( In particular, the exponential map can be used to define abstract index groups.

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.

The exponential map from the Lie algebra M < sub > n </ sub >( R ) of the general linear group GL < sub > n </ sub >( R ) to GL < sub > n </ sub >( R ) is defined by the usual power series:

If G is any subgroup of GL < sub > n </ sub >( R ), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group.

We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers ( because R is the Lie algebra of the Lie group of positive real numbers with multiplication ), for complex numbers ( because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication ) and for matrices ( because M < sub > n </ sub >( R ) with the regular commutator is the Lie algebra of the Lie group GL < sub > n </ sub >( R ) of all invertible matrices ).

Because the exponential map is surjective on some neighbourhood N of e, it is common to call elements of the Lie algebra infinitesimal generators of the group G. The subgroup of G generated by N is the identity component of G.

The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected ( though it does map onto the Lie group for connected groups that are either compact or nilpotent ).

For example, the exponential map of SL < sub > 2 </ sub >( R ) is not surjective.

Also, exponential map is not surjective nor injective for infinite-dimensional ( see below ) Lie groups modelled on C < sup >∞</ sup > Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

Thus, G < sub > t </ sub >( V ) = exp ( tV ) is the exponential map of the vector tV.

* Normal coordinates in differential geometry, local coordinates obtained from the exponential map.

# 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.

The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography.

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

For the element A ( t ) from the algebra ( set g with the non-commutative product *), where t is the " time parameter ", the ordered exponential

Two important special cases of this are the exponential map for a manifold with a Riemannian metric, and the exponential map from a Lie algebra to a Lie group.

In the theory of Lie groups the exponential map is a map from the Lie algebra of a Lie group to the group which allows one to recapture the local group structure from the Lie algebra.

The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of non-zero real numbers ( whose Lie algebra is the additive group of all real numbers ).

It is EXPTIME-complete because, roughly speaking, we can use it to determine if a machine solving an EXPTIME problem accepts in an exponential number of steps ; it will not use more.

Radial horns have two surfaces based on an exponential flare rate, and two straight walls that determine the output pattern.

* The exponential map is not, however, a covering map in general – it is not a local diffeomorphism at all points.

However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of T < sub > p </ sub > M on which the exponential map is an embedding ( i. e., the exponential map is a local diffeomorphism ).

Their top-down parsing algorithm also requires polynomial space for potentially exponential ambiguous parse trees by ' compact representation ' and ' local ambiguities grouping '.

It can be shown that this algorithm is guaranteed to converge to a local maximum, and since both posterior distributions are in the exponential family, this local maximum will be a global maximum.

In 1948, the Diocese of Joliet was formally established to meet the demands of the exponential growth of Catholicism in the region, resulting from local post-World War II housing developments and commercial modernization.

The transcension scenario ( vs. expansion scenario ) proposes that once civilizations saturate their local region of space with their intelligence, they need to leave our visible, macroscopic universe in order to continue exponential growth of complexity and intelligence, and thus disappear from this universe, thus explaining the Fermi Paradox.

for all s and t. The operation on the right hand side is the group multiplication in G. The formal similarity of this formula with the one valid for the exponential function justifies the definition

In the case of Lie groups with a bi-invariant metric — a pseudo-Riemannian metric invariant under both left and right translation — the exponential maps of the pseudo-Riemannian structure are the same as the exponential maps of the Lie group.

Taking into consideration the structure of, the exponential operator may be expanded into Taylor series:

In terms of the exponential map of G, in general, only some of the Lie subalgebras of the Lie algebra g of G correspond to closed Lie subgroups H of G. There is no criterion solely based on the structure of g which determines which those are.

A desire for infrequent inspection intervals, combined with the exponential growth of cracks in structure has led to the development of non-destructive testing methods which allow inspectors to look for very tiny cracks which are often invisible to the naked eye.

These are not isomorphic as algebraic groups ; at the Lie algebra level we see the same structure, but this is not enough to make any stronger assertion ( essentially because the exponential map is not an algebraic function ).

The procedure to convert a formula into clausal form can destroy the structure of the formula, and naive translations often causes exponential blowup in the size of the resulting formula.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

According to Technorati, the number of blogs doubles about every 6 months with a total of 35. 3 million blogs. This is an example of the early stages of logistic growth, where growth is approximately exponential, since blogs are a recent innovation.

Note the Semi-log plot | logarithmic vertical scale ; the line corresponds to exponential growth with transistor count doubling every two years.

This exponential improvement has dramatically enhanced the impact of digital electronics in nearly every segment of the world economy.

where the last right-hand term of the previous equation is easily obtained by remembering that the Fourier transform of a Dirac pulse is a complex exponential function, an eigenfunction of every linear system.

The exponential rate of change of frequency is shown as a function of time, in this case from nearly 0 up to 8 kHz repeating every second.

This resettlement was one of Leto's goals for mankind's ultimate survival: the exponential growth in human numbers and colonized planets, combined with the dissemination of Siona's genes which render their bearer invisible to prescience, ensuring humanity's survival by making it impossible for any one force to track down every human in the universe or to control them all through prescience ( which forces the future to happen according to the vision of the prescient ).

Because of the exponential function, the current increases rapidly with temperature when kT is less than W. ( For essentially every material, melting occurs well before kT

An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.

Weaker results relating the degree of a graph to unavoidable sets of cycle lengths are known: there is a set S of lengths, with | S | = O ( n < sup > 0. 99 </ sup >), such that every graph with average degree ten or more contains a cycle with its length in S, and every graph whose average degree is exponential in the iterated logarithm of n necessarily contains a cycle whose length is a power of two.

By the power series representation of the exponential function, we have for every natural number m ( including zero )

A report based on the 2012 BP Statistical Review shows an exponential growth in global solar generation from 2001 to end 2011, with an approximate doubling of generation every two years.

In fact, this map is surjective which means that every invertible matrix can be written as the exponential of some other matrix ( for this, it is essential to consider the field C of complex numbers and not R ).

then its exponential can be obtained by just exponentiating every entry on the main diagonal:

# M is geodesically complete ; that is, for every p in M, the exponential map exp < sub > p </ sub > is defined on the entire tangent space T < sub > p </ sub > M.

Such one-parameter groups are of basic importance in the theory of Lie groups, for which every element of the associated Lie algebra defines such a homomorphism, the exponential map.

While Gordon Moore first observed in 1965 that the transistor densities of integrated circuits were doubling every two years, an extended analysis shows that computers have been experiencing exponential improvements to their cost-performance ( maximum number of calculations per second per $ 1, 000 ) since at least 1900, when the very first electromechanical computers were invented.

Lovász and Plummer conjectured that every cubic bridgeless graph has an exponential number of perfect matchings.

Copying video from one VCR to another VCR displayed compromised playback reliability, and exponential levels of head-switching noise, chroma smearing and noise, compounded with every generation.

The computational complexity of the software testing problem is of exponential order (, ) and every fuzzer takes shortcuts to find something interesting in a timeframe that a human cares about.

The line shows the best-fit exponential to the production for the most recent 10 years, indicating a doubling of production every 2 years.