We will denote the values of f{t} on different components by Af.

We will use the notation to denote the multiplicative inverse of, it is defined exactly when and are coprime ; the following construction explains why the coprimality condition is needed.

We call the set C the pair of A and B, and denote it

We usually denote this set using set-builder notation as

We write hom ( a, b ) ( or hom < sub > C </ sub >( a, b ) when there may be confusion about to which category hom ( a, b ) refers ) to denote the hom-class of all morphisms from a to b. ( Some authors write Mor ( a, b ) or simply C ( a, b ) instead.

We denote ( and would continue to denote ) noise from as "".

We can see this from the Bayesian update rule: letting U denote the unlikely outcome of the random process and M the proposition that the process has occurred many times before, we have

We use the shorthand notation to denote the joint probability of by.

We denote the subgroup of principal fractional ideals by Prin ( R ).

We are now left with the word tawdry to denote, essentially, the inferior garb worn by the poor and not a general put-down for St Ives itself, which remains a charming and attractive small river town.

We know from Plutarch that Xenocrates, if he did not explain the Platonic construction of the world-soul as Crantor after him did, nevertheless drew heavily on the Timaeus ; and further that he was at the head of those who, regarding the universe as unoriginated and imperishable, looked upon the chronological succession in the Platonic theory as a form in which to denote the relations of conceptual succession.

Suppose X is a normed vector space over R or C. We denote by its continuous dual, i. e. the space of all continuous linear maps from X to the base field.

We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by and, respectively.

We denote the eigenstates of by.

We denote the diagonal components of the density matrices for the canonical distributions for and in this basis as:

We denote the unitary operators we get by U ( a, A ), and these give us a continuous, unitary and true representation in that the collection of U ( a, A ) obey the group law of the inhomogeneous SL ( 2, C ).

We will denote this space by C. We define

We can rephrase the above proof, using partitions, which we denote as:

We denote the spectrum of a commutative ring A by Spec ( A ).

" is reporting the sentence " We shall all sin from time to time " ( assuming the archbishop is including himself in the proposition ), where shall is used to denote simple futurity.

We end up with a sequence of numbers which denote the height of each throw to be made.

Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell ( to be defined precisely ) corresponding to S by DS.

We can then define the differential map d: C < sup >∞</ sup >( M ) → T < sub > x </ sub >< sup >*</ sup > M at a point x as the map which sends f to df < sub > x </ sub >.

We can produce a map of the abundance of these molecules to produce an understanding of the varying composition of the clouds.

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

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over.

Mayor Walter Maddox was quoted saying that " We have neighborhoods that have been basically removed from the map.

We will refer to this as the " genotype-phenotype map ".

We blew the oil storage tanks of them off the map.

We also have the following universal property: if a continuous map to a totally disconnected space, then it uniquely factors into where is continuous.

We see that we can easily read off the index of the linear map T from the involved spaces, without any need to analyze T in detail.

We can map each in A to in B by the rule.

We have now traced on a spherical surface the area in which we say the inhabited world is situated ; and the man who would most closely approximate the truth by constructed figures must necessarily take for the earth a globe like that of Crates, and lay off on it the quadrilateral, and within the quadrilateral put down the map of the inhabited world.

Heuristically, if we have a space M for which each point m ∈ M corresponds to an algebro-geometric object U < sub > m </ sub >, then we can assemble these objects into a topological family U over M. ( For example, the Grassmanian G ( k, V ) carries a rank k bundle whose fiber at any point ∈ G ( k, V ) is simply the linear subspace L ⊂ V .) We say that such a family is universal if any family of algebro-geometric objects T over any base space B is the pullback of U along a unique map B → M. A fine moduli space is a space M which is the base of a universal family.

We can use the existence of the splitting field to define reduced norm and reduced trace for a CSA A. Map A to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively.

We have an invertible linear map from the tangent bundle TM to T. This map is the vierbein.

:" We put a big map of the world on a wall, Douglas stuck a pin in everywhere he fancied going, I stuck a pin in where all the endangered animals were, and we made a journey out of every place that had two pins.

We could choose this subset arbitrarily, but if we're going to want a reconstruction formula R that is also a linear map, then we have to choose an n-dimensional linear subspace of.

We also have a problem, if, the largest prime divisor of the order of the Jacobian, is equal to the characteristic of By a different injective map we could then consider the DLP in the additive group instead of DLP on the Jacobian.

We say f is proper or is proper over if ( i ) f is an adic morphism ( i. e., maps the ideal of definition to the ideal of definition ) and ( ii ) the induced map is proper, where and K is the ideal of definition of.

" We are on the map!

" The phrase " We are on the map!

We will denote the Bohr compactification of G by Bohr ( G ) and the canonical map by