Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

Abstractly, a hierarchy can be modelled mathematically as a rooted tree: the root of the tree forms the top level, and the children of a given vertex are at the same level, below their common parent.

Abstractly, this is equivalent to stabilising a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis and the resulting flag < math > 0 <

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly programmable toolpath guidance began with mechanical solutions, such as in musical box cams and Jacquard looms.

Abstractly, we can identify the chain rule as a 1-cocycle.

Abstractly, 1 / 8-schisma tuning may be considered the analog, among schismatic tunings, of 1 / 4-comma meantone among meantone tunings, as it also has pure interval ratios of 2: 1 and 5: 4, though with much more accurate interval ratios of 3: 2 and 6: 5 ( less than a quarter of a cent off from just intonation ) than its meantone counterpart.

Abstractly, a parametric equation defines a relation as a set of equations.

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

Abstractly it is assumed that the economy consists only of capitalist production and that the capitalist economy is equal to capitalist society.

Abstractly, the probability model for a classifier is a conditional model

Abstractly, it can be defined as the integer triples ( a, b, c ), associated to 3 < sup > a </ sup > 5 < sup > b </ sup > 7 < sup > c </ sup >, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm

Abstractly we can say that this is a projective line in the space of all conics, on which we take

Abstractly, the method is as follows:

She said, `` My name is Songau and these girls are Ponkob and Piwen.

This is puzzling to an outsider conscious of the classic tradition of liberalism, because it is clear that these Democrats who are left-of-center are at opposite poles from the liberal Jefferson, who held that the best government was the least government.

And the great majority of these people are of Anglo-Saxon or Celtic descent.

A national consensus of near unanimity exists that these governmental efforts are desirable as well as necessary.

I have just asked these questions in the Pentagon, in the White House, in offices of key scientists across the country and aboard the submarines that prowl for months underwater, with neat rows of green launch tubes which contain Polaris missiles and which are affectionately known as `` Sherwood Forest ''.

Only one rule prevailed in my conversations with these men: The more highly placed they are -- that is, the more they know -- the more concerned they have become.

For these centers, too, are on the gold circuit.

What these fragments are and how they activate the go order may not be revealed.

Often, too, the social institutions are housed in these pavilions and palaces and bridges, for these great structures are not simply `` historical monuments '' ; ;

But more important, and the thing which the casual traveler and the blind sojourner often do not see, is that these places and activities are often the settings in which Persians exercise their extraordinary aesthetic sensibilities.

Water, air, fruit, poetry, music, the human form -- these things are important to Persians, and they experience them with an intense and discriminating awareness.

The line of an eyebrow, the color of the skin, a ghazal from Hafiz, the purity of spring water, the long afternoon among the boughs which crowd the upper story of a pavilion -- these things are noticed, judged, and valued.

Where are the writers to treat these changes??

Lucretius has remarked: `` The reason why all Mortals are so gripped by fear is that they see all sorts of things happening in the earth and sky with no discernable cause, and these they attribute to the will of God ''.

We have staved off a war and, since our behavior has involved all these elements, we can only keep adding to our ritual without daring to abandon any part of it, since we have not the slightest notion which parts are effective.

And Zen Buddhism, though it is extremely difficult to understand how these internal contradictions are reconciled, helps them in their struggle to achieve personal salvation through sexual release.

Whatever pole of this contrast one emphasizes and whatever the tension between these two approaches to understanding the artistic imagination, it will be readily seen that they are not mutually exclusive, that they belong together.

Underlying these conceptions of mimesis are certain presuppositions concerning the nature of primary human experience which require some exposition before the main argument can proceed.

Later abstractive and rational processes may indicate errors of judgment in these apprehensions of value, but the apprehensions themselves are the primary stuff of experience.

Of these there are surely few that would be more rewarding discoveries than Verner Von Heidenstam, the Swedish poet and novelist who received the award in 1916 and whose centennial was celebrated two years ago.

Somewhat uneven in interest for an average reader, eight or ten of these are among the finest of their kind in literature.

Carvey believes that reapportionment, which left many Democratic clubs split by these new district boundaries, actually will increase CDC membership.

What is left to traditional systems of philosophy is, in effect, only the history of these fields prior to their becoming rigorous enough to abide by the canons of scientific method.

that is, by being placed in such relation to the forms depicted within the illusion that these forms left no room for the typography except near the surface.

I shall first indicate a couple of weaknesses in Fromm's analysis, then argue that, granted these weaknesses, he still has much left that is valuable, and, finally, raise the general question of a philosophical versus a sociological approach to the question of alienation.

By 1926, when the mighty Yanks were at their mightiest, only a few of these were left but they still shone brightest, even beside able and agile rookies like Tony Lazzeri ( who managed never to have one of his epileptic fits on the field ), Mark Koenig, Lou Gehrig, George Pipgras, and gray-thatched Earl Combs.

Upon most of these Christianity had left an impress and through them had had a share in making the individual what he was.

Pope Pius the Sixth, at Rome, in April, 1778, wrote the following: `` The faithful should be excited to the reading of the Holy Scriptures: For these are the most abundant sources which ought to be left open to everyone, to draw from them purity of morals and of doctrine, to eradicate errors which are so widely disseminated in these corrupt times ''.

By 454, the Delian League could be fairly characterized as an Athenian Empire ; at the start of the Peloponnesian War, only Chios and Lesbos were left to contribute ships, and these states were by now far too weak to secede without support.

Amongst these is Psyche, with a butterfly, which is placed on the left hand, and held by the wings with the right.

Some instances of exploring these " nearby " nodes require traversing up to 2 × log ( n ) links ( particularly when moving from the rightmost leaf of the root's left subtree to the leftmost leaf of the root's right subtree ).

Almost all of Bulgaria's 500, 000-man standing army was positioned against these two countries, on two fronts – western and southern, while the borders with Romania and the Ottoman Empire were left almost unguarded.

Campbell-Bannerman tried to keep these forces together at the head of a moderate Liberal rump, but in 1901 he delivered a speech on the government's " methods of barbarism " in South Africa that pulled him further to the left and nearly tore the party in two.

to their left are four more anchored ships and to the left of these vessels is a distant shoreline.

The burden of proof should be on the people who make these statements, to show where they got their information from, to see if their conclusions and interpretations are valid, and if they have left anything out.

For all these reasons Costa Rica was by and large unappreciated and overlooked by the Spanish Crown and left to develop on its own.

A great triumph of these laborious months was the decipherment of the palimpsest Codex Ephraemi Syri Rescriptus, of which the New Testament part was printed before he left Paris, and the Old Testament in 1845.

Some opponents of these decisions left the Conservative movement to form the Union for Traditional Judaism.

As a result, these first releases were followed by further distributions across much of the region, although their effectiveness on other crops, such as cabbages, has been questioned ; when the toads were released at Wau, the cabbages provided insufficient shelter and the toads rapidly left the immediate area for the superior shelter offered by the forest.

This left the Rump basically a conservative body whose vested interests in the existing land ownership and legal systems made them unlikely to want to reform these.

Some people were left out from these four caste classifications, and were called panchama ( literally, the fifth ).

However, the Mongol invasion left Korea in disarray and anomie, and these groups became nomadic.

The alternative majority in these issues was because the Social liberal Party ( Radikale Venstre ) supported the governing majority in economic policy issues, but was against certain NATO policies and voted with the left in these issues.