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, the probability model for a classifier is a conditional model

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

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

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

Abstractly, the method is as follows:

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

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

Abstractly, if one starts with a complex vector space W and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of W and its conjugate:

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

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

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

As Lipton puts it: `` The Eros is felt in the magic circle of marijuana with far greater force, as a unifying principle in human relationships, than at any other time except, perhaps, in the mutual metaphysical orgasms.

The magic circle is, in fact, a symbol of and preparation for the metaphysical orgasm ''.

This is less than the length of a jet runway -- well within the circle of destruction.

Like ours, the economy of the space merchants must constantly expand in order to survive, and, like ours, it is based on the principle of `` ever increasing everybody's work and profits in the circle of consumption ''.

First and foremost: No one -- no, not anyone -- in the family is allowed to issue blanket invitations to his or her own circle.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

Whosever fault, it is evident that Brumidi intended to fill out the whole frieze with his `` histories '' and come full circle with the scene of the discovery of California gold.

the fire of love is dead, and Hardy stands, as the speaker does in the last poem of the sequence, over the burnt circle of charred sticks, and thinks of past happiness and present grief, honest and uncomforted.

For circular fibers in a closely packed hexagonal array, the packing efficiency is given by: Af where Af, and 0.906 is the ratio of the area of a circle to that of the circumscribed hexagon.

The hotly debated plan for the capital's Franklin D. Roosevelt Memorial, a circle of huge tablets engraved with his speeches ( and promptly dubbed by one of its critics, `` Instant Stonehenge '' ), is another of Udall's headaches, since as supervisor of the National Parks Commission he will share in the responsibility for building it.

Its rotunda forms a perfect circle whose diameter is equal to the height from the floor to the ceiling.

The region north of this circle is known as the Arctic, and the zone just to the south is called the Northern Temperate Zone.

The equivalent polar circle in the Southern Hemisphere is called the Antarctic Circle.

In contrast, the largest North American community north of the circle, Sisimiut ( Greenland ), has approximately 5, 000 inhabitants, while between Canada and the USA, Barrow, Alaska is the largest settlement with circa 4, 000 inhabitants.

Since the widespread adoption of digital setting circles, any classical engraved setting circle is now specifically identified as an " analog setting circle " ( ASC ).

A computerized setting circle is called a " digital setting circle " ( DSC ).

An aspect ratio of 1: 1 is a circle.

With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen like atom, a Bohr electron " wavelength " could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength ( this historically incorrect Bohr model is still occasionally taught to students ).

The structure is also illustrated as a circle around the inside of the ring to show six electrons floating around in delocalized molecular orbitals the size of the ring itself.

It is used occasionally when it is necessary to limit the turning circle as the yacht swings when it is anchored, such as in a very narrow river or a deep pool in an otherwise shallow area.

The kites that are also cyclic quadrilaterals ( i. e. the kites that can be inscribed in a circle ) are exactly the ones formed from two congruent right triangles.

Consequently the four nine-point centers are cylic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral.

In Euclidean geometry, Brahmagupta's formula finds the area of any cyclic quadrilateral ( one that can be inscribed in a circle ) given the lengths of the sides.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.

The word cyclic is from the Greek kuklos which means " circle " or " wheel ".

( This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic ( inscribed in a circle ).

An equilateral polygon which is cyclic ( its vertices are on a circle ) is a regular polygon ( a polygon that is both equilateral and equiangular ).

In Mithran mystery cults the figure of Mithra being reborn ( one of the things he is famous for ) is sometimes seen wrapped with an ouroboros, indicating his eternal and cyclic nature, and even references which do not mention the ouroboros refer to this circular shape as symbolizing the immortality of the soul or the cyclic nature of Karma, suggesting that the circle retains its meaning even when the details of the image are obscured.

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.

Thus, in Euclidean geometry three non-collinear points determine a circle ( as the circumcircle of the triangle they define ), but four points in general do not ( they do so only for cyclic quadrilaterals ), so the notion of " general position with respect to circles ", namely " no four points lie on a circle " makes sense.

In mathematics, a cyclic order is a way to arrange a set of objects in a circle.

Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line.

As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180 °; conversely, any quadrilateral for which this is true can be inscribed in a circle.

For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral ( one whose vertices all fall on a single circle ) are supplementary.

The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer n > 0, the n < sup > th </ sup > roots of unity form a cyclic group of order n, which is unique up to isomorphism.

Kronecker's theorem may also refer to a result in diophantine approximations applying to several real numbers x < sub > i </ sub >, for 1 ≤ i ≤ N, that generalises the equidistribution theorem, which implies that an infinite cyclic subgroup of the unit circle group is a dense subset.

By drawing a line between two pitch classes when they differ by a generator, we can depict the circle of generators as a cyclic graph, in the shape of a regular polygon.

An example for G infinite cyclic is the circle as X.

) In the case of the circle example, what is being said is that we remark that an infinite cyclic group C acts freely on the real line R, which is contractible.

# The circle is a classifying space for the infinite cyclic group.

The homotopy group π < sub > 1 </ sub >( S < sup > 1 </ sup >) is therefore an infinite cyclic group, and is isomorphic to the group Z of integers under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle.

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral ( a quadrilateral whose vertices lie on a common circle ).