Signed graph

In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.

A signed graph is balanced if the product of edge signs around every cycle is positive.  Three fundamental questions about a signed graph are:  Is it balanced?  What is the largest size of a balanced edge set in it?  What is the smallest number of vertices that must be deleted to make it balanced?  The first question is easy to solve quickly; the second and third are computationally intractable (technically, they are NP-hard).

The name "signed graph" and the notion of balance appeared first in a mathematical paper of Frank Harary in 1953. Dénes Kőnig had already studied equivalent notions in 1936 under a different terminology but without recognizing the relevance of the sign group.
At the Center for Group Dynamics at the University of Michigan, Dorwin Cartwright and Harary generalized Fritz Heider's psychological theory of balance in triangles of sentiments to a psychological theory of balance in signed graphs.

Signed graphs have been rediscovered many times because they come up naturally in many unrelated areas.  For instance, they enable one to describe and analyze the geometry of subsets of the classical root systems.  They appear in topological graph theory and group theory.  They are a natural context for questions about odd and even cycles in graphs.  They appear in computing the ground state energy in the non-ferromagnetic Ising model; for this one needs to find a largest balanced edge set in Σ.  They have been applied to data classification in correlation clustering.

Examples
 The complete signed graph on n vertices with loops, denoted by ±Kno, has every possible positive and negative edge including negative loops, but no positive loops.  Its edges correspond to the roots of the root system Cn; the column of an edge in the incidence matrix (see below) is the vector representing the root.
 The complete signed graph with half-edges, ±Kn', is ±Kn with a half-edge at every vertex.  Its edges correspond to the roots of the root system Bn, half-edges corresponding to the unit basis vectors.
 The complete signed link graph, ±Kn, is the same but without loops.  Its edges correspond to the roots of the root system Dn.
 An all-positive signed graph has only positive edges.  If the underlying graph is G, the all-positive signing is written +G.
 An all-negative signed graph has only negative edges.  It is balanced if and only if it is bipartite because a circle is positive if and only if it has even length.  An all-negative graph with underlying graph G is written −G.
 A signed complete graph has as underlying graph G the ordinary complete graph Kn.  It may have any signs.  Signed complete graphs are equivalent to two-graphs, which are of value in finite group theory.  A two-graph can be defined as the class of vertex sets of negative triangles (having an odd number of negative edges) in a signed complete graph.

Adjacency matrix
The adjacency matrix of a signed graph Σ on n vertices is an n × n matrix A(Σ).  It has a row and column for each vertex.  The entry avw in row v and column w is the number of positive vw edges minus the number of negative vw edges.  On the diagonal, avv = 0 if there are no loops or half-edges; the correct definition when such edges exist depends on the circumstances.

Orientation
A signed graph is oriented when each end of each edge is given a direction, so that in a positive edge the ends are both directed from one endpoint to the other, and in a negative edge either both ends are directed outward, to their own vertices, or both are directed inward, away from their vertices.  Thus, an oriented signed graph is the same as a bidirected graph.  (It is very different from a signed digraph.)

Incidence matrix
The (more correctly, "an") incidence matrix of a signed graph with n vertices and m edges is an n × m matrix, with a row for each vertex and a column for each edge.  It is obtained by orienting the signed graph in any way.  Then its entry ηij is +1 if edge j is oriented into vertex i, −1 if edge j is oriented out of vertex i, and 0 if vertex i and edge j are not incident.  This rule applies to a link, whose column will have two nonzero entries with absolute value 1, a half-edge, whose column has a single nonzero entry +1 or −1, and a loose edge, whose column has only zeroes.  The column of a loop, however, is all zero if the loop is positive, and if the loop is negative it has entry ±2 in the row corresponding to its incident vertex.

Any two incidence matrices are related by negating some subset of the columns.  Thus, for most purposes it makes no difference which orientation we use to define the incidence matrix, and we may speak of the incidence matrix of Σ without worrying about exactly which one it is.

Negating a row of the incidence matrix corresponds to switching the corresponding vertex.

Switching
Switching a vertex in Σ means negating the signs of all the edges incident to that vertex.  Switching a set of vertices means negating all the edges that have one end in that set and one end in the complementary set.  Switching a series of vertices, once each, is the same as switching the whole set at once.

