arXiv subjects

Quanyu Tang

Publications and source records attributed to Quanyu Tang.

At least 19 recordsLinked to original sources

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

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 provides an independent negative answer to a question going back to Nevanlinna's 1925 work that had remained open for over a century. 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

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↗

arXiv2026

Minimum-rank parameters of complements of threshold Kneser graphs

Let $J_{\ge s}(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]$, with two distinct vertices adjacent whenever their intersection has size at least $s$. Equivalently, $J_{\ge s}(n,k)$ is the complement of a threshold Kneser graph. We determine both the symmetric minimum rank over an arbitrary infinite field and the real positive semidefinite minimum rank of this family. Specifically, for $k\ge2$, $1\le s\le k-1$, and $n\ge2k-s$, we prove $$ \operatorname{mr}^{\mathbb F}\left(J_{\ge s}(n,k)\right) = \binom{n-2(k-s)}{s} $$ for every infinite field $\mathbb F$, and $$ \operatorname{mr}_{+}^{\mathbb R}\left(J_{\ge s}(n,k)\right) = \binom{n-2(k-s)}{s}. $$ The lower bound follows from a diagonal submatrix indexed by two carefully chosen families of $k$-subsets. For the upper bound, we construct a symmetric matrix using an exterior power of a bilinear form, a Lagrange interpolation identity, and a generic nonvanishing argument. Over $\mathbb R$, an interlacing choice of parameters makes the bilinear form positive definite and yields a positive semidefinite matrix attaining the required upper bound. As consequences, we answer a question from an American Institute of Mathematics workshop, determine the real faithful orthogonality dimension of all graphs $J_{\ge s}(n,k)$ in the stated range, and recover the known minimum-rank formula for Johnson graphs.

math.CO↗

arXiv2026

The positive and negative square-energy conjecture

Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$ \min\{s^+(G),s^-(G)\}\ge n-1. $$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.

math.CO↗

arXiv2026

A positive square-energy strengthening of Turán's theorem

Let $G$ be an $n$-vertex graph with clique number $ω(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{ω(G)}\right)n. $$ This strengthens Wilf's classical spectral Turán theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin--Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro--Wei greedy algorithm on the complement.

math.CO↗

arXiv2026

Supersaturation in Nosal graphs: Triangles and books

In this paper, we use the spectral surplus $λ(G) - \sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph $G$ with $m\ge 3$ edges and $λ(G) \ge 1 + \sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \vee \tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \ge \lfloor \tfrac{1}{2}(\sqrt{m}-1) \rfloor$ proved by Ning and Zhai, and the second layer $t(G) \ge \tfrac{m-1}{2}$ by Zhang and Zhai. (b) Every $m$-edge graph $G$ satisfies $t(G) \ge m\bigl(λ- \sqrt{m}\,\bigr)$, with equality if and only if $G$ is complete bipartite. Consequently, $λ(G) \ge \sqrt{m} + q$ forces $t(G) > q m$ for every real $q > 0$. This is an edge-spectral counterpart of the Lovász--Simonovits theorem, and it improves the Bollobás--Nikiforov bound $t(G) \ge \tfrac13 λ(λ^2 - m)$ in the range $\sqrt m \le λ(G) \le 1.3\sqrt m $. (c) Every $m$-edge Nosal graph $G$ contains a book of size greater than $\tfrac14 \sqrt{m}$. This improves two recent results on the booksize constant: $\tfrac{1}{24}$ proved by Li, Liu and Zhang, and $\tfrac19$ by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant $\tfrac13$. (d) Every $m$-edge Nosal graph $G$ contains at least $\bigl(\tfrac{1}{8} - o(1)\bigr) m$ copies of the kite $C_4^+=B_2$, and the constant $\tfrac18$ is best possible. This determines the sharp asymptotic constant for counting $C_4^+$ and strengthens the $Ω(m)$ bound of Li, Liu and Zhang.

math.CO↗

arXiv2026

Rational values and non-principality for $\ell$-degree Turán densities of hypergraphs

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ of $\mathcal{F}$ is defined as the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all $k>\ell>k/2$, every rational number $α\in[0,1)$ can be realized as the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ for some finite family $\mathcal{F}$ of $k$-graphs. Furthermore, for any $k>\ell\ge 1$, we construct an explicit infinite sequence of values realized by single forbidden $k$-graphs: for each integer $q\ge2$, there exists a $k$-graph $F$ such that $γ^{(k)}_\ell(F)=1-q^{\ell-k}$. We also establish a strengthened non-principality property for $\ell$-degree Turán densities: for all $k>\ell>1$, there exist two $k$-graphs $F_1$ and $F_2$ satisfying $0< γ^{(k)}_{\ell}(\{F_1,F_2\})<\min\{ γ^{(k)}_\ell(F_1),γ^{(k)}_\ell(F_2)\}$.

math.CO↗