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Injective sheaf

In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext).

There is a further group of related concepts applied to sheaves: flabby (flasque in French), fine, soft (mou in French), acyclic. In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory. The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence.

Injective sheaves
An injective sheaf $$\mathcal{F}$$ is a sheaf that is an injective object of the category of abelian sheaves; in other words, homomorphisms from $$\mathcal{A}$$ to $$\mathcal{F}$$ can always be extended to any sheaf $$\mathcal{B}$$ containing $$\mathcal{A}.$$

The category of abelian sheaves has enough injective objects: this means that any sheaf is a subsheaf of an injective sheaf. This result of Grothendieck follows from the existence of a generator of the category (it can be written down explicitly, and is related to the subobject classifier). This is enough to show that right derived functors of any left exact functor exist and are unique up to canonical isomorphism.

For technical purposes, injective sheaves are usually superior to the other classes of sheaves mentioned above: they can do almost anything the other classes can do, and their theory is simpler and more general. In fact, injective sheaves are flabby (flasque), soft, and acyclic. However, there are situations where the other classes of sheaves occur naturally, and this is especially true in concrete computational situations.

The dual concept, projective sheaves, is not used much, because in a general category of sheaves there are not enough of them: not every sheaf is the quotient of a projective sheaf, and in particular projective resolutions do not always exist. This is the case, for example, when looking at the category of sheaves on projective space in the Zariski topology. This causes problems when attempting to define left derived functors of a right exact functor (such as Tor). This can sometimes be done by ad hoc means: for example, the left derived functors of Tor can be defined using a flat resolution rather than a projective one, but it takes some work to show that this is independent of the resolution. Not all categories of sheaves run into this problem; for instance, the category of sheaves on an affine scheme contains enough projectives.

Acyclic sheaves
An acyclic sheaf $$\mathcal{F}$$ over X is one such that all higher sheaf cohomology groups vanish.

The cohomology groups of any sheaf can be calculated from any acyclic resolution of it (this goes by the name of De Rham-Weil theorem).

Fine sheaves
A fine sheaf over X is one with "partitions of unity"; more precisely for any open cover of the space X we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover.

Fine sheaves are usually only used over paracompact Hausdorff spaces X. Typical examples are the sheaf of germs of continuous real-valued functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings. Also, fine sheaves over paracompact Hausdorff spaces are soft and acyclic.

One can find a resolution of a sheaf on a smooth manifold by fine sheaves using the Alexander–Spanier resolution.

As an application, consider a real manifold X. There is the following resolution of the constant sheaf $$\R$$ by the fine sheaves of (smooth) differential forms:

 * $$0\to\R\to C^0_X \to C^1_X \to \cdots \to C^{\dim X}_X \to 0.$$

This is a resolution, i.e. an exact complex of sheaves by the Poincaré lemma. The cohomology of X with values in $$\R$$ can thus be computed as the cohomology of the complex of globally defined differential forms:

 * $$H^i(X,\R) = H^i(C^\bullet_X(X)).$$

Soft sheaves
A soft sheaf $$\mathcal{F}$$ over X is one such that any section over any closed subset of X can be extended to a global section.

Soft sheaves are acyclic over paracompact Hausdorff spaces.

Flasque or flabby sheaves
A flasque sheaf (also called a flabby sheaf) is a sheaf $$\mathcal{F}$$ with the following property: if $$X$$ is the base topological space on which the sheaf is defined and

 * $$U \subseteq V \subseteq X$$

are open subsets, then the restriction map

 * $$r_{U \subseteq V} : \Gamma(V, \mathcal{F}) \to \Gamma(U, \mathcal{F})$$

is surjective, as a map of groups (rings, modules, etc.).

Flasque sheaves are useful because (by definition) their sections extend. This means that they are some of the simplest sheaves to handle in terms of homological algebra. Any sheaf has a canonical embedding into the flasque sheaf of all possibly discontinuous sections of the étalé space, and by repeating this we can find a canonical flasque resolution for any sheaf. Flasque resolutions, that is, resolutions by means of flasque sheaves, are one approach to defining sheaf cohomology.

Flasque sheaves are soft and acyclic.

Flasque is a French word that has sometimes been translated into English as flabby.

title: Injective sheaf


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Villarceau circles

In geometry, Villarceau circles are a pair of circles produced by cutting a torus obliquely through its center at a special angle.

Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the equatorial plane of the torus and another perpendicular to that plane (these are analogous to lines of latitude and longitude on the Earth). The other two are Villarceau circles. They are obtained as the intersection of the torus with a plane that passes through the center of the torus and touches it tangentially at two antipodal points. If one considers all these planes, one obtains two families of circles on the torus. Each of these families consists of disjoint circles that cover each point of the torus exactly once and thus forms a 1-dimensional foliation of the torus.

The Villarceau circles are named after the French astronomer and mathematician Yvon Villarceau (1813–1883) who wrote about them in 1848.

Example
Consider a horizontal torus in xyz space, centered at the origin and with major radius 5 and minor radius 3. That means that the torus is the locus of some vertical circles of radius three whose centers are on a circle of radius five in the horizontal xy plane. Points on this torus satisfy this equation:

 * $$ 0 = (x^2+y^2+z^2 + 16)^2 - 100(x^2+y^2). \,\! $$

Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82, the outer and inner equator. Slicing with the x = 0 plane produces two side-by-side circles, (y − 5)2 + z2 = 32 and (y + 5)2 + z2 = 32.

Two example Villarceau circles can be produced by slicing with the plane 3y = 4z. One is centered at (+3, 0, 0) and the other at (−3, 0, 0); both have radius five. They can be written in parametric form as

 * $$ (x,y,z) = (+3+5 \cos \vartheta, 4 \sin \vartheta, 3 \sin \vartheta) \,\!$$

and

 * $$ (x,y,z) = (-3+5 \cos \vartheta, 4 \sin \vartheta, 3 \sin \vartheta) \,\!$$

The slicing plane is chosen to be tangent to the torus at two points while passing through its center. It is tangent at (0, 16/5, 12/5) and at (0, -16/5, -12/5). The angle of slicing is uniquely determined by the dimensions of the chosen torus. Rotating any one such plane around the z-axis gives all of the Villarceau circles for that torus.

