• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.487-500
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• 2013

In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra B(X), using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator T, then for each ${\lambda}{\in}{\sigma}_{s-F}(T){\setminus}\overline{{\rho}^{\pm}_{s-F}(T)}$ and ${\in}$ > 0, the ball $B({\lambda},{\in})$ contains a component of ${\sigma}_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459-503] page 462.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.487-496
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• 2021

In this paper we show that if A ∈ L(X) and B ∈ L(Y), X and Y complex Banach spaces, then A ⊕ B ∈ L(X ⊕ Y) is subscalar if and only if both A and B are subscalar. We also prove that if A, Q ∈ L(X) satisfies AQ = QA and Q^p = 0 for some nonnegative integer p, then A has property (C) (resp. property (𝛽)) if and only if so does A + Q (resp. property (𝛽)). Finally, we show that A ∈ L(X, Y) and B, C ∈ L(Y, X) satisfying operator equation ABA = ACA and BA ∈ L(X) is subscalar with property (𝛿) then both Lat(BA) and Lat(AC) are non-trivial.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.655-668
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• 2012

An operator $T\;{\varepsilon}\;B(\mathcal{H})$ is said to be $k$-quasi-paranormal operator if $||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$ for every $x\;{\epsilon}\;\mathcal{H}$, $k$ is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of $k$-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically $k$-quasi-paranormal operator has Bishop's property ($\beta$), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for $f(T)$ for every $f\;{\epsilon}\;H({\sigma}(T))$; (ii) generalized a - Browder's theorem holds for $f(S)$ for every $S\;{\prec}\;T$ and $f\;{\epsilon}\;H({\sigma}(S))$; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.

• Interdisciplinary Bio Central
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• v.5 no.1
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• pp.1.1-1.8
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• 2013

High quality publicly-available transcriptomic data representing relationships in gene expression across a diverse set of biological conditions is used as a context network to explore transcriptomics of the CNS. The context network, 18367Hu-matrix, contains pairwise Pearson correlations for 22,215 human genes across18,637 human tissue samples1. To do this, we compute a network derived from biological samples from CNS cells and tissues, calculate clusters of co-expressed genes from this network, and compare the significance of these to clusters derived from the larger 18367Hu-matrix network. Sorting and visualization uses the publicly available software, MetaOmGraph (http://www.metnetdb.org/MetNet_MetaOm-Graph.htm). This identifies genes that characterize particular disease conditions. Specifically, differences in gene expression within and between two designations of glial cancer, astrocytoma and glioblastoma, are evaluated in the context of the broader network. Such gene groups, which we term outlier-networks, tease out abnormally expressed genes and the samples in which this expression occurs. This approach distinguishes 48 subnetworks of outlier genes associated with astrocytoma and glioblastoma. As a case study, we investigate the relationships among the genes of a small astrocytoma-only subnetwork. This astrocytoma-only subnetwork consists of SVEP1, IGF1, CHRNA3, and SPAG6. All of these genes are highly coexpressed in a single sample of anaplastic astrocytoma tumor (grade III) and a sample of juvenile pilocytic astrocytoma. Three of these genes are also associated with nicotine. This data lead us to formulate a testable hypothesis that this astrocytoma outlier-network provides a link between some gliomas/astrocytomas and nicotine.