Before the reign of Henry VIII, the sovereign always granted his or her assent in person. The sovereign, wearing the Imperial State Crown, would be seated on the throne in the Lords chamber, surrounded by heralds and members of the royal court—a scene that nowadays is repeated only at the annual State Opening of Parliament. The Commons, led by their speaker, would listen from the Bar of the Lords, just outside the chamber. The Clerk of the Parliaments presented the bills awaiting assent to the monarch, save that supply bills were traditionally brought up by the speaker. The Clerk of the Crown, standing on the sovereign's right, then read aloud the titles of the bills (in earlier times, the entire text of the bills). The Clerk of the Parliaments, standing on the sovereign's left, responded by stating the appropriate Norman French formula.


A new device for granting assent was created during the reign of King Henry VIII. In 1542, Henry sought to execute his fifth wife, Catherine Howard, whom he accused of committing adultery; the execution was to be authorised not after a trial but by a bill of attainder, to which he would have to personally assent after listening to the entire text. Henry decided that "the repetition of so grievous a Story and the recital of so infamous a crime" in his presence "might reopen a Wound already closing in the Royal Bosom". Therefore, parliament inserted a clause into the Act of Attainder, providing that assent granted by Commissioners "is and ever was and ever shall be, as good" as assent granted by the sovereign personally. The procedure was used only five times during the 16th century, but more often during the 17th and 18th centuries, especially when George III's health began to deteriorate. Queen Victoria became the last monarch to personally grant assent in 1854.


When granting assent by commission, the sovereign authorises three or more (normally five) lords who are Privy Counsellors to grant assent in his or her name. The Lords Commissioners, as the monarch's representatives are known, wear scarlet parliamentary robes and sit on a bench between the throne and the Woolsack. The Lords Reading Clerk reads the commission aloud; the senior commissioner then states, "My Lords, in obedience to Her Majesty's Commands, and by virtue of the Commission which has been now read, We do declare and notify to you, the Lords Spiritual and Temporal and Commons in Parliament assembled, that Her Majesty has given Her Royal Assent to the several Acts in the Commission mentioned."


During the 1960s, the ceremony of assenting by commission was discontinued and is now only employed once a year, at the end of the annual parliamentary session. In 1960, the Gentleman Usher of the Black Rod arrived to summon the House of Commons during a heated debate and several members protested against the disruption by refusing to attend the ceremony. The debacle was repeated in 1965; this time, when the Speaker left the chair to go to the House of Lords, some members continued to make speeches. As a result, the Royal Assent Act 1967 was passed, creating an additional form for the granting of royal assent. As the attorney-general explained, "there has been a good deal of resentment not only at the loss of Parliamentary time that has been involved but at the breaking of the thread of a possibly eloquent speech and the disruption of a debate that may be caused." The granting of assent by the monarch in person, or by commission, is still possible, but this third form is used on a day-to-day basis.


Under the Royal Assent Act 1967, royal assent can be granted by the sovereign in writing, by means of letters patent, that are presented to the presiding officer of each house of parliament. Then, the presiding officer makes a formal, but simple statement to the house, acquainting each house that royal assent has been granted to the acts mentioned. Thus, unlike the granting of royal assent by the monarch in person or by Royal Commissioners, the method created by the Royal Assent Act 1967 does not require both houses to meet jointly for the purpose of receiving the notice of royal assent. The standard text of the letters patent is set out in The Crown Office (Forms and Proclamations Rules) Order 1992, with minor amendments in 2000. In practice this remains the standard method, a fact that is belied by the wording of the letters patent for the appointment of the Royal Commissioners and by the wording of the letters patent for the granting of royal assent in writing under the 1967 Act ("... And forasmuch as We cannot at this time be present in the Higher House of Our said Parliament being the accustomed place for giving Our Royal Assent...").


When the act is assented to by the sovereign in person, or by empowered Royal Commissioners, royal assent is considered given at the moment when the assent is declared in the presence of both houses jointly assembled. When the procedure created by the Royal Assent Act 1967 is followed, assent is considered granted when the presiding officers of both houses, having received the letters patent from the king or queen signifying the assent, have notified their respective house of the grant of royal assent. Thus, if each presiding officer makes the announcement at a different time (for instance because one house is not sitting on a certain date), assent is regarded as effective when the second announcement is made. This is important because, under British Law, unless there is any provision to the contrary, an act takes effect on the date on which it receives royal assent and that date is not regarded as being the date when the letters patent are signed, or when they are delivered to the presiding officers of each house, but the date on which both houses have been formally acquainted of the assent.


Independently of the method used to signify royal assent, it is the responsibility of the Clerk of the Parliaments, once the assent has been duly notified to both houses, not only to endorse the act in the name of the monarch with the formal Norman French formula, but to certify that assent has been granted. The clerk signs one authentic copy of the bill and inserts the date (in English) on which the assent was notified to the two houses after the title of the act. When an act is published, the signature of the clerk is omitted, as is the Norman French formula, should the endorsement have been made in writing. However, the date on which the assent was notified is printed in brackets.


In Australia, the formal ceremony of granting assent in parliament has not been regularly used since the early 20th century. Now, the bill is sent to the governor-general's residence by the house in which it originated. The governor-general then signs the bill, sending messages to the President of the Senate and the Speaker of the House of Representatives, who notify their respective houses of the governor-general's action. A similar practice is followed in New Zealand, where the governor-general has not personally granted the Royal Assent in parliament since 1875.


In Canada, the traditional ceremony for granting assent in parliament was regularly used until the 21st century, long after it had been discontinued in the United Kingdom and other Commonwealth realms. One result, conceived as part of a string of royal duties intended to demonstrate Canada's status as an independent kingdom, was that King George VI personally assented to nine bills of the Canadian parliament during the 1939 royal tour of Canada—85 years after his great-grandmother, Queen Victoria, had last granted royal assent personally in the United Kingdom. Under the Royal Assent Act 2002, however, the alternative practice of granting assent in writing, with each house being notified separately ( the Speaker of the Senate or a representative reads to the senators the letters from the governor general regarding the written declaration of Royal Assent), was brought into force. As the act also provides, royal assent is to be signified—by the governor general, or, more often, by a deputy, usually a Justice of the Supreme Court, at least twice each calendar year: for the first appropriation measure and for at least one other act, usually the first non-appropriation measure passed. However, the act provides that a grant of royal assent is not rendered invalid by a failure to employ the traditional ceremony where required.


The Royal Assent ceremony takes place in the Senate, as the sovereign is traditionally barred from the House of Commons. On the day of the event, the Speaker of the Senate will read to the chamber a notice from the secretary to the governor general indicating when the viceroy or a deputy thereof will arrive. The Senate thereafter cannot adjourn until after the ceremony. The speaker moves to sit beside the throne, the Mace Bearer, with mace in hand, stands adjacent to him or her, and the governor general enters to take the speaker's chair. The Usher of the Black Rod is then commanded by the speaker to summon the Members of Parliament, who follow Black Rod back to the Senate, the Sergeant-at-Arms carrying the mace of the House of Commons. In the Senate, those from the commons stand behind the bar, while Black Rod proceeds to stand next to the governor general, who then nods his or her head to signify Royal Assent to the presented bills (which do not include appropriations bills). Once the list of bills is complete, the Clerk of the Senate states: "in Her Majesty's name, His [or Her] Excellency the Governor General [or the deputy] doth assent to these bills." If there are any appropriation bills to receive Royal Assent, the Speaker of the House of Commons will read their titles and the Senate clerk repeats them to the governor general, who nods his or her head to communicate Royal Assent. When these bills have all been assented to, the Clerk of the Senate recites "in Her Majesty's name, His [or Her] Excellency the Governor General [or the deputy] thanks her loyal subjects, accepts their benevolence and assents to these bills. The governor general or his or her deputy then depart parliament.