Existence and equations

A proof of the circles’ existence can be constructed from the fact that the slicing plane is tangent to the torus at two points. One characterization of a torus is that it is a surface of revolution. Without loss of generality, choose a coordinate system so that the axis of revolution is the z axis. [See the figure to the right.] Begin with a circle of radius r in the yz plane, centered at (0,R, 0):

 * $$ 0 = (y-R)^2 + z^2 - r^2 \,\!$$

Sweeping this circle around the z-axis replaces y by (x2 + y2)1/2, and clearing the square root produces a quartic equation for the torus:

 * $$ 0 = (x^2+y^2+z^2 + R^2 - r^2)^2 - 4R^2(x^2+y^2) . \,\!$$

The cross-section of the swept surface in the yz plane now includes a second circle, with equation

 * $$ 0 = (y+R)^2 + z^2 - r^2 \,\!$$

This pair of circles has two common internal tangent lines, with slope at the origin found from the right triangle with hypotenuse R and opposite side r (which has its right angle at the point of tangency). Thus, on these tangent lines, z/y equals ±r / (R2 − r2)1/2, and choosing the plus sign produces the equation of a plane bitangent to the torus:

 * $$ y r = z\sqrt{R^2-r^2} \,\!$$

We can calculate the intersection of this plane with the torus analytically, and thus show that the result is a symmetric pair of circles of radius R centered at $$ (\pm r, 0, 0).$$

A parametric description of these circles is

$$ (x,y,z) = \big(\pm r + R \cos \vartheta,\ \sqrt{R^2-r^2}\; \sin\vartheta,\  r \sin\vartheta \big) \,\!$$

These circles can also be obtained by starting with a circle of radius R in the xy-plane, centered at (r,0,0) or (-r,0,0), and then rotating this circle about the x-axis by an angle of arcsin(r/R).

A treatment along these lines can be found in Coxeter (1969).

A more abstract — and more flexible — approach was described by Hirsch (2002), using algebraic geometry in a projective setting. In the homogeneous quartic equation for the torus,

 * $$ 0 = (x^2+y^2+z^2 + R^2w^2 - r^2w^2)^2 - 4R^2w^2(x^2+y^2), \,\!$$

setting w to zero gives the intersection with the “plane at infinity”, and reduces the equation to

 * $$ 0 = (x^2+y^2+z^2)^2 . \,\!$$

This intersection is a double point, in fact a double point counted twice. Furthermore, it is included in every bitangent plane. The two points of tangency are also double points. Thus the intersection curve, which theory says must be a quartic, contains four double points. But we also know that a quartic with more than three double points must factor (it cannot be irreducible), and by symmetry the factors must be two congruent conics, which are the two Villarceau circles.

Hirsch extends this argument to any surface of revolution generated by a conic, and shows that intersection with a bitangent plane must produce two conics of the same type as the generator when the intersection curve is real.

Filling space and the Hopf fibration
The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles, S1, as fibers. When the 3-sphere is mapped to Euclidean 3-space by stereographic projection, the inverse image of a circle of latitude on S2 under the fiber map is a torus, and the fibers themselves are Villarceau circles. Banchoff has explored such a torus with computer graphics imagery. One of the unusual facts about the circles making up the Hopf fibration is that each links through all the others, not just through the circles in its own torus but through the circles making up all the tori filling all of space; Berger has a discussion and drawing.

Further properties
Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Villarceau circles


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Luc d'Achery

Luc d'Achery (1609 – 29 April 1685) was a learned French Benedictine of the Congregation of St. Maur, a specialist in the study and publication of medieval manuscripts.

Life
D'Achery was born at Saint Quentin in Picardy. He entered the Order of St. Benedict at an early age and was professed at the Trinity Abbey, Vendôme, on 4 October 1632, but his health soon obliged him to remove to Paris. He became a member of the monastery of St. Germain des Prés in 1637, which in nearly fifty years he scarcely ever left, and he died there aged about 75.

As librarian of the monastery he was soon acquainted with its rich treasures of medieval history and theology. By a continuous correspondence with other monasteries, both in and out of France, he made himself a bibliographical authority of the first rank, especially in all that pertained to the unedited or forgotten writings of medieval scholars.

Works
His first important work was an edition (Paris, 1645) of the Epistle of Barnabas, whose Greek text had been prepared for the press, before his death, by the Maurist Hugo Menardus. D'Achery's "Asceticorum vulgo spiritualium opusculorum Indiculus" (Paris, 1645) served as a guide to his colleague, Claude Chantelou, in the preparation of the five volumes of his "Bibliotheca Patrum ascetica" (Paris, 1661). In 1648 he published all the works of Blessed Lanfranc of Canterbury.

He published and edited for the first time the works of Abbot Guibert of Nogent (Paris, 1661) with an appendix of minor writings of an ecclesiastical character. In 1656 he edited the "Regula Solitaria" of the ninth century priest Grimlaicus (Grimlaic), a spiritual guide for hermits. His principal work, however, is the famous "Spicilegium, sive Collectio veterum aliquot scriptorum qui in Galliae bibliothecis, maxime Benedictinorum, latuerunt" (Paris, 1655–1677), continued by Baluze and Martène, who produced an enlarged and improved edition (Paris, 1723).

D'Achery collected the historical materials for the "Acta Ordinis S. Benedicti" but Mabillon added so much to it in the way of prefaces, notes, and "excursus" that it is justly accounted as his work.

