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Strict Monotonicity of Numerical Invariants for the Submodules $[(z-w)^k]$ in $H^2(\mathbb D^2)$
Authors:
Yin Liu,
Yufeng Lu,
Chao Zu
Abstract:
For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of the core operator: \[ σ(C_{M_k}) =
\{0,1\} \cup \left\{ \pm\frac{k}{n+k}:n\geq1 \r…
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For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of the core operator: \[ σ(C_{M_k}) =
\{0,1\} \cup \left\{ \pm\frac{k}{n+k}:n\geq1 \right\}. \] In particular, the spectral data determine the parameter $k$.
The determinant and cofactor formulas further yield a unified finite-sum representation for $α_{n,j}^{(k)} =\langle w^jφ_n,z^jψ_n\rangle$, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting $α_{n,j}^{(k)}$ and $α_{n,j+1}^{(k)}$ by means of an explicit telescoping certificate, and show that the corresponding finite-section transformations are strict contractions. Combining these finite-dimensional estimates with the asymptotic behavior of $α_{n,j}^{(k)}$, we prove the strict monotonicity \[ Σ_0(M_k)> Σ_1(M_k)> Σ_2(M_k)> \cdots . \] The cases $k\geq3$ constitute the new part of the analysis, while the previously known cases $k=1,2$ are recovered within the same framework. Consequently, Yang's monotonicity conjecture holds in strict form for the entire family $\{[(z-w)^k]:k\geq1\}$.
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Submitted 18 August, 2026;
originally announced August 2026.
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A Counting Lemma for Somewhat Restricted 3-APs
Authors:
Amey Bhangale,
Subhash Khot,
Yang P. Liu,
Dor Minzer
Abstract:
For a prime $p\geq 3$, a somewhat restricted $3$-AP in $\mathbb{F}_p^n$ is a triplet $(x,x+a,x+2a)$, where $x\in\mathbb{F}_p^n$ and $a\in \{0,1,2\}^n$. We prove a counting lemma for somewhat restricted $3$-APs in dense sets in $\mathbb{F}_p^n$. More precisely, we prove that for all $α>0$, there exists $β>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at…
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For a prime $p\geq 3$, a somewhat restricted $3$-AP in $\mathbb{F}_p^n$ is a triplet $(x,x+a,x+2a)$, where $x\in\mathbb{F}_p^n$ and $a\in \{0,1,2\}^n$. We prove a counting lemma for somewhat restricted $3$-APs in dense sets in $\mathbb{F}_p^n$. More precisely, we prove that for all $α>0$, there exists $β>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at least $α$, then it contains at least $β$ fraction of all somewhat restricted $3$-APs.
Our proof builds on recently developed machinery from [Bhangale, Khot, Minzer, 2026]. Our main new ingredient is an arithmetic regularity lemma for patterns such as somewhat restricted 3-APs. This result is in the spirit of arithmetic regularity lemmas from the theory of Gowers uniformity norms [Green, Tao, 2010] and may be of independent interest.
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Submitted 18 August, 2026;
originally announced August 2026.
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Extremal graphs for a conjecture on the square energy of graphs
Authors:
Fu-Tao Hu,
Ya-Yang Liu,
Yi Wang
Abstract:
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[
\min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, equality for $s^+$ holds exactly for trees, whereas equality for $s^-$ holds ex…
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For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[
\min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, equality for $s^+$ holds exactly for trees, whereas equality for $s^-$ holds exactly for trees and complete graphs. The proof combines the $P_3$-removal lemma in the no-cut-vertex case with a detailed equality analysis of the underlying doubly nonnegative matrix inequality. Every block is forced to be complete, and a minimal-counterexample argument gives an exact rank-one decomposition of the folded matrix $M^c$. The resulting non-edge vanishings, together with $AX=XA$, rule out an interface between a bridge and a nontrivial block.
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Submitted 17 August, 2026;
originally announced August 2026.
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A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $σ_k$-curvature Metrics
Authors:
Jiahuan Li,
Yilu Liu,
Xi-Nan Ma
Abstract:
Let $k\in\mathbb N$ satisfy $1<k<n/2$, and let $Σ^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\setminusΣ$ satisfying $λ(g^{-1}A_g)\in\Gk$ and $σ_k(g^{-1}A_g)=κ>0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth…
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Let $k\in\mathbb N$ satisfy $1<k<n/2$, and let $Σ^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\setminusΣ$ satisfying $λ(g^{-1}A_g)\in\Gk$ and $σ_k(g^{-1}A_g)=κ>0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.
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Submitted 17 August, 2026;
originally announced August 2026.
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zenDot: An LLM-integrated quantum TCAD platform for semiconductor quantum-device design and optimization automation
Authors:
Zeheng Wang,
Yan Liu,
Yue Hao,
Genquan Han
Abstract:
Semiconductor quantum-device design still lacks an integrated Technology Computer-Aided Design (TCAD)-like environment that connects material geometry, quantum many-body simulation, and automated design. Here we introduce zenDot, a large-language model (LLM)-integrated quantum TCAD platform that links a material-labelled device state to a unified condensed-matter physics toolbox. The device and ca…
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Semiconductor quantum-device design still lacks an integrated Technology Computer-Aided Design (TCAD)-like environment that connects material geometry, quantum many-body simulation, and automated design. Here we introduce zenDot, a large-language model (LLM)-integrated quantum TCAD platform that links a material-labelled device state to a unified condensed-matter physics toolbox. The device and calculation components are integrated into a desktop workbench, Python API, and an embedded LLM agent, allowing electrostatics, charge and transport characterization, correlated-state calculations, and qubit modelling to be executed within one reproducible environment. We demonstrate zenDot on a Si/SiO2 double quantum dot, where a single device state reproduces the characterization workflow and supports hybrid, tunnel-charge, and singlet-triplet qubit analyses. A platform-level universal-control scan revises the singlet-triplet operating point and reduces the predicted worst-gate infidelity by nearly 30-fold. Beyond analysis, the LLM agent directly operates the same physics environment as human users, proposing design changes, executing registered simulations, and iterating on solver-returned metrics under physics-aware validation. Across three demonstration tasks it completes 18 validated design iterations, including geometry modification followed by a full re-solve from the material stack. zenDot establishes a machine-operable quantum TCAD workflow that connects device physics with LLM-driven design exploration.
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Submitted 16 August, 2026;
originally announced August 2026.
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On Brezis' open problem 2.2
Authors:
Hong-Ge Chen,
Yong Liu,
Juncheng Wei,
Wen Yang
Abstract:
We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg--Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem list. This is achieved by comparing the radial solution $f$ in the disk with the degree-one radial solution $F$ in…
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We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg--Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem list. This is achieved by comparing the radial solution $f$ in the disk with the degree-one radial solution $F$ in the whole plane. Multiplying a disk competitor by $F/f$ enables us to use the known minimality of the whole-plane vortex without changing the boundary trace. The difference of the two energies can be decomposed into Fourier modes. Every nonzero mode is nonnegative, and the zero mode is then handled by a Picone type identity.
