Thus each orthogonal group falls into two pieces ; and because the projection map splits, O ( n ) is a semidirect product of SO ( n ) by O ( 1 ).

In more abstract terms, the map splits.

Porter ( 1996: 44-45 ) interprets the tail of the terrestrial Yinglong, which " uses its tail to sketch on the land a map of channel-like formations whereby the floodwaters were allowed to drain ", as the tail of the celestial dragon Scorpius, which is " situated precisely where the Milky Way splits into two branches ".

The group later splits and three of them head in a different direction ; Pran continues following the map with one of them.

She splits the map in two, giving one piece to Homer and Bart and keeping one piece for her and Marge.

The user will validate the map, place and route results via timing analysis, simulation, and other verification methodologies.

The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent.

Phonology relates to phonetics via the set of distinctive features, which map the abstract representations of speech units to articulatory gestures, acoustic signals, and / or perceptual representations.

If the tangent space is defined via curves, the map is defined as

The layout of the town as depicted in Speed's map of 1610 would be easily recognisable to present day inhabitants, with the layout of the main axis clearly visible from the castle via the main street, Monnow Street, to the bridge.

1100 ) includes the generations " Caratauc map Cinbelin map Teuhant ", corresponding, via established processes of language change, to " Caratacus, son of Cunobelinus, son of Tasciovanus ", preserving the names of the three historical figures in correct relationship.

If a function is harmonic ( that is, it satisfies Laplace's equation ) over a particular space, and is transformed via a conformal map to another space, the transformation is also harmonic.

The player navigates through the game via two game screens: an overworld map and a level playfield.

* Topographic map of Mullan from USGS via Microsoft Research Maps

The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve

* Topographic map & aerial photo of Timberline Lodge from USGS via Microsoft Research Maps

A high resolution map is also an essential accompaniment so that the route can be followed with accuracy, and escape envisioned via recognised paths in the case of bad weather or injury.

For that, the mission is creating a full-sky map of the CMB, with a 13 arcminute resolution via multi-frequency observation.

# The Cartan – Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R ^ n with n = dim M via the exponential map at any point.

In Scotland, the national dataset of listed buildings and other heritage assets can be searched online via Historic Scotland, or through the map database Pastmap.

It can be searched either by browsing for listed buildings by country, county and parish / locality, or by keyword search or via the online map.

The map, hailed by the WHO as an important step forward in fighting SARS, is shared with scientists worldwide via the GSC website ( see below ).

It is a platform game featuring single and 2-player cooperative modes, allowing players to choose which levels to access via a map, giving them access to various locations in the city.

The story is divided into 27 non-linear missions that can be reached via a city map.

* Topographic map of Marquette Mountain from USGS via Microsoft Research Maps

Note that this means that if the original 3-manifold had in fact a trivial fundamental group, then it is homeomorphic to the 3-sphere ( via the covering map ).

The goal of each player is to accumulate money by moving his or her train token to map destinations which are generated at random via a lookup table.

In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p. This means that for each point x in X, there is associated an ordered pair, ( K, U ), where U is a neighborhood of x and where K is a collection of disjoint open sets in C, each of which gets mapped homeomorphically, via p, to U ( as shown in the image ).

The Saxons were one of the most robust groups in the late tribal culture of the times, and eventually bequeathed their tribe's name to a variety of more and more modern geo-political territories from Old Saxony near the mouth of the Elbe up the river via the Prussian Province of Saxony ( in present-day Saxony-Anhalt ) to Upper Saxony, the Electorate and Kingdom of Saxony from 1806 corresponding with the German Free State of Saxony, which bears the name today though it was not part of the medieval duchy ( see map on the right ).

Furthermore, a map f: A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings.

There is a natural map F: X → X ′′ ( the dual of the dual

The map F: x → F ( x ) is thus a map X → X ′′.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

A Banach *- algebra A is a Banach algebra over the field of complex numbers, together with a map *: A → A called involution which has the following properties:

We can then define the differential map d: C < sup >∞</ sup >( M ) → T < sub > x </ sub >< sup >*</ sup > M at a point x as the map which sends f to df < sub > x </ sub >.

Just as every differentiable map f: M → N between manifolds induces a linear map ( called the pushforward or derivative ) between the tangent spaces

A C *- algebra, A, is a Banach algebra over the field of complex numbers, together with a map *: A → A.

A bounded linear map, π: A → B, between C *- algebras A and B is called a *- homomorphism if

Model example: if U and V are two connected open subsets of R < sup > n </ sup > such that V is simply connected, a differentiable map f: U → V is a diffeomorphism if it is proper and if

The flow now defines a map, the Poincaré map F: S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x < sub > 0 </ sub >.

A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has.

The canonical map ker: X ^ X → Con X, relates the monoid X ^ X of all functions on X and Con X. ker is surjective but not injective.

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

A differential field F is a field F < sub > 0 </ sub > ( rational functions over the rationals Q for example ) together with a derivation map u → ∂ u.