Switching of signed graphs (signed switching) is generalized from Seidel (1976), where it was applied to graphs (graph switching), in a way that is equivalent to switching of signed complete graphs.

Switching equivalence means that two graphs are related by switching, and an equivalence class of signed graphs under switching is called a switching class.  Sometimes these terms are applied to equivalence of signed graphs under the combination of switching and isomorphism, especially when the graphs are unlabeled; but to distinguish the two concepts the combined equivalence may be called switching isomorphism and an equivalence class under switching isomorphism may be called a switching isomorphism class.

Switching a set of vertices affects the adjacency matrix by negating the rows and columns of the switched vertices.  It affects the incidence matrix by negating the rows of the switched vertices.

Fundamental theorem
The sign of a path is the product of the signs of its edges. Thus a path is positive only if there are an even number of negative edges in it (where zero is even). In the mathematical balance theory of Frank Harary, a signed graph is balanced when every cycle is positive. He proves that a signed graph is balanced when (1) for every pair of nodes, all paths between them have the same sign, or (2) the graph partitions into a pair of subgraphs, each consisting of positive edges, but connected by negative edges.  The theorem was published by Harary in 1953. It generalizes the theorem that an ordinary (unsigned) graph is bipartite if and only if every cycle has even length.

A simple proof uses the method of switching.  To prove Harary's theorem, one shows by induction that Σ can be switched to be all positive if and only if it is balanced.

A weaker theorem, but with a simpler proof, is that if every 3-cycle in a signed complete graph is positive, then the graph is balanced. For the proof, pick an arbitrary node n and place it and all those nodes that are linked to n by a positive edge in one group, called A, and all those linked to n by a negative edge in the other, called B. Since this is a complete graph, every two nodes in A must be friends and every two nodes in B must be friends, otherwise there would be a 3-cycle which was unbalanced.  (Since this is a complete graph, any one negative edge would cause an unbalanced 3-cycle.)  Likewise, all negative edges must go between the two groups.

Frustration
Give each vertex a value of +1 or −1; we call this a state of Σ.  An edge is called satisfied if it is positive and both endpoints have the same value, or it is negative and the endpoints have opposite values. An edge that is not satisfied is called frustrated. The smallest number of frustrated edges over all states is called the frustration index (or line index of balance) of Σ.  Finding the frustration index is an NP-hard problem. Aref et al. suggest binary programming models that are capable of computing the frustration index of graphs with up to 105 edges in a reasonable time. One can see the NP-hard complexity by observing that the frustration index of an all-negative signed graph is equivalent to the maximum cut problem in graph theory, which is NP-hard.  The reason for the equivalence is that the frustration index equals the smallest number of edges whose negation (or, equivalently, deletion; a theorem of Harary) makes Σ balanced.  (This can be proved easily by switching.)

The frustration index is important in a model of spin glasses, the mixed Ising model.  In this model, the signed graph is fixed.  A state consists of giving a "spin", either "up" or "down", to each vertex.  We think of spin up as +1 and spin down as −1.  Thus, each state has a number of frustrated edges.  The energy of a state is larger when it has more frustrated edges, so a ground state is a state with the fewest frustrated energy.  Thus, to find the ground state energy of Σ one has to find the frustration index.

Matroid theory
There are two matroids associated with a signed graph, called the signed-graphic matroid (also called the frame matroid or sometimes bias matroid) and the lift matroid, both of which generalize the cycle matroid of a graph.  They are special cases of the same matroids of a biased graph.

The frame matroid (or signed-graphic matroid) M(G) has for its ground set the edge set E.  An edge set is independent if each component contains either no circles or just one circle, which is negative.  (In matroid theory a half-edge acts exactly like a negative loop.)  A circuit of the matroid is either a positive circle, or a pair of negative circles together with a connecting simple path, such that the two circles are either disjoint (then the connecting path has one end in common with each circle and is otherwise disjoint from both) or share just a single common vertex (in this case the connecting path is that single vertex).  The rank of an edge set S is n − b, where n is the number of vertices of G and b is the number of balanced components of S, counting isolated vertices as balanced components.
This matroid is the column matroid of the incidence matrix of the signed graph.
That is why it describes the linear dependencies of the roots of a classical root system.

