The positive matrix factorization problem is for a given positive matrix to determine those factorizations of the given matrix as a product of two positive matrices for which the space of the positive real numbers ove...
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The positive matrix factorization problem is for a given positive matrix to determine those factorizations of the given matrix as a product of two positive matrices for which the space of the positive real numbers over which is factored has the lowest possible dimension. Geometrically the problem is to embed a polyhedral cone in another polyhedral cone which has as few spanning vectors as possible. It is proven that this problem can be reduced to the search for an embedding in either an extremal polyhedral cone or in a facet of the positive orthant. (C) 1999 Elsevier Science Inc. All rights reserved.
We characterize those square partial matrices whose specified entries constitute a rectangular submatrix that may be completed to an inverse,M-matrix. Together with the notion of an interior inverse M-matrix, this is ...
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We characterize those square partial matrices whose specified entries constitute a rectangular submatrix that may be completed to an inverse,M-matrix. Together with the notion of an interior inverse M-matrix, this is used to show that any positive matrix is a sum of inverse M-matrices and to estimate the number of summands needed to represent a given matrix. Nonnegative matrices are also considered. There are substantial differences from the analogous problem of decomposing a positive matrix as a sum of totally positive matrices. In particular, the upper bound on the number of inverse M-matrix summands is much less than that in the totally positive case (although an example is given to show that the number of totally positive summands may be less than the required number of inverse M-matrix summands). (c) 2005 Elsevier Inc. All rights reserved.
To control COVID-19, strict restrictions were implemented in China, leading to a decline in pollutant emissions. However, in the post-pandemic era, pollutants rebounded rapidly, particularly in the Central Plains regi...
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To control COVID-19, strict restrictions were implemented in China, leading to a decline in pollutant emissions. However, in the post-pandemic era, pollutants rebounded rapidly, particularly in the Central Plains region, without PM2.5 concentrations meeting national standard. This study analyzed Water-soluble inorganic ions (WSIIs) concentration and sources across different pandemic periods using long-term high-resolution PM2.5, WSIIs and meteorological data from Zhengzhou (ZZ), Anyang (AY), and Xinyang (XY). Results indicated that during the lockdown, WSIIs concentration in the three cities were significantly lower compared to other periods, but rebounded shortly after lifting of restrictions due to human activity resumption. positive matrix Factorization analysis showed that secondary aerosol sources were dominant in all cities. Simulations revealed a 90 % reduction of secondary aerosol sources in ZZ could lead to an additional decrease of 2.6 mu g & sdot;m-3 of PM2.5 due to pH change. In AY, a 30 % reduction of secondary aerosol sources resulted in an extra reduction of 14.9 mu g & sdot;m-3 in PM2.5. When the contribution of secondary aerosol sources in XY was decreased to 18 %, PM2.5 concentration decreased by 30 %, achieving an additional reduction of 9.5 mu g center dot m-3 . This study offers strategies for achieving PM2.5 compliance and mitigating its impact on health and environment in the post-pandemic era.
We shown among other inequalities that if A(1), B-1, X-1, and Y-1 are nxn complex matrices such that A1 and B1 are positive semidefinite, then s(j)(Y(1)A(1)X(1)-X1B1Y1)<= s(j)(Z circle plus P) for j=1,2,& mldr;...
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We shown among other inequalities that if A(1), B-1, X-1, and Y-1 are nxn complex matrices such that A1 and B1 are positive semidefinite, then s(j)(Y(1)A(1)X(1)-X1B1Y1)<= s(j)(Z circle plus P) for j=1,2,& mldr;,2n, where Z=Z(1)+|Z(2)|, P=P-1+|P-2|, Z(1)=1/2A(1)(1/2)(|Y-1|(2)+|X-1(& lowast;)|(2))A(1)(1/2), Z(2)=1/2B(1)(1/2)((X1Y1)-Y-& lowast;-Y1X1 & lowast;)A(1)(1/2), P-1=1/2B(1)(1/2)(|X-1|(2)+|Y-1 & lowast;|(2))B-1(1/2), and P-2=1/2A(1)(1/2)(Y-1 & lowast;X-1-X1Y1 & lowast;)B-1(1/2). This inequality generalizes a recent inequality due to Audeh.
Utilizing the notion of positive multilinear mappings, we present some matrix inequalities. In particular, the Choi- Davis-Jensen inequality f (Phi(A,B)) <= Phi(f(A), f(B)) does not hold in general for a matrix con...
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Utilizing the notion of positive multilinear mappings, we present some matrix inequalities. In particular, the Choi- Davis-Jensen inequality f (Phi(A,B)) <= Phi(f(A), f(B)) does not hold in general for a matrix convex function f and a positive multilinear mapping Phi. We prove this inequality under certain condition on f. Moreover, some Kantorovich and convexity type inequalities including positive multilinear mappings are presented. (C) 2015 Elsevier Inc. All rights reserved.
For a probability measure of compact support mu on the set P-n of all positive definite matrices and t is an element of (0, 1], let P-t(mu) be the unique positive solution of X = integral(Pn) X (sic)(t)Z d mu(Z). In t...
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For a probability measure of compact support mu on the set P-n of all positive definite matrices and t is an element of (0, 1], let P-t(mu) be the unique positive solution of X = integral(Pn) X (sic)(t)Z d mu(Z). In this paper, we show that if mu and nu are probability measures on P-n and P-m, respectively, and Phi : M-n xM(m) -> M-k is a unital positive bilinear map, then Phi(P-t(mu), P-t(nu)) = P-t(lambda) for all t is an element of [-1, 1]\positive, where lambda is the push-forward of the product measure mu x nu by the mapping Phi vertical bar P(n)xP(m). vertical bar n particular, Pt(mu) circle times P-t(nu) <= P-t(lambda). Moreover, we show that Pt(mu) <= I double right arrow P-t/p (nu) <= P-t(mu), for every p >= 1, where nu(Z) = mu(Z(1/p)) and I denotes the identity matrix.
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