Searcharxiv⌕ Search

arXiv subjects

Quanyu Tang

Publications and source records attributed to Quanyu Tang.

At least 19 recordsLinked to original sources
arXiv2026

Constructing Casimirs for the quantum reflection equation

We explore the quantum reflection equation in the case of a full-dimensional $n\times n$ matrix $\mathbb A$. Using the technique of $q$-determinants, we find closed expressions for the quantum Pfaffian (in the case of even $n$) and for new quantum Casimirs that are ratios of special quantum (Sklyanin) minors of $\mathbb A$. Special attention is paid to the self-adjointness property of the newly constructed Casimirs. We put forward a conjecture that, adopting a natural ansatz for a quantum deformation of a Casimir of a classical algebra, self-adjointness implies commutativity. We also explore the relationship to networks of quantum cluster variables.

math-ph↗
arXiv2026

Three omitted values and non-Blaschke point divisors in half-planes

We construct a real meromorphic function $F$ on $\mathbb C$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb R$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in\widehat{\mathbb C}\setminus\{0,1,\infty\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction gives a negative answer to the half-plane bounded-type question arising from Nevanlinna's 1925 work. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.

math.CV↗
arXiv2026

On almost perfect additive complements

Let $A, B$ be infinite sets of non-negative integers with $n\in A+B$ for all sufficiently large integers $n$. We prove that if $A(x)B(x)-x$ does not tend to infinity, then $$\limsup_{x\to\infty}\frac{A(x)B(x)}{x}\in\Big[\frac{3}{2}, 2\Big].$$ The lower bound is sharp, and confirms a conjecture of Chen. Our proof is based on the weak limits and the convolution factorization theorem of Gabardo and Lai. A discrete proof of the same bound was also given by Zhang. Motivated by the known spectrum of perfect additive complements, we conjecture that the interval $(3/2, 8/5)$ is also excluded for almost perfect additive complements.

math.NT↗
arXiv2026

Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains

We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.

math.AP↗
arXiv2026

The number of Laplacian eigenvalues of trees less than one

Let $m_T[0,1)$ denote the number of Laplacian eigenvalues of a tree $T$ that are strictly less than $1$. Guo, Xue and Liu \cite{GXL} proved that every tree $T$ of diameter $d$ satisfies $m_T[0,1)\ge \lceil(d+1)/3\rceil$, and that this bound is sharp when $d\equiv2\pmod3$. It is classical that every graph of diameter $d$ has domination number at least $\lceil(d+1)/3\rceil$. In this paper, for any diameter $d$, $m_T[0,1)=\lceil(d+1)/3\rceil$ if and only if $γ(T)=\lceil(d+1)/3\rceil$; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy $m_T[0,1)\ge\lceil(d+1)/3\rceil+1$.

math.CO↗
arXiv2026

The solution to Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors

In 1967, Kadison asked whether every type $\mathrm{II}_1$ factor admits an orthonormal basis, with respect to its trace, consisting of unitaries. We resolve this problem in full generality and, more broadly, characterize the diffuse finite von Neumann algebras admitting such bases consisting of symmetries. Let $M$ be a diffuse finite von Neumann algebra with a faithful normal tracial state $τ$, let $κ$ be the density character of $L^2(M,τ)$, and identify $M$ with its canonical image in $L^2(M,τ)$. We prove that there exists a family $B\subset {s\in M:s=s^*=s^{-1},\ τ(s)=0}$ such that ${1}\cup B$ is an orthonormal basis of $L^2(M,τ)$ if and only if the density character of $L^2(zM,τ(z)^{-1}τ|_{zM})$ equals $κ$ for every nonzero central projection $z\in Z(M)$. In particular, every type $\mathrm{II}_1$ factor admits an orthonormal basis consisting of unitaries, thereby answering Kadison's question affirmatively. We also provide a Lean 4 formalization of the main results.