D'Achery's correspondence is preserved in the Bibliothèque Nationale at Paris.

title: Luc d'Achery


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Achery

Achery may refer to:

 * Achery, Aisne, a commune in France
 * Luc d'Achery (1609–1685), Maurist librarian and historian

title: Achery


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Talk:Hardy-Weinberg principle/to do

Things to do:

 * 1) Add a de Finetti diagram with line through (I have equations, just need something that'll plot them with same scale on x and y axes.
 * Have added a diagram from the de finetti diagram section and some more info on this, could do with bulking out the maths (from memory I think the HW parabola is m2 = 4ln), I do think however that the graph should be replaced with one with scales on the axis, will try and get something from the site I linked uploaded to replace this. Slack---line 04:43, 15 January 2007 (UTC).

put the fact that there are examples of HUMANS who are ridiculously close to the HW-equilibrium numbers!!! Strike-through text
 * 1) Improve the generalisation section and put in about generalising to n-ploidy
 * 2) Tidy the maths up(!)
 * 3) Add an example of the use of Fisher's exact test.

title: Talk:Hardy-Weinberg principle/to do


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Finnat Már

Finnat Már ("the great", later spellings Innatmar, Ionnadmhar) son of Nia Segamain, was, according to medieval Irish legend and historical tradition, a High King of Ireland. He succeeded to the throne after the death of Rudraige mac Sithrigi of plague, but after a reign of one, three or nine years he was killed by Rudraige's son Bresal Bó-Díbad. The Lebor Gabála Érenn synchronises his reign with that of Ptolemy X Alexander I (110–88 BC) in Egypt. The chronology of Geoffrey Keating's Foras Feasa ar Éirinn dates his reign to 154–151 BC, that of the Annals of the Four Masters to 219–210 BC.

title: Finnat Már


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John Muth

John Fraser Muth (September 27, 1930 – October 23, 2005) was an American economist. He is "the father of the rational expectations revolution in economics", primarily due to his article "Rational Expectations and the Theory of Price Movements" from 1961.

Muth earned his PhD in mathematical economics from Carnegie Mellon University, and was in 1954 the first recipient of the Alexander Henderson Award. He was affiliated with Carnegie Mellon as a research associate from 1956 until 1959, as an assistant professor from 1959 to 1962, and as an associate professor without tenure from 1962 to 1964. He was a full professor at Michigan State University from 1964 to 1969 and a full professor at Indiana University from 1969 until his retirement in 1994.

Muth asserted that expectations "are essentially the same as the predictions of the relevant economic theory." Although he formulated the rational expectations principle in the context of microeconomics it has subsequently become associated with macroeconomics and the work of Robert Lucas, Jr., Finn E. Kydland, Edward C. Prescott, Neil Wallace, Thomas J. Sargent, and others.

Rationalization of Friedman's adaptive expectations model
Phillip Cagan, Milton Friedman and others used the ad hoc updating rule which they labeled adaptive expectations to forecast the hidden state y* (e.g., permanent income). In a 1960 paper, Muth answered the question: for what stochastic process for y will adaptive expectations as postulated by Cagan and Friedman be the optimal forecast of y*. Muth's approach to find recursive optimal linear forecast of a "hidden" state vector, x, given an "observer", y is very similar to the Kalman filter, presented by Rudolf Kálmán in his paper from the same year.

In his paper "Optimal Properties of Exponentially Weighted Forecasts", which was published in the Journal of the American Statistical Association in 1960, Muth rationalized Friedman's adaptive expectations model for permanent income. He did this by reverse engineering a stochastic process for income for which Cagan's expectation formula equals a mathematical expectation of future values conditioned on the infinite history of past incomes. Among Muth's insights was that the stochastic process being forecast should dictate both the distributed lag and the conditioning variables that people use to forecast the future.

Hypothesis of rational expectations
In "Rational Expectations and the Theory of Price Movements", published in 1961, Muth put forward his hypothesis, in contrast to Simon, that "expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory." Muth continued, "At the risk of confusing this purely descriptive hypothesis with a pronouncement as to what firms ought to do, we call such expectations rational."

"Muth's notion was that the professors [of economics], even if correct in their model of man, could do no better in predicting than could the hog farmer or steelmaker or insurance company. The notion is one of intellectual modesty.... The common sense is 'rationality': therefore Muth called the argument 'rational expectations'."

Legacy
Muth's works influenced almost every area of economic research into dynamic problems.

"Of course we knew about [rational expectations]. Muth was a colleague of ours [in the early 1960s]. We just didn't think it was important. The hypothesis was more or less buried during the '60s. Arrow used it in his paper on learning-by-doing in the '60s. Prescott and I used it in that paper of ours on investment. People were aware of it, but I didn't understand then how fundamental a difference it made econometrically. I didn't realize that if you took it seriously you had to rethink the whole question of testing and estimation. I guess no one else did either, except for Muth."

"It must be quite an experience to write papers that radical and have people just pat you on the head and say 'That's interesting,' and nothing happens."

"Muth's role in the history of economics is unusual. Like Hermann Heinrich Gossen, he became famous for one idea, he provided the analytical key to developments that, in the jargon of scientific journalism, were described as revolutionary, and he was virtually ignored by his immediate contemporaries. However, whereas Gossen had no influence on those developments, his key results being independently rediscovered by Jevons and Walras, the rational expectations economics of the 1970s and 1980s was a direct outgrowth of Muth's seminal idea. In fact, Muth's contribution is one of the relatively few instances in which there is no indication that the history of economics would have taken about the same course in its absence. It was a novel and ingenious idea, it was not 'in the air,' and no multiple discovery has yet come to light."