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Submitted 16 August, 2026;
originally announced August 2026.
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Support $τ$-tilting posets and Hochschild reconstruction for matrix centralizer algebras
Authors:
Jiangsheng Hu,
Yu-Zhe Liu,
Tiwei Zhao
Abstract:
Let $R$ be a field, and let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$-algebra of Loewy length $\ell$ with $\operatorname{rad}\mathcal{O}=(π)$. For a nonempty set $P=\{p_1<\cdots<p_s\}\subseteq\{1,\ldots,\ell\}$, set $Λ_{\mathcal{O}}(P):=\operatorname{End}_{\mathcal{O}}(\bigoplus_{p\in P}\mathcal{O}/(π^p))$. We prove that the support $τ$-tilting poset of…
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Let $R$ be a field, and let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$-algebra of Loewy length $\ell$ with $\operatorname{rad}\mathcal{O}=(π)$. For a nonempty set $P=\{p_1<\cdots<p_s\}\subseteq\{1,\ldots,\ell\}$, set $Λ_{\mathcal{O}}(P):=\operatorname{End}_{\mathcal{O}}(\bigoplus_{p\in P}\mathcal{O}/(π^p))$. We prove that the support $τ$-tilting poset of $Λ_{\mathcal{O}}(P)$ is isomorphic to the left weak order on $Σ_{s+1}$ and hence recovers precisely $s$. We also show that $Z(Λ_{\mathcal{O}}(P))\simeq\mathcal{O}/(π^{p_s})$ and, as a module over this center, $HH_0(Λ_{\mathcal{O}}(P))\simeq\bigoplus_{i=1}^s\mathcal{O}/(π^{g_i})$, where $g_1=p_1$ and $g_i=p_i-p_{i-1}$ for $i\ge2$. Applied blockwise, these formulas compute the corresponding invariants of matrix centralizer algebras over arbitrary fields. Finally, a polynomial primary block is Morita equivalent to a split string algebra if and only if its defining irreducible polynomial is linear and its exponent set is $\{p\}$ or $\{p,p+1\}$. It is Morita equivalent to a split gentle algebra if and only if the polynomial is linear and the exponent set is $\{1\}$, $\{2\}$, or $\{1,2\}$. These criteria yield Morita reconstruction within the corresponding classes.
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Submitted 15 August, 2026;
originally announced August 2026.
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On the Positivity of the Products of Positive Primitive Forms
Authors:
Yuhang Liu
Abstract:
We study the positivity of the product of two positive primitive n-forms whose restrictions to every Lagrangian subspace are nonzero. We prove that in dimensions at most six, the positivity of the product is guaranteed, while in higher dimensions, counterexamples are constructed. This answers Question 2.11 in [1]. The main content of the proof is generated by ChatGPT 5.6 and verified by the author…
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We study the positivity of the product of two positive primitive n-forms whose restrictions to every Lagrangian subspace are nonzero. We prove that in dimensions at most six, the positivity of the product is guaranteed, while in higher dimensions, counterexamples are constructed. This answers Question 2.11 in [1]. The main content of the proof is generated by ChatGPT 5.6 and verified by the author.
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Submitted 14 August, 2026;
originally announced August 2026.
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Ideal class maps on projective spaces and Grassmannians
Authors:
Yufan Liu
Abstract:
We study ideal class maps on projective spaces and Grassmannians over a number field. We prove weak approximation in every fixed ideal class fiber on positive-dimensional linear subspaces and on Grassmannians. We also introduce local defects of morphisms of projective spaces and use them to describe the ideal classes arising from the image of a linear subspace as a finite union of cosets in the cl…
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We study ideal class maps on projective spaces and Grassmannians over a number field. We prove weak approximation in every fixed ideal class fiber on positive-dimensional linear subspaces and on Grassmannians. We also introduce local defects of morphisms of projective spaces and use them to describe the ideal classes arising from the image of a linear subspace as a finite union of cosets in the class group.
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Submitted 16 July, 2026;
originally announced August 2026.
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Brauer graph algebras are closed under stable equivalence of Morita type
Authors:
Pengyun Chen,
Nengqun Li,
Yuming Liu,
Bohan Xing
Abstract:
We study stable equivalences of Brauer graph algebras. In particular, we prove that Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type. As a consequence, we reprove the result of Antipov and Zvonareva that Brauer graph algebras are closed under derived equivalence. As a byproduct, we get a solution of the reconstruction problem posed by Rickard and R…
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We study stable equivalences of Brauer graph algebras. In particular, we prove that Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type. As a consequence, we reprove the result of Antipov and Zvonareva that Brauer graph algebras are closed under derived equivalence. As a byproduct, we get a solution of the reconstruction problem posed by Rickard and Rouquier for algebras stably equivalent of Morita type to Brauer graph algebras.
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Submitted 14 August, 2026;
originally announced August 2026.
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TOGEARI: Interaction-Space Preconditioning for Condensed Finite-Element Systems with IPC Contact
Authors:
Yanlin Liu,
Chao Huang,
Kaixiang Yao,
Yao Shen
Abstract:
Condensed finite-element contact systems combine a large material-volumetric core with a thin factorized contact update. We formulate Truncated Operator-Gram Eigenspace Approximation of Relevant Interactions (TOGEARI), a factor-space compression of that update. It selects a low-dimensional space of contact combinations and installs their contribution as a Woodbury right preconditioner. The assembl…
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Condensed finite-element contact systems combine a large material-volumetric core with a thin factorized contact update. We formulate Truncated Operator-Gram Eigenspace Approximation of Relevant Interactions (TOGEARI), a factor-space compression of that update. It selects a low-dimensional space of contact combinations and installs their contribution as a Woodbury right preconditioner. The assembled Newton equation and independent residual test remain fixed. The inverse action requires an invertible core and reduced Woodbury system. Core-response selection additionally assumes a symmetric core and remains available when that core is indefinite. For a positive definite core, the response values have an energy ordering. The same analysis then gives the exact generalized spectrum, a scaled inverse-error identity, and the optimal worst omitted interaction at each dimension. A raw contact-factor selector provides a lower-setup empirical alternative. We derive the split for a quadratic-tetrahedral displacement formulation with locally condensed elementwise-constant pressure and a factorized positive-semidefinite IPC normal tangent. In a frozen system with 325,260 free coordinates, an eight-dimensional subspace of a 94-row contact factor with registered numerical rank 42 reduces the Arnoldi basis from 11 vectors to 5. The core-response and raw spaces are closely aligned on the primary and near-repeat extractions; row-norm and deterministic-random controls require 12 vectors. The retained-dimension sweep reduces the warm median from 2.849 s to 1.629 s. Post-load setup plus first solve changes from 53.885 s to 61.958 s. These results identify a compact interaction correction in one frozen, frictionless regime. Distinct Newton states, evolving contact, contact-rank and mesh scaling, friction, and nested pressure-space performance remain open experimental questions.