In Belgium, the sanction royale has the same legal effect as royal assent; the Belgian constitution requires a theoretically possible refusal of royal sanction to be countersigned—as any other act of the monarch—by a minister responsible before the House of Representatives. The monarch promulgates the law, meaning that he or she formally orders that the law be officially published and executed. In 1990, when King Baudouin advised his cabinet he could not, in conscience, sign a bill decriminalising abortion (a refusal patently not covered by a responsible minister), the Council of Ministers, at the King's own request, declared Baudouin incapable of exercising his powers. In accordance with the Belgian constitution, upon the declaration of the sovereign's incapacity, the Council of Ministers assumed the powers of the head of state until parliament could rule on the King's incapacity and appoint a regent. The bill was then assented to by all members of the Council of Ministers "on behalf of the Belgian People". In a joint meeting, both houses of parliament declared the King capable of exercising his powers again the next day.


The constitution of Jordan grants its monarch the right to withhold assent to laws passed by its parliament. Article 93 of that document gives the Jordanian sovereign six months to sign or veto any legislation sent to him from the National Assembly; if he vetoes it within that timeframe, the assembly may override his veto by a two-thirds vote of both houses; otherwise, the law does not go into effect (but it may be reconsidered in the next session of the assembly). If the monarch fails to act within six months of the bill being presented to him, it becomes law without his signature.


After the House of Representatives has debated the law, it either approves it and sends it to the Senate with the text "The Second Chamber of the States General sends the following approved proposal of law to the First Chamber", or it rejects it and returns it to the government with the text "The Second Chamber of the States General has rejected the accompanying proposal of law." If the upper house then approves the law, it sends it back to the government with the text "To the King, The States General have accepted the proposal of law as it is offered here."


The government, consisting of the monarch and the ministers, will then usually approve the proposal and the sovereign and one of the ministers signs the proposal with the addition of an enacting clause, thereafter notifying the States General that "The King assents to the proposal." It has happened in exceptional circumstances that the government does not approve a law that has been passed in parliament. In such a case, neither the monarch nor a minister will sign the bill, notifying the States General that "The King will keep the proposal under advisement." A law that has received royal assent will be published in the State Magazine, with the original being kept in the archives of the King's Offices.


Articles 77–79 of the Norwegian Constitution specifically grant the monarch of Norway the right to withhold royal assent from any bill passed by the Storting. Should the sovereign ever choose to exercise this privilege, Article 79 provides a means by which his veto may be over-ridden: "If a Bill has been passed unaltered by two sessions of the Storting, constituted after two separate successive elections and separated from each other by at least two intervening sessions of the Storting, without a divergent Bill having been passed by any Storting in the period between the first and last adoption, and it is then submitted to the King with a petition that His Majesty shall not refuse his assent to a Bill which, after the most mature deliberation, the Storting considers to be beneficial, it shall become law even if the Royal Assent is not accorded before the Storting goes into recess."


Title IV of the 1978 Spanish constitution invests the Consentimiento Real (Royal Assent) and promulgation (publication) of laws with the monarch of Spain, while Title III, The Cortes Generales, Chapter 2, Drafting of Bills, outlines the method by which bills are passed. According to Article 91, within fifteen days of passage of a bill by the Cortes Generales, the sovereign shall give his or her assent and publish the new law. Article 92 invests the monarch with the right to call for a referendum, on the advice of the president of the government (commonly referred to in English as the prime minister) and the authorisation of the cortes.


No provision within the constitution grants the monarch an ability to veto legislation directly; however, no provision prohibits the sovereign from withholding royal assent, which effectively constitutes a veto. When the Spanish media asked King Juan Carlos if he would endorse the bill legalising same-sex marriages, he answered "Soy el Rey de España y no el de Bélgica" ("I am the King of Spain and not that of Belgium")—a reference to King Baudouin I of Belgium, who had refused to sign the Belgian law legalising abortion. The King gave royal assent to Law 13/2005 on 1 July 2005; the law was gazetted in the Boletín Oficial del Estado on 2 July and came into effect on 3 July 2005. Likewise, in 2010, King Juan Carlos gave royal assent to a law permitting abortion on demand.


If the Spanish monarch ever refused in conscience to grant royal assent, a procedure similar to the Belgian handling of King Baudouin's objection would not be possible under the current constitution. If the sovereign were ever declared incapable of discharging royal authority, his or her powers would not be transferred to the Cabinet, pending the parliamentary appointment of a regency. Instead, the constitution mandates the next person of age in the line of succession would immediately become regent. Therefore, had Juan Carlos followed the Belgian example in 2005 or 2010, a declaration of incapacity would have transferred power to Felipe, then the heir apparent.


Articles 41 and 68 of the constitution empower the sovereign to withhold royal assent from bills adopted by the Legislative Assembly. In 2010, the kingdom moved towards greater democracy, with King George Tupou V saying that he would be guided by his prime minister in the exercising of his powers. Nonetheless, this does not preclude an independent royal decision to exercise a right of veto. In November 2011, the assembly adopted an Arms and Ammunitions (Amendment) Bill, which reduced the possible criminal sentences for the illicit possession of firearms. The bill was adopted by ten votes to eight. Two members of the assembly had recently been charged with the illicit possession of firearms. The Prime Minister, Lord Tuʻivakanō, voted in favour of the amendment. Members of the opposition denounced the bill and asked the King to veto it, which he did in December.


In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.


Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.


The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right.a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983.aa[›] Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.


The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G of the group (G, •)" or "an element of the underlying set G of the group (G, •)". Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.


These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Composing two of these symmetry functions gives another symmetry function. These symmetries determine a group called the dihedral group of degree 4 and denoted D4. The underlying set of the group is the above set of symmetry functions, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first a and then b is written symbolically from right to left as


The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.


The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.


The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch, impacting many other fields.a[›]


To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.


Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) ∗ a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.


In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]


In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are


This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1 = (g−1)N.e[›]


Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations


Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.


Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.


In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,


For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted


In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.


Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.


Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.


Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.


Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.


Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.


A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.


Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.


Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:


for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.


Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.


In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.


The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine  pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].


What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.


Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.


During the 16th and 17th centuries slave traders began to raid the region as part of the expansion of the Saharan and Nile River slave routes. Their captives were slaved and shipped to the Mediterranean coast, Europe, Arabia, the Western Hemisphere, or to the slave ports and factories along the West and North Africa or South the Ubanqui and Congo rivers. In the mid 19th century, the Bobangi people became major slave traders and sold their captives to the Americas using the Ubangi river to reach the coast. During the 18th century Bandia-Nzakara peoples established the Bangassou Kingdom along the Ubangi River.


In 1920 French Equatorial Africa was established and Ubangi-Shari was administered from Brazzaville. During the 1920s and 1930s the French introduced a policy of mandatory cotton cultivation, a network of roads was built, attempts were made to combat sleeping sickness and Protestant missions were established to spread Christianity. New forms of forced labor were also introduced and a large number of Ubangians were sent to work on the Congo-Ocean Railway. Many of these forced laborers died of exhaustion, illness, or the poor conditions which claimed between 20% and 25% of the 127,000 workers.


