Sentence examples similar to implications hold a from inspiring English sources

In general, the following implications hold: class A ⇒ quasi-class A ⇒ quasi-paranormal; paranormal ⇒ quasi-paranormal ⇒ algebraically quasi-paranormal.

T ⊗ S ∈ B ( H ⊗ K ) denotes the tensor product of T and S; i.e., ( T ⊗ S ) ( x ⊗ y ) = T x ⊗ S y for x ∈ H, y ∈ K. Definition 1.1 T ∈ B ( H ) is called a k-quasi-class A ( n ) operator for positive integers n and k if T ∗ k ( | T 1 + n | 2 1 + n − | T | 2 ) T k ≥ 0. In general, the following implications hold: p -hyponormal ⊆ class  A ⊆ class  A ( n ) ⊆ k -quasi-class  A ( n ).

The analysis is limited to 2D depth-averaged systems, but implications hold for 3D cases.

Note also that the following implications hold.

The same implications hold for the corresponding right notions.

In general, the following implications hold: T polaroid ⇒ T isoloid.

Conversely, it is clear that the following implications hold: Theorem 2.4 ⟹ Theorem 2.3 ⟹ Theorem 2.2.

Remark 2.1 It follows from the proof of the above theorems that the following implications hold: Theorem 2.1 ⟹ Theorem 2.2 ⟹Theorem 2.3 ⟹ Theorem 2.4.

While this rhetoric is the preserve of fringe controversy merchants, it's implications hold sway in mainstream politics.

It is also possible that these implications hold for other biofilm forming bacteria.

Rules 3 and 4 show that the following implications among formulas (a)–(f) are valid: The implications holding among the negations of (a)–(f) follow from these by the law of transposition; e.g., since (a) ⊃ (b) is valid, so is ∼(b) ⊃ ∼(a).