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Submitted 14 August, 2026;
originally announced August 2026.
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Block Repetition of Numerical Invariants for the Submodules $[z^k-w^k]$ in $H^2(\mathbb D^2)$
Authors:
Yin Liu,
Yufeng Lu,
Chao Zu
Abstract:
For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ Σ_0(M_k)=\frac{π^2}{6},\qquad Σ_j(M_k)=Σ_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently,…
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For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ Σ_0(M_k)=\frac{π^2}{6},\qquad Σ_j(M_k)=Σ_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.
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Submitted 13 August, 2026;
originally announced August 2026.
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Sinkhorn Linearization and the Spectral Proxy: Unifying the Statistical and Algorithmic Theory of Feature-Parameterized Inverse Optimal Transport via a Single Spectral Sandwich
Authors:
Han Dong,
Jiaming Li,
Yongqiang Gong,
Ruixi Li,
Yin Liu
Abstract:
We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization -- the implicit-function sensitivity of the entropic OT plan to the cost -- together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent.
The…
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We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization -- the implicit-function sensitivity of the entropic OT plan to the cost -- together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent.
The restricted Hessian on the tangent space satisfies the spectral sandwich (pi_min/epsilon) I <= H_T^{-1} <= (pi_max/epsilon) I, yielding the single core bound sigma_min >= (pi_min/(a_max epsilon)) sqrt(lambda_min(Sigma)) that drives the entire theory. On this core we establish four theorems and one observation.
T1 (identifiability): theta is globally injective on the quotient of the gauge kernel, with dimension bound F <= (K-1)^2. T2 (sparsistency): the l1-penalized estimator recovers the true support under irrepresentability and score concentration, with exponential failure probability. T3 (well-posedness): the feature-moment map M(theta) = Phi^T x_theta is strongly monotone, and the inverse is Lipschitz with constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)). T4 (convergence): local strong convexity with mu >= pi_min^2 lambda_min(Sigma) / epsilon^2 guarantees monotone gradient descent convergence. O5 (misspecification): the estimator converges to the OT-model projection of the truth; the Holder continuity of the projection map is assessed numerically, yielding setting-dependent empirical exponents alpha_eff in (0,1).
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Submitted 13 August, 2026;
originally announced August 2026.
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Katok's intermediate entropy conjecture for amenable group actions
Authors:
Yage Liu,
Ercai Chen,
Xiaoyao Zhou
Abstract:
In this paper, we investigate a broad class of systems for which Katok's intermediate entropy conjecture holds. In particular, we show that a dynamical system with an amenable group action satisfies Katok's intermediate entropy conjecture provided that it has the specification property and is asymptotically entropy expansive.
In this paper, we investigate a broad class of systems for which Katok's intermediate entropy conjecture holds. In particular, we show that a dynamical system with an amenable group action satisfies Katok's intermediate entropy conjecture provided that it has the specification property and is asymptotically entropy expansive.
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Submitted 11 August, 2026;
originally announced August 2026.
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Nonradial stable solutions near the Joseph--Lundgren threshold
Authors:
Shibing Chen,
Yong Liu,
Juncheng Wei,
Wen Yang
Abstract:
We study positive stable solutions of the supercritical Lane--Emden equation in the first Joseph--Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is o…
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We study positive stable solutions of the supercritical Lane--Emden equation in the first Joseph--Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is obtained by matching a polar cap to an inner neck. A sharp expansion of the lowest shifted eigenvalue proves that the resulting singular cone is strictly stable. Finally, a minimal-solution and rescaling argument replaces the cone by a smooth stable entire solution while preserving its sphere variation. To the best of our knowledge, this is the first nontrivial example of nonradial stable solutions for the Emden-Fowler equation.
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Submitted 11 August, 2026;
originally announced August 2026.
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Direct Sum and Direct Product Decompositions of Multivariate Functions
Authors:
Hua-Lin Huang,
Yiming Liu,
Jianhua Xiang,
Yu Ye
Abstract:
This paper addresses the problem of whether or not a vector-valued multivariate functions can be expressed as a sum or a product of vector-valued functions in disjoint sets of variables through a proper invertible linear change of variables. The crux is an invariant algebra, the so-called center, that we introduce for a set of multivariate functions with second order partial derivatives. We thus p…
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This paper addresses the problem of whether or not a vector-valued multivariate functions can be expressed as a sum or a product of vector-valued functions in disjoint sets of variables through a proper invertible linear change of variables. The crux is an invariant algebra, the so-called center, that we introduce for a set of multivariate functions with second order partial derivatives. We thus provide simple criteria and algorithms for simultaneous additive and multiplicative decompositions of any set of multivariate functions with minor analytic conditions. This is applied to the factorization problem of multivariate homogeneous polynomials, in particular those that are products of linear forms.
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Submitted 17 July, 2026;
originally announced August 2026.
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Bijectivity analysis of rational T-spline surfaces via Bernstein representations
Authors:
Jia-Xuan Li,
Ying-Ying Yu,
Ya-Shu Liu,
Xin Li,
Ye Ji,
Chun-Gang Zhu
Abstract:
Ensuring the bijectivity of spline-based parameterizations is fundamental in geometric modeling and isogeometric analysis, as invalid mappings may lead to self-intersections, singular Jacobians, and numerical instability. While T-splines offer enhanced flexibility through local refinement, this flexibility also makes bijectivity verification significantly more challenging. In this work, we propose…
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Ensuring the bijectivity of spline-based parameterizations is fundamental in geometric modeling and isogeometric analysis, as invalid mappings may lead to self-intersections, singular Jacobians, and numerical instability. While T-splines offer enhanced flexibility through local refinement, this flexibility also makes bijectivity verification significantly more challenging. In this work, we propose a rigorous and efficient framework for bijectivity analysis of rational T-spline surfaces based on Bézier extraction. The key idea is to reformulate the T-spline representation into a collection of element-wise rational Bézier patches, on which the Gram determinant of the mapping admits a Bernstein polynomial representation. This enables a coefficient-based analysis of local regularity by exploiting the convex hull and positivity properties of the Bernstein basis. Based on this formulation, we derive a sufficient condition for bijectivity from the nonnegativity of Bernstein coefficients, together with a necessary condition based on the sign consistency of corner coefficients. For cases where these conditions are inconclusive, we introduce a hierarchical subdivision strategy that progressively localizes ambiguous regions and resolves them through refinement. The proposed method provides a certified and adaptive procedure for bijectivity verification that avoids dense numerical sampling and remains computationally efficient. Numerical experiments on complex T-spline geometries demonstrate that the approach accurately detects both valid and near-degenerate configurations, while scaling effectively to large models with thousands of rational Bézier patches. The framework is fully compatible with standard isogeometric analysis workflows.
