# American Institute of Mathematical Sciences

eISSN:
2577-8838

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## Mathematical Foundations of Computing

February 2021 , Volume 4 , Issue 1

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2021, 4(1): 1-13 doi: 10.3934/mfc.2020022 +[Abstract](705) +[HTML](383) +[PDF](4425.14KB)
Abstract:

It is evident that the traditional hard matching of a fixed-length template cannot satisfy the nearly indefinite variations in natural language. This issue mainly results from three major problems of the traditional matching mode: 1) in matching with a short template, the context of natural language cannot be effectively captured; 2) in matching with a long template, serious data sparsity will lead to a low success rate of template matching (i.e., low recall); and 3) due to a lack of flexible matching ability, traditional hard matching is more prone to failure. Therefore, this paper proposed a novel method of stretchable matching of the semantic template (SMOST) to deal with the above problems. We have applied this method to word sense disambiguation in the natural language processing field. In the same case of using only the SemCor corpus, the result of our system is very close to the best result of existing systems, which shows the effectiveness of new proposed method.

2021, 4(1): 15-30 doi: 10.3934/mfc.2020023 +[Abstract](586) +[HTML](316) +[PDF](472.17KB)
Abstract:

In the present paper, we shall investigate the pointwise approximation properties of the \begin{document}$q-$\end{document}analogue of the Bernstein-Schurer operators and estimate the rate of pointwise convergence of these operators to the functions \begin{document}$f$\end{document} whose \begin{document}$q-$\end{document}derivatives are bounded variation on the interval \begin{document}$[0,1+p].$\end{document} We give an estimate for the rate of convergence of the operator \begin{document}$\left( B_{n,p,q}f\right)$\end{document} at those points \begin{document}$x$\end{document} at which the one sided \begin{document}$q-$\end{document}derivatives \begin{document}$D_{q}^{+}f(x)$\end{document} and \begin{document}$D_{q}^{-}f(x)$\end{document} exist. We shall also prove that the operators \begin{document}$\left( B_{n,p,q}f\right) (x)$\end{document} converge to the limit \begin{document}$f(x)$\end{document}. As a continuation of the very recent and initial study of the author deals with the pointwise approximation of the \begin{document}$q-$\end{document}Bernstein Durrmeyer operators [12] at those points \begin{document}$x$\end{document} at which the one sided \begin{document}$q-$\end{document}derivatives \begin{document}$D_{q}^{+}f(x)$\end{document} and \begin{document}$D_{q}^{-}f(x)$\end{document} exist, this study provides (or presents) a forward work on the approximation of \begin{document}$q$\end{document}-analogue of the Schurer type operators in the space of \begin{document}$D_{q}BV$\end{document}.

2021, 4(1): 31-44 doi: 10.3934/mfc.2020024 +[Abstract](580) +[HTML](318) +[PDF](417.25KB)
Abstract:

A deep neural network with invertible hidden layers has a nice property of preserving all the information in the feature learning stage. In this paper, we analyse the hidden layers of residual rectifier neural networks, and investigate conditions for invertibility under which the hidden layers are invertible. A new fixed-point algorithm is developed to invert the hidden layers of residual networks. The proposed inverse algorithms are capable of inverting some residual networks which cannot be inverted by existing inverting algorithms. Furthermore, a special residual rectifier network is designed and trained on MNIST so that it can achieve comparable performance with the state-of-art performance while its hidden layers are invertible.

2021, 4(1): 45-59 doi: 10.3934/mfc.2021001 +[Abstract](922) +[HTML](260) +[PDF](403.91KB)
Abstract:

As a sophisticated and popular age-period-cohort method, the Intrinsic Estimator (IE) and related estimators have evoked intense debate in demography, sociology, epidemiology and statistics. This study aims to provide a more holistic review and critical assessment of the overall methodological significance of the IE and related estimators in age-period-cohort analysis. We derive the statistical properties of the IE from a linear algebraic perspective, provide more precise mathematical proofs relevant to the current debate, and demonstrate the essential, yet overlooked, link between the IE and classical statistical tools that have been employed by scholars for decades. This study offers guidelines for the future use of the IE and related estimators in demographic research. The exposition of the IE and related estimators may help redirect, if not settle, the logic of the debate.

2021, 4(1): 61-71 doi: 10.3934/mfc.2021002 +[Abstract](373) +[HTML](267) +[PDF](375.04KB)
Abstract:

In the present paper, notion of the distance between two intuitionistic fuzzy elements is presented. Using the new distance measure, we extend TOPSIS (a technique for order preference by similarity to ideal solution) to group decision making for the intuitionistic fuzzy set. Also, group preferences are aggregated within the procedure. Two numerical examples concerning supplier selection in a manufacturing company and nurse selection in a hospital are constructed to show the practicability and the usefulness of this extension for group decision making to reach an optimum solution.