Major works

 * Charles C. Holt, Franco Modigliani, John F. Muth, and Herbert A. Simon (1960). Planning Production, Inventories, and Work Force.
 * John F. Muth. (1960). "Optimal Properties of Exponentially Weighted Forecasts", Journal of the American Statistical Association, 55(290), pp. 299–306.
 * John F. Muth. (1961). "Rational Expectations and the Theory of Price Movements", Econometrica 29, pp. 315–335.

title: John Muth


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Achilleid

The Achilleid (Achillēis) is an unfinished epic poem by Publius Papinius Statius that was intended to present the life of Achilles from his youth to his death at Troy. Only about one and a half books (1,127 dactylic hexameters) were completed before the poet's death. What remains is an account of the hero's early life with the centaur Chiron, and an episode in which his mother, Thetis, disguised him as a girl on the island of Scyros, before he joined the Greek expedition against Troy.

Composition
Based upon three references to the poem in the Silvae, the Achilleid seems to have been composed between 94 and 96 CE. At Silvae 4. 7. 21–24, Statius complains that he lacks the motivation to make progress upon his "Achilles" without the company of his friend C. Vibius Maximus who was travelling in Dalmatia (and to whom poem is addressed). Statius apparently overcame this self-described writer's block, for in a poem from the posthumously published fifth book of the Silvae he refers to an upcoming recitation of a section from the Achilleid. This reference is believed to date from the summer of 95, and Statius presumably died later that year or early in the next, leaving the Achilleid unfinished.

Poetic models
Statius' primary models are Homer and the poems of the epic cycle which touch on the life of Achilles. In the opening of the Achilleid, Statius asks that his poem not stop with the death of Hector (nec in Hectore tracto sistere 1.6) as the Iliad does but that it continue through the whole Trojan cycle, invoking these two important models. His style in the Achilleid has been seen as far more reminiscent of Ovid than Virgil, his major influence in the composition of the Thebaid. Statius tried to revise the image of the Homeric Achilles with the Achilleid, just as Ovid did for the Virgilian Aeneas. While doing this, they also exploited the tension between the accepted epic narrative and competing traditions pertaining to the heroes' lives.

On account of its unfinished state, the Achilleid is often referred to as a "fragment", but this is a misleading label. Fragments are typically pieces of writing that have become seriously destroyed in the process of being transmitted to its audience. Statius' Achilleid is a work that is partially completed that had already been polished and presented to the world in his lifetime. The structure of the narrative is deliberate and balanced. The first words of the poem are the pseudo-Homeric patronymic that introduces Achilles through his father's father, while the last word of the poem is mother. Achilles' childhood experiences are then told in the space left vacant by his two absentee parents.

Book 1
Lines 1–13. The introduction states the goals and scope of the epic, including the intention to cover the entire life of Achilles, not simply up to Hector's death as was done in the Iliad.

The Achilleid opens with a traditional epic invocation of the Muses and Apollo, requesting inspiration for the poet's work and outlining the content of the poem to follow. The Muses are the first to be addressed (Ach. 1.1–7):

As in Vergil's Aeneid and Statius' own Thebaid, the very first words present the poem's primary topic, expanded with a clause joined by the Latin enclitic conjunction -que. While the structure of the first line puts the Achilleid within the Vergilian tradition of martial epic, Philip Hardie sees the last line quoted above as an indication of Statius' debt to Ovid. Specifically, the choice of the verb deducere, "to lead down", evokes the invocation in the Metamorphoses in which Ovid asks the gods to lead down (deducite) to his own time a "perpetual song" (perpetuum carmen), with which Hardie also compares Statius' "Troy's whole story" (tota Troia, literally: "all of Troy").

Lines 14–19. Statius praises Domitian and dedicates the epic to this emperor.

Lines 20–94. Thetis, worrying that the Greek troops preparing to head to Troy will soon come to recruit her son and thus greatly endanger his life (for he is fated to die if he goes to Troy), asks Neptune if he will sink the Trojan fleet carrying Paris and Helen. Neptune denies this request, stating that the war is fated.

Lines 95–197. Thetis goes to Thessaly, where the centaur Chiron has been raising and tutoring Achilles. She tells Chiron that she wants to take her son back and then enjoys a night of eating, singing, and drinking with them in his cave.

Lines 198–282. Thetis decides that she must hide Achilles on Lycomedes' island of Scyros and takes him there while he sleeps. Once he wakes, she tries to convince him to hide himself there disguised as a girl. He refuses to accept this plan despite her promises that no one else will ever find out.

Lines 283–396. Achilles is finally convinced to follow his mother's advice when he witnesses the daughters of King Lycomedes performing a dance at a festival of Pallas. Achilles is immediately struck by the outstanding beauty of one of these women, Deidamia, and so agrees to disguise himself as a woman and live among the king's daughters in order to be near her. Thetis dresses her son in women's clothing and teaches him how to act feminine, then presents her "daughter" to King Lycomedes and asks for him to care for and protect her among his own daughters. The king agrees to her request.

One of the main themes up through this section, and as an undercurrent for the rest of the book, is that of maternal anxiety on the part of Thetis. Some authors have made note of the strong emphasis on and significance of Thetis and her concern here, especially as compared to in other Classical works.

Konstan points out that many scholars have interpreted the depiction of Achilles disguised as a girl as symbolizing his maturation from a child living among women to an adult. Konstan believes that Statius in fact focuses more on the humorous aspects of the story, thus providing a comedic contrast to the serious tones of war by "delight[ing] in the naughty humor of the situation and the deflation of epic pretentiousness."

Lines 397–466. The Argives (Greeks) in various regions prepare for the Trojan War.

Lines 467–559. The Greeks muster their forces at Aulis but notice that Achilles is missing. The prophet Calchas sees in a trance that Achilles has been hidden on Scyros, and Ulysses and Diomedes depart to fetch him.

Lines 560–674. Achilles continues to fall in love with Deidamia, who has by now discovered his true identity and is helping him to maintain his disguise. Achilles rapes Deidamia in a sacred grove and she makes the conscious decision to forgive him for this indiscretion and keep it a secret. She becomes pregnant and gives birth to their child (Neoptolemus, although he is never referred to by name in the Achilleid).

