For vectors $X, Yin mathbb{R}^{n}$, we say $X$ is left matrix majorized by $Y$ and write $X prec_{ell} Y$ if for some row stochastic matrix $R, ~X=RY.$ Also, we write $Xsim_{ell}Y,$ when $Xprec_{ell}Yprec_{ell}X.$ A linear operator $Tcolon mathbb{R}^{p}to mathbb{R}^{n}$ is said to be a linear preserver of a given relation $prec$ if $Xprec Y$ on $mathbb{R}^{p}$ implies that $TXprec TY$ on $mathb...

Regardless of rich biodiversity, cytogenetic reports on Pentatomidae refer to less than 400 species. This paper reports, for the first time, course of meiosis of three species of subfamily Pentatominae viz., Tolumnia antennata Distant, Cahara jugatoria (Lethierry) and Cahara confusa (Distant). Species-specific variations observed with respect to association pattern of X and Y during prophase, a...

The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...

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‎Let $mathcal{F}$ be an field of zero characteristic and $N_{infty‎}(‎mathcal{F})$ be the algebra of infinite strictly upper triangular‎ ‎matrices with entries in $mathcal{F}$‎, ‎and $f:N_{infty}(mathcal{F}‎)rightarrow N_{infty}(mathcal{F})$ be a non-additive Lie centralizer of $‎N_{infty }(mathcal{F})$; that is‎, ‎a map satisfying that $f([X,Y])=[f(X),Y]$‎ ‎for all $X,Yin N_{infty}(mathcal{F})...

Let $mathcal {A}$ be an abelian category with enough projective objects and $mathcal {X}$ be a full subcategory of $mathcal {A}$. We define Gorenstein projective objects with respect to $mathcal {X}$ and $mathcal{Y}_{mathcal{X}}$, respectively, where $mathcal{Y}_{mathcal{X}}$=${ Yin Ch(mathcal {A})| Y$ is acyclic and $Z_{n}Yinmathcal{X}}$. We point out that under certain hypotheses, these two G...

Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let X and Y be Banach spaces such that X is weakly compactly generated Asplund space and X has the approximation property (respectively Y is weakly compactly generated Asplund space and Y ∗ has the approximation property). Suppose that L(X, Y ) 6= K(X,Y ) and let 1 < λ < 2. Then X (respective...