arXiv subjects

Chenxiao Tian

Publications and source records attributed to Chenxiao Tian.

5 recordsLinked to original sources

arXiv2026

Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$

For a finite graph $H$, a connectivity graph-code is a family $\mathcal C\subseteq 2^{E(H)}$ such that $A\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\mathcal C$. Let $m(H)$ denote the maximum size of such a family, and let $f(d)$ be the largest integer $q$ for which $m(H)=q$ for infinitely many pairwise nonisomorphic $d$-regular graphs $H$. Restricting codewords to the edges incident with a vertex gives $f(d)\leq 2^d$. Alon proved equality for all sufficiently large $d$ and asked whether it holds for every $d\geq 4$. We answer this question affirmatively. More precisely, for every $d\geq 4$ we construct infinitely many finite simple $d$-regular bipartite graphs carrying a linear connectivity graph-code of dimension $d$. The construction begins with a vector-labelled copy of $K_{d,d}$. For $d\geq 7$, the required labelling follows from a probabilistic count over an irreducible conjugacy class in $\mathrm{GL}_d(2)$; explicit matrices, verified by a short exact exhaustive program, cover $d=4,5,6$. Cyclic voltage lifts then produce the required infinite families.

math.CO↗

arXiv2026

Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples

We study Frankl's union-closed sets conjecture through the height of the inclusion poset. Working in the equivalent empty-set-free formulation, where one seeks an element contained in strictly more than half of the members, we prove the conjecture for every finite union-closed family of height at most four. Equivalently, the usual at-least-half formulation holds for every union-closed family containing the empty set and having height at most five. We also develop a structural theory for the next unresolved case. Assuming a smallest empty-set-free counterexample of height at most five, we show that it has even cardinality $2t$, at least three critical elements of frequency $t$, and satisfies the minimal-counterexample bound $t \geq 2n-1$. Every critical element determines a coatom of the form $U \setminus \{x\}$, while every critical pair satisfies a dichotomy between a full double-avoidance top and a large avoidance fiber admitting a three-layer trace normal form. Coordinate deletion further yields an exact matching-defect obstruction. Finally, introducing the minimum number of join-irreducible members required to cover all critical elements, we exclude the five-cover case and show that this critical join-cover number is either three or four. These results substantially constrain any possible height-five counterexample while leaving the remaining transfer problem explicit.

math.CO↗

arXiv2026

Resolution of Singularities in Positive Characteristic: Frobenius-Hasse Towers and Exceptional-History Descent

Let k be a perfect field of characteristic p>0. We introduce an object-level construction for canonical strong embedded resolution and principalization over k. The construction program replaces monotonicity of pointwise numerical invariants by a well-founded history of addressed comparison factors. Starting from the differential-integral saturation of a marked Rees algebra, we construct total-Hasse activity packets, filtered coefficient cubes, semilinear Frobenius-Hasse sources, and literal transform data for ordinary permissible blowups. A six-row defect calculus routes local problems to certified surface, toroidal-monomial, binomial, and additive-type procedures, after which clean centre portfolios are serialized and descended on a global nerve. The central structure is a global replacement certificate transporting successor addresses, paid quotients, typed traces, displayed parents, reopening data, and terminal truth across macroblocks. We prove that a complete state equipped with this certificate admits a strict multiset replacement in a single dependent well-founded order; hence the iteration terminates and the exhausted state reconstructs a regular strict transform having normal crossings with the ordered boundary. We further formulate an object-level realization of the certificate through rigid generation, cross-generation no-reset, complete wild-capacity control, a centre-or-typed-exit alternative, structured cofibres, displayed-parent allocation, and literal terminal truth. The program page is also available at website https://sites.google.com/view/positive-char-resolution

math.GM↗

arXiv2026

Quantifying the complexity of trajectory ensembles with clustering-weighted multivariate multiscale sample entropy

Across the physical and life sciences, data increasingly appear as ensembles of trajectories, from chaotic flows and satellite constellations to clinical cohorts. Established sample-entropy measures characterize individual time series, while averaging across an ensemble discards population structure and cannot distinguish redundancy from diversity. We introduce clustering-weighted multivariate multiscale sample entropy (CWMMSE), which groups trajectories into behavioral patterns and weights each by its dynamical complexity. CWMMSE is a weighted entropy of the population's pattern distribution. Its empirical plug-in estimator is strongly consistent for a fixed finite partition, and it separates two components that can diverge in real data: individual complexity and population diversity. Both are essential. Averaging ignores diversity, whereas spread alone can mistake a varied but predictable population for a complex one. Across eleven physical, environmental, engineering, and biomedical systems, CWMMSE ranks a calm ocean region above an energetic but individually more complex one, identifies a major earthquake as a collapse in system complexity, and reverses the conclusion from averaging in cardiac cohorts, where disease reduces population diversity. Supported by an open, reproducible implementation, these results show that population complexity should be measured rather than averaged.

eess.SP↗

arXiv2022

Union-closed Sets Conjecture Holds for Height No More Than 3 and Height No Less Than N-1

For each given union-closed family F of n elements and m sets, we discuss the union-closed sets conjecture from height number of the UC family, which is a natural parameter from lattice theory. In this paper, we call it height number of F(n, m), recorded as H(F). we prove that for any given union-closed family F, union-closed sets conjecture holds if its height number H(F) no more than 3 or no less than n-1. Since the height number H(F) is a positive integer which is bounded between 1 to n. As an attempted approach and framework, if we can prove union-closed sets conjecture holds for all possible value of H(F), then union-closed sets conjecture is true.

math.CO↗