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In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of and correspond to the Coxeter diagram Conversely, the cyclic point groups in two dimensions are not generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group.
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two and dimensions
If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.
It may be applied to two characteristic dimensions of a three-dimensional shape.
In this relief, the two principal figures, the stern and courageous pope and the dismayed and frightened Attila, surge forward from the center into three dimensions.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions.
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.
Finally, Dürer discusses the Delian Problem and moves on to the ' construzione legittima ', a method of depicting a cube in two dimensions through linear perspective.
Instead, the ratio of the speed of the wind and that of the cups, the anemometer factor, depends on the dimensions of the cups and arms, and may have a value between two and a little over three.
Each string is a one-dimensional object, and the largest number of dimensions in which two strings will generically intersect ( and, presumably, annihilate ) is three.
After the advent of quantum mechanics, work done by Landau in 1930 predicted the quantization of the Hall conductance for electrons confined in two dimensions.
In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.
Rotating frames and fictitious forces can often reduce the description of motion in two dimensions to a simpler description in one dimension ( corresponding to a co-rotating frame ).
In two dimensions the position vector which has magnitude ( length ) and directed at an angle above the x-axis can be expressed in Cartesian coordinates using the unit vectors and:
The modern Cartesian coordinate system in two dimensions ( also called a rectangular coordinate system ) is defined by an ordered pair of perpendicular lines ( axes ), a single unit of length for both axes, and an orientation for each axis.
In two dimensions, the vector from the origin to the point with Cartesian coordinates ( x, y ) can be written as:
) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.
The data warehouse bus architecture is primarily an implementation of " the bus ", a collection of conformed dimensions and conformed facts, which are dimensions that are shared ( in a specific way ) between facts in two or more data marts.
Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows:
This means that the radio spectrum is divided into physical channels in two dimensions: frequency and time.
In two dimensions, one way to detect if point D lies in the circumcircle of A, B, C is to evaluate the determinant:
A divide and conquer algorithm for triangulations in two dimensions is due to Lee and Schachter which was improved by Guibas and Stolfi and later by Dwyer.
He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers.
In dimension one this is the real line ; in dimension two it is the Cartesian plane ; and in higher dimensions it is a coordinate space with three or more real number coordinates.
In two or more dimensions we must use, the del or gradient operator, which generalises the first derivative, obtaining
two and finite
We note that two such curves C and Af, cannot coincide at more than a finite number of points ; ;
* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )
Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.
The difference between the two definitions is that under the former, finite sets are also considered to be countable, while under the latter definition, they are not considered to be countable.
The function F is called prefix-free if there are no two elements p, p ′ in its domain such that p ′ is a proper extension of p. This can be rephrased as: the domain of F is a prefix-free code ( instantaneous code ) on the set of finite binary strings.
Electric field lines of two opposing charges separated by a finite distance.
The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e. g., fuzzy logic.
It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96 ; in fact there are two nonisomorphic groups of order 96 with this property.
It tends towards a line segment ( see below ) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away.
* Any two finite fields with the same number of elements are isomorphic.
In Jacques Dupuis ’ Who Do You Say I Am ?, he argues that, within the one person of Jesus Christ, we can distinguish between his two natures, human and divine, and thus between the operations of his uncreated divine nature and his created finite human nature.
* In the case of QAM ( quadrature amplitude modulation ), a finite number of at least two phases, and at least two amplitudes are used.
Pushdown automata differ from finite state machines in two ways:
If we allow a finite automaton access to two stacks instead of just one, we obtain a more powerful device, equivalent in power to a Turing machine.
Power law distributions have a well defined mean only if the exponent exceeds 1 and have a finite variance only when the exponent exceeds two.
In the digital QAM case, a finite number of at least two phases and at least two amplitudes are used.
If the material superconducts in the absence of a field, then the superconducting phase free energy is lower than that of the normal phase and so for some finite value of the magnetic field ( proportional to the square root of the difference of the free energies at zero magnetic field ) the two free energies will be equal and a phase transition to the normal phase will occur.
With finite samples, approximation results measure how close a limiting distribution approaches the statistic's sample distribution: For example, with 10, 000 independent samples the normal distribution approximates ( to two digits of accuracy ) the distribution of the sample mean for many population distributions, by the Berry – Esseen theorem.
More generally, any collection of events — possibly more than just two of them — are mutually independent if and only if for every finite subset A < sub > 1 </ sub >, ..., A < sub > n </ sub > of the collection we have
Notice that independence requires this rule to hold for every finite subset of the collection ; see for a three-event example in which and yet no two of the three events are pairwise independent.
; Lambda calculus: A computation consists of an initial lambda expression ( or two if you want to separate the function and its input ) plus a finite sequence of lambda terms, each deduced from the preceding term by one application of Beta reduction.
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