In this note we show how to test whether or not certain algebraic equations derived in the paper Fault Detection and Isolation for State Affine Systems, are solvable and to describe a solution method. Test and solution rely upon elementary linear algebraic techniques.

‎A ring $R$ is strongly clean provided that every element‎ ‎in $R$ is the sum of an idempotent and a unit that commutate‎. ‎Let‎ ‎$T_n(R,sigma)$ be the skew triangular matrix ring over a local‎ ‎ring $R$ where $sigma$ is an endomorphism of $R$‎. ‎We show that‎ ‎$T_2(R,sigma)$ is strongly clean if and only if for any $ain‎ ‎1+J(R)‎, ‎bin J(R)$‎, ‎$l_a-r_{sigma(b)}‎: ‎Rto R$ is surjective‎. ‎Furt...

Budaghyan and Carlet [4] constructed a family of almost perfect nonlinear (APN) hexanomials over a field with r2 elements, and with terms of degrees r + 1, s + 1, rs+ 1, rs+ r, rs+ s, and r + s, where r = 2m and s = 2n with GCD(m,n) = 1. The construction requires a certain technical condition, which was verified empirically in a finite number of examples. Bracken, Tan, and Tan [1] proved the co...

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...

A ring $R$ with an automorphism $sigma$ and a $sigma$-derivation $delta$ is called $delta$-quasi-Baer (resp., $sigma$-invariant quasi-Baer) if the right annihilator of every $delta$-ideal (resp., $sigma$-invariant ideal) of $R$ is generated by an idempotent, as a right ideal. In this paper, we study Baer and quasi-Baer properties of skew PBW extensions. More exactly, let $A=sigma(R)leftlangle x...

We construct embeddings of Bers slices ideal polygon reflection groups into the classical family univalent functions $\Sigma$. This embedding is such that conformal mating group with anti-holomorphic polynomial $z\mapsto\overline{z}^d$ Schwarz map arising from corresponding in characterize image this $\Sigma$ as a rational maps. Moreover, we show limit set every Kleinian closure slice naturally...

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An extension of the classic Enneper–Weierstrass representation for conformally pa-rametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP 1 and CP 2 sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3-and 8-dimensional spaces, respectively. Relations of Weierstrass type systems to the equations of t...