Lines 675–818. Ulysses and Diomedes arrive at Scyros, are entertained by Lycomedes, and set out gifts for his daughters. When Achilles alone is attracted by the shield and helmet and not the more womanly items, his identity is revealed, as Ulysses had intended. Achilles, now convinced to follow the Greek heroes to war, explains for the first time his relationship with Deidamia and their baby son and persuades Lycomedes to allow him to officially marry his daughter.

Lines 927–60. Deidamia sees the future and recites a speech of despair, expressing her hope that Achilles will one day return to her.

Book 2
Lines 1–22. After praying to his mother for forgiveness, Achilles sets sail from Scyros with Ulysses and Diomedes.

Lines 23–48. Deidamia and Achilles each grieve, separately, for the loss of the other. Ulysses tries to take Achilles' mind off his wife.

Lines 49–85. Ulysses tells the story of the events leading up to the war on which they are about to embark and expresses his indignation at Paris' reckless abduction of Helen and the threat that he feels toward society as a whole as a result.

Lines 86–167. Per Diomedes' request, Achilles tells of his youth, his hunting exploits, and the teachings of Chiron. The poem ends with the closure of Achilles' narrative.

Influence of and critical responses to the Achilleid
The Achilleid has generally received far more positive criticism than the Thebaid. One branch of this focuses on comparisons between the two poems; many scholars see a drastic difference between the "serious" and "Iliadic" Thebaid and the playful "Ovidian" Achilleid. Some have seen the Achilleid as Statius' attempt to write an entirely new multi-generic type of epic as a challenge to the Virgilian model. Others have noted the importance of female emotions and feminine characteristics in the poem. Finally, some have interpreted the character of Achilles as a subversive foil for Domitian.

Critics have also said that the Achilleid was a failure because Statius wrote it as an attempt to constitute an alternative epic tradition, which he was unsuccessful in doing. However, it has also been argued that Statius' alternative epic tradition has influenced some of his successors. Claudian's De raptu Proserpinae emulated Statius' alternative epic tradition, leaving his work seemingly unfinished. Claudian believed that the inevitability of Homeric and Virgilian narrative was the cause of Statius' inability to proceed. Other writers such as Dante Alighieri borrowed from Statius and thought highly of his style; Giovanni Boccaccio was inspired by him; and Geoffrey Chaucer studied and imitated Statius. The influences of Statius and the Achilleid are also clearly seen in Edmund Spenser's The Faerie Queene, especially in one Canto of Book III.

Statius' Achilleid also had a great impact in the realm of opera in the late seventeenth and early eighteenth centuries across Europe. These operas raised the issues of transvestitism, biological sex, and social gender. When a woman played the character of Achilles, the audience saw a woman playing the role of a man pretending to be a woman. When a castrato played Achilles, the unveiling of the "girl" forced the observation of a contrast between the fictional character who sheds his false gender identity on Scyros and the singer who cannot. Some directors such as Giulio Strozzi, Ippolito Bentivoglio, and Carlo Capece, chose to embody the spirit of Carnival: the greatest hero of antiquity puts on a female disguise to pursue his love and sexual desires. For later writers such as Pietro Metastasio and Paolo Rolli, the myth teaches that gender is essential, in that the masculinity of Achilles is a primal force of nature that cannot be hidden, and it is a crucial component of his heroism. The first treatment of the "Achilleid" for the operatic stage was La finta pazza, "the woman feigning madness," performed in Venice in 1641. Following that was the opera, Achille in Sciro, first performed in Ferrara in 1663.

Women in the Achilleid
In ancient epic, women have been portrayed through various roles that help, hinder, and protect characters from disaster. Greek poets, such as Homer, have generally illustrated women as victims of conflict, the cause of conflict, negotiators among combatant men, and mourners of the dead. Roman poets, like Virgil, describe women in a similar light, but they also tend to complicate the portrayal of women, often depicting them as hindering a hero's destiny and stirring conflict among men.

In the Achilleid, classicist P. J. Heslin argues that Statius upholds the Roman trend of portraying women as "heroic blockers" with the development of Thetis' character. In the Achilleid, Thetis is a prophet, protector, and hinderer to Achilles. She desperately tries to protect Achilles from going off to fight the Trojan War, knowing that he will die in battle if he goes. Thetis's initial reaction of anger to this knowledge (inspiring her idea to sink Paris's fleet) imitates the classic anger of the goddess Juno. However, her surge in anger does not help her protect Achilles. Thetis's supplication of Neptune mirrors Venus's supplication of Neptune in the Aeneid, except Thetis's attempt fails whereas Venus's succeeds. Thetis's maternal instinct to protect her child from danger fulfills one of the typical roles women play in ancient epic. She also hinders the course of Achilles' fate by trying to change his destiny, which is to become one of the most glorified heroes in Greek history.

The other major female character in the Achilleid is Deidamia. Heslin argues that Achilles rapes Deidamia in order to assert his masculinity because dressing and acting like a woman makes him feel belittled. Deidamia's rape is just another example from epic tales that shows women as property, ultimately in the control of men. Her obedience to Achilles is further exemplified by her silence after the rape. After marrying Achilles, Deidamia then fulfills the role of the faithful wife waiting for her husband to return home from war.

Heslin illuminates how the expectations for the behavior of Roman women during Statius's life can also be seen in the Achilleid through Thetis's instructions on how Achilles should act on Scyros. Thetis criticizes his "masculine" mannerisms and leaves him on Scyros to learn more about how to act in a womanly fashion. Hence, this instruction on "womanliness" can be interpreted as insight into Rome's feminine world during Statius's lifetime.