In September 1940, during the Second World War, pro-Gaullist French officers took control of Ubangi-Shari and General Leclerc established his headquarters for the Free French Forces in Bangui. In 1946 Barthélémy Boganda was elected with 9,000 votes to the French National Assembly, becoming the first representative for CAR in the French government. Boganda maintained a political stance against racism and the colonial regime but gradually became disheartened with the French political system and returned to CAR to establish the Movement for the Social Evolution of Black Africa (MESAN) in 1950.


In the Ubangi-Shari Territorial Assembly election in 1957, MESAN captured 347,000 out of the total 356,000 votes, and won every legislative seat, which led to Boganda being elected president of the Grand Council of French Equatorial Africa and vice-president of the Ubangi-Shari Government Council. Within a year, he declared the establishment of the Central African Republic and served as the country's first prime minister. MESAN continued to exist, but its role was limited. After Boganda's death in a plane crash on 29 March 1959, his cousin, David Dacko, took control of MESAN and became the country's first president after the CAR had formally received independence from France. Dacko threw out his political rivals, including former Prime Minister and Mouvement d'évolution démocratique de l'Afrique centrale (MEDAC), leader Abel Goumba, whom he forced into exile in France. With all opposition parties suppressed by November 1962, Dacko declared MESAN as the official party of the state.


In April 1979, young students protested against Bokassa's decree that all school attendees would need to buy uniforms from a company owned by one of his wives. The government violently suppressed the protests, killing 100 children and teenagers. Bokassa himself may have been personally involved in some of the killings. In September 1979, France overthrew Bokassa and "restored" Dacko to power (subsequently restoring the name of the country to the Central African Republic). Dacko, in turn, was again overthrown in a coup by General André Kolingba on 1 September 1981.


By 1990, inspired by the fall of the Berlin Wall, a pro-democracy movement arose. Pressure from the United States, France, and from a group of locally represented countries and agencies called GIBAFOR (France, the USA, Germany, Japan, the EU, the World Bank, and the UN) finally led Kolingba to agree, in principle, to hold free elections in October 1992 with help from the UN Office of Electoral Affairs. After using the excuse of alleged irregularities to suspend the results of the elections as a pretext for holding on to power, President Kolingba came under intense pressure from GIBAFOR to establish a "Conseil National Politique Provisoire de la République" (Provisional National Political Council, CNPPR) and to set up a "Mixed Electoral Commission", which included representatives from all political parties.[citation needed]


When a second round of elections were finally held in 1993, again with the help of the international community coordinated by GIBAFOR, Ange-Félix Patassé won in the second round of voting with 53% of the vote while Goumba won 45.6%. Patassé's party, the Mouvement pour la Libération du Peuple Centrafricain (MLPC) or Movement for the Liberation of the Central African People, gained a simple but not an absolute majority of seats in parliament, which meant Patassé's party required coalition partners.[citation needed]


Patassé purged many of the Kolingba elements from the government and Kolingba supporters accused Patassé's government of conducting a "witch hunt" against the Yakoma. A new constitution was approved on 28 December 1994 but had little impact on the country's politics. In 1996–1997, reflecting steadily decreasing public confidence in the government's erratic behaviour, three mutinies against Patassé's administration were accompanied by widespread destruction of property and heightened ethnic tension. During this time (1996) the Peace Corps evacuated all its volunteers to neighboring Cameroon. To date, the Peace Corps has not returned to the Central African Republic. The Bangui Agreements, signed in January 1997, provided for the deployment of an inter-African military mission, to Central African Republic and re-entry of ex-mutineers into the government on 7 April 1997. The inter-African military mission was later replaced by a U.N. peacekeeping force (MINURCA).


In the aftermath of the failed coup, militias loyal to Patassé sought revenge against rebels in many neighborhoods of Bangui and incited unrest including the murder of many political opponents. Eventually, Patassé came to suspect that General François Bozizé was involved in another coup attempt against him, which led Bozizé to flee with loyal troops to Chad. In March 2003, Bozizé launched a surprise attack against Patassé, who was out of the country. Libyan troops and some 1,000 soldiers of Bemba's Congolese rebel organization failed to stop the rebels and Bozizé's forces succeeded in overthrowing Patassé.[citation needed]


In 2004 the Central African Republic Bush War began as forces opposed to Bozizé took up arms against his government. In May 2005 Bozizé won a presidential election that excluded Patassé and in 2006 fighting continued between the government and the rebels. In November 2006, Bozizé's government requested French military support to help them repel rebels who had taken control of towns in the country's northern regions. Though the initially public details of the agreement pertained to logistics and intelligence, the French assistance eventually included strikes by Mirage jets against rebel positions.


The Syrte Agreement in February and the Birao Peace Agreement in April 2007 called for a cessation of hostilities, the billeting of FDPC fighters and their integration with FACA, the liberation of political prisoners, integration of FDPC into government, an amnesty for the UFDR, its recognition as a political party, and the integration of its fighters into the national army. Several groups continued to fight but other groups signed on to the agreement, or similar agreements with the government (e.g. UFR on 15 December 2008). The only major group not to sign an agreement at the time was the CPJP, which continued its activities and signed a peace agreement with the government on 25 August 2012.


Michel Djotodia took over as president and in May 2013 Central African Republic's Prime Minister Nicolas Tiangaye requested a UN peacekeeping force from the UN Security Council and on 31 May former President Bozizé was indicted for crimes against humanity and incitement of genocide. The security situation did not improve during June–August 2013 and there were reports of over 200,000 internally displaced persons (IDPs) as well as human rights abuses and renewed fighting between Séléka and Bozizé supporters.


In the southwest, the Dzanga-Sangha National Park is located in a rain forest area. The country is noted for its population of forest elephants and western lowland gorillas. In the north, the Manovo-Gounda St Floris National Park is well-populated with wildlife, including leopards, lions, cheetahs and rhinos, and the Bamingui-Bangoran National Park is located in the northeast of CAR. The parks have been seriously affected by the activities of poachers, particularly those from Sudan, over the past two decades.[citation needed]


There are many missionary groups operating in the country, including Lutherans, Baptists, Catholics, Grace Brethren, and Jehovah's Witnesses. While these missionaries are predominantly from the United States, France, Italy, and Spain, many are also from Nigeria, the Democratic Republic of the Congo, and other African countries. Large numbers of missionaries left the country when fighting broke out between rebel and government forces in 2002–3, but many of them have now returned to continue their work.


In 2006, due to ongoing violence, over 50,000 people in the country's northwest were at risk of starvation but this was averted due to assistance from the United Nations.[citation needed] On 8 January 2008, the UN Secretary-General Ban Ki-Moon declared that the Central African Republic was eligible to receive assistance from the Peacebuilding Fund. Three priority areas were identified: first, the reform of the security sector; second, the promotion of good governance and the rule of law; and third, the revitalization of communities affected by conflicts. On 12 June 2008, the Central African Republic requested assistance from the UN Peacebuilding Commission, which was set up in 2005 to help countries emerging from conflict avoid devolving back into war or chaos.


A new government was appointed on 31 March 2013, which consisted of members of Séléka and representatives of the opposition to Bozizé, one pro-Bozizé individual, and a number representatives of civil society. On 1 April, the former opposition parties declared that they would boycott the government. After African leaders in Chad refused to recognize Djotodia as President, proposing to form a transitional council and the holding of new elections, Djotodia signed a decree on 6 April for the formation of a council that would act as a transitional parliament. The council was tasked with electing a president to serve prior to elections in 18 months.


