Sentence examples for integral version from inspiring English sources

An integral version of the Kantorovich inequality states that, for any integrable function (f:[alpha,beta ] to mathbb {R}) with (m leqslant f(x) leqslant M) for all (x in[alpha,beta ]), we have (see e.g. [5]) begin{aligned} int_{alpha }^{beta } f(x)^{2},dx leqslant frac{(m+M ^{2}}{4mM} biggl( int _{alpha }^{beta } f(x),dx biggr)^{2}.

and is an integral version of (1.11).

An integral version utilizes a measure.

Here we state the integral version of the previous theorem.

An integral version and some interrelated results are also obtained.

Now, we are able to obtain an integral version of Gupta-Saxena result.

Moreover, the integral versions of classical discrete inequalities can be obtained.

We also design a new class of regularizers defined by "integral" versions of stable spline/TC kernels.

In this paper, we present integral versions of some recently proved results which refine the Jensen-Steffensen inequality.

These translations display different degrees of inclusiveness (from integral versions of Aristotle's work to translations of its single books) and literalness (from strict word for word procedures to styles more akin to paraphrase).

The discrete-integral version refers to the connection of the interval barycenter with interval endpoints.