math.OA↗
arXiv2026

Integral Representations and Asymptotics for a Family of Areal Mahler Measures

We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$ P_m(x_1,\ldots,x_m,u) = \prod_{j=1}^m(1+x_j) + u\prod_{j=1}^m(1-x_j), \qquad m\geq1. $$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for $\mathrm m_{\mathbb D}(P_m)$. We prove that, for every fixed $m$, the value $π^m\mathrm m_{\mathbb D}(P_m)$ belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case $m=2$ explicitly in terms of $\operatorname{Li}_4(1/2)$, $ζ(3)$, Catalan's constant, $π$, and $\log2$. We also derive an explicit three-term asymptotic expansion for $\mathrm m_{\mathbb D}(P_m)$ as $m\to\infty$.

math.NT↗
arXiv2026

Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane

Let $Ω\subset\mathbb R^2$ be a bounded connected Lipschitz domain, and let $\{μ_j(Ω)\}_{j\geq1}$ and $\{λ_j(Ω)\}_{j\geq1}$ denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that $$ μ_3(Ω)<λ_1(Ω), $$ thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.

math.SP↗
arXiv2026

The $Δ$-Conjecture for CIS $d$-Graphs

We prove the $Δ$-conjecture, which dates back to Gurvich's 1978 thesis. Specifically, let the edges of a complete graph be colored with colors $1,\ldots,d$, and for each $i$ let $G_i$ be the graph on the same vertex set formed by the edges of color $i$. We prove that if every choice of a maximal stable set $S_i$ of $G_i$, one for each $i\in[d]$, has nonempty intersection, then the coloring contains no rainbow triangle. Together with a result of Andrade, Boros, and Gurvich, this characterizes CIS $d$-graphs as precisely the Gallai $d$-graphs whose chromatic components are ordinary CIS graphs. We also show that every factor in the canonical modular decomposition of a CIS $d$-graph is a CIS $d$-graph whose edge-coloring uses at most two colors.

math.CO↗
arXiv2026

Tilting Completion and Full-Rank Self-Orthogonal Modules

We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.

math.RT↗
arXiv2026

Infinitely Many Components in Auslander--Reiten Quivers of Representation-Infinite Algebras over Perfect Fields

Let $k$ be a perfect field and let $A$ be a representation-infinite finite-dimensional $k$-algebra. We prove that the Auslander--Reiten quiver of $A$ has infinitely many connected components. This establishes, for finite-dimensional algebras over perfect fields, a conjecture of Auslander, Reiten, and Smalø concerning Artin algebras. Over an algebraically closed field, the proof combines a localized polynomial representation embedding with semilinear twists induced by field automorphisms. The passage from a perfect field to its algebraic closure is obtained by separable base change: we prove that if the Auslander--Reiten quiver of $A$ has only finitely many components, then the same holds for the scalar extension to the algebraic closure.

math.RT↗
arXiv2026

A Family of Simultaneously Cospectral Trees for Degree-Distance Matrices

Spectral characterization of graphs for various graph matrices constitutes a central topic in spectral graph theory. Let $G$ be a graph with adjacency matrix $A(G)$, diagonal degree matrix $\Deg(G)$, distance matrix $D(G)$, and transmission matrix \(\Trs(G)\), respectively. Recently, Alfaro and Zapata (2024) introduced the degree-distance matrices \(\Ddegp(G)=\Deg(G)+D(G)\) and \(\Ddeg(G)=\Deg(G)-D(G)\), together with the transmission-adjacency matrices \(\Atrsp(G)=\Trs(G)+A(G)\) and \(\Atrs(G)=\Trs(G)-A(G)\). Based on computational evidence for trees on at most \(20\) vertices, they conjectured that all trees are determined by the spectra of \(\Ddegp\) as well as \(\Ddeg\). In this paper, we disprove these conjectures by constructing an infinite family of pairs of non-isomorphic trees. More precisely, for each integer \(r\ge 3\), we construct a pair of trees on \(17r-15\) vertices which are simultaneously cospectral with respect to the following six matrices \[ A,\quad L,\quad Q,\quad D,\quad \Ddegp,\quad \Ddeg . \] The construction is based on an \(r\)-regularized leaf extension and an equitable-partition reduction. We also record a simple sign-switching observation for transmission-adjacency matrices: if \(G\) is bipartite, then \(\Atrs(G)\) and \(\Atrsp(G)\) are similar via a diagonal \(\{\pm1\}\)-matrix and have the same Smith normal form. Consequently, for trees, the spectral and Smith normal form problems for \(\Atrs\) and \(\Atrsp\) are equivalent.

