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2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves ) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables ( complex surfaces ), though not every 4-manifold admits a complex structure.
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2-dimensional and topology
For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).
have a 2-dimensional Euclidean topology.
The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere.
2-dimensional and can
We can then " compose " these " bimorphisms " both horizontally and vertically, and we require a 2-dimensional " exchange law " to hold, relating the two composition laws.
For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates, but one can make do with a single coordinate ( the polar coordinate angle ), so the circle is 1-dimensional even though it exists in the 2-dimensional plane.
A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces.
For example, a linear map can be represented by a matrix, a 2-dimensional array, and therefore is a 2nd-order tensor.
Traditional 2-dimensional views and drawings can be created by appropriate rotation of the object and selection of hidden line removal via cutting planes.
If each wave is modelled by a vector, then it can be seen that if a number of vectors with random angles are added together, the length of the resulting vector can be anything from zero to the sum of the individual vector lengths — a 2-dimensional random walk, sometimes known as a drunkard's walk.
These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude ( amplitude ) and phase in polar coordinates.
For instance, the problem of flat-foldability ( whether a crease pattern can be folded into a 2-dimensional model ) has been a topic of considerable mathematical study.
The term " Penrose triangle " can refer to the 2-dimensional depiction or the impossible object itself.
In 1961, Wang conjectured that if a finite set of tiles can tile the plane, then there exists also a periodic tiling, i. e., a tiling that is invariant under translations by vectors in a 2-dimensional lattice, like a wallpaper pattern.
Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons.
N-dimensional knots are generally not decomposable into 2-dimensional knots, though they can be projected to superpositions of lower-dimensional knots.
Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis.
The most direct way of representing a board is as a 1 or 2-dimensional array, where elements in the array represent points on the board, and can take on a value corresponding to a white stone, a black stone, or an empty intersection.
By encoding and decoding sound information on a number of channels, a 2-dimensional (" planar ", or horizontal-only ) or 3-dimensional (" periphonic ", or full-sphere ) sound field can be presented.
However it does not eliminate the possibility that the coefficient field is the field of l-adic numbers for some prime l ≠ p, because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space.
Via the exponential map, it now can be precisely defined as the Gaussian curvature of a surface through p determined by the image under exp < sub > p </ sub > of a 2-dimensional subspace of T < sub > p </ sub > M.
The 2-dimensional Riemann tensor has only one independent component and it can be easily expressed
More precisely, he demonstrated that the Hilbert space of states is always finite dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k. Conformal blocks are locally holomorphic and antiholomorphic factors whose products sum to the correlation functions of a 2-dimensional conformal field theory.
For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex.
For example, adding an extra tape to the Turing machine, giving it a 2-dimensional ( or 3 or any-dimensional ) infinite surface to work with can all be simulated by a Turing machine with the basic 1-dimensional tape.
2-dimensional and be
That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.
Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.
A common method is Platt's Sequential Minimal Optimization ( SMO ) algorithm, which breaks the problem down into 2-dimensional sub-problems that may be solved analytically, eliminating the need for a numerical optimization algorithm.
Let be a finite 2-dimensional pseudo-manifold.
After introducing one or more such compounds into tissue via perfusion, injection or gene expression, the 1 or 2-dimensional distribution of electrical activity may be observed and recorded.
Therefore all tangent vectors in a point p span a linear space, called the tangent space at point p. For example, taking a 2-dimensional space, like the ( curved ) surface of the Earth, its tangent space at a specific point would be the flat approximation of the curved space.
SDS-PAGE might also be coupled with urea-PAGE for a 2-dimensional gel.
Monoculars, sometimes called telescopes when used in this capacity, are used wherever a magnified 2-dimensional image of a distant object is required ( Though some may be used to look at objects closer ).
Pixel art comprises a large part of " sprite art " as a whole ; though technological advances since the mid-nineties allowed pre-rendered raytraced imagery, or essentially any 2-dimensional image style to be used as a sprite.
The term " planar " ( on a single plane, i. e. no height, or 2-dimensional ) is used to refer to horizontal-only Ambisonics ; the term " pantophonic " will also be found with the same meaning.
Let P be the 1-dimensional subspaces and L the 2-dimensional subspaces ( vector space dimension ) of this vector space.
2-dimensional and studied
In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane ( a planar curve ) or the 3-dimensional space ( space curve ).
Although Lichtenberg only studied 2-dimensional ( 2D ) figures, modern high voltage researchers study 2D and 3D figures ( electrical trees ) on, and within, insulating materials.
2-dimensional and complex
* The complex numbers form a 2-dimensional unitary associative algebra over the real numbers.
In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.
Viewing the complex plane as a 2-dimensional space over the reals, the 2D similarity transformations expressed in terms of complex arithmetic are and where a and b are complex numbers, a ≠ 0.
Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers.
In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional vector space over the real numbers.
Each circle illustrates the position of a single atom ; note that the actual atomic interactions used in current simulations are more complex than those of 2-dimensional hard spheres.
Indeed each incidence structure gives a spherical building of rank 2 ( see ); and Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building, not necessarily classical.
A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane.
A more general geometric version, due to Zuk and, states that if a discrete group acts properly discontinuously and cocompactly on a contractible 2-dimensional simplicial complex with the same graph theoretic conditions placed on the link at each vertex, then has property ( T ).
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