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, 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, 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 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, 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, 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.

In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces ( see below ), or as a system of parametric equations.

* A parametric equation includes variables which are all functions of one or more common variables ( called parameters ).

A somewhat more detailed description can be found at parametric equation.

* Program in C ++ to draw a sphere using parametric equation

In addition to the standard two-dimensional function plots, it can also produce graphs of parametric equations, polar equations, sequence plots, differential equation fields, and three-dimensional ( two independent variable ) functions.

In parametric form, the equation for the line that passes through the two points is:

* The parametric equation of the fold is then:

And the parametric equation of the fold is:

If we find some vector v that is perpendicular to the line l < sub > 1 </ sub >, then the parametric equation of the fold is:

The term " path " simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i. e. the curve q ( t ), parameterized by time ( see also parametric equation for this concept ).

* As parametric equation:

This may be viewed as an equation parametric in and it demonstrates that the poles lie on an ellipse in s-space centered at s = 0 with a real semi-axis of length and an imaginary semi-axis of length of

Like its predecessor, Voyage 200 is an advanced calculator that supports plotting multiple functions on the same graph, parametric, polar, 3D, and differential equation graphing as well as sequence representations.

A comparametric equation is an equation that describes a parametric relationship between a function and a dilated version of the same function, where the equation does not involve the parameter.

In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation.

In mathematics, parametric equation is a method of defining a relation using parameters.

For a real-valued continuous function f, defined on an interval b ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f ( x ), for x ∈ b.

* In Cartesian coordinates the involute of a circle has the parametric equation:

* In polar coordinates the involute of a circle has the parametric equation:

γ ( t ) = ( x ( t ), y ( t )), then the parametric equation of the evolute can be expressed in terms of the radius of curvature R = 1 / k and the tangential angle φ, which is the angle the tangent to the curve makes with a fixed reference axis x-axis.

In terms of R and φ, the evolute has the parametric equation

Vector line element dr ( green ) in three-dimensional | 3d Euclidean space, where λ is a parametric equation | parameter of the space curve ( yellow ).

This reparametrisation of γ < sub > 1 </ sub > defines the equivalence relation on the set of all parametric C < sup > r </ sup > curves.

So we have to define a suitable equivalence relation on the set of all parametric curves.

We first define a function b{t} as follows: given the set of squares such that each has three corners on C and vertex at t, b{t} is the corresponding set of positive parametric differences between T and the backward corner points.

* A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another.

A simple parametric representation of the figure-eight knot is as the set of all points ( x, y, z ) where

These equations are parametric in d. Setting d equal to any non-zero number and substituting it into these equations will yield one solution set.

The term arises in contexts where the set of all possible population distributions is put in parametric form.

For example, if one makes the parametric assumption that the underlying distribution is a normal distribution, and has a sample set

A Klein bottle which has been defined by a set of parametric equations.

If the likelihood and its prior take on simple parametric forms ( such as 1-or 2-dimensional likelihood functions with simple conjugate priors ), then the empirical Bayes problem is only to estimate the marginal and the hyperparameters using the complete set of empirical measurements.

This parametric model can be estimated using constrained least-squares estimation, using a training set of compounds with experimentally measured partition coefficients.

In the case in which the elements of this set can be indexed by a finite number of real-valued parameters, the model is called a parametric model ; otherwise it is a nonparametric or semiparametric model.

An iterator over a set provides a dynamic and parametric family of microthreads.

As well, while a standalone home theatre subwoofer may contain a range of sound-shaping circuitry, such as a crossover control, a phase inversion switch, and a parametric equalizer, a HTIB subwoofer system usually has its crossover point set at the factory, which means that the user cannot change it.