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We write f: a → b, and we say " f is a morphism from a to b ".
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We and write
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 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.
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 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
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.
Lloyd George was also helped by John Maynard Keynes to write We can Conquer Unemployment, setting out Keynesian economic policies to solve unemployment.
We and f
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 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 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 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 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.
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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