Manuscripts
There are only a few manuscripts of the Achilleid:

 * Trivulziano 792, 1391-1410, Milan, Archivio storico civico e Biblioteca Trivulziana
 * A 5 inf., 1401-1500, Milan, Biblioteca Ambrosiana
 * H 166 inf, 1401-1450, Milan, Biblioteca Ambrosiana
 * AE._XII.52, 1413, Milan, Biblioteca nazionale Braidense
 * It.173/15=alfa.K.3.29, 1401-1650, Modena, Biblioteca Estense
 * Lat.211=alfa.P.6.14, 1401-1500, Modena, Biblioteca Estense
 * Lat.939=alfa.M,9.22, 1401-1500, Modena, Biblioteca Estense
 * Ms.Pal.0069, 1401-1450, Parma, Biblioteca Palatina
 * Manoscritti, 98,1401-1500, Poppi, Biblioteca comunale Riliana
 * Ms.1721, 1401-1500, Rome, Biblioteca Angelica
 * Ms.B 30/1-2, Rome, Biblioteca Vallicelliana
 * Ms.C 95, Rome, Biblioteca Vallicelliana

title: Achilleid


text:
User talk:Krishnanmba

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title: User talk:Krishnanmba


text:
Talk:Ana Ng

The reason I removed the reference is because it originally said it mentions the song, when it in fact doesn't. Small world and small girl, while similar, are not the same. Then again, the song references lots of things, and if we were to mention them all, we might as well just add the lyrics. So I suppose saying it references the song is better than it mentions it. Kbdank71 14:12, 28 Feb 2005 (UTC)

IPA
The fact that this article features non-IPA pronounciation information isn't really bad. Most people I've met have no idea how all the symbols and whatnot from dictionaries work and what the mean. So, honestly this article is just using a clearer way of describing things, even if it isn't "official" or whatever. - Razorhead July 30, 2007
 * Actually it's not clear at all, I can't work out what "aang" is supposed to represent (what vowel sound). IPA should be used for pronunciations because it's unambiguous and it is clear, as all one has to do is read a chart if they don't know what the symbols mean. - Zeibura (Talk) 13:55, 5 August 2007 (UTC)

Smallest woman on earth?
On youtube (http://de.youtube.com/watch?v=7dQLkxz6c2E) I've found in a comment that "Ana Ng was the smallest woman in the world. She was toured round the world in the 1900s. She was even at the crystal palace in London before it burned down." Does anyone know anything about that? --80.171.123.61 (talk) 23:35, 11 January 2008 (UTC)

 * Whether true or not, it's irrelevant. That particular Ana Ng, if she existed, was not an inspiration for this song. It seems more like fan folklore/rumor. Refer to the 1996 Pitchfork Magazine article. Antiaverage (talk) 19:41, 8 February 2010 (UTC)

Math Rock
I haven't heard TMBG described as math rock before, and the cited article is just a passing reference. Does anyone else agree/disagree with that categorization? Improudtoamerica (talk) 02:57, 23 May 2024 (UTC)

title: Talk:Ana Ng


text:
Abrantes

Abrantes is a municipality in the central Médio Tejo subregion of Portugal. The population was 39,325, in an area of 714.69 km2. The municipality includes several parishes divided by the Tagus River, which runs through the middle of the municipality. The urbanized part, the parish of Abrantes (São Vicente e São João) e Alferrarede, located on the north bank of the Tagus, has about 17,000 residents.

History
The Celts are believed to have established the first settlement in Abrantes around 300 BC. The name is derived from Latin Aurantes, perhaps referring to deposits of alluvial gold (Latin: aurum) along the Tagus. Similarly, Roman mosaics, coins, the remains of ancient aqueducts, as well as other antiquities, have been discovered in the vicinity of Abrantes.

The village of Abrantes and the Abrantes Castle were conquered from the Moors by Afonso I of Portugal in 1148. After 1172, Abrantes was a military outpost, under the dominion of the Order of Saint James of Compostela. In 1173, the region, including its castle, were donated to the Order of Saint James of the Sword, a distinct order. Pope Nicholas IV (around 1290) released this Order from the jurisdiction of the Castilian grand master of Santiago de Compostela Order in Uclés. It received its municipal foral in 1179, in compensation for its resistance against the Moroccan Abem Jacob. Its creation in 1179, was based on the concelhos perfeitos ("perfect municipality") model; they were formed on the specific day that settlers were invited to populate them; the administrative regime, the organization of the municipal authority and geographic distribution were planned as part of system of populating uninhabited lands, and extending the political influence of the Kingdom. Its geography contributed to the colonization: "Zone of permanent conflicts, where the Reconquista came to pontual conclusion, the colonization of Monte Abrantino was facilitated by the geographic situation of the mount, situated to the north of the Tagus, which served as a natural frontier and protected it from incursions; the Tagus which was an important source of richness (water, fish, gold, navigation, etc.), a mount situated in the confluence and transition of territories, crossroads of trails between south and norther, free of overflows, with persistent fog, cleansed of good aires, all these were conditions that,...gave the location the conditions necessary to attract colonists, colonization that, it seems, not to have been easy..."

On 24 April 1281, King Dinis of Portugal, conceded the ownership of Abrantes to Queen Elizabeth of Portugal.

Afonso IV of Portugal transferred the title of the religious parish to the Knights Hospitaller, in 1327, and granted Abrantes to Leonor Teles. Partners of the Master of Avis, the Abrantinos were one of the first to support the Lisboete movement of 1383, among them Fernando Álvares de Almeida, protector of the Casa de Abrantes.

King Manuel stayed in Abrantes for a long period; by 1510 he conceded a new foral to the town; and his children, the Infantes Fernando and Luís were born in Abrantes. During the 16th century, Abrantes and its religious parish was one of the largest and more populous lands in the Kingdom, with 3,436 inhabitants, four convents and 13 chapels or churches within its frontiers.

In 1581, in his move to Tomar, Philip I of Portugal stopped in Abrantes for several days, and during his reign was responsible for the reconstruction of Abrantes' municipal buildings. It was the seat of Portuguese marquesses, one of which was an early patron of the Brazilian-born versatile scientist, naturalist and pioneering inventor-aeronaut Bartholomeu Lourenço de Gusmão.

