We can use ƒ and g together to compute as close a rational approximation as we like to the real number they represent.

We can compute the saturation intensity as

We are asked to compute the ratio of female computer science majors to all computer science majors.

We first segment the data by the class, and then compute the mean and variance of in each class.

We may use the theorem to compute the Taylor series of at

< li > We will compute certain multiples ( k times ) of the point P using the standard addition rule on our elliptic curve: given two points P, Q on the curve, their sum can be computed using the formulas given in the group law section of the article on elliptic curves.

We can similarly compute 4! P, and so on, but 8! P requires inverting 599 ( mod 455839 ).

We compute the partials on the right:

We must compute the determinant of

We can now compute the width of a button.

We can also compute the width of a.

We now have enough relations to compute the polynomials of all the links we've encountered, and can use the above equations in reverse order to work up to the cinquefoil knot itself:

We can compute it here using integration by parts.

We then compute the density of the sum of three independent variables, each having the above density.

We apply G to this sequence, chop off the last term, and compute homology to get L < sub > i </ sub > G ( X ).

We may compute &# 8472 ; very rapidly in terms of theta functions ; because these converge so quickly, this is a more expeditious way of computing

* We only know an upper bound on the error ( i. e., ε ≤ V ( f ) D < sub > N </ sub >) and it is difficult to compute and.

We can then stop doing exponentiation, and compute

We then compute various features describing each example ( e. g., does the phrase begin with an upper-case letter ?).

We can compute the likelihood ratio to find the key statistic in this test and its effect on the test's outcome:

We can thus compute the smallest index i such that ( x < sub > i </ sub >) ≥ y as follows:

We then have values for the hyperparameters of the approximating distributions of the posterior parameters, which we can use to compute any properties we want of the posterior — e. g. its mean and variance, a 95 % highest-density region ( the smallest interval that includes 95 % of the total probability ), etc.

Imagine now that the forces and stresses in the-State undergo the displacements and deformations in the-State: We can compute the total virtual ( imaginary ) work done by all forces acting on the faces of all cubes in two different ways:

We can compute the number of labeled spanning trees of this kite with the Matrix-Tree Theorem.

We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.

Using the number 2 would give the symbol for singular " You "; adding the plural indicator ( a small cross at the top ) would produce the pronouns " We " and plural " You ".

Traditionally monarchs, particularly European ones, address each other in formal communications in the singular e. g. ‘ I being desirous ’ but address Presidents and other Heads of State in the majestic plural e. g. ‘ We being desirous ’.

The English translations of the documents of John Paul II dispensed with this practice, using the singular " I ," even though the Latin original usually continued to use the first person plural " We.

Queen Louise wrote to her parents on Marie's marriage that " We see singular things.

We see also that the expression is singular if any of these states have the same energy as state n, which is why we assumed that there is no degeneracy.

We define a simplicial R-tree to be an R-tree whose set of singular points is discrete and closed.

* re-phrasing sentences such that they avoid the need for third-person singular sex-specific pronouns ( e. g. " We decided to eat out ," rather than " She and I decided to eat out.

" We never saw it grow in any other place, nor have I ever since seen it growing wild, in all my travels, from Pennsylvania to Point Coupe, on the banks of the Mississippi, which must be allowed a very singular and unaccountable circumstance ; at this place there are two or of ground where it grows plentifully.

' We have relied on a singular imperative: ' Thou shalt not give Hitler a posthumous victory.

We normally distinguish between 3 persons singular and 3 persons plural, but there are also some languages that have specific words for the dual ( e. g. Slovene ).

We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components.

We may therefore form its right derived functors ; their values are abelian groups and they are denoted by H < sup > n </ sup >( G, M ), " the n-th cohomology group of G with coefficients in M ".

We must show that the cohomology class of this is non-zero.

We write M < sup > G </ sup > for the subgroup of M consisting of all elements of M that are held fixed by G. This is a left-exact functor, and its right derived functors are the group cohomology functors, typically written as H < sup > i </ sup >( G, M ).

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

We need not determine the existence or non-existence of X.

Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom ( X, Y ) of morphisms from X to Y.

Describing his intention, Capa said ' We named this unknown generation, The Generation X, and even in our first enthusiasm we realised that we had something far bigger than our talents and pockets could cope with '.

We can assume without loss of generality that the support of each X < sub > i </ sub > is

We allow X to be empty.

We may define a possibly different topology on X using the continuous ( or topological ) dual space X < sup >*</ sup >.

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 ).

:( 1 ) Speaker X: " We should leave for the show or else we ’ ll be late.

We define Thm ( C ) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod ( X ) be the class of all frames which validate every formula from X.

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

We do this first for, where the desired extension of f: X → is just the projection onto the coordinate in.

* We may give R < sup > N </ sup > the I-adic topology, where I = ( X ) is the ideal generated by X, which consists of all sequences whose first term a < sub > 0 </ sub > is zero.

Strategic goals could be " We want to conquer area X ", or " We want to stop country Y's expansion in world trade in commodity Z "; while tactical decisions range from a general statement, e. g. " We're going to do this by a naval invasion of the North of country X ", " We're going to blockade the ports of country Y ", to a more specific " C Platoon will attack while D platoon provides fire cover ".

We define a chain complex for X by taking A < sub > n </ sub > to be the free abelian group ( or free module ) whose generators are all continuous maps from n-dimensional simplices into X.