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Submitted 10 August, 2026;
originally announced August 2026.
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Input convex neural networks as surrogates in mathematical optimisation
Authors:
Yu Liu,
Jan Kronqvist,
Fabricio Oliveira
Abstract:
Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networ…
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Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.
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Submitted 10 August, 2026;
originally announced August 2026.
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A Local Lewy Theorem for $p$-Harmonic Function with Non-zero Gradient in $R^3$
Authors:
Jiahuan Li,
Yilu Liu,
Xi-Nan Ma
Abstract:
We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space $R^3$. Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(Ω)\) be a weak \(p\)-harmonic function in a domain \(Ω\subset\mathbb R^3\), assume it satisfies \(|Du|>0\), we prove that a locally homeomorphic gradient map \(Du\) must have non-vanishing Hessian determinant. Hence \(Du…
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We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space $R^3$. Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(Ω)\) be a weak \(p\)-harmonic function in a domain \(Ω\subset\mathbb R^3\), assume it satisfies \(|Du|>0\), we prove that a locally homeomorphic gradient map \(Du\) must have non-vanishing Hessian determinant. Hence \(Du\) is a local diffeomorphism.
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Submitted 9 August, 2026;
originally announced August 2026.
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The $Δ$-Conjecture for CIS $d$-Graphs
Authors:
Yinchen Liu,
Quanyu Tang
Abstract:
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 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.
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 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.
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Submitted 9 August, 2026;
originally announced August 2026.
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Kernel Localization and Whole-Trajectory Generalization for Linear Multistep Methods in Deep Learning-Based Discovery of Dynamical Systems
Authors:
Yaru Liu,
Yiqi Gu
Abstract:
Linear multistep methods (LMMs) combined with neural-network approximation provide a high-order framework for learning governing vector fields of dynamical systems from discrete trajectory data. This paper studies two issues in LMM-based discovery that are not resolved by existing grid-level convergence theory. First, in non-auxiliary Adams--Bashforth (A-B) and Adams--Moulton (A-M) discovery syste…
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Linear multistep methods (LMMs) combined with neural-network approximation provide a high-order framework for learning governing vector fields of dynamical systems from discrete trajectory data. This paper studies two issues in LMM-based discovery that are not resolved by existing grid-level convergence theory. First, in non-auxiliary Adams--Bashforth (A-B) and Adams--Moulton (A-M) discovery systems, we observe that zero-residual grid solutions are nonunique but consistent along the trajectory, with differences limited to the boundary layer. We explain this phenomenon through a kernel analysis of the non-auxiliary discovery matrices. Under the corresponding discovery-stability conditions, the differences between zero-residual grid solutions are exponentially localized near the initial indices for A-B schemes, whereas they form two-sided boundary layers near the initial and terminal indices for A-M schemes. Second, we derive whole-trajectory generalization estimates for both auxiliary and non-auxiliary formulations. Once the learned vector field is restricted to a fixed observed trajectory, each component of the error becomes a scalar function of time. For the auxiliary formulation, the trajectory error is $O(h^p)$ under the corresponding grid accuracy, approximation, and trace regularity assumptions. For non-auxiliary formulations, the global estimates contain additional boundary-layer terms. On fixed interior subintervals, these terms are exponentially damped. Numerical experiments illustrate the convergence behavior.
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Submitted 8 August, 2026;
originally announced August 2026.
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The error estimate of entropy-stable discontinuous Galerkin methods for hyperbolic conservation laws
Authors:
Xu-Kun Chen,
Yong Liu,
Chi-Wang Shu
Abstract:
Entropy inequalities are fundamental to the well-posedness of hyperbolic conservation laws, providing the essential criterion for selecting the physically admissible solution among infinitely many weak solutions. Chen and Shu [J. Comput. Phys. 345 (2017)] proposed a unified framework for constructing high-order discontinuous Galerkin (DG) methods that satisfy entropy inequalities for any given ent…
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Entropy inequalities are fundamental to the well-posedness of hyperbolic conservation laws, providing the essential criterion for selecting the physically admissible solution among infinitely many weak solutions. Chen and Shu [J. Comput. Phys. 345 (2017)] proposed a unified framework for constructing high-order discontinuous Galerkin (DG) methods that satisfy entropy inequalities for any given entropy via specific numerical quadrature; however, their accompanying error analysis was limited to the truncation error level, leaving a critical gap in the rigorous convergence theory for these entropy-stable schemes. This paper closes that gap by establishing rigorous a priori error estimates for semi-discrete entropy-stable DG (ESDG) methods on general unstructured meshes for hyperbolic conservation laws. The analysis applies to both scalar equations and systems, and is built upon a finite-difference-type consistency-stability argument carried out directly at the nodal level. Under a polynomial-reconstruction hypothesis and an $L^\infty$ a priori bound, we prove an $O(h^k)$ error estimate in a quadrature-based norm, which is equivalent to the broken $L^2$ norm on the finite-dimensional reconstruction space. We further extend this framework to the entropy-stable oscillation-free DG (ESOFDG) method introduced by Liu, Lu, and Shu [SIAM J. Sci. Comput. 46 (2024)], demonstrating that the additional damping terms do not degrade the convergence order. Numerical experiments suggest that the observed convergence rates may exceed the theoretical prediction by up to half an order.
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Submitted 6 August, 2026;
originally announced August 2026.
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Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation
Authors:
Jiahuan Li,
Yilu Liu,
Xi-Nan Ma,
Guohuan Qiu
Abstract:
Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long con…
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Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.
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Submitted 6 August, 2026;
originally announced August 2026.
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The sequence property for fractal dimensions
Authors:
Kenneth Falconer,
Yuyang Liu
Abstract:
For various definitions of fractal dimension that are finitely stable, including upper box dimension, upper intermediate dimensions and Assouad spectra, we show that, given a compact subset $E$ of a metric space, typically $R^n$, there is a convergent sequence of points contained in $E$ of the same dimension as $E$ itself. Moreover, under certain conditions it is possible for a sequence to witness…
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For various definitions of fractal dimension that are finitely stable, including upper box dimension, upper intermediate dimensions and Assouad spectra, we show that, given a compact subset $E$ of a metric space, typically $R^n$, there is a convergent sequence of points contained in $E$ of the same dimension as $E$ itself. Moreover, under certain conditions it is possible for a sequence to witness the dimension of $E$ for many definitions of dimension simultaneously, for example in a self-affine set $E$ there is a single convergent sequence that has the same Assouad spectrum or intermediate dimension values as $E$ itself.
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Submitted 6 August, 2026;
originally announced August 2026.
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An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$
Authors:
Cheng Lin,
Yu-De Liu,
Ge Xiong
Abstract:
The generalized Busemann--Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann-…
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The generalized Busemann--Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann--Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.
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Submitted 20 August, 2026; v1 submitted 6 August, 2026;
originally announced August 2026.
