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When and subring
* When the subring is isomorphic to the split-complex number plane, then g acts as a squeeze mapping and preserves the hyperbolic angle.
* When the subring is isomorphic to the ordinary complex number plane, then g acts as a rotation and preserves the Euclidean angle.

When and is
When Heidegger and Sartre speak of a contrast between being and existence, they may be right, I don't know, but their language is too philosophical for me.
When they express themselves it is incandescent hatred that shines forth, the rage of repudiation, the ecstasy of negation.
When I try to work out my reasons for feeling that this passage is of critical significance, I come up with the following ideas, which I shall express very briefly here and revert to in a later essay.
When the proprietor dies, the establishment should become a corporation until it is either acquired by another proprietor or the government decides to drop it.
When the reactionary response is thus bolstered by an intellectual defense, the characteristics of that defense are explicable only in terms of the basic attitudes of unanalyzed reaction.
When I take over Taliesin, the first thing I'll do is fire you ''.
When someone in the audience rose and asked how does it feel to be a celebrity, Carl said, `` A celebrity is a fellow who eats celery with celerity ''.
When decision makers act within this frame they determine whether a claim put forward in the name of religion is to be accepted by the larger community as appropriate to religion.
When we consider the disorganized state of the world community, and the legacy of predispositions adversely directed against all who are identified as Jews, it is obvious that the struggle for the minds and muscles of men needs to be prosecuted with increasing vigor and skill.
When we turn to Aristotle's ideas on the moral measure of literature, it is at once apparent that he is at times equally concerned about the influence of the art.
When their faith in civil liberties is tested against strong pressures of social expediency in specific issues, e.g., suppression of `` dangerous ideas '', many waver and give in.
There is, of course, nothing new about dystopias, for they belong to a literary tradition which, including also the closely related satiric utopias, stretches from at least as far back as the eighteenth century and Swift's Gulliver's Travels to the twentieth century and Zamiatin's We, Capek's War With The Newts, Huxley's Brave New World, E. M. Forster's `` The Machine Stops '', C. S. Lewis's That Hideous Strength, and Orwell's Nineteen Eighty-Four, and which in science fiction is represented before the present deluge as early as Wells's trilogy, The Time Machine, `` A Story Of The Days To Come '', and When The Sleeper Wakes, and as recently as Jack Williamson's `` With Folded Hands '' ( 1947 ), the classic story of men replaced by their own robots.
When I first came across Samuel Johnson's pronouncement, `` the remedy for the ills of life is palliative rather than radical '', it seemed to me to sum up the profoundest of political and social truths.
When all else fails, it is argued that open sessions slow down governmental operations.
When a person has thoughtlessly or deliberately caused us pain or hardship it is not always easy to say, `` Just forget it ''.
When a person meets God's requirements for the experience of forgiveness he is forgiven.
When I hold my son he stiffens his whole body in my arms until he is as straight and stiff as a board.
When different colors are used, she is just as likely to color trees purple, hair green, etc..
When we become firm enough to stand for those ideals which we know to be right, when we become hard enough to refuse to aid nations which do not permit self-determination, when we become strong enough to resist any more drifts towards socialism in our own Nation, when we recognize that our enemy is Communism not war, and when we realize that concessions to Communists do not insure peace or freedom, then, and only then will we no longer be `` soft ''.
When one powerful nation strives to emulate the success of another, it is only natural.
When a man is laid to rest, he is entitled to stay put.
When I talked to Ching about it, he said, Everyone can learn, if he is not a Reactionary or lazy.

