Conversely, suppose that M is a simple R-module.

Conversely, if M is positive definite, it has an eigendecomposition P < sup >− 1 </ sup > DP where P is unitary, D diagonal, and all diagonal elements of D are real and positive.

Conversely, any Hermitian positive semidefinite matrix M can be written as M = A < sup >*</ sup > A ; this is the Cholesky decomposition.

Conversely, if ∇ is a connection on E and A is a one-form on M with values in End ( E ), then ∇+ A is a connection on E.

Conversely, one can show that given a manifold M and a principal H-bundle P over M, and a 1-form η with these properties, then P is locally isomorphic as an H-bundle to the principal homogeneous bundle G → G / H.

A section Γ of dπ: TP / G → TM such that Γ: TM → TP / G is a linear morphism of vector bundles over M, can be identified with a principal connection in P. Conversely, a principal connection as defined above gives rise to such a section Γ of TP / G.

Conversely, given a left kG module M, then M is a k vector space, and multiplication with an element g of G yields a k-linear automorphism of M ( since g is invertible in kG ), which describes a group homomorphism G → GL ( M ).

Conversely, he refused to share information in his APL-provided files when Ohio Governor James M. Cox requested it.

Suppose S is an immersed submanifold of M. If the inclusion map i: S → M is closed then S is actually an embedded submanifold of M. Conversely, if S is an embedded submanifold which is also a closed subset then the inclusion map is closed.

Conversely, if the holonomy group of the Riemannian manifold M is contained in Sp ( k ), choose complex structures I < sub > x </ sub >, J < sub > x </ sub > and K < sub > x </ sub > on T < sub > x </ sub > M which make T < sub > x </ sub > M into a quaternionic vector space.

: Conversely, suppose that φ is a smooth real-valued function on M in a neighborhood of p. Since every smooth function has a local coordinate expression, we may express φ as a function in the coordinates.

To summarize, M is a symmetric space G / K with a compact isotropy group K. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way.

Statement 4 then implies that a basis of M / mM lifts to a minimal set of generators of M. Conversely, every minimal set of generators of M is obtained in this way, and any two such sets of generators are related by an invertible matrix with entries in the ring.

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, discretionary behavior is motivated by internal values and is seen as more cooperative and therefore ideal within a group.

Conversely, other British acts such as Wang Chung, Breathe and Murray Head, as well as the Anglo-Australian duo Air Supply, are one-hit wonders in the UK but not in the US, and the American group 3T, best known in their homeland as being the three sons of Tito Jackson of The Jackson 5, along with other bands such as Blind Melon and Deep Blue Something, are one-hit wonders in the US but not in the UK.

Conversely, a group may also devolve to an earlier stage if unable to resolve outstanding issues in a particular stage.

Conversely, it is a consequence of the subsequent conclusions of the theorem that any compact Lie group is isomorphic to a matrix group.

Conversely, any affine linear transformation extends uniquely to a projective linear transformations, so the affine group is a subgroup of the projective group.

Conversely, if n is prime, then there exists a primitive root modulo n, or generator of the group ( Z / nZ )*.

* Conversely, a letter or group of letters can correspond to different phonemes in different contexts ( as th does in English ; originally this stood for a single phoneme, which then split ).

Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.

Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it.

Conversely, if one pottery-type suddenly replaces a great diversity of pottery types in an entire region this might be interpreted as a new group migrating in with this new style.

Conversely, symmetric tensors are defined as invariants: given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially.

Conversely, if G contains a proper non-trivial normal subgroup N, then the composition of the natural surjective group homomorphism G → G / N with the regular representation of G / N produces a representation π of G which has kernel N. Taking χ to be the character of some non-trivial subrepresentation of π, we have a character satisfying the hypothesis in the direct statement above.

Conversely if a Kleinian group has no nontrivial torsion elements then it is the fundamental group of a hyperbolic 3-manifold.

Conversely, the conservative group Americans for Constitutional Action stated Mathias agreed with their positions only 16 % of the time.

Conversely, heavy rain may attenuate the range of even local stations.

