Sentence examples for dual relations from inspiring English sources

The dual relations for left fractional sums and differences were investigated in [12].

For the homogeneous and non-homogeneous Herz spaces, there exist the following dual relations, respectively.

Indeed, the following two lemmas are dual relations between the delta left fractional sums (differences) and the nabla left fractional sums (differences).

Also it is necessary to obtain dual relations between the left and right fractional sums and differences via the Q-operator.

For example, Marsalli and West [2] obtained a series of results including a Riesz factorization theorem, the dual relations between (H^{p}(mathcal{A})) and (H^{q}(mathcal{A})), and Labuschagne [3] proved the universal validity of Szegö's theorem for finite subdiagonal algebras.

Solving from primal dual relation for different values of weights, we get the optimal values of the decision variables which are given in Table 1.

While the existing algorithm is based on the equivalence of G(D) and G P DD), the proposed algorithm uses the dual relation between H(D) and G P DD).

An answer is given in terms of a dual relation which involves positive measures in the disk with bounded Poisson balayage.

The extant commentary on the Categories mentions only a dual relation, that between words and things, whereas other sources attribute to Porphyry a triadic relation between words, concepts and things.

In future, the dual relation between ideal space and kernel space can be further used to switch the computation of kernel space into that of ideal space, thereby realizing the application of commutative algebra to efficiently solve the machine learning problem.

By the dual relation, it is easy to obtain the similar upper estimate of the Kolmogorov's N-widths d N ( T ) of operators from ℓ 2 m into ℓ ∞ m, i.e., d N ( T ) ≤ C ( log ( m N + 1 ) N ) 1 / 2 ∥ T ∥.