The per capita income of the Republic is often listed as being approximately $400 a year, one of the lowest in the world, but this figure is based mostly on reported sales of exports and largely ignores the unregistered sale of foods, locally produced alcoholic beverages, diamonds, ivory, bushmeat, and traditional medicine. For most Central Africans, the informal economy of the CAR is more important than the formal economy.[citation needed] Export trade is hindered by poor economic development and the country's landlocked position.[citation needed]


Agriculture is dominated by the cultivation and sale of food crops such as cassava, peanuts, maize, sorghum, millet, sesame, and plantain. The annual real GDP growth rate is just above 3%. The importance of food crops over exported cash crops is indicated by the fact that the total production of cassava, the staple food of most Central Africans, ranges between 200,000 and 300,000 tonnes a year, while the production of cotton, the principal exported cash crop, ranges from 25,000 to 45,000 tonnes a year. Food crops are not exported in large quantities, but still constitute the principal cash crops of the country, because Central Africans derive far more income from the periodic sale of surplus food crops than from exported cash crops such as cotton or coffee.[citation needed] Much of the country is self-sufficient in food crops; however, livestock development is hindered by the presence of the tsetse fly.[citation needed]


Presently, the Central African Republic has active television services, radio stations, internet service providers, and mobile phone carriers; Socatel is the leading provider for both internet and mobile phone access throughout the country. The primary governmental regulating bodies of telecommunications are the Ministère des Postes and Télécommunications et des Nouvelles Technologies. In addition, the Central African Republic receives international support on telecommunication related operations from ITU Telecommunication Development Sector (ITU-D) within the International Telecommunication Union to improve infrastructure.


The 2009 Human Rights Report by the United States Department of State noted that human rights in CAR were poor and expressed concerns over numerous government abuses. The U.S. State Department alleged that major human rights abuses such as extrajudicial executions by security forces, torture, beatings and rape of suspects and prisoners occurred with impunity. It also alleged harsh and life-threatening conditions in prisons and detention centers, arbitrary arrest, prolonged pretrial detention and denial of a fair trial, restrictions on freedom of movement, official corruption, and restrictions on workers' rights.


Asthma is thought to be caused by a combination of genetic and environmental factors. Environmental factors include exposure to air pollution and allergens. Other potential triggers include medications such as aspirin and beta blockers. Diagnosis is usually based on the pattern of symptoms, response to therapy over time, and spirometry. Asthma is classified according to the frequency of symptoms, forced expiratory volume in one second (FEV1), and peak expiratory flow rate. It may also be classified as atopic or non-atopic where atopy refers to a predisposition toward developing a type 1 hypersensitivity reaction.


There is no cure for asthma. Symptoms can be prevented by avoiding triggers, such as allergens and irritants, and by the use of inhaled corticosteroids. Long-acting beta agonists (LABA) or antileukotriene agents may be used in addition to inhaled corticosteroids if asthma symptoms remain uncontrolled. Treatment of rapidly worsening symptoms is usually with an inhaled short-acting beta-2 agonist such as salbutamol and corticosteroids taken by mouth. In very severe cases, intravenous corticosteroids, magnesium sulfate, and hospitalization may be required.


Asthma is characterized by recurrent episodes of wheezing, shortness of breath, chest tightness, and coughing. Sputum may be produced from the lung by coughing but is often hard to bring up. During recovery from an attack, it may appear pus-like due to high levels of white blood cells called eosinophils. Symptoms are usually worse at night and in the early morning or in response to exercise or cold air. Some people with asthma rarely experience symptoms, usually in response to triggers, whereas others may have marked and persistent symptoms.


A number of other health conditions occur more frequently in those with asthma, including gastro-esophageal reflux disease (GERD), rhinosinusitis, and obstructive sleep apnea. Psychological disorders are also more common, with anxiety disorders occurring in between 16–52% and mood disorders in 14–41%. However, it is not known if asthma causes psychological problems or if psychological problems lead to asthma. Those with asthma, especially if it is poorly controlled, are at high risk for radiocontrast reactions.


Many environmental factors have been associated with asthma's development and exacerbation including allergens, air pollution, and other environmental chemicals. Smoking during pregnancy and after delivery is associated with a greater risk of asthma-like symptoms. Low air quality from factors such as traffic pollution or high ozone levels, has been associated with both asthma development and increased asthma severity. Exposure to indoor volatile organic compounds may be a trigger for asthma; formaldehyde exposure, for example, has a positive association. Also, phthalates in certain types of PVC are associated with asthma in children and adults.


The hygiene hypothesis attempts to explain the increased rates of asthma worldwide as a direct and unintended result of reduced exposure, during childhood, to non-pathogenic bacteria and viruses. It has been proposed that the reduced exposure to bacteria and viruses is due, in part, to increased cleanliness and decreased family size in modern societies. Exposure to bacterial endotoxin in early childhood may prevent the development of asthma, but exposure at an older age may provoke bronchoconstriction. Evidence supporting the hygiene hypothesis includes lower rates of asthma on farms and in households with pets.


Family history is a risk factor for asthma, with many different genes being implicated. If one identical twin is affected, the probability of the other having the disease is approximately 25%. By the end of 2005, 25 genes had been associated with asthma in six or more separate populations, including GSTM1, IL10, CTLA-4, SPINK5, LTC4S, IL4R and ADAM33, among others. Many of these genes are related to the immune system or modulating inflammation. Even among this list of genes supported by highly replicated studies, results have not been consistent among all populations tested. In 2006 over 100 genes were associated with asthma in one genetic association study alone; more continue to be found.


Asthma is the result of chronic inflammation of the conducting zone of the airways (most especially the bronchi and bronchioles), which subsequently results in increased contractability of the surrounding smooth muscles. This among other factors leads to bouts of narrowing of the airway and the classic symptoms of wheezing. The narrowing is typically reversible with or without treatment. Occasionally the airways themselves change. Typical changes in the airways include an increase in eosinophils and thickening of the lamina reticularis. Chronically the airways' smooth muscle may increase in size along with an increase in the numbers of mucous glands. Other cell types involved include: T lymphocytes, macrophages, and neutrophils. There may also be involvement of other components of the immune system including: cytokines, chemokines, histamine, and leukotrienes among others.


While asthma is a well recognized condition, there is not one universal agreed upon definition. It is defined by the Global Initiative for Asthma as "a chronic inflammatory disorder of the airways in which many cells and cellular elements play a role. The chronic inflammation is associated with airway hyper-responsiveness that leads to recurrent episodes of wheezing, breathlessness, chest tightness and coughing particularly at night or in the early morning. These episodes are usually associated with widespread but variable airflow obstruction within the lung that is often reversible either spontaneously or with treatment".


There is currently no precise test with the diagnosis typically based on the pattern of symptoms and response to therapy over time. A diagnosis of asthma should be suspected if there is a history of: recurrent wheezing, coughing or difficulty breathing and these symptoms occur or worsen due to exercise, viral infections, allergens or air pollution. Spirometry is then used to confirm the diagnosis. In children under the age of six the diagnosis is more difficult as they are too young for spirometry.


Spirometry is recommended to aid in diagnosis and management. It is the single best test for asthma. If the FEV1 measured by this technique improves more than 12% following administration of a bronchodilator such as salbutamol, this is supportive of the diagnosis. It however may be normal in those with a history of mild asthma, not currently acting up. As caffeine is a bronchodilator in people with asthma, the use of caffeine before a lung function test may interfere with the results. Single-breath diffusing capacity can help differentiate asthma from COPD. It is reasonable to perform spirometry every one or two years to follow how well a person's asthma is controlled.


