An alternating magnetic field is applied at the atomizer ( graphite furnace ) to split the absorption line into three components, the π component, which remains at the same position as the original absorption line, and two σ components, which are moved to higher and lower wavelengths, respectively ( see Zeeman Effect ).

The inverse limit, A, comes equipped with natural projections π < sub > i </ sub >: A → A < sub > i </ sub > which pick out the ith component of the direct product for each i in I.

The intuitive idea is that ω ( X ) provides a vertical component of X, using the isomorphism of the fibers of π with H to identify vertical vectors with elements of.

Here, one component has an electron promoted from the HOMO ( π bonding ) to the LUMO ( π * antibonding ).

For example, the " Little Dipper " asterism of the constellation Ursa Minor has ten stars with Bayer designation, α UMi to π UMi.

Class B has a maximum theoretical efficiency of π / 4.

* Triplet oxygen, the lowest energy molecular oxygen species, has two unpaired electrons in antibonding π * molecular orbitals.

This is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees.

Because of the connection between the Riemann zeta function and π ( x ), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.

* For Ω > 1 ( the sphere, see diagram ) the ratio falls below π: indeed, a great circle on a sphere has circumference only twice its diameter.

) This is the reason for the N + 1 in the denominator of the sine function: the equivalent DFT has 2 ( N + 1 ) points and has 2π / 2 ( N + 1 ) in its sinusoid frequency, so the DST-I has π /( N + 1 ) in its frequency.

* If π < sub > 1 </ sub >( M ) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H < sup > 2 </ sup >× R or the universal cover of SL ( 2, R ).

If it is compact, then the 2 geometries can be distinguished by whether or not π < sub > 1 </ sub >( M ) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else.

* If π < sub > 1 </ sub >( M ) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.

The resulting molecular orbital has π symmetry.

* The statement that if the function has a limit at infinity, then the limit is 1 ( where π is the prime-counting function ).

Each double bond has two additional localized π electrons, while each triple bond has four additional π electrons ( two pairs ) between the bonded atoms.

* π is nondegenerate, that is the space of vectors π ( x ) ξ is dense as x ranges through A and ξ ranges through H. Note that if A has an identity, nondegeneracy means exactly π is unit-preserving, i. e. π maps the identity of A to the identity operator on H.

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.

Numbers such as π that are not algebraic are said to be transcendental ; almost all real and complex numbers are transcendental.

Originally Gaussian frequency-shift keying ( GFSK ) modulation was the only modulation scheme available ; subsequently, since the introduction of Bluetooth 2. 0 + EDR, π / 4-DQPSK and 8DPSK modulation may also be used between compatible devices.

In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6 / π < sup > 2 </ sup > ( see pi ), which is about 61 %.

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (- π, π ) on the x-axis ; then bend the ends of this interval upwards ( in positive y-direction ) and move them towards each other, until you get a circle with one point ( the topmost one ) missing.

A bijective *- homomorphism π is called a C *- isomorphism, in which case A and B are said to be isomorphic.

They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3.

Charge-transfer complexes can also be formed between iodine and a metal ion, in which electrons in the filled π * orbitals of iodine molecules were donated to the empty, low-lying 5s and 5d orbitals, meanwhile back donation from the metal occurs, which weakens and lengthens the I — I bond.

By contrast, the Maxwell equations describing electromagnetism can be understood to be the Hodge equations of a principal U ( 1 )- bundle or circle bundle π: P → M with fiber U ( 1 ).

* The projection of to the horizontal subspace of the tangent space at point p ∈ P must be isomorphic to the metric g on M at π ( p ).

An alternative representation uses degrees and minutes, where parts of a minute are expressed in decimal notation with a fraction, thus: 23 ° 27. 500 ′ E. Degrees may also be expressed as a decimal fraction: 23. 45833 ° E. For calculations, the angular measure may be converted to radians, so longitude may also be expressed in this manner as a signed fraction of π ( pi ), or an unsigned fraction of 2π.

For a bonding MO with π-symmetry the orbital can be labeled π < sub > u </ sub > because inversion through the center of symmetry for Figure 5 ( c ) would produce a sign change.

However, for an antibonding MO with π-symmetry the orbital can be labeled π < sub > g </ sub > because inversion through the center of symmetry for Figure 5 ( d ) would not produce a sign change.

( The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.

Let π ( x ) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x.

So, the prime number theorem can also be written as π ( x ) ~ Li ( x ).

If the phase difference is 180 degrees ( π radians ), then the two oscillators are said to be in antiphase.

The Cartesian coordinates x and y can be converted to polar coordinates r and θ with r ≥ 0 and θ in the interval (− π, π

In organic chemistry, a nucleophilic addition reaction is an addition reaction where in a chemical compound a π bond is removed by the creation of two new σ bonds by the addition of a nucleophile.

Sigmatropic rearrangements are concisely described by an order term, which is defined as the migration of a σ-bond adjacent to one or more π systems to a new position ( i-1 ) and ( j-1 ) atoms removed from the original location of the σ-bond.

In the case of C *- algebras, any *- homomorphism π between C *- algebras is non-expansive, i. e. bounded with norm ≤ 1.

In this case the probability that the gcd equals d is d < sup >− 2 </ sup >/ ζ ( 2 ), and since ζ ( 2 ) = π < sup > 2 </ sup >/ 6 we have

* In the case of the dimensionless quantity π, being the ratio of a circle's circumference to its diameter, the number would be constant regardless of what unit is used to measure a circle's circumference and diameter ( e. g. centimetres, miles, light-years, etc.

In this case, the solutions are the pairs of the form ( 5, 2kπ ) and (− 5 ,( 2k + 1 ) π ), where k ranges over all integers.

A periodic function is the special case k = 0, and an antiperiodic function is the special case k = π / P.

If X is a connected cell complex with homotopy groups π < sub > n </ sub >( X ) = 0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K ( G, 1 ) for G = π < sub > 1 </ sub >( X ).

In the trivial case, E is just B × F, and the map π is just the projection from the product space to the first factor.

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π ( g ) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact ( Hausdorff ) topological group and the representations are strongly continuous.

is norm dense in H, in which case π is called a cyclic representation.

In this case the phases are the pure solvent at pressure P and the solution at total pressure P + π.

In that case, a and b are π / 2 − φ < sub > 1, 2 </ sub > ( i. e., 90 ° − latitude ), C is the longitude separation Δλ, and c is the desired d / R.

The solution, in the case where the needle length is not greater than the width of the strips, can be used to design a Monte Carlo-style method for approximating the number π.

* For X a compact smooth manifold, π < sup > p </ sup >( X ) is isomorphic to the set of homotopy classes of smooth maps X → S < sup > p </ sup >; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.

We identify canonical coordinates ( such as x in the example above, or a field φ ( x ) in the case of quantum field theory ) and canonical momenta π < sub > x </ sub > ( in the example above it is p, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time ):