Let $A$ be a real square and invertible matrix.I would like to find $$ s(A) = min_U \rho(U A), $$ Where $U$ is orthonormal, i.e., $U U^T = I$ and $\rho(A)$ is the spectral radius, i.e., the largest eigenvalue of $A$ in absolute values. For normal matrix $A$, it is $s(A) = \rho(A)$. I am interes...

I'm curious to know about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia) is feasible for Beatty primes. As in the first slide of [2] we define the set of Beatty primes as the set $$\mathbb{P}_{\alpha,\beta}=\{\text{primes}\}\cap\{\lfloor\alpha n+\b...

An element $x$ of an algebraic ring $S$ is said to be right quasiregular if there exists another element $y\in S$ such that $x\circ y=x+y+xy=0$. DEFINITION- A topological ring is said to be a $D_r$ - ring if its right quasiregular (r.q.r.) elements form a d-open set (union of open $F_{\s...

Let $A$ be an abelian group and let $n \geq 2$. For any connected CW complex $X$, it is standard that a fibration $f\colon E \rightarrow X$ whose fibers are homotopy equivalent to a $K(A,n)$ is fiberwise homotopy equivalent to the pull-back of the loop-space fibration over a $K(A,n+1)$ if and on...

Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let $$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\times n}$$ be a diagonal matrix of size $n\times n$. Let $U$ and $V$ be two independent random ma...

Given a morphism $f:\Bbb{A}^n_{\Bbb{C}}\rightarrow\Bbb{A}^m_{\Bbb{C}}$ of complex affine spaces, Chevalley's theorem states that the image of a closed subvariety is constructible, i.e. could be described with finitely many equations $f(y_1,\dots,y_m)=0$ and inequations $g(y_1,\dots,y_m)\neq 0$ co...