Alternatively, the path integral formulation of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields.

Alternatively, the decay of quantum information caused by the coupling of the system to the environment is referred to as decoherence.

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U ( t ): H → H such that

Alternatively, a roll can be considered as a group of notes of identical pitch and duration with different articulations.

Alternatively, if the facticity of the social world and the structures of consciousness prove to be culturally and historically relative, then we are at an impasse in regard to any meaningful scientific understanding of the social world which is not subjective ( as opposed to being objective and grounded in nature, or inter subjective and grounded in the structures of consciousness ), and relative to the cultural and idealization formations of particular concrete individuals living in a particular socio-historical group.

e. Alternatively, T is also injective since, if g * x = g * x implies g = g ' ( by post-multiplying with the inverse of x, which exists because G is a group ).

Alternatively a user of a system may be assigned to an " Administrators " group, membership of which grants them the same privilege as the Administrator user.

Alternatively, a straw-lined shed may house a larger group ( i. e., not batched ) in age groups.

Alternatively, a group of disadvantaged children could be tested to identify the ones with most college potential.

Alternatively, rather that representing an alternative name for all of Dál Riata, it has been suggested Corcu Réti was the name given to the kin group which later divided to form the Cenél nGabráin of Kintyre and the Cenél Comgaill of Cowal, thus excluding the Cenél nÓengusa of Islay and the Cenél Loairn of middle and northern Argyll.

Alternatively, various groups within the larger group may be dancing different, but complementary, parts of the larger dance.

Alternatively, the 7 points of the plane correspond to the 7 non-identity elements of the group ( Z < sub > 2 </ sub >)< sup > 3 </ sup > = Z < sub > 2 </ sub > × Z < sub > 2 </ sub > × Z < sub > 2 </ sub >.

Alternatively, an intertwiner for representations of G over a field K is the same thing as a module homomorphism of K-modules, where K is the group ring of G.

( Alternatively, Euclidean space is a principal homogeneous space or torsor under the group of translations, which is a subgroup of the affine group.

Alternatively, this dinosaur may be a very early thyreophoran, a member of the group including the armored stegosaurians and ankylosaurians.

Alternatively, if he has taken the explosives across town, Tanner aids a group of gangsters left in the lurch in Chinatown.

Alternatively, the action of a group G on a complex vector space V

Alternatively the private owner choosing to do his own purchasing ( usually from cash and carries ) may carry advertising for a local paper, national news group or soft drinks brand externally.

Alternatively, it is possible a written exam and then an oral examination in front of a group of professors ( who are registered in the Professional College of that specific profession ).

* Alternatively contact FESA to find your local group,

Alternatively, the hydroxyl group and the carboxylic acid ( or their derivatives ) may be within the same molecule, as in the case of caprolactone.

Alternatively, in terms of the bipartite Kneser graph, the symmetric group on five points acts separately on the two-element and three-element subsets of the five points, and complementation of subsets forms a group of order two that transforms one type of subset into the other.

Alternatively one can define the Grothendieck group using a similar universal property: An abelian group G together with a mapping is called the Grothendieck group of iff every " additive " map from into an abelian group X (" additive " in the above sense, i. e. for every exact sequence we have ) factors uniquely through φ.

Alternatively, we could control the Hamiltonian of the system, and thus the time evolution operator U ( for a time independent Hamiltonian,, where is called the generator of translations in time ) up to some fixed time interval, which yields a unitary operator.

Alternatively, when purchasing a zero coupon bond issued by a U. S. state or local government entity, the imputed interest is free of U. S. federal taxes, and in most cases, state and local taxes, too.

In fact, the fraction fields of the rings of regular functions on any open set will be the same, so we define, for any U, K < sub > X </ sub >( U ) to be the common fraction field of any ring of regular functions on any open affine subset of X. Alternatively, one can define the function field in this case to be the local ring of the generic point.

Alternatively, the American label Radikal Records issued a limited edition titled " The U. S. Remixes ".

Alternatively, the State Department may argue that U. S. citizenship has been surrendered due to the person making an oath of allegiance to a foreign state voluntarily and with the intent to relinquish U. S. citizenship, such that:

Alternatively, students may spend five months studying at their U. S. sponsoring universities and seven months working and taking classes at the Walt Disney World Resort.

Alternatively, DUNS numbers for companies that do business with the U. S. Federal government can be found by looking up the company name via advanced search at.

Alternatively a therm is used representing 100, 000 or 10 < sup > 5 </ sup > Btu, and a quad as 10 < sup > 15 </ sup > Btu.

( Alternatively, for a preorder "", a relation "<" can be defined as a < b if and only if ( a b and a ≠ b ).

Alternatively, it follows from () that if the polar coordinates of the point p are ( r < sub > 1 </ sub >, θ < sub > 1 </ sub >) and those of q are ( r < sub > 2 </ sub >, θ < sub > 2 </ sub >), then the distance between the points is

Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x < sub > 0 </ sub > in its domain

Alternatively, the mother may prefer to be known as " The Dowager Lady < Bloggs >".

Alternatively, linear independence can be directly defined as follows: a set of vectors is linearly independent if and only if the only representations of the zero vector as linear combinations of its elements are trivial solutions, i. e., whenever a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > are scalars such that

Alternatively a dilute solution of ammonium fluoride ( NH < sub > 4 </ sub > F ) can be used.

Alternatively, the energy source can be another chemical gradient already in place, as in the Ca < sup > 2 +</ sup >/ Na < sup >+</ sup > antiporter.

Alternatively, the right-hand side expression can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1, in other words that all coefficients except the one of x < sup > 0 </ sup > vanish.

Alternatively, some materials with the formula MTiO < sub > 3 </ sub > crystallize in the motif known as Perovskite ( structure ), which is also the name of the mineral form of calcium titanate ( CaTiO < sub > 3 </ sub >).

Alternatively in terms of free current, the line integral of the magnetic H-field ( in ampere per metre, Am < sup >− 1 </ sup >) around closed curve C equals the free current I < sub > f, enc </ sub > through a surface S:

Alternatively, one can work entirely in a local coordinate system ( x < sup > 1 </ sup >, ..., x < sup > n </ sup >).

Alternatively, if f < sub > r </ sub > is reduced by a factor of 40 to compensate, it becomes 312. 5 Hz, which is much too low to give good filtering and lock performance characteristics.

Alternatively, an uncountable cardinal κ is supercompact if for every A such that | A | ≥ κ there exists a normal measure over < sup >< κ </ sup >.

Alternatively, a Jordan curve is the image of a continuous map φ: → R < sup > 2 </ sup > such that φ ( 0 ) = φ ( 1 ) and the restriction of φ to