Sentence examples for as a direct sum from inspiring English sources
Their intertwined tale, which toward its end sings praises of the intricate Book of Kells, is knotty with words like "pubococcygeal," sentences like "According to Margenau's theory, human consciousness could be compared to a field of probabilities in a Fock space, defined as a direct sum of Hilbert spaces," and pages of tendentious analogy between quantum physics and biological processes.
If we write (mathrm{Soc}(M =oplus _{i=1}^n C_i) as a direct sum of indecomposable modules, then (1) (nge 2).
For infinite countable couniserial dimension one can show under some condition that the module can be represented as a direct sum of uniform modules.
where μ ( M ) is the Krull Remak Schmidt number (the number of direct summands when M is written as a direct sum of indecomposable modules.
Moreover, von Neumann showed in his reduction-theory paper that all von Neumann algebras that are not factors can be decomposed as a direct sum (or integral) of factors.
Under a mild technical assumption, we prove that every isometry between two such algebras decomposes as a direct sum of a unitary multiple of an isometric algebra isomorphism and a unitary multiple of an isometric algebra anti-isomorphism.
As an application of this result to the reductive groups viewed as symmetric spaces, we are able to realize any Harish-Chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups, the inducing representations being trivial on the unipotent radical.
However the proofs seem to be technically much more complicated since one has to represent the space of harmonic functions on as a quotient of a direct sum of the complete tensor product of spaces of harmonic functions with fewer variables, while we produced here a, somehow, direct proof.
We use a simple and transparent grouping of similar types of sources and a direct sum of their contributions as a score for each drug drug-event association.
On the other hand, as R is right perfect, the projective cover P of M is a direct sum of indecomposable direct summands.
As we will show, the parity superselection rule induces a decomposition into a direct sum of two irreducible state-space components exploiting H F ′ ∩ ρ F ′ = 〈 1, P 〉.
