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 denote this map by Ad < sub > g </ sub >:

We will denote this space by C. We define

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

" 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 also show that the possible values of the observable A in any state must belong to the spectrum of A.

In an interview, Davey Havok described the band's influences: " We have many, many influences that span the musical spectrum.

We might note that if T is multiplication by f, then the spectrum of T is just the essential range of f.

We now say that x ∈ V is an eigenvector of M if x is an eigenvector of T. Similarly, λ ∈ K is an eigenvalue of M if it is an eigenvalue of T and the spectrum of M, written σ < sub > M </ sub >, is the set of all such eigenvalues.

Interviewed following that transaction by the Financial Times, Mr Kyriakou said: “ We will be looking at a variety of assets across the broad spectrum of the media world and across Europe .”

We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum ( corresponding to spacetime translations ) lies on and in the positive light cone.

: We want to become an egalitarian, populist party slightly to the right of center in the political spectrum

We also note that a perfect point source radiator will not only radiate a uniform spectrum of propagating plane waves, but a uniform spectrum of exponentially decaying ( evanescent ) waves as well, and it is these which are responsible for resolution finer than one wavelength ( see Fourier optics ).

We can define the ( stable ) homotopy groups of a spectrum to be those given by

We define the generalized homology theory of a spectrum E by

We seek to create a positive and inclusive campus that is safe and welcoming for people from across the spectrum of sexuality and gender.

At one end of the spectrum George P. Elliot has drawn a parallel between Pound and Adolph Eichmann based on their anti-Semitism, while at the other Marjorie Perloff places Pound's anti-Semitism in a wider context by examining the political views of many of his contemporaries, arguing that " We have to try to understand why " anti-Semitism was widespread in the early twentieth century, " and not say let's get rid of Ezra Pound, who also happens to be one of the greatest poets of the 20th C ."

We need first to convert apparent magnitudes into fluxes of photons ; this clearly depends on the spectrum of the source, but we will ignore that initially.

CMJ reviewer Tracey John explained that the album " continues to unabashedly straddle both ends of the hardcore / emo spectrum ", however, the band " doesn't show much of a progression from 2003's The Fiction We Live.

We say that a function is stable with respect to D if its spectrum is in D. Here, the notion of spectrum is defined on a case by case basis, as explained below.

Renck said that " It's not about the music, but the bodily expression, [...] We wanted to show the whole spectrum, be it krumping, breakdancing, jazz or disco.

We then define the extreme response spectrum ( ERS ), similar to the shock response spectrum ;

We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only ; these also have the benefit of working for matrices with entries in any commutative ring.

We can rewrite these conditions using following commutative diagrams: