arXiv subjects

Xinan Dai

Publications and source records attributed to Xinan Dai.

13 recordsLinked to original sources

arXiv2026

Topological isomorphism of procountable groups is universal analytic

We prove that topological isomorphism of procountable groups is a universal analytic equivalence relation, answering a question of Gao, Nies, and Paolini. The same conclusion follows for non-Archimedean Polish groups. More strongly, there is one countable group $H$ for which universality already holds among inverse limits of sequences of surjective endomorphisms of $H$. Thus all the classification complexity can be carried by the bonding maps. We encode permutative equivalence of unconditional basic sequences by integer weights with prescribed symmetries. An extension of Gao, Nies, and Paolini's tree construction recovers bounded weight differences from uniform continuity while allowing these symmetries to act. Przeździecki's almost-full functor transfers the resulting inverse graph systems to groups. Identifying the stage graphs with a single graph makes the passage from weights to bonding endomorphisms continuous.

math.GR↗

arXiv2026

The Borel complexity of conjugacy for Cantor minimal systems

We prove that conjugacy of minimal homeomorphisms of the Cantor space is Borel bireducible with isomorphism of countable graphs, answering the Cantor minimal case of a question of Foreman. We obtain the lower bound by encoding countably based profinite groups. Finite quotient homomorphisms are represented by factor maps between a common family of minimal subshifts. Amalgamation makes the resulting inverse limit independent, up to conjugacy, of the quotient presentation. Conversely, finite-stage factorization recovers the group from any conjugacy of these inverse limits.

math.DS↗

arXiv2026

Centralizer Excess as an Obstruction to Carlson's Depth Conjecture

Let $G=\operatorname{SmallGroup}(128,859)$ and $k=\overline{k}$. The cohomology ring $H^*(G;k)$ has depth two, and we prove that the minimum quotient dimension of an associated prime is exactly three. Okuyama's theorem shows that an integer $r$ occurs as such a dimension exactly when there is an elementary abelian subgroup $E\leq G$ of rank $r$ with $\operatorname{depth} H^*(C_G(E);k)=r$. We use this equivalence to define the centralizer excess. If $d=\operatorname{depth} H^*(K;k)$, Carlson's equality holds precisely when some rank-$d$ subgroup has zero excess. For $G$, all rank-two centralizers have positive excess. A complete enumeration of the thirty-one actual rank-three elementary abelian subgroups finds six zero-excess witnesses. Hence $ω_a\bigl(H^*(G;k)\bigr)=3$. Since $H^*(G\times(C_2)^n;k)\cong H^*(G;k)[u_1,\dots,u_n]$, the standard behavior of associated primes under polynomial extension gives $ω_a\bigl(H^*(G\times(C_2)^n;k)\bigr)=n+3$ for $n\geq0$. We also study the class $α_0=g+fc\in H^3(G;\mathbb F_2)$. It is killed by two degree-one classes but restricts nontrivially to a rank-four elementary abelian subgroup. It follows that $\dim H^*(G;\mathbb F_2)/\operatorname{ann}(α_0)=4$. Thus two explicit linear annihilators do not force a two-dimensional cyclic support. The assertion is about Krull dimension; it does not say that the support is the whole spectrum.

math.GR↗

arXiv2026

Bounded Representatives in Critical Sobolev-Hodge Spaces

Let $n\geq 2$, $1\leq \ell\leq n-1$, and $1<p<\infty$. We prove that every $v\in \dot W^{n/p,p}(\mathbb{R}^n;Λ^\ell)$ has a representative $u\in \dot W^{n/p,p}\cap L^\infty$ with $du=dv$ and $\max{|u|{\dot W^{n/p,p}},|u|{L^\infty}}\lesssim |v|{\dot W^{n/p,p}}$. Equivalently, $d[\dot W^{n/p,p}Λ^\ell]=d[(\dot W^{n/p,p}\cap L^\infty)Λ^\ell]$ with equivalent quotient norms. The proof reduces the selection problem to an endpoint graph estimate for a Riesz potential and the exact Hodge projection. Its main analytic input is a finite-dimensional-input Maz'ya--$Φ$ inequality for operator-valued homogeneous kernels. For the Riesz/Hodge pair, a nonlinear spherical profile built from the projected kernel has exact atomic cancellation and is coercive by the identity $\int{S^{n-1}}P(θ),d\barσ=(\ell/n)\operatorname{Id}$. Frequency-localized graph closure and Hahn--Banach duality then return a bounded representative.