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Lorenz Links that are not Horseshoe and Rossler Links
Authors:
Thiago de Paiva,
Yi Liu
Abstract:
We compare the periodic orbit types of the Lorenz, horseshoe, and Rössler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the Rössler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this p…
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We compare the periodic orbit types of the Lorenz, horseshoe, and Rössler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the Rössler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this paper, Rössler links and horseshoe links are treated as belonging to the same template class.
We show that the relationship between Lorenz links and horseshoe/Rössler links has two complementary sides. First, we prove that the overlap between the two families is nontrivial by constructing infinite families of links which can be embedded in both templates. This verifies, for these families, a conjecture stated by Kofman that horseshoe links should also be Lorenz links. On the other hand, we prove that the two families are far from being the same. Holmes and Williams showed that many Lorenz torus knots cannot be embedded in the horseshoe template: if the torus knot \(T(p,q)\), with \(p<q\), is a horseshoe knot, then \(3p\leq 2q\). We show that this phenomenon is much broader. We extend the Holmes--Williams obstruction from torus knots to torus links, and we construct infinitely many hyperbolic Lorenz knots and links, as well as infinitely many satellite Lorenz knots and links, which cannot be embedded in the horseshoe template. Consequently, these examples are Lorenz links which are not horseshoe links and hence not Rössler links.
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Submitted 5 August, 2026;
originally announced August 2026.
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Persistence of invariant graphs for twist maps under analytic perturbations
Authors:
Qi Li,
Yi Liu,
Lin Wang
Abstract:
We consider the persistence for invariant graphs of twist maps that exhibit the strongest possible dynamics, namely those real-analytically conjugate to rigid rotations, under Gevrey-$γ$ ($γ\in [0,1]$) perturbations. By enhancing the regularity of the perturbation itself, we show that invariant graphs with the strongest dynamics can persist even when the size of the perturbation and the constraint…
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We consider the persistence for invariant graphs of twist maps that exhibit the strongest possible dynamics, namely those real-analytically conjugate to rigid rotations, under Gevrey-$γ$ ($γ\in [0,1]$) perturbations. By enhancing the regularity of the perturbation itself, we show that invariant graphs with the strongest dynamics can persist even when the size of the perturbation and the constraints on the frequency go beyond the requirements of classical KAM theory and the theory of normally hyperbolic invariant manifolds. The proofs of these results are based on a parameterized direct KAM method.
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Submitted 5 August, 2026;
originally announced August 2026.
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Nondegeneracy and Morse Index of Ginzburg--Landau Vortices
Authors:
Manuel del Pino,
Yong Liu,
Monica Musso,
Juncheng Wei,
Wen Yang
Abstract:
We prove that the standard degree-two and degree-three vortex solutions of the Ginzburg-Landau equation are nondegenerate. Their Morse indices are also computed. The proof relies on new explicit upper and lower bounds of the modulus of these solutions and a comparison argument. It is expected that our method can be generalized to study higher degree solutions.
We prove that the standard degree-two and degree-three vortex solutions of the Ginzburg-Landau equation are nondegenerate. Their Morse indices are also computed. The proof relies on new explicit upper and lower bounds of the modulus of these solutions and a comparison argument. It is expected that our method can be generalized to study higher degree solutions.
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Submitted 5 August, 2026;
originally announced August 2026.
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Barbell twists are natural
Authors:
Yi Liu
Abstract:
For any oriented smooth $4$--manifold $X$ diffeomorphic to $(S^2\times D^2)^{\natural n}$ ($n\geq0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney--Gabai barbell twist…
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For any oriented smooth $4$--manifold $X$ diffeomorphic to $(S^2\times D^2)^{\natural n}$ ($n\geq0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney--Gabai barbell twist $\varphi\in\mathrm{Mod}(\mathcal{N},\partial\mathcal{N})$ is identified with a generator of the factor subgroup $\wedge^2H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}$. Up to boundary-fixing diffeotopy, the barbell spines of $\mathcal{N}$ are completely classified by the bases of $H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}^2$, forming a homogeneous set modeled on the group $\mathrm{GL}(H_2(\mathcal{N};\mathbb{Z}))\cong\mathrm{GL}(2,\mathbb{Z})$. Any barbell spine of $\mathcal{N}$ gives rise to an implanted barbell twist equal to $\varphi$ or $\varphi^{-1}$ in $\mathrm{Mod}(\mathcal{N},\partial \mathcal{N})$, according to the sign of the homological basis orientation.
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Submitted 3 August, 2026;
originally announced August 2026.
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Binary X-rays of doubly stochastic matrices
Authors:
Bangzheng Li,
Yuewei Liu,
Yuan Yao
Abstract:
The X-ray of a permutation is a sequence of sums along each diagonal of the associated permutation matrix. They satisfy certain necessary constraints on distribution of the values, which are conjectured to be sufficient when the sequence is binary. By re-expressing the constraints in a form that allows for real-valued relaxations, we prove that these binary sequences are always X-rays of doubly st…
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The X-ray of a permutation is a sequence of sums along each diagonal of the associated permutation matrix. They satisfy certain necessary constraints on distribution of the values, which are conjectured to be sufficient when the sequence is binary. By re-expressing the constraints in a form that allows for real-valued relaxations, we prove that these binary sequences are always X-rays of doubly stochastic matrices.
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Submitted 2 August, 2026;
originally announced August 2026.
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High-order WENO-based semi-implicit Newton-type fast sweeping methods for static Hamilton-Jacobi equations
Authors:
Yuan Liu,
Jianliang Qian
Abstract:
In this paper, we propose high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods for solving the generalized Eikonal equation arising in wave propagation through a moving fluid. Building upon the Newton-type framework of Li and Qian (2020), which updates the solution line-wise using Newton's method with a tridiagonal…
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In this paper, we propose high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods for solving the generalized Eikonal equation arising in wave propagation through a moving fluid. Building upon the Newton-type framework of Li and Qian (2020), which updates the solution line-wise using Newton's method with a tridiagonal and strictly diagonally dominant Jacobian, we extend the local solver to fifth-, seventh-, and ninth-order accuracy by incorporating high-order WENO approximations of the spatial derivatives into the numerical Hamiltonian. Three alternating sweeping strategies, namely column-wise, row-wise, and column-row-wise, are considered within the Gauss-Seidel iteration framework. Numerical examples in both two and three spatial dimensions demonstrate the efficiency and accuracy of the proposed schemes.
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Submitted 2 August, 2026;
originally announced August 2026.