Other supportive evidence includes: a ≥20% difference in peak expiratory flow rate on at least three days in a week for at least two weeks, a ≥20% improvement of peak flow following treatment with either salbutamol, inhaled corticosteroids or prednisone, or a ≥20% decrease in peak flow following exposure to a trigger. Testing peak expiratory flow is more variable than spirometry, however, and thus not recommended for routine diagnosis. It may be useful for daily self-monitoring in those with moderate to severe disease and for checking the effectiveness of new medications. It may also be helpful in guiding treatment in those with acute exacerbations.


Asthma is clinically classified according to the frequency of symptoms, forced expiratory volume in one second (FEV1), and peak expiratory flow rate. Asthma may also be classified as atopic (extrinsic) or non-atopic (intrinsic), based on whether symptoms are precipitated by allergens (atopic) or not (non-atopic). While asthma is classified based on severity, at the moment there is no clear method for classifying different subgroups of asthma beyond this system. Finding ways to identify subgroups that respond well to different types of treatments is a current critical goal of asthma research.


Although asthma is a chronic obstructive condition, it is not considered as a part of chronic obstructive pulmonary disease as this term refers specifically to combinations of disease that are irreversible such as bronchiectasis, chronic bronchitis, and emphysema. Unlike these diseases, the airway obstruction in asthma is usually reversible; however, if left untreated, the chronic inflammation from asthma can lead the lungs to become irreversibly obstructed due to airway remodeling. In contrast to emphysema, asthma affects the bronchi, not the alveoli.


Exercise can trigger bronchoconstriction both in people with or without asthma. It occurs in most people with asthma and up to 20% of people without asthma. Exercise-induced bronchoconstriction is common in professional athletes. The highest rates are among cyclists (up to 45%), swimmers, and cross-country skiers. While it may occur with any weather conditions it is more common when it is dry and cold. Inhaled beta2-agonists do not appear to improve athletic performance among those without asthma however oral doses may improve endurance and strength.


Asthma as a result of (or worsened by) workplace exposures, is a commonly reported occupational disease. Many cases however are not reported or recognized as such. It is estimated that 5–25% of asthma cases in adults are work–related. A few hundred different agents have been implicated with the most common being: isocyanates, grain and wood dust, colophony, soldering flux, latex, animals, and aldehydes. The employment associated with the highest risk of problems include: those who spray paint, bakers and those who process food, nurses, chemical workers, those who work with animals, welders, hairdressers and timber workers.


Many other conditions can cause symptoms similar to those of asthma. In children, other upper airway diseases such as allergic rhinitis and sinusitis should be considered as well as other causes of airway obstruction including: foreign body aspiration, tracheal stenosis or laryngotracheomalacia, vascular rings, enlarged lymph nodes or neck masses. Bronchiolitis and other viral infections may also produce wheezing. In adults, COPD, congestive heart failure, airway masses, as well as drug-induced coughing due to ACE inhibitors should be considered. In both populations vocal cord dysfunction may present similarly.


Chronic obstructive pulmonary disease can coexist with asthma and can occur as a complication of chronic asthma. After the age of 65 most people with obstructive airway disease will have asthma and COPD. In this setting, COPD can be differentiated by increased airway neutrophils, abnormally increased wall thickness, and increased smooth muscle in the bronchi. However, this level of investigation is not performed due to COPD and asthma sharing similar principles of management: corticosteroids, long acting beta agonists, and smoking cessation. It closely resembles asthma in symptoms, is correlated with more exposure to cigarette smoke, an older age, less symptom reversibility after bronchodilator administration, and decreased likelihood of family history of atopy.


The evidence for the effectiveness of measures to prevent the development of asthma is weak. Some show promise including: limiting smoke exposure both in utero and after delivery, breastfeeding, and increased exposure to daycare or large families but none are well supported enough to be recommended for this indication. Early pet exposure may be useful. Results from exposure to pets at other times are inconclusive and it is only recommended that pets be removed from the home if a person has allergic symptoms to said pet. Dietary restrictions during pregnancy or when breast feeding have not been found to be effective and thus are not recommended. Reducing or eliminating compounds known to sensitive people from the work place may be effective. It is not clear if annual influenza vaccinations effects the risk of exacerbations. Immunization; however, is recommended by the World Health Organization. Smoking bans are effective in decreasing exacerbations of asthma.


Avoidance of triggers is a key component of improving control and preventing attacks. The most common triggers include allergens, smoke (tobacco and other), air pollution, non selective beta-blockers, and sulfite-containing foods. Cigarette smoking and second-hand smoke (passive smoke) may reduce the effectiveness of medications such as corticosteroids. Laws that limit smoking decrease the number of people hospitalized for asthma. Dust mite control measures, including air filtration, chemicals to kill mites, vacuuming, mattress covers and others methods had no effect on asthma symptoms. Overall, exercise is beneficial in people with stable asthma. Yoga could provide small improvements in quality of life and symptoms in people with asthma.


For those with severe persistent asthma not controlled by inhaled corticosteroids and LABAs, bronchial thermoplasty may be an option. It involves the delivery of controlled thermal energy to the airway wall during a series of bronchoscopies. While it may increase exacerbation frequency in the first few months it appears to decrease the subsequent rate. Effects beyond one year are unknown. Evidence suggests that sublingual immunotherapy in those with both allergic rhinitis and asthma improve outcomes.


The prognosis for asthma is generally good, especially for children with mild disease. Mortality has decreased over the last few decades due to better recognition and improvement in care. Globally it causes moderate or severe disability in 19.4 million people as of 2004 (16 million of which are in low and middle income countries). Of asthma diagnosed during childhood, half of cases will no longer carry the diagnosis after a decade. Airway remodeling is observed, but it is unknown whether these represent harmful or beneficial changes. Early treatment with corticosteroids seems to prevent or ameliorates a decline in lung function.


As of 2011, 235–330 million people worldwide are affected by asthma, and approximately 250,000–345,000 people die per year from the disease. Rates vary between countries with prevalences between 1 and 18%. It is more common in developed than developing countries. One thus sees lower rates in Asia, Eastern Europe and Africa. Within developed countries it is more common in those who are economically disadvantaged while in contrast in developing countries it is more common in the affluent. The reason for these differences is not well known. Low and middle income countries make up more than 80% of the mortality.


From 2000 to 2010, the average cost per asthma-related hospital stay in the United States for children remained relatively stable at about $3,600, whereas the average cost per asthma-related hospital stay for adults increased from $5,200 to $6,600. In 2010, Medicaid was the most frequent primary payer among children and adults aged 18–44 years in the United States; private insurance was the second most frequent payer. Among both children and adults in the lowest income communities in the United States there is a higher rates of hospital stays for asthma in 2010 than those in the highest income communities.


In 1873, one of the first papers in modern medicine on the subject tried to explain the pathophysiology of the disease while one in 1872, concluded that asthma can be cured by rubbing the chest with chloroform liniment. Medical treatment in 1880, included the use of intravenous doses of a drug called pilocarpin. In 1886, F.H. Bosworth theorized a connection between asthma and hay fever. Epinephrine was first referred to in the treatment of asthma in 1905. Oral corticosteroids began to be used for this condition in the 1950s while inhaled corticosteroids and selective short acting beta agonist came into wide use in the 1960s.


Although the format was capable of offering higher-quality video and audio than its consumer rivals, the VHS and Betamax videocassette systems, LaserDisc never managed to gain widespread use in North America, largely due to high costs for the players and video titles themselves and the inability to record TV programming. It also remained a largely obscure format in Europe and Australia. By contrast, the format was much more popular in Japan and in the more affluent regions of Southeast Asia, such as Hong Kong, Singapore, and Malaysia, being the prevalent rental video medium in Hong Kong during the 1990s. Its superior video and audio quality did make it a somewhat popular choice among videophiles and film enthusiasts during its lifespan.


