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

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

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

Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

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

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

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

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

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

This can mean that where it is the defendant who appeals, the name of the case in the law reports reverses ( in some cases twice ) as the appeals work their way up the court hierarchy.

Alexander was fascinated by Pseudo-Dionysian hierarchy of angels and in how their nature can be understood, given Aristotelian metaphysics.

Moore, there can be no end to the process of set formation, and thus no such thing as the totality of all sets, or the set hierarchy.

For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).

Finally, the natural hierarchy of Perl module names ( such as " Apache :: DBI " or " Lingua :: EN :: Inflect ") can sometimes be used to browse modules in the CPAN.

Even outside the Acme :: hierarchy, some modules are still written largely for amusement ; one example is Lingua :: Romana :: Perligata, which can be used to write Perl programs in a subset of Latin.

Interaction with memory hierarchy effects can make this overhead intolerable in circumstances that are hard to predict or to detect in routine testing.

Even the protoctists can be elevated, although their new position in the hierarchy is not clearly defined.

The species lives in a male-dominated, strict hierarchy, which means disputes can generally be settled without the need for violence.

Traditionally listed as a part of the major branch of philosophy known as metaphysics, ontology deals with questions concerning what entities exist or can be said to exist, and how such entities can be grouped, related within a hierarchy, and subdivided according to similarities and differences.

Each node knows only about some number of other nodes that it can reach directly ( its conceptual " neighbors "), but any node can be a neighbor to any other ; no hierarchy or other structure is intended.

Cases can be ranked in the following hierarchy, where a language that does not have a given case will tend to not have any cases to the right of the missing case:

Regarding the file system, a suitable program can be designated as a translator for a single file or a whole directory hierarchy.

A hierarchy ( sometimes abbreviated HR ) can link entities either directly or indirectly, and either vertically or horizontally.

The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies.

All parts of the hierarchy which are not linked vertically to one another nevertheless can be " horizontally " linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again.

For this reason, it can even be argued that one's inheritance places them in a specific social class position that requires a level of participation in certain activities that promote the oppression of lower-class individuals in terms of the social hierarchy and system of stratification.

Species can be placed in a ranked hierarchy, starting with either domains or kingdoms.

As a result of differences in notation, history, use of enumeration, hierarchy, and facets, classification systems can differ in the following ways:

The archaeology of shipwrecks can be divided into a three-tier hierarchy, of which the first tier considers the wrecking process itself: how does a ship break up, how does a ship sink to the bottom, and how do the remains of the ship, cargo and the surrounding environment evolve over time?

" He argues this " implies that any conceivable parallel universe theory can be described at Level IV " and " subsumes all other ensembles, therefore brings closure to the hierarchy of multiverses, and there cannot be say a Level V ."

This may be a formal hierarchy, or they may be vague, overlapping terms, or a combination of both.

Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.

Minsk has a digital metropolitan network ; waiting lists for telephones are long ; fixed line penetration is improving although rural areas continue to be undeserved ; intercity-Belarus has developed fibre-optic backbone system presently serving at least 13 major cities ( 1998 ); Belarus's fibre optics form synchronous digital hierarchy rings through other countries ' systems ; an inadequate analogue system remains operational.

The enemy may be the cosmopolitan intellectual, but the paranoid will outdo him in the apparatus of scholarship ... the Ku Klux Klan imitated Catholicism to the point of donning priestly vestments, developing an elaborate ritual and an equally elaborate hierarchy.

In this way a hierarchy may be established between any two record types, e. g., type A is the owner of B.

Although the worship of images would eventually fall out of favour ( and be replaced by the iconoclastic fire temples ), the lasting legacy of the Achaemenids was a vast, complex hierarchy of Yazatas ( modern Zoroastrianism's Angels ) that were now not just evident in the religion, but firmly established, not least because the divinities received dedications in the Zoroastrian calendar, thus ensuring that they were frequently invoked.

To deconstruct the opposition, first of all, would be to overturn the hierarchy at a given moment.

Although it never uses the term, The Church of Jesus Christ of Latter-day Saints ( The Mormon Church ) is episcopal, rather than presbyterian or congregational, in the sense that it has a strict hierarchy of leadership from the local bishop up to a single prophet / president, believed to be personally authorized and guided by Jesus Christ.

If a significant amount of greens work is required in a film, then the Greens may be an identifiable sub-department, with its own team-often of a size numbering double figures-and hierarchy ( eg.

This strict yet flexible hierarchy exemplifies how the Hudson ’ s Bay Company was able to be so successful while still having its central management and trade posts located so far apart.

All citizens of Haiti, regardless of skin color, to be known as " Black " ( this was an attempt to eliminate the multi-tiered racial hierarchy that had developed in Haiti, with full or near full-blooded Europeans at the top, various levels of light to brown skin in the middle, and dark skinned " Kongo " from Africa at the bottom ).

Her reference to herself as a member of the " weaker sex " and her rather constant belittling of women, though at first seemingly problematic, must be considered within the context of the patriarchal church hierarchy.