The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group.

The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center.

* From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal.

This ringing is a mixture of normal modes with discrete frequencies and periods of an hour or longer.

" The results indicate even exceptional skills previously attributed to aberrant sources are variations of normal cerebral dynamics associated with intuition and may involve small but discrete changes in proximal energy.

This type of recording is often called " multi-unit recording ", and is often used in conscious animals to record changes in the activity in a discrete brain area during normal activity.

A hypothesis test using normal approximation for discrete data.

Colonic propagating pressure wave sequences ( PSs ) are responsible for discrete movements of the bowel contents and are vital for normal defecation.

Audio on vinyl records was problematic because one of the systems was based on discrete sound channels ( allowing for full separation of the four original recorded channels, albeit with restricted high-frequency response and reduced record life ), while two others were matrix encoded into two tracks that would also play back in standard, two-channel, stereo on normal audio equipment ( so-called ' compatible ' quadraphonic ).

a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.

Typically H will be the conjugate prior of F. The two most common choices of F are Gaussian aka " normal " ( for real-valued observations ) and categorical ( for discrete observations ).

Legislation aimed at discrete and insular minorities, who lack the normal protections of the political process, should be an exception to the presumption of constitutionality, and a heightened standard of judicial review should be applied.

Let G be a covering group of H. The kernel K of the covering homomorphism is just the fiber over the identity in H and is a discrete normal subgroup of G. The kernel K is closed in G if and only if G is Hausdorff ( and if and only if H is Hausdorff ).

Going in the other direction, if G is any topological group and K is a discrete normal subgroup of G then the quotient map p: G → G / K is a covering homomorphism.

If G is connected then K, being a discrete normal subgroup, necessarily lies in the center of G and is therefore abelian.

Moreover, given any two connected locally isomorphic groups H < sub > 1 </ sub > and H < sub > 2 </ sub >, there exists a topological group G with discrete normal subgroups K < sub > 1 </ sub > and K < sub > 2 </ sub > such that H < sub > 1 </ sub > is isomorphic to G / K < sub > 1 </ sub > and H < sub > 2 </ sub > is isomorphic to G / K < sub > 2 </ sub >.

The bone that results occurs independently of the normal skeleton, forming its own discrete skeletal elements.

* A discrete barrier is one for which the barrier event is considered at discrete times, rather than the normal continuous barrier case.

" This referred to the famous Footnote Four in United States v. Carolene Products in which the Supreme Court had suggested that heightened judicial scrutiny might be appropriate in three types of cases: those where a law was challenged as a deprivation of a specifically enumerated right ( such as a challenge to a law because it denies " freedom of speech ," a phrase specifically included in the Bill of Rights ); those where a challenged law made it more difficult to achieve change through normal political processes ; and those where a law impinged on the rights of " discrete and insular minorities.

Adaptive clothing typically incorporates flat seams to reduce friction, discrete adaptations to make the clothing look as normal as possible, easy access with snaps, Velcro, or stretchy fabric, roomy design to accommodate incontinence aids, longer rise in the back to accommodate sitting in wheelchairs, elastic waist for increased comfort, and reduced effort when dressing.

* Lattice ( discrete subgroup ), a discrete subgroup of a topological group with finite covolume

The global structure of a Lie group is not determined by its Lie algebra ; for example, if Z is any discrete subgroup of the center of G then G and G / Z have the same Lie algebra ( see the table of Lie groups for examples ).

A geometric structure on a manifold M is a diffeomorphism from M to X / Γ for some model geometry X, where Γ is a discrete subgroup of G acting freely on X.

The motivation behind this extension to the differential operator is that the semigroup of powers D < sup > a </ sup > will form a continuous semigroup with parameter a, inside which the original discrete semigroup of D < sup > n </ sup > for integer n can be recovered as a subgroup.

The adelic ring contains the number or function field as a discrete co-compact subgroup.

A connected metrizable surface is a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

In the case of a cover space, when the fiber is discrete, we have that π < sub > n </ sub >( E ) is isomorphic to π < sub > n </ sub >( B ) for all n greater than 1, that π < sub > n </ sub >( E ) embeds injectively into π < sub > n </ sub >( B ) for all positive n, and that the subgroup of π < sub > 1 </ sub >( B ) that corresponds to the embedding of π < sub > 1 </ sub >( E ) has cosets in bijection with the elements of the fiber.

The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group ( see Fuchsian group and Kleinian group ).

A particularly important discrete subgroup of the Möbius group is the modular group ; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations.

A subgroup G of H acts properly discontinuously on X if and only if G is a discrete subgroup of H.

The Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group.

A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one.

For example, the integers, Z, form a discrete subgroup of the reals, R ( with the standard metric topology ), but the rational numbers, Q, do not.

The identity component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself.

It follows that every finite subgroup of a Hausdorff group is discrete.

A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.

Prior to the advent of the microprocessor, a computer was usually built in a card-cage case or mainframe with components connected by a backplane consisting of a set of slots themselves connected with wires ; in very old designs the wires were discrete connections between card connector pins, but printed circuit boards soon became the standard practice.

It held that neurons are discrete cells ( not connected in a meshwork ), acting as metabolically distinct units.

Monolithic ICs consist of a single chip as opposed to a chip and discrete parts ( a discrete IC ) or multiple chips bonded and connected on a circuit board ( a hybrid IC ).

Christopher Pelling asserts that Athena Parthenos may have constituted a discrete cult of Athena, intimately connected with, but not identical to, that of Athena Polias.

If the bulk is connected to the source ( as is generally the case with discrete devices ) it is sometimes angled to meet up with the source leaving the transistor.

The capacitors are shown connected with dotted lines to indicate that the capacitance is actually distributed along the line rather than the discrete elements shown.

* Blob discovery & manipulation: inspecting an image for discrete blobs of connected pixels ( e. g. a black hole in a grey object ) as image landmarks.

The correct way to define the constant sheaf F < sub > A </ sub > ( associated to for instance a set A ) on a topological space is to require sections on an open set U to be continuous maps from U to A equipped with the discrete topology ; then in particular F < sub > A </ sub >( U ) = A for connected U.

For every x in X, the fiber over x is a discrete subset of C. On every connected component of X, the fibers are homeomorphic.

If X is connected, there is a discrete space F such that for every x in X the fiber over x is homeomorphic to F and, moreover, for every x in X there is a neighborhood U of x such that its full pre-image p < sup >− 1 </ sup >( U ) is homeomorphic to U x F. In particular, the cardinality of the fiber over x is equal to the cardinality of F and it is called the degree of the cover p: C → X.

* pinnatifid and pinnatipartite: leaves with pinnate lobes that are not discrete, remaining sufficiently connected to each other that they are not separate leaflets.

In effect, space as traversed by this type of technology is not the equivalent of a wide-open ocean where ships can choose from countless possible routes, but is similar to a series of discrete narrow seas connected by specific channels which can be easily blockaded by a superior naval force.

The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis ( discrete Fourier transform ).

In terms of the natural topology on the fundamental group, a space is semi-locally simply connected if and only if its topological fundamental group is discrete.

Since BGL ( R )< sup >+</ sup > is path connected and K < sub > 0 </ sub >( R ) discrete, this definition doesn't differ in higher degrees and also holds for n = 0.

** Pinnatifid and pinnatipartite: Leaves with pinnate lobes that are not discrete, remaining sufficiently connected to each other that they are not separate leaflets.

GPUs may be an integrated part of the computer's motherboard, the most common solution in laptops, or come packaged with a discrete graphics card with a supply of dedicated Video RAM, connected to the motherboard through either an AGP or PCI-Express port.