A neurodynamic approach to compute the generalized eigenvalues of symmetric positive matrix pair

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

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A general inverse matrix series relation and associated polynomials – II

Mathematica Slovaca ◽

10.1515/ms-2017-0469 ◽

2021 ◽

Vol 71 (2) ◽

pp. 301-316

Author(s):

Reshma Sanjhira

Keyword(s):

Matrix Polynomial ◽

Combinatorial Identities ◽

Inverse Matrix ◽

Matrix Polynomials ◽

Special Cases ◽

Series Representations ◽

The Matrix ◽

Associated Polynomials ◽

Matrix Pair ◽

Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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Geochemical Mapping, Risk Assessment, and Source Identification of Heavy Metals in Road Dust Using Positive Matrix Factorization (PMF)

Atmosphere ◽

10.3390/atmos12050614 ◽

2021 ◽

Vol 12 (5) ◽

pp. 614

Author(s):

Muhammad Faisal ◽

Zening Wu ◽

Huiliang Wang ◽

Zafar Hussain ◽

Chenyang Shen

Keyword(s):

Heavy Metals ◽

Risk Assessment ◽

Matrix Factorization ◽

Anthropogenic Activities ◽

Contamination Factor ◽

Road Dust ◽

Positive Matrix Factorization ◽

Prevention Measures ◽

Positive Matrix ◽

The City

Heavy metals in road dust pose a significant threat to human health. This study investigated the concentrations, patterns, and sources of eight hazardous heavy metals (Cr, Ni, Cu, Zn, As, Cd, Pb, and Hg) in the street dust of Zhengzhou city of PR China. Fifty-eight samples of road dust were analyzed based on three methods of risk assessment, i.e., Geo-Accumulation Index (Igeo), Potential Ecological Risk Assessment (RI), and Nemerow Synthetic Pollution Index (PIN). The results exhibited higher concentrations of Hg and Cd 14 and 7 times higher than their background values, respectively. Igeo showed the risks of contamination in a range of unpolluted (Cr, Ni) to strongly polluted (Hg and Cd) categories. RI came up with the contamination ranges from low (Cr, Ni, Cu, Zn, As, and Pb) to extreme (Cd and Hg) risk of contamination. The risk of contamination based on PIN was from safe (Cu, As, and Pb) to seriously high (Cd and Hg). The results yielded by PIN indicated the extreme risk of Cd and Hg in the city. Positive Matrix Factorization was used to identify the sources of contamination. Factor 1 (vehicular exhaust), Factor 2 (coal combustion), Factor 3 (metal industry), and Factor 4 (anthropogenic activities), respectively, contributed 14.63%, 35.34%, 36.14%, and 13.87% of total heavy metal pollution. Metal’s presence in the dust is a direct health risk for humans and warrants immediate and effective pollution control and prevention measures in the city.

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Development of a versatile source apportionment analysis based on positive matrix factorization: a case study of the seasonal variation of organic aerosol sources in Estonia

Atmospheric Chemistry and Physics ◽

10.5194/acp-19-7279-2019 ◽

2019 ◽

Vol 19 (11) ◽

pp. 7279-7295 ◽

Cited By ~ 2

Author(s):

Athanasia Vlachou ◽

Anna Tobler ◽

Houssni Lamkaddam ◽

Francesco Canonaco ◽

Kaspar R. Daellenbach ◽

...

Keyword(s):

Matrix Factorization ◽

Organic Aerosol ◽

Positive Matrix Factorization ◽

Water Soluble ◽

Oxidation Products ◽

Positive Matrix ◽

Bootstrap Analysis ◽

Seasonal Behavior ◽

Yearly Cycle ◽

Aerosol Mass

Abstract. Bootstrap analysis is commonly used to capture the uncertainties of a bilinear receptor model such as the positive matrix factorization (PMF) model. This approach can estimate the factor-related uncertainties and partially assess the rotational ambiguity of the model. The selection of the environmentally plausible solutions, though, can be challenging, and a systematic approach to identify and sort the factors is needed. For this, comparison of the factors between each bootstrap run and the initial PMF output, as well as with externally determined markers, is crucial. As a result, certain solutions that exhibit suboptimal factor separation should be discarded. The retained solutions would then be used to test the robustness of the PMF output. Meanwhile, analysis of filter samples with the Aerodyne aerosol mass spectrometer and the application of PMF and bootstrap analysis on the bulk water-soluble organic aerosol mass spectra have provided insight into the source identification and their uncertainties. Here, we investigated a full yearly cycle of the sources of organic aerosol (OA) at three sites in Estonia: Tallinn (urban), Tartu (suburban) and Kohtla-Järve (KJ; industrial). We identified six OA sources and an inorganic dust factor. The primary OA types included biomass burning, dominant in winter in Tartu and accounting for 73 % ± 21 % of the total OA, primary biological OA which was abundant in Tartu and Tallinn in spring (21 % ± 8 % and 11 % ± 5 %, respectively), and two other primary OA types lower in mass. A sulfur-containing OA was related to road dust and tire abrasion which exhibited a rather stable yearly cycle, and an oil OA was connected to the oil shale industries in KJ prevailing at this site that comprises 36 % ± 14 % of the total OA in spring. The secondary OA sources were separated based on their seasonal behavior: a winter oxygenated OA dominated in winter (36 % ± 14 % for KJ, 25 % ± 9 % for Tallinn and 13 % ± 5 % for Tartu) and was correlated with benzoic and phthalic acid, implying an anthropogenic origin. A summer oxygenated OA was the main source of OA in summer at all sites (26 % ± 5 % in KJ, 41 % ± 7 % in Tallinn and 35 % ± 7 % in Tartu) and exhibited high correlations with oxidation products of a-pinene-like pinic acid and 3-methyl-1, 2, 3-butanetricarboxylic acid (MBTCA), suggesting a biogenic origin.

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