Distributional versions of Littlewood's Tauberian theorem

We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Ces\`{a}ro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.Comment: 15 page

Fuzzy Self-Learning Controllers for Elasticity Management in Dynamic Cloud Architectures

Cloud controllers support the operation and quality management of dynamic cloud architectures by automatically scaling the compute resources to meet performance guarantees and minimize resource costs. Existing cloud controllers often resort to scaling strategies that are codified as a set of architecture adaptation rules. However, for a cloud provider, deployed application architectures are black-boxes, making it difficult at design time to define optimal or pre-emptive adaptation rules. Thus, the burden of taking adaptation decisions often is delegated to the cloud application. We propose the dynamic learning of adaptation rules for deployed application architectures in the cloud. We introduce FQL4KE, a self-learning fuzzy controller that learns and modifies fuzzy rules at runtime. The benefit is that we do not have to rely solely on precise design-time knowledge, which may be difficult to acquire. FQL4KE empowers users to configure cloud controllers by simply adjusting weights representing priorities for architecture quality instead of defining complex rules. FQL4KE has been experimentally validated using the cloud application framework ElasticBench in Azure and OpenStack. The experimental results demonstrate that FQL4KE outperforms both a fuzzy controller without learning and the native Azure auto-scalin

The k-metric dimension of a graph

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems

Response of a Fermi gas to time-dependent perturbations: Riemann-Hilbert approach at non-zero temperatures

We provide an exact finite temperature extension to the recently developed Riemann-Hilbert approach for the calculation of response functions in nonadiabatically perturbed (multi-channel) Fermi gases. We give a precise definition of the finite temperature Riemann-Hilbert problem and show that it is equivalent to a zero temperature problem. Using this equivalence, we discuss the solution of the nonequilibrium Fermi-edge singularity problem at finite temperatures.Comment: 10 pages, 2 figures; 2 appendices added, a few modifications in the text, typos corrected; published in Phys. Rev.