Page "Isomorphism of categories" ¶ 11
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## Some Related Sentences

Conversely and given
Conversely, a current of one ampere is one coulomb of charge going past a given point per second:
Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.
Conversely, there are other stars that never rise above the horizon, as seen from any given point on the Earth's surface ( except exactly on the equator ).
Conversely, given a groupoid G in the algebraic sense, let G < sub > 0 </ sub > be the set of all elements of the form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as the set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists.
Conversely, given central idempotents a < sub > 1 </ sub >,..., a < sub > n </ sub > in R that are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra < sub > 1 </ sub >,…, Ra < sub > n </ sub >.
Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.
Conversely, only the wiki users interested in a given project need look at its associated wiki pages, in contrast to high-traffic mailing lists which may burden subscribers with many messages, regardless of their relevance.
Conversely, given a harmonic function, it is the real part of an analytic function, ( at least locally ).
Each convex set containing X must ( by the assumption that it is convex ) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the sets given by these two definitions must be equal.
Conversely, surveys conducted among living donors postoperatively and in a period of five years following the procedure have shown extreme regret in a majority of the donors, who said that given the chance to repeat the procedure, they would not.
Conversely, given an ordered tree, and conventionally draw the root at the top, then the child nodes in an ordered tree can be drawn left-to-right, yielding an essentially unique planar embedding ( up to embedded homotopy, i. e., moving the edges and nodes without crossing ).
Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are characteristic curves.
Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally.
Conversely, if ( A, m, e, inv ) is a group object in one of those categories, then m necessarily coincides with the given operation on A, e is the inclusion of the given identity element on A, inv is the inversion operation and A with the given operation is an abelian group.
Conversely, if the Turing Machine is expected polynomial-time ( for any given x ), then a considerable fraction of the runs must be polynomial-time bounded, and the coin sequence used in such a run will be a witness.
Conversely, suppose we are given ( U, V, W ) satisfying
Conversely, given any ring, we can form a category by taking objects A < sub > n </ sub > indexed by the set of natural numbers ( including zero ) and letting the hom-set of morphisms from to be the set of-by-matrices over, and where composition is given by matrix multiplication.
Conversely, it is hard to believe, given the length and intensity of the struggle between Máel Sechnaill and Brian, that the High King would surrender his title without a fight.
Conversely, an algorithm to test for solvability in arbitrary integers could be used to test a given equation for solvability in natural numbers by applying that supposed algorithm to the equation obtained from the given equation by replacing each unknown by the sum of the squares of four new unknowns.
Conversely, a disease that is easily transmitted but has a short duration might spread widely during 2002 but is likely to have a low prevalence at any given point in 2003 ( due to its short duration ) but a high incidence during 2002 ( as many people develop the disease ).
Conversely, given a commutative diagram, it defines a poset category:

Conversely and left
Conversely the equilibrium position is said to be " far to the left " if hardly any product is formed from the reactants.
Conversely, if the same is done with the left hand, a left-handed system results.
Conversely women take one step backwards and then slide to the left passing behind of their neighbour, then step forwards and slide right in front of their neighbour.
Conversely in rings with unity, any simple left R module arises this way.
Conversely, if F is left adjoint to G, and G is naturally isomorphic to Gthen F is also left adjoint to G ′.
As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G: C D from every object of D, then G has a left adjoint.
" Conversely, they also explained that fascism was of the right, not of the left.
Conversely, the left-hand side is everything on the left side.
Conversely, a natural, military or ecological disaster might move the PPF to the left, in response to a reduction in an economy's productivity.
Conversely, the pulmonary veins, carrying blood from the lungs to the left atrium of the heart, are the only veins carrying oxygenated blood.
Conversely, a hypertrophied left ventricle has a lower afterload.
Conversely, if the left eye is covered, the pendulum would appear to be revolving clockwise-from-top, appearing closer as it swings toward the left and farther as it swings toward the right.
Conversely, in Japan-where the Matrix is treated simply as a reservoir of pure energy-the shell was left hidden on Earth to slowly recharge its depleted energy.
Conversely the United Empire Loyalists left what was to be the United States and moved North to remain under British rule.
Conversely, his sister wrote in the December 1952 issue of Modern Screen Magazine that he left Norma Jeane because she decided to pursue modeling, and he didn't like the attention it created.

Conversely and module
Conversely, if a highest weight module is integrable, then its highest weight vector v satisfies, where are complex numbers such that
A sublinear modulus of continuity can easily found for any uniformly function which is a bounded perturbations of a Lipschitz function: if is a uniformly continuous function with modulus of continuity, and is a Lipschitz function with uniform distance from, then admits the sublinear module of continuity Conversely, at least for real-valued functions, any bounded, uniformly continuous perturbation of a Lipschitz function is a special uniformly continuous function ; indeed more is true as shown below.
Conversely, a crossed module with surjective boundary defines a central extension.
Conversely, different types of hulks exist including one that simply " swallows " the probe without yielding a module, requiring the player to purchase a new probe ( and possibly sell useful modules to raise the required funds ).
Conversely, a PXI module installed in a CompactPCI chassis would not utilize the additional clocking and triggering features of the PXI module.

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