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We and write
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 and f
We then define f ( x, y ) to be this z.
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 say that f is a diffeomorphism if it is bijective, smooth, and if its inverse is smooth.
We define the inverse limit of the inverse system (( A < sub > i </ sub >)< sub > i ∈ I </ sub >, ( f < sub > ij </ sub >)< sub > i ≤ j ∈ I </ sub >) as a particular subgroup of the direct product of the A < sub > i </ sub >' s:
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 often wish to describe the behavior of a function f ( x ), as either the argument x or the function value f ( x ) gets " very big " in some sense.
We do this first for, where the desired extension of f: X is just the projection onto the coordinate in.
We can then take y ′< sub > n </ sub >= f ′( x ′< sub > n </ sub >).
We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f ; similarly, their coequaliser is the cokernel of their difference.
We have refrained from using the term ' electromotive force ' or ' e. m. f.
We say that the number x is a periodic point of period m if f < sup > m </ sup >( x ) = x ( where f < sup > m </ sup > denotes the composition of m copies of f ) and having least period m if furthermore f < sup > k </ sup >( x ) ≠ x for all 0 < k < m. We are interested in the possible periods of periodic points of f. Consider the following ordering of the positive integers:

We and
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.
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 have lim < sub > x c </ sub > a < sub > n </ sub > = L if and only if for every neighborhood Y of L, the net is eventually in Y.
We can write xy for the operation " first do y, then do x "; so that ab is the operation RGB RBG BRG, which could be described as " move the first two blocks one position to the right and put the third block into the first position ".
We define a sheaf Γ ( Y / X ) on X by setting Γ ( Y / X )( U ) equal to the sections U Y, that is, Γ ( Y / X )( U ) is the set of all functions s: U Y such that fs = id < sub > U </ sub >.
We say that V < sub > 1 </ sub > and V < sub > 2 </ sub > are isomorphic, and write V < sub > 1 </ sub > ≅ V < sub > 2 </ sub >, if there are regular maps φ: V < sub > 1 </ sub > V < sub > 2 </ sub > and ψ: V < sub > 2 </ sub > V < sub > 1 </ sub > such that the compositions ψ ° φ and φ ° ψ are the identity maps on V < sub > 1 </ sub > and V < sub > 2 </ sub > respectively.
We define the representation Ψ: R B ( L < sup > 2 </ sup >( R )) by Ψ ( r )
We therefore define the sum of maps f, g: < sup > n </ sup > X by the formula ( f + g )( t < sub > 1 </ sub >, t < sub > 2 </ sub >, ... t < sub > n </ sub >) = f ( 2t < sub > 1 </ sub >, t < sub > 2 </ sub >, ... t < sub > n </ sub >) for t < sub > 1 </ sub > in and ( f + g )( t < sub > 1 </ sub >, t < sub > 2 </ sub >, ... t < sub > n </ sub >) = g ( 2t < sub > 1 </ sub > − 1, t < sub > 2 </ sub >, ... t < sub > n </ sub >) for t < sub > 1 </ sub > in.
* We say that x and y can be separated by a function if there exists a continuous function f: X ( the unit interval ) with f ( x )
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 summarize this lifting property as follows: a module P is projective if and only if for every surjective module homomorphism f: N ↠ M and every module homomorphism g: P M, there exists a homomorphism h: P N such that fh = g. ( We don't require the lifting homomorphism h to be unique ; this is not a universal property.
We have a forgetful functor Ord Set which assigns to each preordered set the underlying set, and to each monotonic function the underlying function.
We can also define L ( M ) in terms of Δ *: Q × Σ * P ( Q ) such that:
extends to an injective continuous map H < sub > 1 </ sub > H. We regard H < sub > 1 </ sub > as a subspace of H.
We can form an associated sphere bundle Sph ( E ) B by taking the one-point compactification of each fiber separately.

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