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

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, the method is as follows:

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

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

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

It is not possible to reconstruct fully the arrangements whereby these honors lists were then made up or even how the names that they contained assumed the order in which we find them.

The soldiers themselves cannot stage a successful rebellion, it is assumed: but will their discontent spread to the officer class??

`` The human ego being what it is '', I put in, `` science fiction has always assumed that the creatures on the planets of a thousand larger solar systems than ours must look like gigantic tube-nosed fruit bats.

The infrared emission could then be assumed to originate at the surface of the moon, while the radio emission originates at some depth beneath the surface, where the temperature variation due to solar radiation is reduced in amplitude and shifted in phase.

The apparent black-body disk temperature is the temperature which must be assumed for the black body in order that the intensity of its radiation should equal that of the observed radiation.

If it is assumed that the formula given by Lodge of cosec Af applies, the pressure difference along the major axes can be calculated from the angle of inclination of the major axis, and from this the interfacial tension can be calculated.

The structure appears to be unique among OOH compounds, but is the same as that assumed by Af.

It is assumed that the flux values increase by a factor of 2.512 per magnitude, in accordance with the opinion that the total mass flux in each unit range in magnitude is constant.

It is assumed that the iodine released from the iodotyrosines remains in the iodide pool of the thyroid, where it is oxidised and re-incorporated into thyroglobulin.

He showed convincingly that anxiety is a learned ( conditioned ) reaction and is the basis of experimental and clinical neuroses and assumed, therefore, that the neuronal changes which underlie the neuroses are functional and reversible.

From this presumption it is an easy step to the conclusion that any observed increases in the basic wage rate must be due to union behavior different and more aggressive than assumed in our model.

That is to say, an allowance for `` cost of capital '' will be assumed to be included directly in the cost apportionment.

So far these remarks, like most criticisms of Hardy, have tacitly assumed that his poetry is all of a piece, one solid mass of verse expressing a sensibility at a single stage of development.

he further reasoned that frequent formulas in epic verse indicate oral composition, and assumed the slightly less likely corollary that oral epic is inclined towards the use of formulas.

One of the greatest Homerists of our time, Frederick M. Combellack, argues that when it is assumed The Iliad and The Odyssey are oral poems, the postulated single redactor called Homer cannot be either credited with or denied originality in choice of phrasing.

Proponents of single elements tend to ensure predominance of that element without determining if it is justified, and the element with the most enthusiastic and vociferous proponents has assumed the greatest importance.

Communist target areas can be assumed, but there is no certainty that such assumptions coincide with Soviet intentions.

In general, P is a function of V and the current density, but it shall here be assumed as a constant.

It is assumed that the gyros are designed with electrical torquers so that a torque can be applied about their output axes.

He contends this idea doesn't conflict with experiments on which the principle of conservation of matter and energy is based because some slight error must be assumed in such experiments.

Random error is typically assumed to be normally distributed with zero mean and a constant variance.

The Brønsted-Lowry definition is the most widely used definition ; unless otherwise specified acid-base reactions are assumed to involve the transfer of a proton ( H < sup >+</ sup >) from an acid to a base.

-- liberal considers that the need for a national economy with controls that will assure his conception of social justice is so great that individual and local liberties as well as democratic processes may have to yield before it.

He knows that the economy of life in the `` outback '' is awful.

The field, then, is ripe for new Southerners to step to the fore and write of this twentieth-century phenomenon, the Southern Yankeefication: the new urban economy, the city-dweller, the pains of transition, the labor problems ; ;

William Styron, while facing the changing economy with a certain uneasy reluctance, insists he is not to be classified as a Southern writer and yet includes traditional Southern concepts in everything he publishes.

Lacking the pioneer spirit necessary to write of a new economy, these writers seem to be contenting themselves with an old one that is now as defunct as Confederate money.

The international economy of 1960 is markedly different from that of the early postwar years.

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 ''.

What is wrong with advertising is not only that it is an `` outrage, an assault on people's mental privacy '' or that it is a major cause for a wasteful economy of abundance or that it contains a coercive tendency ( which is closer to the point ).

But it would greatly strengthen any Mayor's executive powers, remove the excuse in large degree that he is a captive of inaction in the Board of Estimate, increase his budget-making authority both as to expense and capital budgets, and vest in him the right to reorganize city departments in the interest of efficiency and economy.

It is this loss of initiative in how we conduct our economy which may lead to the loss of initiative in how we conduct our political affairs.

The purchase of compact ( economy ) cars is being made currently on a test basis.

The Merchant Marine is the `` Fourth Arm of Defense '', for a strong and effective American Merchant Marine is essential to the economy and security of our Nation.

While this may well be true in general, I believe it is also important to keep in mind that some recent developments suggest that over the next year or so military electronics may be one of the most strongly growing areas in an economy which is not expanding rapidly in other directions.

But make no mistake about it, the first reason people turn to camping is one of economy.

We should encourage the governments to develop their own technical assistance to communities, state and provincial governments, rural communities, and other smaller groups, making certain that no important segment of the economy is neglected.

It is difficult to see any powerful sources of strength on the horizon at this time which would give the economy a new upward thrust.

Accordingly, during the Sixties our national economy is likely to grow at as fast a rate as in the Fifties and, in the process, to require enormous amounts of capital funds.

And in the economy of the book it is not peripheral but central.

To see how important this economy is, let us suppose that there are M operating variables at each stage and that the state is specified by N variables ; ;