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 $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alp\"oge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.
For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)\ge \frac{\operatorname{per}K}{6}$ for every planar convex body.
Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ imply $\int_Kf\leq C_n\int_Lf$. In an earlier paper the authors proved that $C_n\leq\sqrt n$. In this paper, we prove the matching lower bound $C_n\geq c\sqrt n$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\geq c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.
For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.
Feihu Liu, Jinlong Tang, Guoce Xin et al.· 0 citations
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 has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.
Álvaro Castañeda, Gerardo Honorato, Francisco Valenzuela-Henríquez· 0 citations
Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p>1$ and $0<\theta<1$.