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Dynamic-Threshold Algorithms for the Continuous Quadratic Knapsack Problem: Reset Mechanisms and Complexity
Authors:
Yong-Jin Liu,
Peicheng Xie,
Chuan Yang
Abstract:
Condat's algorithm is an efficient dynamic-threshold method for projection onto the simplex, but its extension to weighted equality constraints and the algorithmic roles of resetting and removal have received limited analysis. We develop a dynamic-threshold algorithm (DTA) for a continuous quadratic knapsack problem with a weighted equality constraint. DTA maintains a threshold invariant through t…
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Condat's algorithm is an efficient dynamic-threshold method for projection onto the simplex, but its extension to weighted equality constraints and the algorithmic roles of resetting and removal have received limited analysis. We develop a dynamic-threshold algorithm (DTA) for a continuous quadratic knapsack problem with a weighted equality constraint. DTA maintains a threshold invariant through three operations--addition, reset, and removal--and we establish its finite termination and correctness. A sufficient condition under which reset cannot occur motivates a simpler no-reset variant, NDTA. We construct instances for which DTA runs in $Θ (n)$ time whereas NDTA requires $Θ(n^2)$ time, although both algorithms have quadratic worst-case complexity. We further show that, when the weight ratio and the number of deletions per removal pass are bounded, a linear number of passes with positive threshold increments requires the minimum nonzero gap between input values, normalized by the data range, to be at most $\exp [- Θ(n \log n)]$. Numerical experiments with up to 10^7 variables demonstrate that DTA and NDTA achieve approximately linear empirical scaling, and outperform Secant, WMVA, Variable Fixing, Newton, Median Search, Heap, and Sort in running time.
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Submitted 1 August, 2026;
originally announced August 2026.
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A counterexample to Han's conjecture
Authors:
Bochao Kong,
Yeqin Liu,
Yu Shen
Abstract:
There is a finite dimensional $\mathbb{C}$-algebra with $\mathrm{gldim}(A)=\infty$ and $\mathrm{HH}_{n}(A)=0$ for $n\geq 1$.
There is a finite dimensional $\mathbb{C}$-algebra with $\mathrm{gldim}(A)=\infty$ and $\mathrm{HH}_{n}(A)=0$ for $n\geq 1$.
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Submitted 31 July, 2026;
originally announced August 2026.
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On isotrivial cone structures of highest weight type
Authors:
Yingqi Liu
Abstract:
We study the projective-geometric criterion of Hwang-Li for isotrivial cone structures admitting characteristic conic connections. We give a complete classification of flag varieties satisfying this criterion, and in the process characterize flag varieties with an injective Gaussian map. As an application, we classify germs of minimal rational curves whose VMRT at a general point is isomorphic to…
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We study the projective-geometric criterion of Hwang-Li for isotrivial cone structures admitting characteristic conic connections. We give a complete classification of flag varieties satisfying this criterion, and in the process characterize flag varieties with an injective Gaussian map. As an application, we classify germs of minimal rational curves whose VMRT at a general point is isomorphic to that of a non-homogeneous smooth projective symmetric variety of Picard number one, apart from two exceptional cases.
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Submitted 31 July, 2026;
originally announced July 2026.
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Skeletons and Toric Extensions of Maximally Short Complexity One Spaces
Authors:
Yichen Liu
Abstract:
Complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,ω,Φ)$ such that $\frac{1}{2}\dim M -\dim T=1$. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we prove that the moment image of the skeleton of a compact, connected maximally short complexity one $T$-space, which is in fact a GKM…
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Complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,ω,Φ)$ such that $\frac{1}{2}\dim M -\dim T=1$. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we prove that the moment image of the skeleton of a compact, connected maximally short complexity one $T$-space, which is in fact a GKM space, is connected. The proof relies on the well-known fact that each connected component of regular values of a proper moment map is a convex locally polyhedral set. We also gave an elementary proof of that fact along the way. Then we use the connectedness result to estimate the number of symplectic toric $(T \times S^1)$-manifolds whose underlying complexity one $T$-space is the same as the given maximally short complexity one $T$-space.
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Submitted 30 July, 2026;
originally announced July 2026.
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On Sirakov's equal-frequency uniqueness conjecture
Authors:
Hong-Ge Chen,
Yong Liu,
Juncheng Wei,
Wen Yang
Abstract:
Let $N\in\{2,3\}$, $0<μ_1\leqμ_2$, and $0<β<μ_1$. We prove that the equal-frequency two-component cubic Schrödinger system \[
-Δu+u=μ_1u^3+βuv^2,
\qquad
-Δv+v=μ_2v^3+βu^2v
\quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the…
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Let $N\in\{2,3\}$, $0<μ_1\leqμ_2$, and $0<β<μ_1$. We prove that the equal-frequency two-component cubic Schrödinger system \[
-Δu+u=μ_1u^3+βuv^2,
\qquad
-Δv+v=μ_2v^3+βu^2v
\quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the synchronized state constructed from the unique positive radial solution of $-Δw+w=w^3$ in $\mathbb{R}^N$. This settles Sirakov's equal-frequency uniqueness conjecture throughout the weak-coupling range.
The main difficulty in the proof is to exclude radial solutions for which the ratio of the normalized components is nonconstant. After normalization, the two components satisfy scalar equations with a common potential. We construct a weighted Pohozaev functional for the system together with a correction term and prove that both the corrected functional and the associated weighted functional are strictly positive. Combining these sign properties with a radial flux identity and an auxiliary quotient associated with the component ratio forces synchronization.
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Submitted 30 July, 2026;
originally announced July 2026.
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Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations
Authors:
Andrew Beveridge,
Yufan Hu,
Yucheng Liu
Abstract:
Let $S_n(32\text{-}1)$ and $S_n(3\text{-}21)$ denote the sets of $n$-permutations avoiding the vincular patterns $32\text{-}1$ and $3\text{-}21$, respectively. Using Lehmer codes, we realize these families as weighted posets $L_n(32\text{-}1)$ and $L_n(3\text{-}21)$, where the weight of a code is the inversion number of its permutation. We show that the maximal elements of each of these posets,…
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Let $S_n(32\text{-}1)$ and $S_n(3\text{-}21)$ denote the sets of $n$-permutations avoiding the vincular patterns $32\text{-}1$ and $3\text{-}21$, respectively. Using Lehmer codes, we realize these families as weighted posets $L_n(32\text{-}1)$ and $L_n(3\text{-}21)$, where the weight of a code is the inversion number of its permutation. We show that the maximal elements of each of these posets, $\operatorname{Max} L_n(32\text{-}1)$ and $\operatorname{Max} L_n(3\text{-}21)$, are enumerated by the Fibonacci numbers. We demonstrate that the classical reverse-complement map on permutations restricts to a natural bijection between these two sets of maximal elements, revealing a deep symmetry between their underlying poset structures.
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Submitted 29 July, 2026;
originally announced July 2026.
