In this paper we consider multivariate Lagrange MEAN-VALUE interpolation problem, where interpolation parameters are INTEGRALs over spheres. We have concentric spheres. Indeed, we consider the problem in three variables when it is not correct.

Since deterministic topology optimization (TO) does not consider the uncertainties in the structure, including materials, loading and geometric dimensions, it may provide the optimal designs with the lowest state of reliability and safety. To solve this problem, reliability-based topology optimization (RBTO) is used, which is actually a combination of TO METHODs with reliability-based design METHODs (RBDO) based on a mathematical framework and process. In this article, by considering four TO METHODs including moving iso-surface threshold (MIST), SIMP, evolutionary structural optimization-extended finite element (XFEM-ESO) and level-set (LS), and considering a constraint or objective function, an optimal volume fraction (Vf) is obtained for TO by the bisection METHOD. Then, with the aid of MEAN volume fraction, TO is performed and its optimized results are applied by an advanced reliability analysis METHOD, i.e. accelerated dynamical MEAN VALUE (ADMV), taking into account the uncertainties and their standard deviations to extract the most probable probability point (MPP). Having the MPP and the constraint, the bisection algorithm is used again and the optimized volume fraction for the RBTO model, and thereby the optimal layout for the RBTO solution, is achieved. Several examples are presented to validate and highlight the optimization capability of the RBTO METHOD using a structural model and the mentioned TO METHODs, and the results are compared together. Based on the results, it is shown that the combination of RBDO and TO approach is able to result in powerful, stable, safe, and reliable structures completely different from the TO results.

In the present research, we will deal with MEAN square INTEGRAL inequalities for preinvex stochastic process and η-convex stochastic process in the setting of beta function. Further, we will present some novel results for improved H¨older INTEGRAL inequality. The results given in this present paper are generalizations of already existing results in the literature.

The use of variance as a risk measure is limited by its non-coherentnature. On the other hand, standard deviation has been demonstrated as acoherent and effective measure of market volatility. This paper suggests theuse of standard deviation in portfolio optimization problems with cardinalityconstraints and short selling, specifically in the MEAN-conditional VALUE-at riskframework. It is shown that, subject to certain conditions, this approach leadsto lower standard deviation. Empirical results obtained from experiments onthe SP index data set from 2016-2021 using various numbers of stocks andconfidence levels indicate that the proposed model outperforms existing modelsin terms of Sharpe ratios.