Narrow Results

This paper aims to provide the topological approaches in the characterization of the algebraic structure of fuzzy multiset finite automaton (FMFA). We have shown how the topological concepts of open sets and continuous functions can be utilized to characterize the algebraic structure of FMFA. The topological definition of a continuous function on fuzzy multiset finite automata and its structure preserving qualities are investigated. The operation preserving function between two fuzzy multiset finite automata is introduced, such functions are analogous to a homomorphism between algebraic structures but have an even stronger structure preserving nature than the continuous functions.

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are $1$. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is $1$ also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be $1$ when $f(x)$ is self-similar.

In this paper, we mainly construct two continuous functions defined on $I=[0,1]$ which have only one unbounded variation point, and show that both Box and Hausdorff dimensions of these functions are 1. We prove that their Riemann–Liouville fractional integrals are bounded variation function. We also show that these functions repair inaccuracies of the example in [N. Liu and K. Yao, Fractal dimension of a special continuous function, Fractals26(4) (2018) 1850048; Y. S. Liang, Definition and classification of one-dimensional continuous functions with unbounded variation, Fractals25(5) (2017) 1750048]. In the proofs, we mainly use Leibniz’s alternating series test and the properties of logarithmic function.

The focus of this paper is to study unbounded variation functions from the perspective of Hölder conditions. Three special unbounded variation functions have been constructed. The first is a continuous function of unbounded variation that satisfies the Hölder condition of a given order and the second is a continuous function of unbounded variation that does not satisfy the Hölder condition of any order. The third is a continuous function of unbounded variation defined on any sub-interval of the interval $I$. Then, specific fractal dimension analysis of the above functions and relevant conclusions have been investigated. Finally, combining functional analysis and reinforcement learning, the convergence of reinforcement learning algorithms can be proved in unified framework.

In this paper, for a typical function $\mathrm{\Phi }\in \mathcal{𝒞}(Y)$ defined in an uncountable compact metric space $Y$, we give the lower Hewitt–Stromberg dimension of graphs $G_{\mathrm{\Phi }}(Y)=\{(y,\mathrm{\Phi }(y))|y\in Y\}$ of the function $\mathrm{\Phi }$. Moreover, we investigate the decomposition of functions within $\mathcal{𝒞}([0,1])$ based on the lower box dimension and the lower Hewitt–Stromberg dimension, revealing significant disparities compared to the context of the packing dimension. Second, we present some results on the lower Hewitt–Stromberg dimension of graphs of sums and products of continuous functions. The main proof is that for a given real number $1\le \beta \le 2$, some real-valued continuous functions in $\mathcal{𝒞}([0,1])$ can be decomposed into the sum and product of two continuous real-valued functions, and the lower Hewitt–Stromberg dimension of the graph for each function is $\beta$.

The Guide to the Expression of Uncertainty in Measurement (GUM) deals with physical quantities whose values are constant in time. However, in many applications the value of the measurand shows a significant time dependence, and the uncertainty evaluation requires that a continuous function is considered as the measurand, which is not covered by the GUM. We show that a corresponding extension of the methods of the GUM and of its supplement 1 (GUM-S1) can be realised consistently within the framework of stochastic processes. It follows that the propagation of (co-)variances (GUM) and the propagation of distributions (GUM-S1) then carry over to the propagation of covariance functions and stochastic processes, respectively. The proposed extension enables the uncertainty evaluation for continuous functions and particularly for continuous-time measurements.