The extended lift matroid L0(G) has for its ground set the set E0 the union of edge set E with an extra point, which we denote e0.  The lift matroid L(G) is the extended lift matroid restricted to E.  The extra point acts exactly like a negative loop, so we describe only the lift matroid.  An edge set is independent if it contains either no circles or just one circle, which is negative.  (This is the same rule that is applied separately to each component in the signed-graphic matroid.)  A matroid circuit is either a positive circle or a pair of negative circles that are either disjoint or have just a common vertex.  The rank of an edge set S is n − c + ε, where c is the number of components of S, counting isolated vertices, and ε is 0 if S is balanced and 1 if it is not.

Other kinds of "signed graph"
Sometimes the signs are taken to be +1 and −1.  This is only a difference of notation, if the signs are still multiplied around a circle and the sign of the product is the important thing.  However, there are two other ways of treating the edge labels that do not fit into signed graph theory.

The term signed graph is applied occasionally to graphs in which each edge has a weight, w(e) = +1 or −1.  These are not the same kind of signed graph; they are weighted graphs with a restricted weight set.  The difference is that weights are added, not multiplied.  The problems and methods are completely different.

The name is also applied to graphs in which the signs function as colors on the edges.  The significance of the color is that it determines various weights applied to the edge, and not that its sign is intrinsically significant.  This is the case in knot theory, where the only significance of the signs is that they can be interchanged by the two-element group, but there is no intrinsic difference between positive and negative.  The matroid of a sign-colored graph is the cycle matroid of the underlying graph; it is not the frame or lift matroid of the signed graph.   The sign labels, instead of changing the matroid, become signs on the elements of the matroid.

In this article we discuss only signed graph theory in the strict sense.  For sign-colored graphs see colored matroids.

Signed digraph
A signed digraph is a directed graph with signed arcs.  Signed digraphs are far more complicated than signed graphs, because only the signs of directed cycles are significant.  For instance, there are several definitions of balance, each of which is hard to characterize, in strong contrast with the situation for signed undirected graphs.

Signed digraphs should not be confused with oriented signed graphs.  The latter are bidirected graphs, not directed graphs (except in the trivial case of all positive signs).

Coloring
As with unsigned graphs, there is a notion of signed graph coloring. Where a coloring of a graph is a mapping from the vertex set to the natural numbers, a coloring of a signed graph is a mapping from the vertex set to the integers.
The constraints on proper colorings come from the edges of the signed graph. The integers assigned to two vertices must be distinct if they are connected by a positive edge. The labels on adjacent vertices must not be additive inverses if the vertices are connected by a negative edge. There can be no proper coloring of a signed graph with a positive loop.

When restricting the vertex labels to the set of integers with magnitude at most a natural number k, the set of proper colorings of a signed graph is finite. The relation between the number of such proper colorings and k is a polynomial in k. This is analogous to the chromatic polynomial of unsigned graphs.

Applications

Social psychology
In social psychology, signed graphs have been used to model social situations, with positive edges representing friendships and negative edges enmities between nodes, which represent people. Then, for example, a positive 3-cycle is either three mutual friends, or two friends with a common enemy; while a negative 3-cycle is either three mutual enemies, or two enemies who share a mutual friend. According to balance theory, positive cycles are balanced and supposed to be stable social situations, whereas negative cycles are unbalanced and supposed to be unstable.  According to the theory, in the case of three mutual enemies, this is because sharing a common enemy is likely to cause two of the enemies to become friends.  In the case of two enemies sharing a friend, the shared friend is likely to choose one over the other and turn one of his or her friendships into an enemy.

Antal, Krapivsky and Reder consider social dynamics as the change in sign on an edge of a signed graph. The social relations with previous friends of a divorcing couple are used to illustrate the evolution of a signed graph in society. Another illustration describes the changing international alliances between European powers in the decades before the First World War. They consider local triad dynamics and constrained triad dynamics, where in the latter case a relationship change is made only when the total number of unbalanced triads is reduced. The simulation presumed a complete graph with random relations having a random unbalanced triad selected for transformation. The evolution of the signed graph with N nodes under this process is studied and simulated to describe the stationary density of friendly links.