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Contact set of solutions to gradient flow of Landau-de Gennes energy with a singular entropy potential
Authors:
Yuning Liu,
Xiang Xu
Abstract:
This paper investigates the convexity structure and contact set of the $Q$-tensor flow with anisotropic Landau--de Gennes elastic energy and a singular entropy potential $f(Q)$. The flow naturally exhibits a coercivity property, yielding the $H^1$ regularity of $f(Q)$ for almost every time. Combining this $H^1$ regularity with spherical averaging and capacity theory, the contact set…
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This paper investigates the convexity structure and contact set of the $Q$-tensor flow with anisotropic Landau--de Gennes elastic energy and a singular entropy potential $f(Q)$. The flow naturally exhibits a coercivity property, yielding the $H^1$ regularity of $f(Q)$ for almost every time. Combining this $H^1$ regularity with spherical averaging and capacity theory, the contact set $$\mathcal{C}(Q)\triangleq \{x\mid f(Q(x))=\infty\}$$ is characterized. In particular, it is shown to be empty in two spatial dimensions and to have Hausdorff dimension at most one in three spatial dimensions.
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Submitted 28 July, 2026;
originally announced July 2026.
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Construction of entropy satisfying Active Flux-type methods
Authors:
Remi Abgrall,
Yongle Liu
Abstract:
This paper is devoted to the analysis of the entropy stability properties of Active Flux\yolo{-type} scheme for a hyperbolic system equipped with one entropy inequality. This type of scheme evolves two sets of degrees of freedom: point values that are chosen on the boundary of the elements that cover the computational domain, and the average of the solution in these elements. We show that the only…
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This paper is devoted to the analysis of the entropy stability properties of Active Flux\yolo{-type} scheme for a hyperbolic system equipped with one entropy inequality. This type of scheme evolves two sets of degrees of freedom: point values that are chosen on the boundary of the elements that cover the computational domain, and the average of the solution in these elements. We show that the only thing to do is to get an entropy inequality for the average values, the point values degrees of freedom do not play any role. We construct a monolithic scheme which is bound preserving of \cite{BP_Pampa_VEM}, non oscillatory following \cite{PampaDG}, and entropy diminishing. The entropy condition is implemented in Tadmor's framework\cite{TadmorEntropy}, i.e. for the semi-discrete scheme only. The scheme is tested on the Kurganov-Popov-Petrova test case \cite{KPP} which is known to \yolo{be} sensitive to the satisfaction of an entropy inequality. We show that our entropy correction is effective: if we do not activate the bound-preserving nor the non oscillatory condition, we get the correct solution with some spurious wiggles, as expected. Though the development, implementation and tests are done with the triangle version of the scheme, the same method can be used for polygonal meshes, following \cite{BP_Pampa_VEM}.
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Submitted 30 July, 2026; v1 submitted 27 July, 2026;
originally announced July 2026.
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Strict positivity property of inhomogeneous subdiffusion equations and its application to coupled subdiffusion systems
Authors:
Yimeng Tian,
Yikan Liu
Abstract:
The positivity of solutions to subdiffusion equations has been widely studied, mainly in the context of homogeneous problems for single equations. In this article, we fill the missing strict positivity for inhomogeneous subdiffusion equations with nonnegative and nontrivial source terms by connecting Green's functions for fractional and classical diffusion equations via special functions. As a dir…
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The positivity of solutions to subdiffusion equations has been widely studied, mainly in the context of homogeneous problems for single equations. In this article, we fill the missing strict positivity for inhomogeneous subdiffusion equations with nonnegative and nontrivial source terms by connecting Green's functions for fractional and classical diffusion equations via special functions. As a direct application, we further investigate the strict positivity property of coupled subdiffusion systems with nonnegative and partially nontrivial initial values or sources. Under suitable cooperativeness and connectivity conditions, the strict positivity turns out to propagate not only in time but also across different components of the system, reflecting the intrinsic interactions induced by the coupling structure. These results provide a unified framework for understanding positivity properties of both scalar and coupled subdiffusion equations, offering new insights beyond the classical maximum principle approach.
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Submitted 24 July, 2026;
originally announced July 2026.
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Weighted Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Methods for Conservation Laws
Authors:
Yongjie Bi,
Yan Jiang,
Yong Liu
Abstract:
In this paper, we propose a weighted inverse Lax-Wendroff (WILW) boundary treatment for the discontinuous Galerkin (DG) method on unfitted meshes to efficiently solve hyperbolic conservation laws in complex geometries. The proposed method employs the standard DG scheme for interior cells and reconstructs high-order approximation polynomials via the ILW principle for cut cells near boundaries to im…
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In this paper, we propose a weighted inverse Lax-Wendroff (WILW) boundary treatment for the discontinuous Galerkin (DG) method on unfitted meshes to efficiently solve hyperbolic conservation laws in complex geometries. The proposed method employs the standard DG scheme for interior cells and reconstructs high-order approximation polynomials via the ILW principle for cut cells near boundaries to impose numerical boundary conditions, effectively eliminating the time-step restriction typically caused by small cut cells. In particular, to address the sensitivity of numerical errors to the geometric size of cut cells in the basic ILW scheme, we raise the reconstruction order at the boundary, ensuring that accuracy becomes independent of the cut-cell size. Furthermore, it incorporates a weighted least-squares reconstruction to reduce the need for complex high-order boundary derivatives during construction. As a result, the method maintains high-order accuracy while significantly improving computational efficiency for multi-dimensional problems. Finally, the stability of the proposed method is theoretically validated through linear stability analysis, and the effectiveness and robustness of the proposed scheme are numerically verified through a series of one-dimensional and two-dimensional numerical experiments for scalar and system equations.
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Submitted 21 July, 2026;
originally announced July 2026.
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A positive square-energy strengthening of Turán's theorem
Authors:
Yinchen Liu,
Quanyu Tang,
Shengtong Zhang
Abstract:
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…
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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.
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Submitted 20 July, 2026;
originally announced July 2026.
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The positive and negative square-energy conjecture
Authors:
Yinchen Liu,
Quanyu Tang,
Shengtong Zhang
Abstract:
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 pos…
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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.
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Submitted 20 July, 2026;
originally announced July 2026.
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Cremona invariance of filtered Varchenko--Gelfand algebras
Authors:
Ye Liu
Abstract:
We prove that the filtered Varchenko--Gelfand algebra is invariant under a natural Cremona operation on a class of real hyperplane arrangements. Namely, suppose that an arrangement contains all coordinate hyperplanes and that every remaining defining form is supported on two coordinates. Swapping the two coefficients in each such form produces its Cremona transform. Coordinatewise inversion gives…
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We prove that the filtered Varchenko--Gelfand algebra is invariant under a natural Cremona operation on a class of real hyperplane arrangements. Namely, suppose that an arrangement contains all coordinate hyperplanes and that every remaining defining form is supported on two coordinates. Swapping the two coefficients in each such form produces its Cremona transform. Coordinatewise inversion gives a chamber bijection and an isomorphism of the corresponding filtered Varchenko--Gelfand algebras over every commutative coefficient ring. As an application, we exhibit two arrangements of eight central planes in $\mathbb{R}^3$ with isomorphic filtered Varchenko--Gelfand algebras but non-isomorphic tope graphs. This disproves a conjecture of Yagi--Yoshinaga on reconstructing tope graphs from filtered Varchenko--Gelfand algebras.
