Stone (Ph.D.\ thesis, Department of Operations Research, Stanford University, 1981) proved that every matrix in $U \cap Q_0$ is a $P_0$-matrix and conjectured that the same conclusion holds for the larger class $E_0^f \cap Q_0$ of fully semimonotone $Q_0$-matrices. Murthy and Parthasarathy [SIAM J.\ Matrix Anal.\ Appl.\ 16 (1995), 1268--1286] verified the conjecture for matrices of order up to $4 \times 4$, for $5 \times 5$ and $6 \times 6$ matrices under additional hypotheses, and for several special subclasses of arbitrary order, but the conjecture remains open in general. In this paper we prove that every $E_0^f$-matrix with positive determinant is a $P_0$-matrix, for matrices of arbitrary order $n$; our proof proceeds by induction on $n$, via an algebraic analysis of principal minors under principal pivotal transforms. We further exhibit a matrix $A \in E_0^f$ with $\det A>0$ that fails to belong to $Q_0$, showing that the hypothesis $\det A>0$ used in our theorem cannot, by itself, be deduced from membership in $Q_0$, and hence does not on its own yield a proof of Stone's conjecture. Stone's conjecture itself remains open.
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomi...
Let $M=\widetilde H^*(MO(2);\mathbb F_2)$ and let $\mathcal A(1)$ be the subalgebra of the mod $2$ Steenrod algebra generated by $\operatorname{Sq}^1$ and $\operatorname{Sq}^2$. In this work, we determine the quotient $\mathbb F_2\otimes_{\mathcal A(1)}M$ in every internal degree $h\geq0$, where internal degree $h$ cor...
Let $k$ be a field of characteristic zero and let $S=k[x_1,x_2,x_3,x_4]$. We prove Fr\"oberg's predicted Hilbert series for ideals generated by $r$ general forms of equal degree $d$ for every $r\geq1$ in each of the two cases $d=5$ and $d=7$. Relative to the classical cases $r\leq5$ and the equal-degree theorem through...
Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all compl...
Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^...
André Volkmann· 3 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