Balance theory has been severely challenged, especially in its application to large systems, on the theoretical ground that friendly relations tie a society together, while a society divided into two camps of enemies would be highly unstable.
Experimental studies have also provided only weak confirmation of the predictions of structural balance theory.

Spin glasses
In physics, signed graphs are a natural context for the general, nonferromagnetic Ising model, which is applied to the study of spin glasses.

Complex systems

Using an analytic method initially developed in population biology and ecology, but now used a similar notion in many scientific disciplines, analysis of the properties of signed digraphs has found application in applied formal causal reasoning about the behavior of complex causal systems. Such analyses answer questions about feedback at given levels of the system, and about the direction of variable response given a perturbation to a system at one or more points, variable correlations given such perturbations, the distribution of variance across the system, and the sensitivity or insensitivity of particular variables to system perturbations.

Data clustering
Correlation clustering looks for natural clustering of data by similarity.  The data points are represented as the vertices of a graph, with a positive edge joining similar items and a negative edge joining dissimilar items.

Generalizations
A signed graph is a special kind of gain graph, where the gain group has order 2.  The pair (G, B(Σ)) determined by a signed graph Σ is a special kind of biased graph.

Notes

References
.
.

Category:Matroid theory
Category:Extensions and generalizations of graphs
Category:Oriented matroids


Compton Branch

Compton Branch is a stream in Washington County in the U.S. state of Missouri. It is a tributary of Hazel Creek.

It is unknown why the name "Compton" was applied to this stream.

See also
List of rivers of Missouri

References

Category:Rivers of Washington County, Missouri
Category:Rivers of Missouri


James Thompson (crime writer)

James Thompson (16 October 1964 – 2 August 2014)  was an American-Finnish crime writer based in Helsinki. He had a master's degree in English philology from The University of Helsinki, where he also studied Finnish, in which he was fluent. He studied six languages. He published four crime novels with the Finnish inspector Kari Vaara as the protagonist.

Helsinki Noir was published by Akashic Books November 2014. It is an anthology edited by Thompson and it includes one story he wrote.

Thompson died unexpectedly in his home town of Lahti, Finland in August 2014.

Inspector Vaara series

Kari Vaara, police chief in the town of Kittilä, Lapland, debuted in Thompson's first novel, Snow Angels. In the second novel, Lucifer's Tears, he moves to Helsinki. While he is portrayed as a good cop in the first two books, in the next installment, Helsinki White, we see Vaara turning into a rogue and corrupt cop. Helsinki Blood takes on further along the same line, with Vaara trying to tie all loose ends and redeem himself.

Bibliography

Inspector Vaara series
 Snow Angels (2009)
 Lucifer's Tears (2011)
 Helsinki White (2012)
 Helsinki Blood (2013)
 Helsinki Dead was scheduled for release in 2014 but remained unfinished at the time of Thompson's death.

Other literary endeavors
James Thompson was first reviewed and later became a reviewer at New York Journal of Books. Following his very early death, New York Journal of Books added to his reviewer biography, "James Thompson passed away in August 2014. The world of literature has lost a tremendous talent. And those at New York Journal of Books who came to know the person he was have lost a treasured friend who was blessed with talent and humanity in equal measure. He is and always will be greatly missed."

Books only published in Finnish
 Jerusalemin veri (2008)
 Jumalan nimeen (2010)

Other works
Thompson was also editor of the anthology Helsinki Noir (November 2014, ).

References

External links

 James Thompson on Akashic Books' website
 James Thompson on Penguin's website

Category:Finnish writers
Category:University of Helsinki alumni
Category:Deaths by drowning
Category:2014 deaths
Category:1964 births


Eagle (Middle-earth)

In J. R. R. Tolkien's fictional universe of Middle-earth, the eagles were immense flying birds that were sapient and could speak. Often emphatically referred to as the Great Eagles, they appear, usually and intentionally serving as agents of eucatastrophe or deus ex machina, in various parts of his legendarium, from The Silmarillion and the accounts of Númenor to The Hobbit and The Lord of the Rings. Just as the Ents *The Giant Redwoods* are guardians of plant life, the giant eagles are the guardians of animal life.

These creatures are usually thought to have been similar to actual eagles (for example, as an independent species of the subfamily Buteoninae), but much larger. In The Silmarillion, Thorondor is said to have been the greatest of them and of all birds, with a wingspan of . Elsewhere, the eagles have varied in nature and size both within Tolkien's writings and in later visualisations and films.

