Let $SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices and set $SM(\mathbb{R})^g = \cup_n SM_n(\mathbb{R})^g$. A free quadrilateral is the collection tuples $X \in SM(\mathbb{R})^2$ which have positive semidefinite evaluation on linear equations defining a classical quadrilateral. Such closed under rich class convex combinations called matrix combination. That is, gi...

We provide a nontrivial bound on the rank of any tensor $T$ over quaternions $\mathbb{H}$ in $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. describe decomposition into $3$ simple tensors $2\times 2\times 2$ case. also show that upper is best possible for some cases, and we various partial results involving decompositions $\mathbb{C}$ $\mathbb{H}$.

Let $R$ be a commutative ring with identity and $M$ be an unitary $R$-module. The intersection graph of an $R$-module $M$, denoted by $Gamma(M)$, is a simple graph whose vertices are all non-trivial submodules of $M$ and two distinct vertices $N_1$ and $N_2$ are adjacent if and only if $N_1cap N_2neq 0$. In this article, we investigate the concept of a planar intersection graph and maximal subm...

We consider the minimal see-saw extension of Standard Model with two right-handed singlet fermions $N_{1,2}$ mass at GeV scale, augmented by an effective dipole operator between sterile states. firstly review current bounds on this interaction from fixed-target and collider experiments as well astrophysical cosmological observations. then highlight prospects for testing decay $N_2 \to N_1 \gamm...