math.FA↗

arXiv2026

The Minimum Cardinality of a Dependent Finite Gabor System Is Four

Recent work produced a linearly dependent system of twelve time--frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $α=\frac13+10^{-12}\sqrt2$ and $β=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $λ\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(α,β/2)f=λf$, where $W$ denotes the Weyl time--frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb R)$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.

math.FA↗

arXiv2026

A Finite E-Group of Nilpotency Class Three

A group is an E-group if every element commutes with each of its endomorphic images. Caranti asked whether a finite E-group can have nilpotency class three. We prove that the $3$-group of order $3^{84}$ introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later shown by Abdollahi, Faghihi, Linton, and O'Brien to have the corresponding automorphism property, is an E-group. Let $P$ denote this group and put $V=P/Φ(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrowΛ^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteqΛ^2 U$. Since the image induced by any endomorphism of $P$ on $V$ has precisely this closure property, every endomorphism acts on $V$ either invertibly or trivially. The invertible case is the known A-group case. In the trivial case the image first lies in $Φ(P)=P'$, and the power relations then force it into $Ω_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

math.GR↗

arXiv2026

L-packet multiplicity and integral structure in the K-theory of real inner forms

Let $G$ be a connected linear real semisimple group with finite centre and discrete series, and let $G_c$ be a fixed compact inner form. We isolate an integral structure behind the discrete-series character identity. There are natural homomorphisms $\mathcal{C}_G:K_0(C_r^*(G))\longrightarrow R(G_c)$ and $\mathcal{J}_G:R(G_c)\longrightarrow K_0(C_r^*(G))$, characterised, respectively, by stable and ordinary elliptic orbital integrals. The first sends a noncompact Dolbeault--Dirac index to its compact counterpart; the second sends an irreducible representation of $G_c$ to the signed sum of the $K$-theory classes in the corresponding discrete-series $L$-packet. We prove $\mathcal{C}_G\mathcal{J}_G=[W_G:W_K]\,\mathrm{id}_{R(G_c)}$. Thus the packet cardinality is the precise integral cost of splitting stable orbital averaging. Writing $S_G=\operatorname{im}\mathcal{J}_G$ and $U_G=\ker\mathcal{C}_G$, we obtain the exact obstruction sequence $0\longrightarrow S_G\oplus U_G\longrightarrow K_0(C_r^*(G))\longrightarrow R(G_c)/[W_G:W_K]R(G_c)\longrightarrow 0$. After inverting the packet cardinality, this yields a canonical stable projector and functorial transfers between the stable $K$-theory lattices of real inner forms. For $\operatorname{SL}(2,\mathbb{R})$ the obstruction is $(\mathbb{Z}/2\mathbb{Z})[z+z^{-1}]$. For the inner forms of type $C_n$, the relevant multiplier is $2^n$ for $\operatorname{Sp}(2n,\mathbb{R})$ and $\binom{n}{p}$ for $\operatorname{Sp}(p,n-p)$.

math.KT↗

arXiv2026

An Explicit Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets

In Problem 6 of his 1988 paper on differential posets, Stanley asked for the least possible cardinality of a fixed rank of an $r$-differential poset and suggested that the minimum should be attained by $Y^r$, the $r$-fold Cartesian power of Young's lattice. We disprove the resulting universal coefficientwise lower bound. For every $r\geq 3$, we construct an infinite $r$-differential poset $P^{(r)}$ satisfying $\lvert P^{(r)}_4\rvert=\lvert (Y^r)_4\rvert-\lfloor r/3\rfloor$. For $r=3$, the construction replaces thirteen rank-four lower-cover blocks of $Y^3$ by twelve blocks with the same point and pair incidence multiplicities, producing the initial rank sequence $1,3,9,22,50$ instead of $1,3,9,22,51$. A reflection extension then yields an infinite differential poset. The construction does not address the cases $r=1$ and $r=2$.

math.CO↗

arXiv2026

A Negative Answer to Stanley's Problem 4 on Differential Posets

In Problem 4 of his 1988 paper on differential posets, Stanley asked whether the weighted $k$-multichain series of an arbitrary differential poset is always a rational multiple of the $k$th power of its rank series. We answer this question in the negative. Already for $k=2$, we construct a locally finite $1$-differential poset $P$ for which $M_{P,2}(q)/F_P(q)^2$ is not rational over any field of characteristic zero. The construction uses Wagner's crown reflection. At each sufficiently high rank there are two valid one-rank extensions with the same required Gram matrix and with new rank sizes differing by one. We show that the first quotient coefficient affected by that rank distinguishes the two choices. Choosing successively between them and diagonalizing against the rational power series produces the required poset. The same coefficient argument also yields continuum many distinct quotient series, of which continuum many are nonrational.

