Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the even-dimensional case for $n\ge1000$. We complete the remaining even-dimensional cases. Starting from the structural identities in Frankel--Urschel Lemma 2.1, we derive an exact global defect budget and combine binary rounding with Gram projection. A refined ten-row obstruction handles every even $n\ge66$; a finite exact calculation handles $4\le n\le64$, $n\ne6$; and a separate multi-column energy argument treats $n=6$. The order-two case follows from a direct calculation. The new even-dimensional proof has been formalized in Lean 4, with Frankel--Urschel Lemma 2.1 as its sole external mathematical input. Together with Cheng's odd-dimensional theorem, this proves the S-matrix conjecture in every dimension.
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...
Two permutations $\sigma,\tau\in S_n$ are called disjoint if the composition $\sigma\tau^{-1}$ has no fixed point. If a family $\mathcal F\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\mathcal F|\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized...
We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of or...
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
The Johnson--Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameter...
We show that the weak Markus--Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix...
Álvaro Castañeda, Gerardo Honorato, Francisco Valenzuela-Henríquez· 1 citation
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