Appearances
The difference between "common" eagles and Great Eagles is prominently described in The Hobbit:
Eagles are not kindly birds. Some are cowardly and cruel. But the ancient race of the northern mountains were the greatest of all birds; they were proud and strong and noble-hearted.

First Age
Throughout The Silmarillion, the Eagles are particularly associated with Manwë, the ruler of the sky and Lord of the Valar. It is stated that "spirits in the shape of hawks and eagles" brought news from Middle-earth to his halls upon Taniquetil, the highest mountain in Valinor, although later in the book the same is said of birds in general, and in the Valaquenta of "all swift birds, strong of wing". (On the different conceptions implied by these and similar passages, see Nature below.)

Upon their first appearance in the main narrative, it is stated that the Eagles had been "sent forth" to Middle-earth by Manwë, to live in the mountains north of the land of Beleriand, in order to "watch upon" Morgoth, and to help the exiled Noldorin Elves "in extreme cases". The Eagles were ruled by Thorondor, who dwelt (apparently with the majority of his folk) in the Encircling Mountains to the west of Dorthonion.

When the Hidden City of Gondolin was built by Turgon, the eagles of Thorondor became his allies, bringing him news and keeping spies off the borders. Therefore the Orcs of Morgoth were unable to approach either the nearby mountains, or the important ford of Brithiach to the south; the eagles' watch had been redoubled after the coming of Tuor, enabling Gondolin to remain undiscovered the longest of all Elven realms. When the city fell at last, the eagles of Thorondor protected the fugitives, from the orcs that ambushed them at Cirith Thoronath north of Gondolin.

The Eagles fought alongside the army of the Valar, Elves, and Men during the War of Wrath at the end of the First Age, when Morgoth was overthrown. In The Silmarillion it is recounted that after the appearance of winged dragons, "all the great birds of heaven" gathered under the leadership of Thorondor to Eärendil, and destroyed the majority of the dragons during a battle in the air.

Second Age
Tolkien mentioned the eagles in his accounts of the island of Númenor during the Second Age. He stated that three eagles guarded the summit of Meneltarma, appearing whenever one approached the hallow and staying in the sky during the Three Prayers. The Númenóreans called them "the Witnesses of Manwë" and believed that these eagles had been "sent by him from Aman to keep watch upon the Holy Mountain and upon all the land".

There was another eyrie upon the tower of the King's House in the capital Armenelos, always inhabited by a pair of eagles, until the days of Tar-Ancalimon and the coming of Shadow to Númenor. In addition, it is stated that many eagles lived upon the hills around Sorontil in the north of the island, although in the last case it is unclear whether these were "great" or "common" eagles.

When the Númenóreans had finally forsaken their former beliefs and began to speak openly against the Ban of the Valar, it was in the way of eagle-shaped storm clouds, called the "Eagles of the Lords of the West", that Manwë tried to reason or threaten them.

Third Age
By the end of the Third Age, a colony of Eagles lived in the northern parts of the Misty Mountains, as described in The Hobbit, upon the eastward slopes not far from the High Pass leading from Rivendell, and thus in the direct vicinity of the Goblin-town beneath the Mountains. It is stated that these Eagles often afflicted the goblins and "stopped whatever wickedness they were doing"; however, their relationship with the local Woodmen was only cool, as the eagles often hunted their sheep.

During the events of the book, eagles of this colony rescued Thorin's company from a band of goblins and Wargs, ultimately carrying the dwarves to the Carrock. Later, having espied the mustering of goblins all over the Mountains, a great flock of Eagles participated in the Battle of the Five Armies.

In The Lord of the Rings it is stated that the Eagles of the Misty Mountains helped the Elves of Rivendell and the Wizard Radagast in gathering news about the Orcs. In addition, a prominent (though behind-the-scene) role is played by Gwaihir, and the Eagles appear in great numbers towards the end of the book. In a parallel to The Hobbit, they arrived at the Battle of the Morannon, helping the Host of the West against the Nazgûl. Several of them rescued Frodo Baggins and Samwise Gamgee from Mount Doom after the One Ring had been destroyed.