During 1640, Abrantes was one of the first lands to declare their support for John IV of Portugal. From the 17th and 18th centuries onwards, its ancient military importance was confirmed, as it was used by Portuguese and foreign armies to garrison or concentrate their forces. The village of Abrantes was captured on 24 November 1807 by French general Jean-Andoche Junot, who was awarded a ducal title for his victory by Napoleon I in 1808, and used as base for further marshaling of French forces. In the Convention of Cintra (22 August 1808) the town was returned to the Portuguese.

During the reign of Joseph I (1750–1777) the village became the center of the silk industry, which prospered until 1800, designated the Academia Tubuciana.

In 1820, Abrantes supported the Liberal Revolution and celebrated the 1822 Constitution; although Miguel of Portugal did find many supporters in the municipality, no outright manifestation of support occurred in light of the Vilafrancada. In November 1862, the Santerém-Abrantes lines was opened, and by March 1868, a bridge between Abrantes and Rossio to the southern Tagus region was well underway.

An active republican center, Abrantes was the place of preparatory meetings for the 5 October 1910 Revolution, which may have contributed to the village's elevation to the status of city on 14 June 1916.

Physical geography
Abrantes lies in a central enclave within the Portuguese territory and a transition zone between the Ribatejo, Beira Baixa and Alto Alentejo. This location allowed development of a diversified culture, influenced by many cultures around it; Sardoal and Vila de Rei, to the north; Mação and Gavião, to the east; Ponte de Sôr and Chamusca, in the south; and Constância and Tomar in the west. Abrantes is also located along many important road and rail links throughout the country, including the Madrid-Badajoz-Lisbon railway, being at the junction with the Guarda-Abrantes line, and the Lisbon-Porto access (the A23 and Beira Baixa roadways).

Abrantes commands the highway along the Tagus valley west from Belver (in the municipality of Gavião) until Constância, occupying the crest of a hill covered with olive woods, gardens and vines; historically, the fortified town has been regarded as an important military position.

The municipality is also center of two major hydrological resources: the Tagus River, which cross the territory for 30 kilometers and the Albufeira from Castelo do Bode.

Climate
Abrantes has a Mediterranean climate with hot, dry summers and mild, wet winters. Its location in the Tagus floodplain enables very high diurnal temperature variations (taken from a station in Alvega) making nights cool, with an average yearly low of 8.7 C, and days warm to hot, with an average yearly high of 23.3 C.

Human geography
Administratively, the municipality is divided into 13 civil parishes (freguesias), responsible for local area development and administration of services:

 * Abrantes (São Vicente e São João) e Alferrarede
 * Aldeia do Mato e Souto
 * Alvega e Concavada
 * Bemposta
 * Carvalhal
 * Fontes
 * Martinchel
 * Mouriscas
 * Pego
 * Rio de Moinhos
 * São Facundo e Vale das Mós
 * São Miguel do Rio Torto e Rossio ao Sul do Tejo
 * Tramagal

It has grown dramatically since the early 20th century: from a base population of 7,255 in 1900, it had more than 41,000 inhabitants in 2001.

Notable people

 * Jorge de Lencastre, Duke of Coimbra (1481–1550) a prince, illegitimate son of King John II of Portugal
 * Luís of Portugal, Duke of Beja (1506–1555) the second son of King Manuel I of Portugal
 * Ferdinand of Portugal, Duke of Guarda 1507–1534) a prince, the son of King Manuel I of Portugal
 * António Leal Moreira (1758–1819) a Portuguese classical composer and organist
 * Joaquim Marques Esparteiro (1895–1976) a Portuguese navy officer and colonial administrator
 * Maria de Lourdes Pintasilgo (1930–2004) a chemical engineer and politician; the first and to date the only woman Prime Minister of Portugal

Sport

 * Luís Mena e Silva (1902–1963) a Portuguese horse rider; bronze medallist at the 1936 & 1948 Summer Olympics
 * Hélder Catalão (born 1955) a former footballer with 335 club caps

title: Abrantes


text:
Flathead National Forest

The Flathead National Forest is a national forest in the western part of the U.S. state of Montana. The forest lies primarily in Flathead County, south of Glacier National Park. The forest covers 2404935 acres of which about 1 e6acre is designated wilderness. It is named after the Flathead Native Americans who live in the area.

Description
The forest lies primarily in Flathead County (about 73% of its acreage), but smaller areas extend into five other counties. In descending order of land area they are Powell, Missoula, Lake, Lewis and Clark, and Lincoln counties. Forest headquarters are located in Kalispell, Montana. There are local ranger district offices in Bigfork, Hungry Horse, and Whitefish.

The Flathead National Forest is bordered by Glacier National Park and Canada to the north, the Lewis and Clark National Forest and Glacier to the east, the Lolo National Forest to the south, and the Kootenai National Forest to the west. The wilderness areas in the forest are the Bob Marshall Wilderness Area, Great Bear Wilderness Area, and Mission Mountains Wilderness Area. Other specially-designated areas in the forest include Flathead Wild and Scenic River, Jewel Basin Hiking Area, and the Coram Experimental Forest. Some 270000 acres of non-federal land are also included in the boundaries drawn for the national forest. This includes private land, commercial forest and part or all of Swan River State Forest, Stillwater State Forest and Coal Creek State Forest.

The forest contains 1700 mi of roads, many of them primitive fire roads and 2800 mi of hiking trails. Approximately 38 mi of the Pacific Northwest National Scenic Trail are located within the Flathead National Forest.

Ecology
The forest is located in the Rocky Mountains with elevations ranging from 4500 to 8500 ft. The forest provides habitat for approximately 250 species of wildlife and 22 species of fish. This includes bald eagle, wolverine, beaver, elk, porcupine, cougar, moose, bobcat, white-tailed deer, coyote, grizzly bear, timber wolf, two species of fox, mountain goat, Canadian lynx, woodchuck, bighorn sheep and bull trout.

