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Conversely, ( M < sub > pq </ sub >) uniquely determines f ( x, y ).
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Conversely and M
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, 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 x ≤ y to mean y − x ∈ 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.
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.
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, 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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