We would write to one another and make a definite plan.

We have just observed that we can write Af where D is diagonalizable and N is nilpotent, and where D and N not only commute but are polynomials in T.

We now write Af where Af are distinct complex numbers.

We write this Af.

We shall write the extension of the spring at a time t as x ( t ).

We see this as diminishing of pressure on the outer shell ( which is used in the ideal gas law ), so we write ( something ) instead of.

When a general said in a meeting " We should throw in a nuke once in a while to keep the other side guessing ," Dyson became alarmed and obtained permission to write an objective report discussing the pros and cons of using such weapons from a purely military point of view.

We write f: x → y to indicate that f is an element of G ( x, y ).

We write gf for, where f ' G ( x, y ), and g ' G ( y, z );

We write for the entry in row, column in matrix with 1 being the first index.

Siege's goal was maximum velocity: " We would listen to the fastest punk and hardcore bands we could find and say, ' Okay, we're gonna deliberately write something that is faster than them '", drummer Robert Williams recalled.

We can then write a many-body wavefunction,

We write to represent the negation of, which can be thought of as the denial of.

We write it, and it is read " or ".

We write for the substitution of for in, in a capture-avoiding manner.

We often omit p or ‖·‖ and just write V for a space if it is clear from the context what ( semi ) norm we are using.

A subgroup, N, of a group, G, is called a normal subgroup if it is invariant under conjugation ; that is, for each element n in N and each g in G, the element gng < sup >− 1 </ sup > is still in N. We write

We can therefore write the magnetic energy stored in such a cylindrical coil as shown below.

Sondheim said of the project, " two people and what goes into their relationship ... We ’ ll write for a couple of months, then have a workshop.

We can now write simply p = E / c as the relationship between momentum, energy, and speed of light.

We can write the left hand side as:

" We write songs rather than riffs with statements ," Westerberg later stated.

We write the Lagrangian function as

Lloyd George was also helped by John Maynard Keynes to write We can Conquer Unemployment, setting out Keynesian economic policies to solve unemployment.

We can now write,

We can use it as basis to say, " a < b " and " b > a ", two judgments which designate the same state of affairs.

We can replace the division by a ( possibly faster ) right bit shift: nP ( key ) >> b.

" We Go Together " b / w " Rosie Lane " ( Dore 555 )-BB No. 53, CB No. 39-( AA )

* We are allowed to " add inequalities ": If a ≤ b and c ≤ d, then a + c ≤ b + d

* We are allowed to " multiply inequalities with positive elements ": If a ≤ b and 0 ≤ c, then ac ≤ bc.

This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say

We make two observations: ( a ) is a linear combination of vectors in the row space of, which implies that belongs to the row space of, and ( b ) since = 0, is orthogonal to every row vector of and, hence, is orthogonal to every vector in the row space of.

Later, after Sokal's self-exposure of his pseudoscientific hoax article in the journal Lingua Franca, the Social Text editors explained in a published essay that they had requested editorial changes that Sokal refused to make, and had had concerns about the quality of the writing, stating " We requested him ( a ) to excise a good deal of the philosophical speculation and ( b ) to excise most of his footnotes.

We write f: a → b, and we say " f is a morphism from a to b ".

:: b. We know < u > who he is trying to avoid </ u >?

We can prove the cancellation law easily using Euclid's lemma, which generally states that if an integer b divides a product rs ( where r and s are integers ), and b is relatively prime to r, then b must divide s. Indeed, the equation

* Finale at JFK Stadium: a ) Bob Dylan, Keith Richards and Ronnie Wood – " Ballad of Hollis Brown ", " When the Ship Comes In ", Blowin ' In The Wind " ( JFK 03: 39 ), b ) USA for Africa ( led by Lionel Richie ) – " We Are the World " ( JFK 3: 55 )

We will redo the record, open up the doors for it to get on the r & b charts and make the black stations to play the record ...

We claim that, as a field, the quotient is isomorphic to the complex numbers, C. A general complex number is of the form, where a and b are real numbers and Addition and multiplication are given by

We could, alternatively, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring < M >.

We use the stronger statement that every odd ( antipode-preserving ) mapping h: S < sup > n-1 </ sup > → S < sup > n-1 </ sup > has odd degree.

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 list the elements of A effectively, n < sub > 0 </ sub >, n < sub > 1 </ sub >, n < sub > 2 </ sub >, n < sub > 3 </ sub >, ...

We repeat this procedure to find m < sub > 2 </ sub > > m < sub > 1 </ sub >, etc.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

We define the periodic Bernoulli functions P < sub > n </ sub > by

We are interested in the following set of continuous functions called loops with base point x < sub > 0 </ sub >.

The ( unique ) representable functor F: → is the Cayley representation of G. In fact, this functor is isomorphic to and so sends to the set which is by definition the " set " G and the morphism g of ( i. e. the element g of G ) to the permutation F < sub > g </ sub > of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group

We must also prevent the decoder from using the last code in the upper block, 2 < sup > n + 1 </ sup > − 1, because when the decoder fills that slot, it will increase the code width.

#* We extend this result to more and more complex and lengthy sentences, D < sub > n </ sub > ( n = 1, 2 ...), built out from B, so that either any of them is refutable and therefore so is φ, or all of them are not refutable and therefore each holds in some model.

#* We finally use the models in which the D < sub > n </ sub > hold ( in case all are not refutable ) in order to build a model in which φ holds.

where ω is implicitly a function of k. We assume that the wave packet α is almost monochromatic, so that A ( k ) is nonzero only in the vicinity of a central wavenumber k < sub > 0 </ sub >.

If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.