerian trail, or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian.


title: Eulerian path

text: An Eulerian cycle, also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree. For finite connected graphs the two definitions are equivalent, while a possibly unconnected graph is Eulerian in the weaker sense if and only if each connected component has an Eulerian cycle.


title: Eulerian path

text: For directed graphs, "path" has to be replaced with "directed path" and "cycle" with "directed cycle".


title: Eulerian path

text: The definition and properties of Eulerian trails, cycles and graphs are valid for multigraphs as well.


title: Eulerian path

text: An Eulerian orientation of an undirected graph "G" is an assignment of a direction to each edge of "G" such that, at each vertex "v", the indegree of "v" equals the outdegree of "v". Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of "G" and then orienting the edges according to the tour. Every Eulerian orientation of a connected graph is a strong orientation, an orientation that makes the resulting directed graph strongly connected.


title: Eulerian path

text: Fleury's algorithm is an elegant but inefficient algorithm that dates to 1883. Consider a graph known to have all edges in the same component and at most two vertices of odd degree. The algorithm starts at a vertex of odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge left at the current vertex. It then moves to the other endpoint of that edge and deletes the edge. At the end of the algorithm there are no edges left, and the sequence from which the edges were chosen forms an Eulerian cycle if the graph has no vertices of odd degree, or an Eulerian trail if there are exactly two vertices of odd degree.


title: Eulerian path

text: While the "graph traversal" in Fleury's algorithm is linear in the number of edges, i.e. formula_1, we also need to factor in the complexity of detecting bridges. If we are to re-run Tarjan's linear time bridge-finding algorithm after the removal of every edge, Fleury's algorithm will have a time complexity of formula_2. A dynamic bridge-finding algorithm of allows this to be improved to formula_3, but this is still significantly slower than alternative algorithms.


title: Eulerian path

text: Hierholzer's 1873 paper provides a different method for finding Euler cycles that is more efficient than Fleury's algorithm:


title: Eulerian path

text: By using a data structure such as a doubly linked list to maintain the set of unused edges incident to each vertex, to maintain the list of vertices on the current tour that have unused edges, and to maintain the tour itself, the individual operations of the algorithm (finding unused edges exiting each vertex, finding a new starting vertex for a tour, and connecting two tours that share a vertex) may be performed in constant time each, so the overall algorithm takes linear time, formula_1.


title: Eulerian path

text: This algorithm may also be implemented with a deque. Because it is only possible to get stuck when the deque represents a closed tour, one should rotate the deque by removing edges from the tail and adding them to the head until unstuck, and then continue until all edges are accounted for. This also takes linear time, as the number of rotations performed is never larger than formula_5 (intuitively, any "bad" edges are moved to the head, while fresh edges are added to the tail)


title: Eulerian path

text: The number of Eulerian circuits in "digraphs" can be calculated using the so-called BEST theorem, named after de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte. The formula states that the number of Eulerian circuits in a digraph is the product of certain degree factorials and the number of rooted arborescences. The latter can be computed as a determinant, by the matrix tree theorem, giving a polynomial time algorithm.


title: Eulerian path

text: BEST theorem is first stated in this form in a "note added in proof" to the Aardenne-Ehrenfest and de Bruijn paper (1951). The original proof was bijective and generalized the de Bruijn sequences. It is a variation on an earlier result by Smith and Tutte (1941).


title: Eulerian path

text: Counting the number of Eulerian circuits on "undirected" graphs is much more difficult. This problem is known to be #P-complete. In a positive direction, a Markov chain Monte Carlo approach, via the "Kotzig transformations" (introduced by Anton Kotzig in 1968) is believed to give a sharp approximation for the number of Eulerian circuits in a graph, though as yet there is no proof of this fact (even for graphs of bounded degree).


title: Eulerian path

text: The asymptotic formula for the number of Eulerian circuits in the complete graphs was determined by McKay and Robinson (1995):


title: Eulerian path

text: Eulerian trails are used in bioinformatics to reconstruct the DNA sequence from its fragments. They are also used in CMOS circuit design to find an optimal logic gate ordering. There are some algorithms for processing trees that rely on an Euler tour of the tree (where each edge is treated as a pair of arcs). The de Bruijn sequences can be constructed as Eulerian trails of de Bruijn graphs.


title: Eulerian path

text: In an infinite graph, the corresponding concept to an Eulerian trail or Eulerian cycle is an Eulerian line, a doubly-infinite trail that covers all of the edges of the graph. It is not sufficient for the existence of such a trail that the graph be connected and that all vertex degrees be even; for instance, the infinite Cayley graph shown, with all vertex degrees equal to four, has no Eulerian line. The infinite graphs that contain Eulerian lines were characterized by . For an infinite graph or multigraph to have an Eulerian line, it is necessary and sufficient that all of the following conditions be met:


title: Eulerian path

text: Euler stated a necessary condition for a graph to be Eulerian as all vertices must have even degree. Hierholzer proved this is a sufficient condition in a paper published in 1873. This leads to the following necessary and sufficient statement for what a graph must have to be Eulerian: An undirected connected graph is Eulerian if and only if every vertex of G has even degree.


title: Eulerian path

text: The following result was proved by Veblen in 1912: An undirected connected graph is Eulerian if and only if it is the disjoint union of some cycles.Hierholzer developed a linear time algorithm for constructing an Eulerian tour in an undirected graph.