LaserDisc was first available on the market, in Atlanta, Georgia, on December 15, 1978, two years after the introduction of the VHS VCR, and four years before the introduction of the CD (which is based on laser disc technology). Initially licensed, sold, and marketed as MCA DiscoVision (also known as simply "DiscoVision") in North America in 1978, the technology was previously referred to internally as Optical Videodisc System, Reflective Optical Videodisc, Laser Optical Videodisc, and Disco-Vision (with a dash), with the first players referring to the format as "Video Long Play".


Pioneer Electronics later purchased the majority stake in the format and marketed it as both LaserVision (format name) and LaserDisc (brand name) in 1980, with some releases unofficially referring to the medium as "Laser Videodisc". Philips produced the players while MCA produced the discs. The Philips-MCA cooperation was not successful, and discontinued after a few years. Several of the scientists responsible for the early research (Richard Wilkinson, Ray Dakin and John Winslow) founded Optical Disc Corporation (now ODC Nimbus).


By the early 2000s, LaserDisc was completely replaced by DVD in the North American retail marketplace, as neither players nor software were then produced. Players were still exported to North America from Japan until the end of 2001. The format has retained some popularity among American collectors, and to a greater degree in Japan, where the format was better supported and more prevalent during its life. In Europe, LaserDisc always remained an obscure format. It was chosen by the British Broadcasting Corporation (BBC) for the BBC Domesday Project in the mid-1980s, a school-based project to commemorate 900 years since the original Domesday Book in England. From 1991 up until the early 2000s, the BBC also used LaserDisc technology to play out the channel idents.


The standard home video LaserDisc was 30 cm (12 in) in diameter and made up of two single-sided aluminum discs layered in plastic. Although appearing similar to compact discs or DVDs, LaserDiscs used analog video stored in the composite domain (having a video bandwidth approximately equivalent to the 1-inch (25 mm) C-Type VTR format) with analog FM stereo sound and PCM digital audio. The LaserDisc at its most fundamental level was still recorded as a series of pits and lands much like CDs, DVDs, and even Blu-ray Discs are today. However, while the encoding is of a binary nature, the information is encoded as analog pulse width modulation with a 50% duty cycle, where the information is contained in the lengths and spacing of the pits. In true digital media the pits, or their edges, directly represent 1s and 0s of a binary digital information stream. Early LaserDiscs featured in 1978 were entirely analog but the format evolved to incorporate digital stereo sound in CD format (sometimes with a TOSlink or coax output to feed an external DAC), and later multi-channel formats such as Dolby Digital and DTS.


As Pioneer introduced Digital Audio to LaserDisc in 1985, they further refined the CAA format. CAA55 was introduced in 1985 with a total playback capacity per side of 55 minutes 5 seconds, reducing the video capacity to resolve bandwidth issues with the inclusion of Digital Audio. Several titles released between 1985 and 1987 were analog audio only due to the length of the title and the desire to keep the film on one disc (e.g., Back to the Future). By 1987, Pioneer had overcome the technical challenges and was able to once again encode in CAA60, allowing a total of 60 minutes 5 seconds. Pioneer further refined CAA, offering CAA45, encoding 45 minutes of material, but filling the entire playback surface of the side. Used on only a handful of titles, CAA65 offered 65 minutes 5 seconds of playback time per side. There are a handful of titles pressed by Technidisc that used CAA50. The final variant of CAA is CAA70, which could accommodate 70 minutes of playback time per side. There are not any known uses of this format on the consumer market.


Sound could be stored in either analog or digital format and in a variety of surround sound formats; NTSC discs could carry two analog audio tracks, plus two uncompressed PCM digital audio tracks, which were (EFM, CIRC, 16-bit and 44.056 kHz sample rate). PAL discs could carry one pair of audio tracks, either analog or digital and the digital tracks on a PAL disc were 16-bit 44.1 kHz as on a CD; in the UK, the term "LaserVision" is used to refer to discs with analog sound, while "LaserDisc" is used for those with digital audio. The digital sound signal in both formats are EFM-encoded as in CD. Dolby Digital (also called AC-3) and DTS—which are now common on DVD titles—first became available on LaserDisc, and Star Wars: Episode I – The Phantom Menace (1999) which was released on LaserDisc in Japan, is among the first home video releases ever to include 6.1 channel Dolby Digital EX Surround. Unlike DVDs, which carry Dolby Digital audio in digital form, LaserDiscs store Dolby Digital in a frequency modulated form within a track normally used for analog audio. Extracting Dolby Digital from a LaserDisc required a player equipped with a special "AC-3 RF" output and an external demodulator in addition to an AC-3 decoder. The demodulator was necessary to convert the 2.88 MHz modulated AC-3 information on the disc into a 384 kbit/s signal that the decoder could handle. DTS audio, when available on a disc, replaced the digital audio tracks; hearing DTS sound required only an S/PDIF compliant digital connection to a DTS decoder.


In the mid to late 1990s many higher-end AV receivers included the demodulator circuit specifically for the LaserDisc players RF modulated Dolby Digital AC-3 signal. By the late 1990s with LaserDisc players and disc sales declining due to DVD's growing popularity the AV receiver manufacturers removed the demodulator circuit. Although DVD players were capable of playing Dolby Digital tracks, the signal out of DVD player were not in a modulated form and not compatible with the inputs designed for LaserDisc AC-3. Outboard demodulators were available for a period that convert the AC-3 signal to standard Dolby Digital signal that was compatible with the standard Dolby Digital/PCM inputs on capable AV receivers. Another type marketed by Onkyo and others converted the RF AC-3 signal to 6-channel analog audio.


At least where the digital audio tracks were concerned, the sound quality was unsurpassed at the time compared to consumer videotape, but the quality of the analog soundtracks varied greatly depending on the disc and, sometimes, the player. Many early and lower-end LD players had poor analog audio components, and many early discs had poorly mastered analog audio tracks, making digital soundtracks in any form most desirable to serious enthusiasts. Early DiscoVision and LaserDisc titles lacked the digital audio option, but many of those movies received digital sound in later re-issues by Universal, and the quality of analog audio tracks generally got far better as time went on. Many discs that had originally carried old analog stereo tracks received new Dolby Stereo and Dolby Surround tracks instead, often in addition to digital tracks, helping boost sound quality. Later analog discs also applied CX Noise Reduction, which improved the signal-noise ratio of their audio.


Both AC-3 and DTS surround audio were clumsily implemented on LaserDiscs, leading to some interesting player- and disc-dependent issues. A disc that included AC-3 audio forfeited the right analog audio channel to the modulated AC-3 RF stream. If the player did not have an AC-3 output available, the next most attractive playback option would be the digital Dolby Surround or stereo audio tracks. The reason for this is the RF signal needs to bypass the audio circuitry in order to be properly processed by the demodulator. If either the player did not support digital audio tracks (common in older players), or the disc did not include digital audio tracks at all (uncommon for a disc which is mastered with an AC-3 track), the only remaining option was to fall back to a monophonic presentation of the left analog audio track. However, many older analog-only players not only failed to output AC-3 streams correctly, but were not even aware of their potential existence. Such a player will happily play the analog audio tracks verbatim, resulting in garbage (static) output in the right channel.


