Sentence examples for be the partial order from inspiring English sources

Let ⪯ be the partial order on X induced by the cone P in X.

Theorem 1 Let ⪯ ˜ be an integral stochastic order on P and ⪯ be the partial order on X determined by ⪯ ˜.

Proposition 4 Let ⪯ ˜ be an integral stochastic order on P and ⪯ be the partial order on X determined by ⪯ ˜.

Let ⪯ be the partial order on (mathbb{E}) induced by the cone (mathbb{P}) in (mathbb{E}).

Let ⪯ be the partial order on X defined by (x,y), z,w in X, quad (x,y preceq z,w quad Longleftrightarrowquad xleq z,quad yleq w.

Next, we prove that P ⪯ g Q implies P ⪯ ˜ Q. Proposition 3 Let ⪯ ˜ be an integral stochastic order on P. Let ⪯ be the partial order on X determined by ⪯ ˜.

Let ((A,preceq )) be a partially ordered set with (A=[0,1]) and the usual relation ordering (le) on real numbers be the partial ordering (preceq).

Let (mathcal{A}) be a Banach algebra with a unit e, P be a cone in (mathcal{A}), and ⪯ be the partial ordering generated by P. Suppose that x is invertible and that (x^{-1}succtheta) implies (xsucctheta).

Let ((X,d)) be a complete cone 2-metric space, P be a cone in the Banach algebra (mathcal{A}) and ⪯ be the partial ordering generated by P. Let ({x_{n}}) and ({y_{n}}) be two sequences in X with (lim_{nrightarrowinfty}x_{n}=x) and (lim_{nrightarrow infty}y_{n}=y).

The dual of ≽ is the partial order ≽∗ on (X^) defined as follows: phisucccurlyeq^psiquad mbox{iff}quad langle phi-psi,xrangle geqslant0 quad mbox{for every }xin X_. (2.1) It is well known that ((X^,succcurlyeq^)) is a Banach lattice, which is called the dual of ((X,succcurlyeq)).

Then the operator T λ h defined by (2.10) is a monotonic increasing operator, i.e., if u 1 ( t ) ≤ u 2 ( t ), then T λ h u 1 ≤ T λ h u 2, where '≤' is the partial order defined on K. Lemma 8 Let (H1) and (H2) hold true.