A Perron-Frobenius-type Theorem for Positive Matrix Semigroups

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

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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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A Korovkin-type theorem for sequences of positive linear operators on function spaces

Positivity ◽

10.1007/s11117-021-00861-2 ◽

2021 ◽

Author(s):

Abderraouf Dorai

Keyword(s):

Type Theorem ◽

Function Spaces ◽

Linear Operators ◽

Positive Linear Operators ◽

Korovkin Type Theorem

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B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

Open Mathematics ◽

10.1515/math-2020-0033 ◽

2020 ◽

Vol 18 (1) ◽

pp. 715-730

Author(s):

Javanshir J. Hasanov ◽

Rabil Ayazoglu ◽

Simten Bayrakci

Keyword(s):

Differential Operator ◽

Singular Integral ◽

Riesz Potential ◽

Type Theorem ◽

Integral Operators ◽

Morrey Spaces ◽

Singular Integral Operators ◽

Sobolev Type ◽

Riesz Potentials ◽

Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

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