When and isomorphic
When molten antimony is slowly cooled, metallic antimony crystallizes in a trigonal cell, isomorphic with that of the gray allotrope of arsenic.
When n = 0, does not have zero elements ; rather, it is isomorphic to, since.
When n = 2, the dicyclic group is isomorphic to the quaternion group Q.
When the dimension of H is infinite, then S < sub > 1 </ sub >( H ) ( the trace class ) is not reflexive, because it contains a subspace isomorphic to ℓ < sup > 1 </ sup >, and S < sub >∞</ sub >( H ) = L ( H ) ( the bounded operators ) is not reflexive, because it contains a subspace isomorphic to ℓ < sup >∞</ sup > ( in both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of H ).
When the base field is an algebraically closed field of characteristic p, the n-torsion is still isomorphic to ( Z / nZ )< sup > 2g </ sup > when n and p are coprime.
When spoken about graph labeling with unique labels, commonly taken from the integer range 1 ,..., n, where n is the number of the vertices of the graph, two labeled graphs are said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic.
When we say that any two models of the above axioms are isomorphic, we mean that for any two models ( R, 0 < sub > R </ sub >, 1 < sub > R </ sub >, +< sub > R </ sub >, *< sub > R </ sub >, ≤< sub > R </ sub >) and ( S, 0 < sub > S </ sub >, 1 < sub > S </ sub >, +< sub > S </ sub >, *< sub > S </ sub >, ≤< sub > S </ sub >), there is a bijection f: R → S preserving both the field operations and the order.
When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is of particular interest: A simple module over k-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. This is in general stronger than being irreducible over the field k, and implies the module is irreducible even over the algebraic closure of k.
When n is even the algebra Cℓ < sub > n </ sub >( C ) is central simple and so by the Artin-Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars ( degree n elements ) as well.
When Γ is isomorphic to the fundamental group of a hyperbolic 3-manifold, then the quotient space H < sup > 3 </ sup >/ Γ becomes a Kleinian model of the manifold.
When k is strictly less than n then the special orthogonal group SO ( n ) also acts transitively on V < sub > k </ sub >( R < sup > n </ sup >) with stabilizer subgroup isomorphic to SO ( n − k ) so that
When Q = P, however, the line can be any line through P. These lines correspond to the space of directions through P, which is isomorphic to P < sup > 1 </ sup >.

When and dual
When the grammatical dual form of Ajax is used in the Iliad, it was once believed that it indicated the lesser Ajax fighting side-by-side with Telamonian Ajax, but now it is generally thought that that usage refers to the Greater Ajax and his brother Teucer.
When applied to vector spaces of functions ( which typically are infinite-dimensional ), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces.
When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.
The name ' Rocabarraigh ' is also used in Scottish Gaelic folklore for a mythical rock which is supposed to appear three times, the last being at the end of the world: " Nuair a thig Rocabarra ris, is dual gun tèid an Saoghal a sgrios " ( When Rocabarra returns, the world will likely come to be destroyed ).
When electrification came, Americans began to speak of trolleycars or later, trolleys, believed to derive from the troller, a four-wheeled device that was dragged along dual overhead wires by a cable that connected the troller to the top of the car and collected electrical power from the overhead wires, sometimes simply strung, sometimes on a catenary.
When powered from a bipolar ( dual rail ) supply,
When referring to individual lanes on dual carriageways, one does not consider traffic travelling the opposite direction.
When Georgi Shonin demonstrated an unacceptable level of g-Force susceptibility in the centrifuge he was replaced by Komarov in May 1962 for planned dual Vostok missions.
When her third child Margriet was born, the Governor General of Canada, Alexander Cambridge, Earl of Athlone, granted Royal Assent to a special law declaring Princess Juliana's rooms at the Ottawa Civic Hospital as extraterritorial so that the infant would have exclusively Dutch, not dual nationality.
When two arteries or their branches join, the area of the myocardium receives dual blood supply.
When a noun or pronoun appears in dual form, it is interpreted as referring to precisely two of the entities ( objects or persons ) identified by the noun or pronoun.
When the highest performance is required, a technique called " dual reflector shaping " may be used.
When X is a locally convex topological vector space, the continuous dual can be equipped with the strong topology, the topology of uniform convergence on bounded subsets of X.
When the canonical embedding J of X into the dual of is bijective, then X is said to be semi-reflexive.
When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i. e.,
When one wants to work over a base ring R ( commutative ), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra.
When AMORC reorganized in April, 1990, the dual function of the Office was merged into one position, that of President of the worldwide AMORC organization.
When Congress re-authorized CDBG in 1978, they instituted a dual formula to strengthen controls on how money was spent and to better serve communities with different types of problems.
When using a single saber, the player can choose between three stances which affect the speed and power of attacks ; if using dual sabers, the player can switch off the second saber and use the " Fast "- style single-saber stance.
When equipped with dual exhaust, the 400 Turbo jet was rated at.
When Turnbull succeeded to the dual role on 2 September 2002, Prime Minister Tony Blair asked him to focus on the management of the civil service, and to make its reorganisation his priority.
When the B1230 was originally the A63, the main road through Gilberdyke was a dual carriageway for about three miles to the other side of Newport.

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