The idea of the Eagles transporting the Ring to Mount Doom, or at least part of the way, is not discussed in The Lord of the Rings and Tolkien himself apparently never specifically addressed it, except in an oblique manner. In The Letters of J. R. R. Tolkien, he stated: "The Eagles are a dangerous 'machine'. I have used them sparingly, and that is the absolute limit of their credibility or usefulness. The alighting of a Great Eagle of the Misty Mountains in the Shire is absurd; it also makes the later capture of G. [Gandalf] by Saruman incredible, and spoils the account of his escape" (Letter 210).

Thorondor
The Lord of Eagles in the First Age, said in The Silmarillion to be the "mightiest of all birds that have ever been", with a wingspan of thirty fathoms (54.9 meters, or 180 feet) and a beak of gold. His name translates from Sindarin, an Elven language devised by Tolkien, as 'King of Eagles'; its cognate form in Quenya, another Elven language, is Sorontar. He led the eagles during most of their appearances in The Silmarillion, and has a significant role of his own.

Thorondor first enters the narrative when he helped the Elven-prince Fingon rescue his kinsman Maedhros from imprisonment upon Thangorodrim. After the Dagor Bragollach, he saved Fingolfin's body from defilement by his slayer Morgoth, giving the Dark Lord a scar on his face and carrying the Elven-king's corpse to the Encircling Mountains north of Gondolin, where it was buried by Turgon. Shortly afterwards, Thorondor espied Húrin and Huor at the feet of the Mountains, and sent two of his servants to fetch them and bear them to Gondolin, fulfilling thus the intentions of the Vala Ulmo. Thorondor and two other eagles rescued Lúthien and the wounded Beren from the doors of Angband during their Quest of the Silmaril, taking them to Doriath.

Lord of the Eagles
While in The Silmarillion the title "Lord of the Eagles" applies to Thorondor, in The Hobbit it evidently has another significance. No eagles are identified by name in this book, and titles "the Lord of the Eagles" or "the Great Eagle" distinguish their leader from others. It is stated that once he had been healed from an arrow-wound by Gandalf, and that it was in the memory of this service that his eagles helped the dwarves. After his participation in the Battle of Five Armies, he was given the title King of All Birds and wore a golden crown.

Many readers assume that it was Gwaihir that led the eagles in this story; but in The Return of the King Gandalf said that Gwaihir had carried him twice before the Battle of the Morannon, while the proper count would have been three or four times if Gwaihir and the Lord of the Eagles had been the same individual.

Gwaihir
An eagle from the Misty Mountains that helped Gandalf before and during the War of the Ring; his name means Windlord in Sindarin, and he is said to have been a descendant of Thorondor and the greatest and the swiftest of the Eagles of the North by the end of the Third Age. When the Eagles heard about Gollum's escape from Mirkwood, they sent Gwaihir to bear the news to Isengard, as they had been told by Radagast; the eagle espied Gandalf imprisoned upon the top of the tower and carried him to Edoras. Next time, Gwaihir was sent to seek for Gandalf by Galadriel; he found the wizard, who had recently defeated the Balrog, upon the summit of Celebdil and took him to Lothlórien. Upon Gandalf's orders, Gwaihir watched the river Anduin and brought him news about the Company of the Ring. The eagle participated in the Battle of the Morannon, and when Mount Doom erupted, he carried Gandalf to it, in order to save Frodo and Sam.

Eagles named "Gwaihir" and "Landroval" (or, in even earlier texts, "Gwaewar" and "Lhandroval") also appear in J. R. R. Tolkien's manuscripts of The Silmarillion, where they are stated to have been the two vassals of Thorondor who helped to bear Beren and Lúthien from Angband, several thousand years before the War of the Ring. The passage was removed from the published Silmarillion by Christopher Tolkien to escape the seeming discrepancy with The Lord of the Rings, although later he admitted that he was unable to interpret his father's intentions and regretted the suppression.

Landroval
An eagle who helped to carry Frodo and Sam from Mount Doom, said in The Lord of the Rings to have been the brother of Gwaihir and descendant of Thorondor. His name means 'wide-wing' in Sindarin, and it was also used for an eagle of the First Age, either the same as that of the Third or not (see above).

