Related papers: The space of all p-th roots of a nilpotent complex…

A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with anisotropic planes in the…

Suppose A and B are unital C*-algebras and A is separable. Let Rep(A,B) denote the set of all unital *-homomorphisms from A to B with the topology of pointwise convergence. We consider the problem of when the closure of the unitary orbit of…

We prove that every nonderogatory matrix $A$ over a field of positive odd characteristic $p,$ that is sum of a $p$-potent matrix and a nilpotent matrix, has a decomposition $A=E+V,$ such that $E^p=E,$ and $V^3=0.$

We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$…

We prove that, over a field $\mathbb{F}$ of odd characteristic $p$, a companion matrix $C$ is the sum of $E$ and $N$, with $E$ $p$-potent (i.e. $E^p = E$,) and $N$ nilpotent, if and only if the trace of $C$ is an integer multiple of unity…

For a complex nilpotent finite dimensional Lie algebra of matrices,and a Jordan-H\"older basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

If $H$ is a Hilbert space, the Stiefel manifold $St(n,H)$ is formed by all the independent $n$-tuples in $H$. In this article, we contribute to the topological study of Stiefel manifolds by proving a path-connectedness result. We prove that…

In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The…

A topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected first countable space is the image of a nonseparably connected complete metric space…

Let $A$ be a unital simple separable exact C$^*$-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed Cuntz class is path connected. This result applies in particular…

We study the structure of nilpotent completely positive maps in terms of Choi-Kraus coefficients. We prove several inequalities, including certain majorization type inequalities for dimensions of kernels of powers of nilpotent completely…

It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix $A$ over an arbitrary field, there is a…

Given an nxn nilpotent matrix over an algebraically closed field K, we prove some properties of the set of all the nxn nilpotent matrices over K which commute with it. Then we give a proof of the irreducibility of the variety of all the…

We prove that in the graded commutative ring $K_{*}(\mathbb{S})$, all positive degree elements are multiplicatively nilpotent. The analogous statements also hold for $TC_{*}(\mathbb{S};\mathbb{Z}^{\wedge}_p)$ and $K_{*}(\mathbb{Z})$.

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.

In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We…

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern. We develop ways to construct some…

Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a…

It is known that strongly nilpotent matrices over a division ring are linearly triangularizable. We describe the structure of such matrices in terms of the strong nilpotency index. We apply our results on quasi-translation x + H such that…

It is proven that if $ (X,d) $ is an arbitrary metric space and $ U $ is a path-connected subset of $ X $ with $M:=\{x_i:\ i\in\{1,2,\dots,k\}\}\subset int(U) $, then the property of path-connectedness is also preserved in the resulting set…