Conversely, in places such as the United States, where commercial broadcasters are the norm, independent radio is sometimes used to refer to non-commercial educational radio stations that are primarily supported by listener contributions and are thus independent of commercial advertising concerns.

Conversely, WPBA is also one of the few PBS stations not to broadcast any digital subchannels, possibly because some of the bitrate is occupied by datacasting ( WGTV operates two subchannels on its frequency, serving the area ).

Conversely, it is allowing the U. S. to eliminate TV channels 52 to 69, freeing up 108 MHz ( from approximately 700 to 800 MHz ) for emergency services and to be auctioned off to the highest bidder, while still retaining ( although moving ) all existing TV stations.

Conversely, stations that use " Move Closer to Your World " do not necessarily use the " Action News " name or format either.

Conversely, the vowel marks of the Tigrinya abugida and the Amharic abugida ( ironically, the original source of the term " abugida ") have been so completely assimilated into their consonants that the modifications are no longer systematic and have to be learned as a syllabary rather than as a segmental script.

Conversely the use of true brass seems to have declined in Western Europe during this period in favour of gunmetals and other mixed alloys but by the end of the first Millennium AD brass artefacts are found in Scandinavian graves in Scotland, brass was being used in the manufacture of coins in Northumbria and there is archaeological and historical evidence for the production of brass in Germany and The Low Countries areas rich in calamine ore which would remain important centres of brass making throughout the medieval period, especially Dinant – brass objects are still collectively known as dinanterie in French.

Conversely, an annual grown under extremely favorable conditions may have highly successful seed propagation, giving it the appearance of being biennial or perennial.

Conversely, the existence of Bletchley Park, and of the decrypting achievements there, was never officially shared with the Soviet Union, whose war effort would have greatly benefited from regular decrypting of German messages relating to the Eastern Front.

Conversely, some of the Allied infantry who had just dealt a crushing defeat to the French at the Battle of Waterloo fully expected to have to fight again the next day ( at the Battle of Wavre ).

Conversely if you do have a better product suddenly you have these businesses that grow like topsy.

Conversely, some men have cross-dressed to escape from mandatory military service or as a disguise to assist in political or social protest, as men did in the Rebecca Riots.

Conversely, some organisms have special techniques to prevent crystallization from occurring, such as antifreeze proteins.

Conversely, foxes and coyotes have sometimes been seen feeding together.

Conversely, states such as Virginia and Massachusetts in the United States of America did not have sovereign status, but had significant political and economic independence coupled with weak federal control, reducing the incentive to secede.

Conversely, a column with a low reflux ratio must have a large number of stages, thus requiring a taller column.

If neither A nor B includes the idea of existence, then " some A are B " simply adjoins A to B. Conversely, if A or B do include the idea of existence in the way that " triangle " contains the idea " three angles equal to two right angles ", then " A exists " is automatically true, and we have an ontological proof of A's existence.

Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range.

Conversely, French directors have had prolific and influential careers in other countries, such as Luc Besson, Jacques Tourneur, or Francis Veber in the United States.

Conversely, languages that grammaticalise aspect can have tense as a secondary use of an aspect.

Conversely, Salinger is reported to have considered the story a " high point of his writing " and made tentative steps to have it reprinted ; these efforts came to nothing however.

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 >.

Conversely, the reasons internalist answers the question in the negative (" No, Sasha does not have a reason not to steal from that poor person, though others might .").

Conversely, John Calvin, Albert Barnes Adam Clarke, John Gill, and John Wesley amongst many other have affirmed this as divine truth and found no problem with its historicity.

Conversely, Henig acknowledges that if the US had been a member, its reluctance to engage in war with European states and to enact economic sanctions may have hampered the ability of the League to deal with international incidents.

Conversely, words containing Lu, as in the word Lugh itself, or lo or le have appeared for millennia always meaning light or sun or sun god.

Conversely, Chinese, Japanese, and Korean do not have single words for a thousand to the second, third, fifth power, etc., unlike English and many other European languages.