Let $n\in \mathbb{N}, n\geq 2.$ An element $x=(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x)|=\|T\|,$ where ${\mathcal denotes the space all continuous $n$-linear forms on $E.$For we define set} $T$
 \centerline{$\qopname\relax o{Norm}(T)=\Big\{(x_1, E^n: (x_1, x_n)~\mbox{is point of}~T\Big\}.$}
 By $i=(i...

While deep neural networks (DNNs) have proven to be efficient for numerous tasks, they come at a high memory and computation cost, thus making them impractical on resource-limited devices. However, these are known contain large number of parameters. Recent research has shown that their structure can more compact without compromising performance.In this paper, we present sparsity-inducing regula...

It is shown that the Orlicz–Lorentz spaces $\ell ^n_{M,a}$, $n\in \mathbb {N}$, with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if norm $\| \cdot \|_{M,a}$ satisfies certain Hardy-type inequalities. This includes embedding some Lorentz $\mathrm {d}^n(a,p)$. The approach based on combinatorial averaging techniques, a new result independent intere...

‎In the present paper we investigate the $L_1$-weak ergodicity of‎ ‎nonhomogeneous continuous-time Markov processes with general state‎ ‎spaces‎. ‎We provide a necessary and sufficient condition for such‎ ‎processes to satisfy the $L_1$-weak ergodicity‎. ‎Moreover‎, ‎we apply‎ ‎the obtained results to establish $L_1$-weak ergodicity of quadratic‎ ‎stochastic processes‎.

Quantum coherence has wide-ranging applications from quantum thermodynamics to metrology, channel discrimination and even biology. Thus, detecting quantifying are two fundamental problems in resource theory. Here, we introduce feasible methods detect estimate the by constructing witnesses for any finite-dimensional states. Our coherent states testing whether expectation value of witness is nega...

The detection and quantification of quantum coherence play significant roles in information processing. We present an efficient way tomographic witnessing for both theoretical experimental coherence. prove that a witness is optimal if only all its diagonal elements are zero. Naturally, we obtain bona fide homographic measure given by the sum absolute values real imaginary parts non-diagonal ent...

‎in the present paper we investigate the $l_1$-weak ergodicity of‎ ‎nonhomogeneous continuous-time markov processes with general state‎ ‎spaces‎. ‎we provide a necessary and sufficient condition for such‎ ‎processes to satisfy the $l_1$-weak ergodicity‎. ‎moreover‎, ‎we apply‎ ‎the obtained results to establish $l_1$-weak ergodicity of quadratic‎ ‎stochastic processes‎.

In this paper, we simultaneously determine the optimal sensor precision and observer gain, which achieves specified accuracy in state estimates. Along with unknown formulation parameterizes scaling of exogenous inputs that correspond to noise. Reciprocal is defined as precision, sparseness achieved by minimizing $l_1$ norm vector. The optimization performed constraints guaranteeing estimates, a...