The proliferation of distributed energy resources (DERs), located at the Distribution System Operator (DSO) level, bring new opportunities as well challenges to operations within grid, specifically, when it comes interaction with Transmission (TSO). To enable interoperability, while ensuring higher flexibility and cost-efficiency, DSOs TSO need be efficiently coordinated. Difficulties behind cr...

This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as geometrical of Banach space ensures the have Mazur–Ulam property. We prove that X enjoys this if and only C(K, X) also show same result holds $$L_\infty (\mu ,X)$$ $$L_1(\mu .

A discrete Gibbsian line ensemble $\mathfrak{L} = (L_1,\dots,L_N)$ consists of $N$ independent random walks on the integers conditioned not to cross one another, i.e., $L_1 \geq \cdots L_N$. In this paper we provide sufficient conditions for convergence a sequence suitably scaled ensembles $f^N (f_1^N,\dots,f_N^N)$ as number curves tends infinity. Assuming log-concavity and KMT-type coupling wa...

We generalize the classical Lebesgue's theorem to multi-dimensional functions. prove that Cesàro means of Fourier series function $f\in L_1(\log L)^{d-1}(\mathbb{T}^d)\supset L_p(\mathbb{T}^d) (1<p<\infty)$ converge $f$ at each strong Lebesgue point.

Given a perfect coloring of graph, we prove that the $L_1$ distance between two rows adjacency matrix graph is not less than corresponding parameter coloring. With help an algebraic approach, deduce corollaries this result for $2$-colorings, colorings in distance-$l$ graphs and distance-regular graphs. We also provide examples when obtained property reject several putative matrices infinite