Activities
Commercial activities in the non-wilderness sections of the forest include timber harvesting, two downhill ski resorts and a small amount of cattle grazing. Individuals can pick less than 10 gal of berries without a permit. Larger amounts of berries, firewood and Christmas tree cutting, and mushroom or mineral gathering in wilderness areas require permits.

While camping is allowed almost anywhere within national forests without a permit, Flathead National Forest also has 31 campgrounds with some facilities. The largest campground has only 40 sites and 2 campgrounds have only one site each. Most campgrounds do not have running water. There are also 14 cabins for rent in the forest.

Points of interest

 * Hungry Horse Dam is a 564 ft tall dam that creates the 34 mi long Flathead Reservoir.
 * Bob Marshall Wilderness Complex spans over 1.5 e6acres, comprising the Bob Marshall Wilderness, the Great Bear Wilderness and the Scapegoat Wilderness.
 * Whitefish Mountain Resort is a ski resort located on Big Mountain in the Flathead National Forest.
 * Chinese Wall is a 22 mi long, 1000 ft tall escarpment located in the Bob Marshall Wilderness Complex.
 * Big Creek Ranger Station is listed on the National Register of Historic Places.
 * Coram Experimental Forest is an 800 acre experimental western larch forest located in the Flathead National Forest.

title: Flathead National Forest


text:
Antonius Johannes Jurgens

Antonius Johannes Jurgens, also known as Anton Jurgens (8 February 1867 – 12 March 1945) was a Dutch businessman.

He was born in Oss, the grandson of Joanna Lemmens and of Antoon Jurgens, who founded a butter company in 1867 and, in 1871, the first margarine factory in the world.

History
The Jurgens family had moved from the neighborhood of Heerlen, in the south of the Netherlands and close to the German border, to Oss, where the family company was founded; they were known as the moffen, a derogatory term for Germans. Anton Jurgens became the pater familias of the company, which would retain its family structure until the fusion with Van den Bergh in 1927.

Business
Jurgens succeeded his father in business at the age of 21, and was one of the main European margarine and soap manufacturers in the early twentieth century with Pieter-Eduard Leverd.

His whaling monopoly also led Jurgens to be called the "prince of whales".

Short political career
From 1919 to 1921, he also served in the Senate as a member of the Roman Catholic State Party; he was forced to give up his seat because of the demands of his business.

Death
He died in 1945 Torquay, in England, and was buried there and later reinterred in Poissy, in France, beside his wife and his son Henri, who had died in an accident in 1933.

Former politician Erik Jurgens is a family member.

title: Antonius Johannes Jurgens


text:
Northern Highway (Victoria)

Northern Highway is a rural highway in northern Victoria, linking Echuca on the banks of the Murray River with Beveridge a short distance north of the northern suburban fringes of Melbourne. In conjunction with McIvor Highway, it provides an important link between Melbourne and Bendigo. It forms a significant freight route providing access to markets and ports in Melbourne and the rural primary production areas of the Murray Valley and southern New South Wales, and serves a number of agricultural and tourism-related industries along its length.

Route
Northern Highway commences at the state border with New South Wales as a continuation of Cobb Highway into Victoria and heads in a southerly direction as a two-lane, single carriageway rural highway through the western part of the regional town of Echuca, running concurrent with Murray Valley Highway for a short period, before continuing south through flat open country through Rochester, Elmore and Heathcote, then traverses moderately hilly terrain through Kilmore and Wallan, before eventually terminating at the interchange with Hume Freeway a short distance south of Wallan in Beveridge.

History
The passing of the Country Roads Act of 1912 through the Parliament of Victoria provided for the establishment of the Country Roads Board (later VicRoads) and their ability to declare Main Roads, taking responsibility for the management, construction and care of the state's major roads from local municipalities. Kilmore-Heathcote(-Bendigo) Road was declared a Main Road, from Kilmore to Pyalong on 30 November 1914, and from Pyalong and Heathcote (and continuing west to Bendigo) on 28 May 1915; Heathcote-Elmore Road was declared a Main Road, between Tooleen and Elmore on 17 March 1915, and between Heathcote and Tooleen on 28 May 1915; and Bendigo-Echuca Road between north-eastern Bendigo and Elmore was declared a Main Road on 17 May 1915.

The passing of the Highways and Vehicles Act of 1924 provided for the declaration of State Highways, roads two-thirds financed by the State government through the Country Roads Board. Northern Highway was declared a State Highway on 1 July 1925, cobbled from a collection of roads from Bendigo through Elmore and Rochester to Echuca (for a total of 55 miles), subsuming the original declaration of Bendigo-Echuca Road. In the 1947/48 financial year, its southern end was rerouted south of Elmore: instead of running to Bendigo, it was extended further south via Heathcote to meet Hume Highway at Kilmore, subsuming the original declarations of Heathcote-Elmore Road and Kilmore-Heathcote-Bendigo Road as Main Roads; the previous alignment of Northern Highway from Elmore to Bendigo was subsumed into Midland Highway. Northern Highway was extended south one last time when the Kilmore bypass was opened in 1976, from Kilmore to Beveridge (just south of Wallan) along the former alignment of Hume Highway.

Northern Highway was later signed National Route 75 in 1955; with Victoria's conversion to the newer alphanumeric system in the late 1990s, this was updated to route B75.

The passing of the Road Management Act 2004 granted the responsibility of overall management and development of Victoria's major arterial roads to VicRoads: in 2022, VicRoads re-declared this road as Northern Highway (Arterial #6540), beginning in Echuca and ending in Beveridge.

Attractions

 * Heathcote-Graytown National Park

title: Northern Highway (Victoria)


text: User talk:70.109.138.4

title: User talk:70.109.138.4