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Conversely and M
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, Austereo's Triple M group of stations have suffered heavily from increased competition from new station Vega 95. 3 and Vega 91. 5 in Sydney and Melbourne respectively, with the stations in Melbourne and Sydney recording some of their lowest ratings of all time.
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 and <
Conversely, some elements do not maintain distinct allotropes in different phases – for example phosphorus has numerous solid allotropes, which all revert to the same P < sub > 4 </ sub > form when melted to the liquid state.
Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.
Conversely, if the measured potential is negative, the electrode is more oxidizing than the SHE ( such as Zn in ZnSO < sub > 4 </ sub > where the standard electrode potential is − 0. 76 V ).
Conversely, if the data are sparse — that is, if only K out of N Fourier coefficients are nonzero — then the complexity can be reduced to O ( K log N log ( N / K )), and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K > 32 in a large-N example ( N = 2 < sup > 22 </ sup >) using a probabilistic approximate algorithm ( which estimates the largest K coefficients to several decimal places ).
Conversely, when one wants to compute an arbitrary number ( N ) of discrete samples of one cycle of a continuous DTFT, it can be done by computing the relatively simple DFT of s < sub > N </ sub >, as defined above.
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, a high pH, low CO < sub > 2 </ sub >, or low 2, 3 BPG favors the relaxed form which can better bind oxygen.
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 we have a b if and only if a < b or a ~ b. This is the reason for using the notation ""; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a
Conversely, if dφ < sub > x </ sub > is an isomorphism then there is an open neighborhood U of x such that φ maps U diffeomorphically onto its image.
Conversely every positive functional φ gives a corresponding inner product < ƒ, g >< sub > φ </ sub > = φ ( g * ƒ ).
Conversely, to switch off a p-channel device requires V < sub > GS </ sub > positive.
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, for the previous range from 2 < sup > 51 </ sub > to 2 < sup > 52 </ sub >, the spacing is 0. 5, etc.

Conversely and 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 xy to mean yx ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Conversely and >
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 and >)
A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and ( x < sub > 1 </ sub >, e < sub > 1 </ sub >) are connected with an edge if and only if vertex x < sub > 1 </ sub > is contained in edge e < sub > 1 </ sub > in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above.
Conversely, assume that ψ (< u >#( β (#( β )))</ u >) holds.
which are linearly independent at each point q in U. Conversely, given such a coframe, there is a unique moving frame X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > which is dual to it, i. e., satisfies the duality relation α < sup > i </ sup >( X < sub > j </ sub >) = δ < sup > i </ sup >< sub > j </ sub >, where δ < sup > i </ sup >< sub > j </ sub > is the Kronecker delta function on U.

Conversely and uniquely
Conversely, any affine linear transformation extends uniquely to a projective linear transformations, so the affine group is a subgroup of the projective group.
Conversely, a function u: E < sub > 1 </ sub > → E < sub > 2 </ sub > defined on a basis of E < sub > 1 </ sub > can be uniquely extended to a linear function.
Conversely, for a given set of natural numbers satisfying the above inequality, there exists a uniquely decodable code over an alphabet of size with those codeword lengths.

Conversely and determines
Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain.
Conversely, mapping the data flow between the components determines the logical topology of the network.
for all vectors w. Conversely, each covector α determines a unique vector v by this equation.
Conversely, a normal subgroup of index p determines a non-trivial map to up to a choice of " which coset maps to which shows that this map is a bijection.
Conversely, the Ricci form determines the Ricci tensor by
Conversely, minimal compliance with a safety statute does not always absolve a defendant if the trier of fact determines that the reasonable person should have taken actions beyond and in excess of what the statute required.
Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series.
Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.
Conversely, a connection on E determines a connection on F ( E ), and these two constructions are mutually inverse.
produces a unique composition of n. Conversely, every composition of n determines an assignment of pluses and commas.
Conversely, a decomposition with these extra properties determines an involution on that is on and on.
Conversely, when local politics determines that it is no longer a crime, they may be decriminalized.
Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of density functional theory.

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