math.CO↗

arXiv2026

Self-complementary completions on six vertices

Let $\cthreshold(n)$ be the largest integer $q$ such that every loopless digraph on $n$ vertices with at most $q$ arcs is isomorphic to a spanning subdigraph of a self-complementary digraph of order $n$. We prove that $\cthreshold(6)=7$. The upper bound is witnessed by \[ \bK{3}\dunion (x\longrightarrow y\longrightarrow z), \] and follows from a direct argument with a self-complementing permutation. We also determine the complete eight-arc obstruction layer: it consists of five isomorphism classes, or three after converse digraphs are identified. All five are arc-minimal. Each nevertheless packs with an isomorphic copy of itself, so ordinary packing is strictly weaker than same-order self-complementary completion already at this first failure layer.

math.CO↗

arXiv2026

An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128

In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let \[ G=\SG{128}{859},\qquad k=\kbar. \] An exact presentation certificate proves that $\depth H^*(G;k)=2$. Okuyama's associated-prime theorem would convert an associated prime of dimension two into a rank-two elementary abelian subgroup $E\leq G$ satisfying $\depth H^*(C_G(E);k)=2$. We enumerate all $75$ rank-two elementary abelian subgroups of $G$ and obtain six centralizer types. Duflot's theorem gives depth at least three for four types, while exact ideal-quotient certificates exhibit regular sequences of length three for the remaining two. Hence every rank-two centralizer has cohomological depth at least three, so $H^*(G;k)$ has no associated prime of dimension two. The finite group presentation, the three cohomology-ring presentations, the enumeration summary, and the exact algebraic certificates are included for independent verification.

math.GR↗

arXiv2025

Extreme Solar Storm Reveals Causal Interactions in Space Weather

Solar storms perturb Earth's magnetosphere, triggering geomagnetic storms that threaten space-based systems and infrastructure. Despite advances in spaceborne and ground-based observations, the causal chain driving solar-magnetosphere-ionosphere dynamics remains elusive due to multiphysics coupling, nonlinearity, and cross-scale complexity. This study presents an information-theoretic framework to decipher interaction mechanisms in extreme solar geomagnetic storms across intensity levels within space weather causal chains, using 1980-2024 datasets. Unexpectedly, we uncover auroral spatial causality patterns associated with space weather threats in the Arctic during May 2024 extreme storms. By integrating causal consistency constraints into spatiotemporal modeling, SolarAurora outperforms existing frameworks, achieving superior accuracy in forecasting May/October 2024 events. These results advance understanding of space weather dynamics and establish a promising framework for scientific discovery and forecasting extreme space weather events.

physics.space-ph↗

arXiv2025

DeepMath-Creative: A Benchmark for Evaluating Mathematical Creativity of Large Language Models

To advance the mathematical proficiency of large language models (LLMs), the DeepMath team has launched an open-source initiative aimed at developing an open mathematical LLM and systematically evaluating its mathematical creativity. This paper represents the initial contribution of this initiative. While recent developments in mathematical LLMs have predominantly emphasized reasoning skills, as evidenced by benchmarks on elementary to undergraduate-level mathematical tasks, the creative capabilities of these models have received comparatively little attention, and evaluation datasets remain scarce. To address this gap, we propose an evaluation criteria for mathematical creativity and introduce DeepMath-Creative, a novel, high-quality benchmark comprising constructive problems across algebra, geometry, analysis, and other domains. We conduct a systematic evaluation of mainstream LLMs' creative problem-solving abilities using this dataset. Experimental results show that even under lenient scoring criteria -- emphasizing core solution components and disregarding minor inaccuracies, such as small logical gaps, incomplete justifications, or redundant explanations -- the best-performing model, O3 Mini, achieves merely 70% accuracy, primarily on basic undergraduate-level constructive tasks. Performance declines sharply on more complex problems, with models failing to provide substantive strategies for open problems. These findings suggest that, although current LLMs display a degree of constructive proficiency on familiar and lower-difficulty problems, such performance is likely attributable to the recombination of memorized patterns rather than authentic creative insight or novel synthesis.

cs.AI↗