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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $\mathbf d=(d_j)_{j\in\mathbb I_m}\in\mathbb N^m$ be a finite sequence(of dimensions) and $\alpha=(\alpha_i)_{i\in\mathbb I_n}$ be a sequence ofpositive numbers (of weights), where $\mathbb I_k=\{1,\ldots,k\}$ for$k\in\mathbb N$. We introduce the $(\alpha\, , \,\mathbf d)$-designs i.e.,$m$-tuples $\Phi=(\mathcal F_j)_{j\in\mathbb I_m}$ such that $\mathcalF_j=\{f_{ij}\}_{i\in\mathbb I_n}$ is a finite sequence in $\mathbb C^{d_j}$,$j\in\mathbb I_m$, and such that the sequence of non-negative numbers$(\|f_{ij}\|^2)_{j\in\mathbb I_m}$ forms a partition of $\alpha_i$,$i\in\mathbb I_n$. We characterize the existence of $(\alpha\, , \, \mathbfd)$-designs with prescribed properties in terms of majorization relations. Weshow, by means of a finite-step algorithm, that there exist $(\alpha\, , \,\mathbf d)$-designs $\Phi^{\rm op}=(\mathcal F_j^{\rm op})_{j\in\mathbb I_m}$that are universally optimal; that is, for every convex function$\varphi:[0,\infty)\rightarrow [0,\infty)$ then $\Phi^{\rm op}$ minimizes thejoint convex potential induced by $\varphi$ among $(\alpha\, , \, \mathbfd)$-designs, namely $$ \sum_{j\in\mathbb I_m}\text{P}_\varphi(\mathcal F_j^{\rmop})\leq \sum_{j\in \mathbb I_m}\text{P}_\varphi(\mathcal F_j) $$ for every$(\alpha\, , \, \mathbf d)$-design $\Phi=(\mathcal F_j)_{j\in\mathbb I_m}$,where $\text{P}_\varphi(\mathcal F)=tr(\varphi(S_{\mathcal F}))$; inparticular, $\Phi^{\rm op}$ minimizes both the joint frame potential and thejoint mean square error among $(\alpha\, , \, \mathbf d)$-designs. We show thatin this case $\mathcal F_j^{\rm op}$ is a frame for $\mathbb C^{d_j}$, for$j\in\mathbb I_m$. This corresponds to the existence of optimalencoding-decoding schemes for multitasking devices with energy restrictions.

Optimal frame designs for multitasking devices with weight restrictions