Only one 5.1 surround sound option exists on a given LaserDisc (either Dolby Digital or DTS), so if surround sound is desired, the disc must be matched to the capabilities of the playback equipment (LD Player and Receiver/Decoder) by the purchaser. A fully capable LaserDisc playback system includes a newer LaserDisc player that is capable of playing digital tracks, has a digital optical output for digital PCM and DTS audio, is aware of AC-3 audio tracks, and has an AC-3 coaxial output; an external or internal AC-3 RF demodulator and AC-3 decoder; and a DTS decoder. Many 1990s A/V receivers combine the AC-3 decoder and DTS decoder logic, but an integrated AC-3 demodulator is rare both in LaserDisc players and in later A/V receivers.


PAL LaserDiscs have a slightly longer playing time than NTSC discs, but have fewer audio options. PAL discs only have two audio tracks, consisting of either two analog-only tracks on older PAL LDs, or two digital-only tracks on newer discs. In comparison, later NTSC LDs are capable of carrying four tracks (two analog and two digital). On certain releases, one of the analog tracks is used to carry a modulated AC-3 signal for 5.1 channel audio (for decoding and playback by newer LD players with an "AC-3 RF" output). However, older NTSC LDs made before 1984 (such as the original DiscoVision discs) only have two analog audio tracks.


In March 1984, Pioneer introduced the first consumer player with a solid-state laser, the LD-700. It was also the first LD player to load from the front and not the top. One year earlier Hitachi introduced an expensive industrial player with a laser diode, but the player, which had poor picture quality due to an inadequate dropout compensator, was made only in limited quantities. After Pioneer released the LD-700, gas lasers were no longer used in consumer players, despite their advantages, although Philips continued to use gas lasers in their industrial units until 1985.


During its development, MCA, which co-owned the technology, referred to it as the Optical Videodisc System, "Reflective Optical Videodisc" or "Laser Optical Videodisc", depending on the document; changing the name once in 1969 to Disco-Vision and then again in 1978 to DiscoVision (without the hyphen), which became the official spelling. Technical documents and brochures produced by MCA Disco-Vision during the early and mid-'70s also used the term "Disco-Vision Records" to refer to the pressed discs. MCA owned the rights to the largest catalog of films in the world during this time, and they manufactured and distributed the DiscoVision releases of those films under the "MCA DiscoVision" software and manufacturing label; consumer sale of those titles began on December 15, 1978, with the aforementioned Jaws.


Philips' preferred name for the format was "VLP", after the Dutch words Video Langspeel-Plaat ("Video long-play disc"), which in English-speaking countries stood for Video Long-Play. The first consumer player, the Magnavox VH-8000 even had the VLP logo on the player. For a while in the early and mid-1970s, Philips also discussed a compatible audio-only format they called "ALP", but that was soon dropped as the Compact Disc system became a non-compatible project in the Philips corporation. Until early 1980, the format had no "official" name. The LaserVision Association, made up of MCA, Universal-Pioneer, IBM, and Philips/Magnavox, was formed to standardize the technical specifications of the format (which had been causing problems for the consumer market) and finally named the system officially as "LaserVision".


Pioneer Electronics also entered the optical disc market in 1977 as a 50/50 joint-venture with MCA called Universal-Pioneer and manufacturing MCA designed industrial players under the MCA DiscoVision name (the PR-7800 and PR-7820). For the 1980 launch of the first Universal-Pioneer player, the VP-1000 was noted as a "laser disc player", although the "LaserDisc" logo displayed clearly on the device. In 1981, "LaserDisc" was used exclusively for the medium itself, although the official name was "LaserVision" (as seen at the beginning of many LaserDisc releases just before the start of the film). However, as Pioneer reminded numerous video magazines and stores in 1984, LaserDisc was a trademarked word, standing only for LaserVision products manufactured for sale by Pioneer Video or Pioneer Electronics. A 1984 Ray Charles ad for the LD-700 player bore the term "Pioneer LaserDisc brand videodisc player". From 1981 until the early 1990s, all properly licensed discs carried the LaserVision name and logo, even Pioneer Artists titles.


During the early years, MCA also manufactured discs for other companies including Paramount, Disney and Warner Bros. Some of them added their own names to the disc jacket to signify that the movie was not owned by MCA. After Discovision Associates shut down in early 1982, Universal Studio's videodisc software label, called MCA Videodisc until 1984, began reissuing many DiscoVision titles. Unfortunately, quite a few, such as Battlestar Galactica and Jaws, were time-compressed versions of their CAV or CLV Disco Vision originals. The time-compressed CLV re-issue of Jaws no longer had the original soundtrack, having had incidental background music replaced for the video disc version due to licensing cost (the music would not be available until the THX LaserDisc box set was released in 1995). One Universal/Columbia co-production issued by MCA Disco Vision in both CAV and CLV versions, The Electric Horseman, is still not available in any other home video format with its original score intact; even the most recent DVD release has had substantial music replacements of both instrumental score and Willie Nelson's songs. An MCA release of Universal's Howard the Duck, sees only the start credits shown in widescreen before changing to 4:3 for the rest of the film. For many years this was the only disc-based release of the film, until widescreen DVD formats were released with extras. Also, the LaserDisc release of E.T. the Extra-Terrestrial is the only format to include the cut scene of Harrison Ford playing the part of the school headmaster telling off Elliott for letting the frogs free in the biology class.


LaserDisc had a number of advantages over VHS. It featured a far sharper picture with a horizontal resolution of 425 TVL lines for NTSC and 440 TVL lines for PAL discs, while VHS featured only 240 TVL lines with NTSC. It could handle analog and digital audio where VHS was mostly analog only (VHS can have PCM audio in professional applications but is uncommon), and the NTSC discs could store multiple audio tracks. This allowed for extras like director's commentary tracks and other features to be added onto a film, creating "Special Edition" releases that would not have been possible with VHS. Disc access was random and chapter based, like the DVD format, meaning that one could jump to any point on a given disc very quickly. By comparison, VHS would require tedious rewinding and fast-forwarding to get to specific points.


LaserDiscs were initially cheaper than videocassettes to manufacture, because they lacked the moving parts and plastic outer shell that are necessary for VHS tapes to work, and the duplication process was much simpler. A VHS cassette has at least 14 parts including the actual tape while LaserDisc has one part with five or six layers. A disc can be stamped out in a matter of seconds whereas duplicating videotape required a complex bulk tape duplication mechanism and was a time-consuming process. However, by the end of the 1980s, average disc-pressing prices were over $5.00 per two-sided disc, due to the large amount of plastic material and the costly glass-mastering process needed to make the metal stamper mechanisms. Due to the larger volume of demand, videocassettes quickly became much cheaper to duplicate, costing as little as $1.00 by the beginning of the 1990s.


LaserDiscs potentially had a much longer lifespan than videocassettes. Because the discs were read optically instead of magnetically, no physical contact needs to be made between the player and the disc, except for the player's clamp that holds the disc at its center as it is spun and read. As a result, playback would not wear the information-bearing part of the discs, and properly manufactured LDs would theoretically last beyond one's lifetime. By contrast, a VHS tape held all of its picture and sound information on the tape in a magnetic coating which is in contact with the spinning heads on the head drum, causing progressive wear with each use (though later in VHS's lifespan, engineering improvements allowed tapes to be made and played back without contact). Also, the tape was thin and delicate, and it was easy for a player mechanism, especially on a low quality or malfunctioning model, to mishandle the tape and damage it by creasing it, frilling (stretching) its edges, or even breaking it.


