In this contribution, we introduce gradual numbers and intervals of gradual numbers to be able to prove the strong law of large numbers for random variables that are valued by such elements. The results show a novel point of view in the investigation of strong law of large numbers for random variables that are valued by non-exactly specified numbers.

In this paper, we reformulate the multivariate fuzzy transform (Fm-transform) of higher degree proposed originally for an approximation of continuous functions to the discrete case. We introduce the Fm-transform components as multivariate polynomials using the matrix algebra, the inverse Fm-transform is established to obtain a continuous function. We provide an analysis of basic properties of Fm-transform.

In this paper, we reformulate the fuzzy transform of higher degree (Fm-transform) proposed originally for an approximation of continuous functions to the discrete case. We introduce two types of Fm-transform which components are defined using polynomials in the first case and using specific values of these polynomials in the second case. We provide an analysis of basic properties of Fm-transform.

We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single‐processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below 2 and show several lower bounds on the competitive ratios. First, we give a barely random $5/3$‐competitive algorithm that uses only one random bit. We also show a lower bound of $3/2$ on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to $8/5$. Second, we give a deterministic $3/2$‐competitive algorithm in the model that allows restarts, and we show that in this model the ratio $3/2$ is optimal. For randomized algorithms with restarts we show a lower bound of $6/5$.

We consider the following buffer management problem arising in QoS networks: Packets with specified weights and deadlines arrive at a network switch and need to be forwarded so that the total weight of forwarded packets is maximized. Packets not forwarded before their deadlines are lost. The main result of the article is an online 64/33 ≈ 1.939-competitive algorithm, the first deterministic algorithm for this problem with competitive ratio below 2. For the 2-uniform case we give an algorithm with ratio ≈ 1.377 and a matching lower bound.