For a given weighted mean $\mathscr{M}$ defined on subinterval of $\mathbb{R}_+$ and sequence weights $\lambda=(\lambda_n)_{n=1}^\infty$ we define Hardy constant $\mathscr H(\lambda)$ as the smallest extended real number such that $$ \sum_{n=1}^\infty \lambda_n \mathscr{M}\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le \mathscr H(\lambda) \cdot x_n \text{ for all }x \in \ell^1(\lambda...

In this work, we initially develop a parallelized multiscale, multidomain modeling scheme that directly reduces computation time requirements without compromising the accuracy of established chemical models for a batch protein crystallization process. Then, a double exponentially weighted moving average (dEWMA)-based model predictive controller (MPC) is applied to a multiscale model of a batch ...

In 1882 – 1883, Čebyšev [1] and [2] proved that if a and b are monotonic in the same (opposite) sense, then (1.2) Tn (p;a,b) ≥ (≤) 0. In the special case p = a ≥ 0, it appears that the inequality (1.2) has been obtained by Laplace long before Čebyšev (see for example [5, p. 240]). The inequality (1.2) was mentioned by Hardy, Littlewood and Pólya in their book [3] in 1934 in the more general set...

Weighted averages of Kiefer–Wolfowitz-type procedures, which are driven by larger step lengths than usual, can achieve the optimal rate of convergence. A priori knowledge of a lower bound on the smallest eigenvalue of the Hessian matrix is avoided. The asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a Newton-type adaptive algorithm. Several di...

Nowadays, the Fuzzy C-Means method has become one of the most popular clustering methods based on minimization of a criterion function. However, the performance of this clustering algorithm may be significantly degraded in the presence of noise. This paper presents a robust clustering algorithm called Bilateral Weighted Fuzzy CMeans (BWFCM). We used a new objective function that uses some kinds...