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 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:

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 say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if

We assume again that the averages of H and in the canonical ensemble defined by are the same:

We then invert both sets of formulae to obtain the old canonical coordinates as functions of the new canonical coordinates.

We then invert both sets of formulae to obtain the old canonical coordinates as functions of the new canonical coordinates.

We then invert both sets of formulae to obtain the old canonical coordinates as functions of the new canonical coordinates.

We then invert both sets of formulae to obtain the old canonical coordinates as functions of the new canonical coordinates.

We find it laid down in the pontificate of Archbishop Ecgbert of York, A. D. 732-766, and referred to as a canonical rule in a capitulary of Charlemagne, and it was finally established as a law of the church in the pontificate of Pope Gregory VII, ca 1085.

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

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 ):

We might also quote the words of Bishop Thomas De Burgo, who, in his " Hibernia Dominicana ", does not hesitate to say that St. Patrick was a canon regular, and that, having preached the Christian faith in Ireland, he established there many monasteries of the canonical institute.

We repeat, that the test of a violation of 7 is whether, at the time of suit, there is a reasonable probability that the acquisition is likely to result in the condemned restraints.

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

We want to study the probability function of this random variable.

We shall find a formula for the probability of exactly X successes for given values of P and N.

We can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff bound.

We can see from the above that, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2, 097, 152.

We would expect to find the total probability by multiplying the probabilities of each of the actions, for any chosen positions of E and F. We then, using rule a ) above, have to add up all these probabilities for all the alternatives for E and F. ( This is not elementary in practice, and involves integration.

We then have a better estimation for the total probability by adding the probabilities of these two possibilities to our original simple estimate.

We are allowed to keep the Schrödinger expression for the current, but must replace by probability density by the symmetrically formed expression

We obtain the same variation in probability amplitudes by allowing the time at which the photon left the source to be indeterminate, and the time of the path now tells us when the photon would have left the source, and thus what the angle of its " arrow " would be.

We need to stop when the state vector passes close to ; after this, subsequent iterations rotate the state vector away from, reducing the probability of obtaining the correct answer.

We consider a special case of this theorem for a binary symmetric channel with an error probability p.

We can then infer that the probability that it has between 600 and 1400 words ( i. e. within k = 2 SDs of the mean ) must be more than 75 %, because there is less than chance to be outside that range, by Chebyshev ’ s inequality.

We use Lagrange multipliers to find the point of maximum entropy,, across all discrete probability distributions on.

We can construct a classical continuous random field that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory ( measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible — in quantum mechanical terms they always commute ).

We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of the " hidden variable " λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ ( λ ), the integral of which over the complete hidden variable space is 1.

There is, nevertheless, a small chance that we are unlucky and hit an a which is a strong liar for n. We may reduce the probability of such error by repeating the test for several independently chosen a.

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

We separate the case in which the measure space is a probability space from the more general case because the probability case is more accessible for the general reader.