Here the maps involving π < sub > 0 </ sub > are not group homomorphisms because the π < sub > 0 </ sub > are not groups, but they are exact in the sense that the image equals the kernel.

Here π < sub > n </ sub > is a homotopy group, GL ( R ) is the direct limit of the general linear groups over R for the size of the matrix tending to infinity, B is the classifying space construction of homotopy theory, and the < sup >+</ sup > is Quillen's plus construction.

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

The area of a disk ( the region inside a circle ), often incorrectly called the area of a circle, is πr < sup > 2 </ sup > when the circle has radius r. Here the symbol π ( Greek letter pi ) denotes, as usual, the constant ratio of the circumference of a circle to its diameter.

Here P represents the value of the stream in state P and Q the cost of operating the stage with conditions Q.

Here highbit ( S ) denotes the most significant bit of S ; the '< tt >*</ tt >' operator denotes unsigned integer multiplication with lost overflow ; '< tt >^</ tt >' is the bitwise exclusive or operation applied to words ; and P is a suitable fixed word.

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Here is how you reach net profit on a P & L ( Profit & Loss ) account:

* Note: Here P ( s ) is any polynomial in s.

Here is the general form, with the true premise P:

( Here primitive is used in the sense that the highest common factor of the set of coefficients of P is 1 ; this is weaker than requiring the coefficients to be pairwise relatively prime.

* Hett, Benjamin C. " Goak Here: A. J. P.

Here P is the loop invariant.

Here, P ( x ) is a ( possibly zero ) polynomial, and the A < sub > ir </ sub >, B < sub > ir </ sub >, and C < sub > ir </ sub > are real constants.

* 1994: We're Rooted Here and They Can't Pull Us Up: Essays in African Canadian Women's History ( with Peggy Bristow, Linda Carty, Afua P. Cooper, Sylvia Hamilton, and Adrienne Shadd ).

( P + dP * Q / dQ ) can also be written as P ( 1 + dPQ / dQP ). Here, ( dP / dQ ) ( Q / P ) is 1 / E < sub > P </ sub >, where E < sub > P </ sub > is price elasticity of demand.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Here, S ( P, Ei ) represents the similarity between the prototype and the ith exemplar, k is the subscript for the dimensions ( k

Here one can prove directly that P ( X ) is irreducible, and not separable.

* Conformal connections: Here G = SO ( p + 1, q + 1 ), and P is the stabilizer of a null ray in R < sup > n + 2 </ sup >.

Here, we have < tt >( λx. xx )( λx. xx ) → ( xx ):= λx. xx = ( x := λx. xx )( x := λx. xx ) = ( λx. xx )( λx. xx )</ tt >.

Here e ( x → a ) is the evaluation which gives x the

Here FG: D → D and GF: C → C, denote the respective compositions of F and G, and I < sub > C </ sub >: C → C and I < sub > D </ sub >: D → D denote the identity functors on C and D, assigning each object and morphism to itself.

Here if f, g: X → Y are group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set

Here a function type is often denoted A → B, following mathematical convention, or B < sup > A </ sub >, based on the fact that there exist exactly B < sup > A </ sub > ( exponentially many ) set-theoretic functions mapping A to B.

Here I: Mon → Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon → Grp the functor sending every monoid to the Grothendieck group of that monoid.

Here, (( α → β ) → α ) is the type of the function f, which can either return a value of type α directly or apply an argument to the continuation of type ( α → β ).

Here, the choice is mostly between Latin letters with diacritics, as used in many Latin-based orthographies of other Slavic languages, and digraph combinations, as used in English: ж → ž / zh, ч → č / ch, ш → š / sh, щ → št / ŝ / sht.

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain.

Here we define the distance in B ( X, M ) in terms of the distance in M as

Here, X is a set, and Σ is a sigma-algebra on X, so that the pair ( X, Σ ) is a measurable space.

Here is the center of the ellipse, and is the angle between the < math > X </ Math >- axis and the major axis of the ellipse.

Here X represents the space of messages transmitted, and Y the space of messages received during a unit time over our channel.

Notable story arcs of this decade are " Revolution " ( 2000 ), " Eve of Destruction ," " E Is For Extinction " ( 2001 ), " Planet X ," " Here Comes Tomorrow ," " Gifted ," ( 2004 ) X-Men: Phoenix-Endsong, " House of M ," " Decimation " ( 2005 ), Deadly Genesis ( 2005 – 2006 ), " Endangered Species " ( 2007 ), " Messiah Complex " ( 2007 – 2008 ), " Divided We Stand " ( 2008 ), " Manifest Destiny " ( 2008 – 2009 ), X-Infernus, " Messiah War ," " Utopia ," " Nation X " and " Necrosha " ( 2009 ).

Here Mavrodes ' worry about X = " to make something its maker cannot lift " will no longer be a problem because " God does X " is not logically consistent.

Here C < sub > b </ sub >( X ) denotes the C *- algebra of all continuous bounded functions on X with sup-norm.

(" Here lies very high and powerful excellent prince Charles X by name king, by the grace of God, of France and of Navarre ")

The American Journal of Human Genetics released an article in 2007 stating " Here we show, by using 86 complete mitochondrial genomes, that all Indigenous American haplogroups, including Haplogroup X ( mtDNA ), were part of a single founding population.

Here A < sub > n </ sub > is the free abelian group or module whose generators are the n-dimensional oriented simplexes of X.

Here one starts with some covariant additive functor F and some module X.

Here the X < sub > i </ sub > are the generalized forces corresponding to the external variables x < sub > i </ sub >.

In 1999, DMX released his best-selling album ... And Then There Was X, which featured the hit single " Party Up ( Up in Here )".

Here Q < sub > Y | X </ sub >( y | x ), sometimes called a test channel, is the conditional probability density function ( PDF ) of the communication channel output ( compressed signal ) Y for a given input ( original signal ) X, and I < sub > Q </ sub >( Y ; X ) is the mutual information between Y and X defined as

Here, each vector Y < sub > j </ sub > of the f ' basis is a linear combination of the vectors X < sub > i </ sub > of the f basis, so that

Here Var ( X ) is the variance of X, defined as: