We determine all equality cases in the Tu--Deng bound $|S_{t,k}|\le 2^{k-1}$. If the $k$-bit cyclic word of $t$ has $R$ ones, $Z$ zeros, and cyclic one-gap lengths $g_1,\ldots,g_Z$, then equality holds if and only if $g_i\ge Z-1$ for every $i$. This resolves Conjecture~3.20 of Flori, Randriambololona, Cohen and Mesnager, and we also enumerate all equality parameters. For $R\ge Z$ we determine the sharp first stability gap and all extremal words, while for $R<Z$ we obtain an exact quantization of the deficit and an explicit run-sensitive lower bound. The proofs are structural: an explicit matrix conjugation identifies the auxiliary enumerators in the two recent complete proofs of the Tu--Deng conjecture. We then develop a rooted coarsening model for all coefficients, prove one-sided deletion rigidity and an exact Macaulay-flux identity, and derive a Macaulay--M\"obius formula from the bounded simplex at the highest cyclic level.
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...
We present a unified presentation of the fixed-size adjacency-spectral extremal problem for odd wheels $W_{2k+1}$, where $k\geq2$ and $W_{2k+1}=K_1\vee C_{2k}$. The exceptional case $W_5$ and the general case $W_{2k+1}$, $k\ge3$, share the same dense-core reduction and edge-spectral stability, but have different rigidi...
Longfei Fang, M. Zhai, Yuhan Zhang· 2 citations· ⚡2
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 eve...
For every odd integer $t\ge17$, we prove that an explicit circulant graph on $5t-10$ vertices is doubly saturated $R(3,t)$-good. The graph is triangle-free and has independence number $t-1$. Adding any nonedge creates a triangle, whereas deleting any edge creates an independent set of order $t$. This settles Conjecture...
Abhishek Saigal, Akaash R. Parthasarathy· 0 citations
In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For o...
Chen Shang, Jiahe Shen, Jiyue Zeng et al.· 1 citation
We study Krasner quotient hyperfields arising from finite fields, $F_q/G_r$, where $G_r\le F_q^\times$ has index $r$. Building on the structure theorem of Baker--Jin, we determine the characteristic and C-characteristic of all sufficiently large such quotients: they depend only on the parity of $r$ and, when $r$ is eve...
Alessandro Linzi· 0 citations
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