Sentence examples for connected with the relation from inspiring English sources

Theorem 4.3 Suppose that the entries of the infinite matrices A = ( a n k ) and B = ( b n k ) are connected with the relation (4.1) and μ be any given sequence space.

Theorem 4.2 Suppose that the entries of the infinite matrices E = ( e n k ) and F = ( f n k ) are connected with the relation e n k = ∑ j = k ∞ u j v k f n j  or  f n k = 1 u k ( e n k v k − e n, k + 1 v k + 1 ) (4.9).

Since the spaces λ and λ B ( r ˜, s ˜ ) are linearly isomorphic, one can easily observe that x = ( x k ) ∈ λ B ( r ˜, s ˜ ) if and only if y = ( y k ) ∈ λ, where the sequences x = ( x k ) and y = ( y k ) are connected with the relation (3.1).

Suppose that the elements of the four-dimensional matrices (A=(a_{mnkl})) and (H=(h_{mnkl})) are connected with the relation h_{mnkl}=sum_{i,j=k,l}^{m,n}b_{mnij}(r,s,t,u e_{ijkl} quad textit{for all }m,n,k,linmathbb{N}, (4.7) where the four-dimensional matrix (E=(e_{mnkl})) is defined as in (4.3).

Let λ and μ be as in Theorem  4.1, and let the elements of the four-dimensional matrices (A=(a_{mnkl})) and (G=(g_{mnkl})) be connected with the relation g_{mnkl}=sum_{i,j=0}^{m,n}b_{mnij}(r,s,t,u a_{ijkl} quad textit{for all } m,n,k,linmathbb{N}.

A conceptual input output analysis is used to highlight measurement issues connected with the inter-relations between information categories, particularly with regard to scale, and the application of the framework is illustrated by reference to issues affecting marine ecosystems included in a Europewide study for the European Commission.

The semantics of theoretical entities, therefore, is connected with the semantics of theoretical relations and functions, which will be dealt with in the next subsection.

The method followed is based on Rankine–Hugoniot relations connected with the flame propagation conditions.

It may be assumed that the response to any mood stabilizer is connected with the interaction of multiple genes, as demonstrated in relation to lithium response.

Due to its relation to both titles, its lineage is connected with the earliest recognized world heavyweight championship.

Matthias' political relations with the papacy and the Italian states were connected with the interests in Turkish wars.