Sun, S. Todorovic, S. Goodison, Local Learning Based Feature Selection for High-dimensional Data Analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.

* High-dimensional space

High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret.

High-dimensional topology means manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above, while low-dimensional topology, concerning questions of dimensions up to four, or embeddings in codimension up to 2.

This process can occur in very small spaces, such as within the cell wall of a plant cell.

Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

The empty spaces that occur from the reduction of oxygen are filled in by carbon molecules in the atmosphere of the container, which makes the piece blacker in spots where more oxygen was retracted .< ref > Arbuckle.

Use of water in fire fighting should also take into account the hazards of a steam explosion, which may occur when water is used on very hot fires in confined spaces, and of a hydrogen explosion, when substances which react with water, such as certain metals or hot carbon such as coal, charcoal, coke graphite, decompose the water, producing water gas.

The most familiar examples of this construction occur when considering vector spaces ( modules over a field ) and abelian groups ( modules over the ring Z of integers ).

Cybersex can occur either within the context of existing or intimate relationships, e. g. among lovers who are geographically separated, or among individuals who have no prior knowledge of one another and meet in virtual spaces or cyberspaces and may even remain anonymous to one another.

The spread of a malignancy into body cavities can occur via seeding the surface of the peritoneal, pleural, pericardial, or subarachnoid spaces.

Cleared spaces also occur naturally in tall forests when an old tree falls, or dies and loses its foliage.

In tectonic strike-slip settings, accommodation spaces occur as transpressional, transtensional or transrotational basins according to the motion of the plates along the fault zone and the local topography pull-apart basins.

* Similarly, not all inspired air goes to areas of the lung where gas exchange can occur ( the anatomic and the physiological dead spaces ), and so is wasted.

Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple Lie groups.

Also, network address translation may occur on the edge of different IP spaces, such as a local area network and the Internet.

The specialization order is often considered in applications in computer science, where T < sub > 0 </ sub > spaces occur in denotational semantics.

Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces ( i. e. the theory of unitary representations ).

Often 15 or so air changes per hour ( ACHs ) occur in spaces served by evaporative coolers, a relatively high rate of air exchange.

The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces ( often with hundreds or thousands of dimensions ) that do not occur in low-dimensional settings such as the physical space commonly modeled with just three dimensions.

The most common of these infections occur in the web ( intertriginous ) spaces of the toes and at the base of the fingernail or toenail.

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In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated contraction ( or trace ) map.

The joint spaces then become smaller, and new bone growth may occur in the area.

This fluid is forced into the interstitial spaces of the granite and pools at the upper margins, where boiling and alteration occur.

If the entire reserve is exhausted however, it is not replenished ; spaces that occur after this point have to be filled with cards from the waste pile or, if a wastepile has not been made yet, the stock.

In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments.

Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

In mathematics, any vector space, V, has a corresponding dual vector space ( or just dual space for short ) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors.

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects.

In mathematics, a linear map, linear mapping, linear transformation, or linear operator ( in some contexts also called linear function ) is a function between two modules ( including vector spaces ) that preserves the operations of module ( or vector ) addition and scalar multiplication.

Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces.

* Module ( mathematics ) over a ring, a generalization of vector spaces

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

In mathematics, " spaces " are examined with different numbers of dimensions and with different underlying structures.

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.

The branch of mathematics that studies topological spaces in their own right is called topology.

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects.

Internationally, the university is known for research relating to the genome of the Populus tree ( Life sciences ), contributions to the Gleason problem and function spaces on fractals ( mathematics ) and its school of industrial design which gives degree programs in English open to students from all of the world.