LaserDisc was a composite video format: the luminance (black and white) and chrominance (color) information were transmitted in one signal, separated by the receiver. While good comb filters can do so adequately, these two signals cannot be completely separated. On DVDs, data is stored in the form of digital blocks which make up each independent frame. The signal produced is dependent on the equipment used to master the disc. Signals range from composite and split, to YUV and RGB. Depending upon which format is used, this can result in far higher fidelity, particularly at strong color borders or regions of high detail (especially if there is moderate movement in the picture) and low-contrast details like skin tones, where comb filters almost inevitably smudge some detail.


In contrast to the entirely digital DVD, LaserDiscs use only analog video. As the LaserDisc format is not digitally encoded and does not make use of compression techniques, it is immune to video macroblocking (most visible as blockiness during high motion sequences) or contrast banding (subtle visible lines in gradient areas, such as out-of-focus backgrounds, skies, or light casts from spotlights) that can be caused by the MPEG-2 encoding process as video is prepared for DVD. Early DVD releases held the potential to surpass their LaserDisc counterparts, but often managed only to match them for image quality, and in some cases, the LaserDisc version was preferred. However, proprietary human-assisted encoders manually operated by specialists can vastly reduce the incidence of artifacts, depending on playing time and image complexity. By the end of LaserDisc's run, DVDs were living up to their potential as a superior format.


LaserDisc players can provide a great degree of control over the playback process. Unlike many DVD players, the transport mechanism always obeys commands from the user: pause, fast-forward, and fast-reverse commands are always accepted (barring, of course, malfunctions). There were no "User Prohibited Options" where content protection code instructs the player to refuse commands to skip a specific part (such as fast forwarding through copyright warnings). (Some DVD players, particularly higher-end units, do have the ability to ignore the blocking code and play the video without restrictions, but this feature is not common in the usual consumer market.)


Damaged spots on a LaserDisc can be played through or skipped over, while a DVD will often become unplayable past the damage. Some newer DVD players feature a repair+skip algorithm, which alleviates this problem by continuing to play the disc, filling in unreadable areas of the picture with blank space or a frozen frame of the last readable image and sound. The success of this feature depends upon the amount of damage. LaserDisc players, when working in full analog, recover from such errors faster than DVD players. Direct comparison here is almost impossible due to the sheer size differences between the two media. A 1 in (3 cm) scratch on a DVD will probably cause more problems than a 1 in (3 cm) scratch on a LaserDisc, but a fingerprint taking up 1% of the area of a DVD would almost certainly cause fewer problems than a similar mark covering 1% of the surface of a LaserDisc.[citation needed]


Similar to the CD versus LP sound quality debates common in the audiophile community, some videophiles argue that LaserDisc maintains a "smoother", more "film-like", natural image while DVD still looks slightly more artificial. Early DVD demo discs often had compression or encoding problems, lending additional support to such claims at the time. However, the video signal-to-noise ratio and bandwidth of LaserDisc are substantially less than that of DVDs, making DVDs appear sharper and clearer to most viewers.


Another advantage, at least to some consumers, was the lack of any sort of anti-piracy technology. It was claimed that Macrovision's Copyguard protection could not be applied to LaserDisc, due to the format's design. The vertical blanking interval, where the Macrovision signal would be implemented, was also used for the internal timing on LaserDisc players, so test discs with Macrovision would not play at all. There was never a push to redesign the format despite the obvious potential for piracy due to its relatively small market share. The industry simply decided to engineer it into the DVD specification.


LaserDisc's support for multiple audio tracks allowed for vast supplemental materials to be included on-disc and made it the first available format for "Special Edition" releases; the 1984 Criterion Collection edition of Citizen Kane is generally credited as being the first "Special Edition" release to home video,[citation needed] and for setting the standard by which future SE discs were measured. The disc provided interviews, commentary tracks, documentaries, still photographs, and other features for historians and collectors.


The space-consuming analog video signal of a LaserDisc limited playback duration to 30 minutes (CAV) or 60 minutes (CLV) per side because of the hardware manufacturer's refusal to reduce line count for increased playtime. After one side was finished playing, a disc has to be flipped over in order to continue watching a movie, and some titles fill two or more discs. Many players, especially units built after the mid-1980s, can "flip" discs automatically by rotating the optical pickup to the other side of the disc, but this is accompanied by a pause in the movie during the side change. If the movie is longer than what could be stored on two sides of a single disc, manually swapping to a second disc is necessary at some point during the film. One exception to this rule is the Pioneer LD-W1, which features two disc platters. In addition, perfect still frames and random access to individual still frames is limited only to the more expensive CAV discs, which only had a playing time of approximately 30 minutes per side. In later years, Pioneer and other manufacturers overcame this limitation by incorporating a digital memory buffer, which "grabbed" a single frame from a CLV disc.


The analog information encoded on LaserDiscs does not include any form of built-in checksum or error correction. Because of this, slight dust and scratches on the disc surface can result in read-errors which cause various video quality problems: glitches, streaks, bursts of static, or momentary picture interruptions. In contrast, the digital MPEG-2 format information used on DVDs has built-in error correction which ensures that the signal from a damaged disc will remain identical to that from a perfect disc right up until the point at which damage to the disc surface is so substantial that it prevents the laser from being able to identify usable data.


In addition, LaserDisc videos sometimes exhibit a problem known as "crosstalk". The issue can arise when the laser optical pickup assembly within the player is out of alignment or because the disc is damaged or excessively warped, but it could also occur even with a properly functioning player and a factory-new disc, depending on electrical and mechanical alignment problems. In these instances, the issue arose due to the fact that CLV discs require subtle changes in rotating speed at various points during playback. During a change in speed, the optical pickup inside the player might read video information from a track adjacent to the intended one, causing data from the two tracks to "cross"; the extra video information picked up from that second track shows up as distortion in the picture which looks reminiscent of swirling "barber poles" or rolling lines of static.


Assuming the player's optical pickup is in proper working order, crosstalk distortion normally does not occur during playback of CAV format LaserDiscs, as the rotational speed never varies. However, if the player calibration is out of order or if the CAV disc is faulty or damaged, other problems affecting tracking accuracy can occur. One such problem is "laser lock", where the player reads the same two fields for a given frame over and over again, causing the picture to look frozen as if the movie were paused.


Another significant issue unique to LaserDisc is one involving the inconsistency of playback quality between different makers and models of player. On most televisions, a given DVD player will produce a picture that is visually indistinguishable from other units. Differences in image quality between players only becomes easily apparent on large televisions and substantial leaps in image quality are generally only obtained with expensive, high-end players that allow for post-processing of the MPEG-2 stream during playback. In contrast, LaserDisc playback quality is highly dependent on hardware quality. Major variances in picture quality appear between different makers and models of LD players, even when tested on a low to mid-range television. The obvious benefits of using high quality equipment has helped keep demand for some players high, thus also keeping pricing for those units comparably high. In the 1990s, notable players sold for anywhere from US$200 to well over $1,000, while older and less desirable players could be purchased in working condition for as little as $25.


Many early LDs were not manufactured properly; sometimes a substandard adhesive was used to sandwich together the two sides of the disc.[citation needed] The adhesive contained impurities that were able to penetrate the lacquer seal layer and chemically attack the metalized reflective aluminium layer, causing it to oxidize and lose its reflective characteristics. This was a problem that was termed "laser rot" among LD enthusiasts, also called "color flash" internally by LaserDisc-pressing plants. Some forms of laser rot could appear as black spots that looked like mold or burned plastic which cause the disc to skip and the movie to exhibit excessive speckling noise. But, for the most part, rotted discs could actually appear perfectly fine to the naked eye.