Meneldor
The third companion of Gwaihir and Landroval in the rescue of Frodo and Sam. His name means 'Sky-king' in Sindarin, and he is given a sobriquet "young and swift".

Concept and creation
The Great Eagles ruled by "Thorondor" [sic] already appeared in the first tale about Middle-earth that Tolkien wrote in late 1910s, The Fall of Gondolin, published in The Book of Lost Tales. The role of Thorondor was expanded in stages, with the successive introduction of the relevant plot elements; and after the conception of Númenor entered in 1930s, the notion that the eagles were the messengers of Manwë was further elaborated. Soon after, Tolkien introduced the eagles into The Hobbit and The Lord of the Rings, repeating in the latter some plot elements and names present in older writings.

In several early texts Tolkien wrote that, before moving to Crissaegrim after the death of Fingolfin, the eagles of Thorondor nested upon the peaks of Thangorodrim above Morgoth's fortress of Angband; Christopher Tolkien assumes that this idea was later abandoned. Another rejected proposal was that after Beren's death Lúthien would not pass of grief, but would be carried to Valinor by Thorondor who would have been "summoned" by Melian the Maia.

The eagles possessed a notable characteristic that distinguished them from other birds in early writings. Tolkien originally described that Eä, the World, was bounded by the Walls of Night, and that the space above the surface of the Earth up to the Walls was divided into three regions; common birds could keep aloft only within the lower layer, while the Eagles of Manwë could fly "beyond the lights of heaven to the edge of darkness". The conception of the limited world and of the layers of the firmament was rejected during the writing of The Lord of the Rings.

The eagle-shaped clouds that appeared in Númenor were one of Tolkien's recurring associations with the downfall of the island, just like the images of a sloping mountain and of an overwhelming wave; they were also introduced by him into two abandoned time-travel stories, The Lost Road and The Notion Club Papers. In a sketch for the former, Tolkien projected that it would be "Sorontur" (Thorondor) himself that appeared in Númenor to the protagonist of the story.

Tolkien's painting of an eagle on a crag appears in some editions of The Hobbit. According to Christopher Tolkien, the author based this picture on a painting by Archibald Thorburn of an immature golden eagle, which Christopher found for him in The Birds of the British Isles by Thomas Coward. However, Tolkien's use of this model does not necessarily mean that his birds were ordinary golden eagles.

Nature
The question of the Great Eagles' nature was faced by Tolkien with apparent hesitation. In early writings there was no need to define it precisely, since he imagined that, beside the Valar, "many lesser spirits... both great and small" had entered the Eä upon its creation; and such sapient creatures as the Eagles or Huan the Hound, in Tolkien's own words, "have been rather lightly adopted from less 'serious' mythologies". The phrase "spirits in the shape of hawks and eagles" in The Silmarillion derives from that stage of writing.

After the completion of The Lord of the Rings, Tolkien moved toward a more carefully defined "system" of creatures, with few if any exceptions:
 Incarnates or the "Children of Ilúvatar": Elves, Men, Dwarves, and Orcs,—those who possessed fëar or souls, with the defining characteristic of being able to speak;
 Self-incarnates or the Valar and Maiar—"angelic" spirits that "arrayed" themselves in bodily forms of the Incarnates or of animals, able to communicate both by thought and speech;
 Animals, unable to speak.

For some time Tolkien considered the Eagles as bird-shaped Maiar; however, later he realised that the statement about Gwaihir and Landroval's descent from Thorondor had already appeared in print in The Lord of the Rings, while the notion of the "Children" of the Valar and Maiar had been rejected by him long before. In the last of his notes on this topic, dated by his son to late 1950s, Tolkien decided that the Great Eagles were common animals that had been "taught language by the Valar, and raised to a higher level—but they still had no fëar."

A different conception may be present in a yet later essay on the origin of the Ents which, according to Christopher Tolkien, is likely to derive from 1963 and was included into the published Silmarillion. Contemporary J. R. R. Tolkien's notes define the Ents as "either souls sent to inhabit trees, or else that slowly took to the likeness of trees"; the essay agrees in this, adding that the Ents appeared shortly after the Awakening of the Elves, when "the thought of Yavanna ... [summoned] spirits from afar".

