English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Although there doesn't exist the Lebesgue measure in the ball $M$ of $C[0,1]$with $p-$norm, the average values (expectation) $EY$ and variance $DY$ of somefunctionals $Y$ on $M$ can still be defined through the procedure of limitationfrom finite dimension to infinite dimension. In particular, the probabilitydensities of coordinates of points in the ball $M$ exist and are derived outeven though the density of points in $M$ doesn't exist. These densities includehigh order normal distribution, high order exponent distribution. This also canbe considered as the geometrical origins of these probability distributions.Further, the exact values (which is represented in terms of finite dimensionalintegral) of a kind of infinite-dimensional functional integrals are obtained,and specially the variance $DY$ is proven to be zero, and then the nonlinearexchange formulas of average values of functionals are also given. Instead ofmeasure, the variance is used to measure the deviation of functional from itsaverage value. $DY=0$ means that a functional takes its average on a ball withprobability 1 by using the language of probability theory, and this is just theconcentration without measure. In addition, we prove that the average valuedepends on the discretization.

Average values of functionals and concentration without measure