math.CO↗
arXiv2026

Nearly permanental cospectral graphs

Let $G$ be a simple graph of order $n$ with adjacency matrix $A= (a_{ij})$. The \emph{determinant} and the \emph{permanen}t of the matrix $A$ are defined as \[\mathrm{det}A= \sum_{σ\in S_n}\mathrm{sgn}(σ) \prod_{i=1}^n a_{iσ(i)}\quad\text{and}\quad\mathrm{per}A= \sum_{σ\in S_n} \prod_{i=1}^n a_{iσ(i)},\]respectively. The polynomials $ϕ(G;x) =\mathrm{det}(xI-A(G))$ and $π(G;x) =\mathrm{per}(xI-A(G))$ are called the \emph{characteristic polynomial} and the \emph{permanental polynomial} of $G$, respectively. Two graphs are said to be \emph{nearly cospectral} with respect to the determinant (resp. permanent) if the difference of their characteristic (resp. permanental) polynomials is a constant. Lv et al. introduced the nearly cospectral graphs problem with respect to the determinant, and provided partial results in the case modulo 4. In this paper, we mainly prove that the corresponding results also hold for the nearly cospectral graphs problem with respect to the permanent. The determinant and permanent are the immanants corresponding to the irreducible characters $(1^n)$ and $(n)$ of the symmetric group $ S_n $, respectively. Here, the \emph{immanant} $d_λ(A)$ of $A$ is defined as \[d_λ(A) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n a_{iσ(i)},\] where $χ_λ$ is the irreducible character of $ S_n $ indexed by the partition $ λ$. The immanantal polynomial of $G$ associated with $ χ_λ$ is given by $ ϕ_λ(G;x)=d_λ(xI-A) $. In this paper, we also establish a similar result for nearly immanantal cospectral graphs in $\mathbb{F}_2[x]$ for all irreducible characters $χ_λ$.

math.CO↗
arXiv2026

Strictly Convex Steklov-Isospectral Plane Domains

We construct pairs of noncongruent bounded Euclidean plane domains with identical Steklov spectra, including multiplicities. This gives a negative answer to the planar Steklov analogue of Kac's question ``Can one hear the shape of a drum?'' The domains are simply connected and strictly convex, have real-analytic boundaries, and may be chosen arbitrarily close to a disk in the $C^\infty$ topology.

math.SP↗
arXiv2026

Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture

Let $λ_k(Ω)$ denote the $k$th Dirichlet Laplacian eigenvalue of a bounded planar domain $Ω$, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne--Pólya--Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer $m\geq3$, we construct a bounded planar domain $Ω_m$ with $C^\infty$ boundary such that $$ \frac{λ_{2m}(Ω_m)}{λ_m(Ω_m)} >\frac{13}{5} >\frac{j_{1,1}^2}{j_{0,1}^2}, $$ where $j_{ν,1}$ is the first positive zero of $J_ν$, the Bessel function of the first kind of order $ν$. For $m=3$, the annulus $\{x\in\mathbb{R}^2:1/10<|x|<1\}$ already provides such a counterexample.

math.SP↗
arXiv2026

Eventually greedy best Egyptian underapproximations of rational numbers via optimal control

We prove that every positive rational number has eventually greedy best Egyptian underapproximations, both when repetitions of the denominators are allowed and when the denominators are required to be distinct. This answers affirmatively a problem originating with Erdős and Graham and later revisited by Nathanson, and yields an application concerning the maximal asymptotic growth of denominators in unit fraction series converging to a given rational number. We reformulate the question as an optimal control problem for a dynamical system, construct an appropriate payoff function, and study properties of the associated Bellman function. We also answer another question of Nathanson by constructing an irrational number with unique and greedy best Egyptian underapproximations.

math.NT↗
arXiv2026

Graph Eigenvalues and Projection Constants

For an integer $k\ge2$, let $λ_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ λ_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $γ(r)$ due to Deręgowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of König and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.

math.CO↗