let $r$ be a commutative ring and let $m$ be an $r$-module. in this article, we introduce the concept of the zariski socles of submodules of $m$ and investigate their properties. also we study modules with noetherian second spectrum and obtain some related results.

a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...

Propionic acid (PA) is a fungicide and bactericide, registered to control fungi and bacteria in stored grains, hay, grain storage areas, poultry litter, and drinking water for livestock and poultry. European Union (EU) certifies PA as the great of grain preserver and most efficient in controlling Salmonella and other pathogens. Recently it is used as feed additive in poultry and non-ruminant pr...

in this paper we study the concept of latin-majorizati-on. geometrically this concept is different from other kinds of majorization in some aspects. since the set of all $x$s latin-majorized by a fixed $y$ is not convex, but, consists of :union: of finitely many convex sets. next, we hint to linear preservers of latin-majorization on $ mathbb{r}^{n}$ and ${m_{n,m}}$.

For a rank-1 matrix A = a ⊗ b over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T (A) = U ⊗ A ⊗ V , or T (A) = U ⊗ A ⊗ V with ...

we define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital banach algebra a and show that the δ-approximate spectrum σ_δ (a) of a is compact. the relation between the δ-approximate spectrum and the usual spectrum is investigated. also an analogue of the classical gleason-kahane-zelazko theorem is established: for each ε>0, there is δ>0 such that if ϕ is...

IN THEIR PERSPECTIVE “The trouble with negative emissions” (14 October, p. 182), K. Anderson and G. Peters assert that negative-emissions technologies are an “unjust and high-stakes gamble.” This characterization would sideline negativeemissions technologies and remove potentially important options from the portfolio for mitigating and ameliorating climate change. As Anderson and Peters acknowl...