The same origin for the Great Eagles may implied by the speech of Manwë within the essay: "... before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West. ... In the mountains the Eagles shall house, and hear the voices of those who call upon [the Valar]." However, this is in tension with the statement that the spirits summoned by Yavanna come to Arda only after the Awakening of the Elves, while the eagles, according to Manwë, already would exist "before the Children awake”. This could be taken to indicate that the eagles would not have been among the spirits summoned by Yavanna in that paragraph, suggesting that Tolkien could still have held the view that the eagles are animals that have no fëar, but other readings are possible.

Adaptations and influences

Different adaptations of Tolkien's books treated both the nature of the Eagles and their role in the plots with varying level of faithfulness to originals. The first scenario for an animated motion-picture of The Lord of the Rings proposed to Tolkien in 1957 was turned down because of several cardinal deviations, among which Humphrey Carpenter recorded that "virtually all walking was dispensed with in the story and the Company of the Ring were transported everywhere on the backs of eagles".

The Rankin-Bass animated version of The Hobbit portrayed the eagles as similar in physique and appearance to harpy, crowned or monkey-eating eagles of the tropics (subfamilies Circaetinae and Harpiinae), while Jackson's trilogy provided a more traditional interpretation, with birds similar to the golden eagle.

In The Lord of the Rings film trilogy directed by Peter Jackson, these creatures are 6 m (20 ft) tall with a maximum wingspan of 23 m (75 ft). A notable deviation from the book is that Gandalf summons Gwaihir to Orthanc with the aid of a by-passing moth (the role of Radagast was not included in the film). The same moth also appears to him before the Eagles arrive at the Battle of the Morannon, and a similar sequences of events (though with a different moth) is played out in Jackson's first installment of The Hobbit. According to fantasy artist Larry Dixon, the digitally animated eagles in the trilogy were based on a stuffed golden eagle he had offered to Weta Workshop for use in the project.

Although giant birds of prey appear in legends of many cultures, Tolkien's Great Eagles may have been the direct inspiration for similar creatures in various modern fantasy genres, such as the Giant eagles in the role-playing game Dungeons & Dragons.

A genus of Diapriidae wasps in Australia was named Gwaihiria in 1982.

In The Lord of the Rings: War in the North, an eagle named Beleram acts as a supporting character, aiding the players in battle by attacking random, often powerful enemies such as trolls, giants or Uruk-hai, and carrying the heroes across areas of Middle-Earth. In the final battle against Agandaur, the game's antagonist and final boss, the player can decide whether to attack Agandaur with Beleram. If Beleram is summoned, he inflicts tremendous damage to Agandaur, but Agandaur will kill him. Preventing Beleram from attacking Agandaur allows him to live to the game's conclusion.

References

General references

External links

Category:Middle-earth races
Category:Middle-earth animals
Category:Fictional birds of prey
Category:Talking animals in fiction


Emmanuel Ija

Dr. Emmanuel Ija is a South Sudanese physician and politician. As of 2011, he is the Minister of Health of Central Equatoria.

References

Category:South Sudanese politicians
Category:Living people
Category:South Sudanese physicians
Category:People from Central Equatoria
Category:Year of birth missing (living people)


Endonuclease V

.

Endonuclease V (endoV) is a highly conserved endonuclease enzyme family. The primary function of endoV differs significantly in prokaryotes and eukaryotes, as suggested by studies on the E. coli and human orthologs. In prokaryotes endoV is primarily a deoxyribonuclease involved in DNA repair of deoxyinosine introduced into the genome by deamidation of adenine bases. However, it has broad substrate specificity and can also act on other types of DNA lesions as well as on inosine-containing RNA.

In eukaryotes endoV is primarily a ribonuclease and cleaves single-stranded RNA at the 3' position relative to an inosine base, which may be present due to RNA editing by deaminase enzymes. The human endoV localizes to the cytoplasm and nucleoli, suggesting a possible role in processes involving ribosomal RNA.

References

Category:Ribonucleases
Category:EC 3.1.21


Jessica Reid

Jessica Reid may refer to:

Jessica Reid, character in Always Goodbye
Jessica Reid, character in The Lottery (2010 film)
Jessica Reid (dancer), partner of Joshua Adams

See also
Jessie Reid, baseball player
Jessica Reed, see List of Law & Order characters