# American Institute of Mathematical Sciences

eISSN:
2577-8838

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## Mathematical Foundations of Computing

August 2021 , Volume 4 , Issue 3

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2021, 4(3): 145-165 doi: 10.3934/mfc.2021009 +[Abstract](633) +[HTML](243) +[PDF](2298.61KB)
Abstract:

The recent progress in learning image feature representations has opened the way for tasks such as label-to-image or text-to-image synthesis. However, one particular challenge widely observed in existing methods is the difficulty of synthesizing fine-grained textures and small-scale instances. In this paper, we propose a novel Global-Affine and Local-Specific Generative Adversarial Network (GALS-GAN) to explicitly construct global semantic layouts and learn distinct instance-level features. To achieve this, we adopt the graph convolutional network to calculate the instance locations and spatial relationships from scene graphs, which allows our model to obtain the high-fidelity semantic layouts. Also, a local-specific generator, where we introduce the feature filtering mechanism to separately learn semantic maps for different categories, is utilized to disentangle and generate specific visual features. Moreover, we especially apply a weight map predictor to better combine the global and local pathways considering the highly complementary between these two generation sub-networks. Extensive experiments on the COCO-Stuff and Visual Genome datasets demonstrate the superior generation performance of our model against previous methods, our approach is more capable of capturing photo-realistic local characteristics and rendering small-sized entities with more details.

2021, 4(3): 167-184 doi: 10.3934/mfc.2021010 +[Abstract](570) +[HTML](196) +[PDF](687.6KB)
Abstract:

Canonical correlation analysis (CCA) has gained great success for fault detection (FD) in recent years. However, it cannot preserve the prior information of the underlying process. To cope with these difficulties, this paper proposes an improved CCA-based FD scheme using a novel multivariate statistical technique, called sparse collaborative regression (SCR). The core of the proposed method is to take the prior information as a supervisor, and then integrate it with CCA. Further, the \begin{document}$\ell_{2,1}$\end{document}-norm is employed to reduce redundancy and avoid overfitting, which facilitates its interpretability. In order to solve the proposed SCR, an efficient alternating optimization algorithm is developed with convergence analysis. Finally, some experimental studies on a simulated example and the benchmark Tennessee Eastman process are conducted to demonstrate the superiority over the classical CCA in terms of the false alarm rate and fault detection rate. The detection results indicate that the proposed method is promising.

2021, 4(3): 185-191 doi: 10.3934/mfc.2021011 +[Abstract](412) +[HTML](166) +[PDF](300.14KB)
Abstract:

Based on the results given by Ahuja and Enneking, we show that the generating function of the associated Lah numbers having only real zeros, and further obtain the asymptotic normality of the associated Lah numbers. As application, we get the asymptotic normality of the signless Lah numbers.

2021, 4(3): 193-208 doi: 10.3934/mfc.2021012 +[Abstract](354) +[HTML](157) +[PDF](359.79KB)
Abstract:

In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.

2021, 4(3): 209-219 doi: 10.3934/mfc.2021013 +[Abstract](647) +[HTML](180) +[PDF](296.21KB)
Abstract:

The volterra-fredholm integral equation in all forms are arose from physics, biology and engineering problems which is derived from differential equation modelling. On the other hand, the trained programming algorithm by the fuzzy artificial neural networks has effective solution to find the best answer. In this article we try to estimate the equation and its answer by developed fuzzy artificial neural network to fuzzy volterra-fredholme integral. Our attempts would lead to benchmark other extended forms of this type of equation.