Page "Alternative algebra" ¶ 15
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Conversely and any
Conversely, let G be any polynomial such that Af.
Conversely, any algebra for which this is true is clearly alternative.
Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.
Conversely the equilibrium position is said to be " far to the left " if hardly any product is formed from the reactants.
* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.
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, corresponding to any partition of X, there exists an equivalence relation ~ on X.
Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain.
Conversely, Derek Bickerton and others argue that the advent of abstract words provided a mental basis for analyzing higher-order relations, and that any communication system that remotely resembles human language utterly relies on cognitive architecture that co-evolved alongside language.
Conversely any planar graph can be formed from a map in this way.
Conversely, a person may feel like he belongs to one state without having any legal relationship to it.
Conversely, the interior of the box has a constant, zero potential energy .< ref > Actually, any constant, finite potential can be specified within the box.
Conversely, extremely thin jewelry can cause the same tearing in what is commonly referred to as the " cheese cutter effect ", either during sudden torsion or over a long period of wearing, especially if the thin jewelry bears any weight.
Conversely, any Hermitian positive semidefinite matrix M can be written as M = A < sup >*</ sup > A ; this is the Cholesky decomposition.
Conversely, if a double sharp were added to any other note not containing a sharp or flat as indicated by the key signature then the note will be raised two semitones or a whole step with respect to the chromatic scale.
Conversely in rings with unity, any simple left R module arises this way.
Conversely, supplementation with L-phenylalanine, L-tyrosine, L-DOPA, or any of the cofactors will increase their respective concentrations.
Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ < sub > i </ sub > yields a non-singular multivariate normal distribution.
Conversely, in the absence of spatial continuity can take any value.
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 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.
It is a fact that the ring is a principal ideal ring ; that is, for any ideal I in, there exists an integer n in I such that every element of I is a multiple of n. Conversely, the set of all multiples of an arbitrary integer n is necessarily an ideal, and is usually denoted by ( n ).

Conversely and algebra
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, many theorems that hold in universal algebra do not generalise all the way to category theory.
Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i. e., so the resulting multiplication satisfies the algebra laws.
Conversely, for any topology the clopen ( i. e. closed and open ) subsets yield a Boolean algebra.
Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it.
Conversely, given any topological space X, the collection of subsets of X that are clopen ( both closed and open ) is a Boolean algebra.
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, the normed division algebras immediately give rise to trialities by taking each V < sub > i </ sub > equal to the division algebra, and using the inner product on the algebra to dualize the multiplication into a trilinear form.
Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra.
Conversely, given any Lie algebra with a direct sum decomposition satisfying these three conditions, the linear map σ, equal to the identity on and minus the identity on, is an involutive automorphism.

Conversely and whose
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, since Imhotep was known as the " Son of Ptah ," his mother was sometimes claimed to be Sekhmet, the patron of Upper Egypt whose consort was Ptah.
Conversely, there have been a number of popes whose reign lasted less than a month.
Conversely, citizens may find themselves ineligible because they do not currently reside within the United States ; exceptions apply for children of military parent ( s ) who were born overseas, were disabled or became blind overseas, or first applied for benefits overseas and for students studying abroad who were eligible for SSI in the month prior to leaving the US, whose absence will be for less than 1 year, and who are studying to enhance their ability to perform substantial gainful activity, sponsored by an educational institution in the US, and would not be available to the individual in the US.
Conversely, nominating the party leader whose party maintains a plurality and who are already familiar with their party manifesto facilitates a smoother nomination process.
Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d ( p, q ) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function ( taken in a suitable manner ).
Conversely, nominating the party leader whose party maintains a plurality and who are already familiar with their party manifesto facilitates a smoother nomination process.
* Conversely, a wave whose phase drifts quickly will have a short coherence time.
Conversely, the valuation on a discrete valuation ring can be extended to a valuation on the quotient field giving a discrete valued field, whose associated discrete valuation ring is just.
Conversely this book, whose discovery is touted in 2 Kings, was therefore believed to have been of an early enough authorship to validate, not only its inclusion, but the book's ultimate placement as the 5th of the " Five Books of Moses ".
Conversely, Clover points out that the villain of slasher films is often a male whose masculinity, and sexuality more generally, are in crisis.
To a graph G there corresponds a signed complete graph Σ on the same vertex set, whose edges are signed negative if in G and positive if not in G. Conversely, G is the subgraph of Σ that consists of all vertices and all negative edges.
Conversely, in the United States Navy, the term " shipmate " can be the derogatory term used by officers and senior enlisted to refer to a sailor whose name they have not bothered to remember or read off of their uniform ( Example: Hey!

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