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Submitted 17 July, 2026;
originally announced July 2026.
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Block-exoticity of reduction simple fusion systems on $S(n,p)$
Authors:
Jun Liao,
Yikun Liu
Abstract:
In this paper, we prove that for $p\geq 13$ and $5\leq n\leq p-3$, every reduction simple fusion system on the finite $p$-group $S(n,p)$ is block-exotic. Furthermore, we establish the block-exoticity of three infinite families of reduction simple exotic fusion systems.
In this paper, we prove that for $p\geq 13$ and $5\leq n\leq p-3$, every reduction simple fusion system on the finite $p$-group $S(n,p)$ is block-exotic. Furthermore, we establish the block-exoticity of three infinite families of reduction simple exotic fusion systems.
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Submitted 11 July, 2026;
originally announced July 2026.
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Orlik--Solomon sheaf homology of geometric lattices
Authors:
Ye Liu
Abstract:
We associate the Orlik--Solomon sheaf with a finite geometric lattice and compute its sheaf homology. We show that this homology concentrates in top degree, admitting a convolution-type decomposition into a principal ideal OS piece tensoring with a principal filter complement poset homology. Applications to uniform matroids provide interesting representations of symmetric groups.
We associate the Orlik--Solomon sheaf with a finite geometric lattice and compute its sheaf homology. We show that this homology concentrates in top degree, admitting a convolution-type decomposition into a principal ideal OS piece tensoring with a principal filter complement poset homology. Applications to uniform matroids provide interesting representations of symmetric groups.
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Submitted 9 July, 2026;
originally announced July 2026.
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Forgetting-Factor Regret for Online Zero-Sum Games
Authors:
Yuhang Liu,
Zi'ang Yan,
Wenjun Mei,
Wenxiao Zhao
Abstract:
This paper studies dynamic equilibrium tracking in online two-player zero-sum games with time-varying convex-concave payoff functions. Existing regret metrics for online saddle-point problems usually aggregate historical payoffs with uniform weights, and hence may fail to characterize the real-time tracking performance with respect to the current Nash equilibrium (NE). To address this issue, we in…
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This paper studies dynamic equilibrium tracking in online two-player zero-sum games with time-varying convex-concave payoff functions. Existing regret metrics for online saddle-point problems usually aggregate historical payoffs with uniform weights, and hence may fail to characterize the real-time tracking performance with respect to the current Nash equilibrium (NE). To address this issue, we introduce a zero-sum game regret function with a forgetting factor, which assigns exponentially decaying weights to past saddle gaps and emphasizes recent performance. This metric directly links regret minimization to the tracking of time-varying NEs. Within this framework, we investigate three online algorithms under different computational and information settings. For first-order feedback, we analyze projected gradient descent-ascent and design a projection-free online Frank-Wolfe method to reduce the computational cost of projections. For zeroth-order feedback, we develop a deterministic finite-difference method that only uses function-value queries. For all three algorithms, we establish forgetting-factor regret bounds that explicitly characterize the effects of NE variation, payoff variation, and gradient-estimation error. We further provide sufficient conditions under which the proposed regret converges to zero, thereby certifying asymptotic tracking of time-varying NEs. The numerical example validates the theoretical results and illustrates the tracking advantage of the proposed regret metric.
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Submitted 8 July, 2026;
originally announced July 2026.
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Changes in the Seidel energy of blow-up graphs under edge deletion
Authors:
Yayang Liu,
Yi Wang
Abstract:
Let $S(G)$ denote the Seidel matrix of a simple graph $G$, and let $E_S(G)$ be the Seidel energy of $G$, defined as the sum of the absolute values of the eigenvalues of $S(G)$. In this paper, we study the change of Seidel energy under edge deletion. For an independent-set blow-up graph $G=H[n_1,\ldots,n_p]$, we establish a general structural criterion within the framework of independent-set blow-u…
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Let $S(G)$ denote the Seidel matrix of a simple graph $G$, and let $E_S(G)$ be the Seidel energy of $G$, defined as the sum of the absolute values of the eigenvalues of $S(G)$. In this paper, we study the change of Seidel energy under edge deletion. For an independent-set blow-up graph $G=H[n_1,\ldots,n_p]$, we establish a general structural criterion within the framework of independent-set blow-up graphs. More precisely, if the endpoints of the deleted edge $e$ belong to blow-up parts of sizes $n_a$ and $n_b$, respectively, then $E_S(G-e)>E_S(G)$ whenever both $n_a,n_b$ are at least $4$, or one is $3$ and the other is at least $6$, or one is $2$ and the other is at least $15$.
As applications, we obtain the following consequences. First, for every Turán graph $T(n,r)$ with $r\geq4$ and $n\geq4r$, deleting any edge strictly increases the Seidel energy. Second, for complete multipartite graphs, we derive an exact reduced-order spectral criterion for the remaining cases not covered by the structural result. This criterion determines whether the Seidel energy increases, decreases, or remains unchanged after deleting an edge, by using matrices whose orders depend only on the number of partite sets. These results provide affirmative answers to two problems proposed by Tian et al. [\textit{Linear and Multilinear Algebra} 70 (19) (2022), 4597--4614].
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Submitted 7 July, 2026;
originally announced July 2026.
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Simple restricted modules over the deformative Schrödinger-Virasoro algebra
Authors:
Haibo Chen,
Yongtao Liu,
Xiaoqing Yue
Abstract:
This paper investigates simple restricted modules over the deformed Schrödinger-Virasoro algebra $\mathcal{G}_{λ,μ}$, which gives a complete classification of them for some $λ,μ\in\mathbb{C}$. More precisely, we provide a systematic construction of these modules, including highest weight modules and Whittaker modules, by inducing simple modules from the positive part's quotient algebras. We prove…
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This paper investigates simple restricted modules over the deformed Schrödinger-Virasoro algebra $\mathcal{G}_{λ,μ}$, which gives a complete classification of them for some $λ,μ\in\mathbb{C}$. More precisely, we provide a systematic construction of these modules, including highest weight modules and Whittaker modules, by inducing simple modules from the positive part's quotient algebras. We prove that any simple restricted $\mathcal{G}_{λ,μ}$-module satisfying certain injective conditions is isomorphic to such an induced module. As an application, we obtain some simple weak $V(c)$-modules over vertex algebras associated to $\mathcal{G}_{λ,μ}$ for some $λ,μ\in\mathbb{C}$. Note that our results include the Schrödinger-Virasoro algebra and the deformed $\mathfrak{bms}_3$ algebra as special cases, thereby improving upon some of the previously reported results of [5,Theorem 3.4] and [6,Theorem 2]. This work effectively classifies and generalizes the representation theory of the deformed family.
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Submitted 2 July, 2026;
originally announced July 2026.
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Total positivity of transformation matrices for uniform subdivisions
Authors:
Yanxin Liu,
Jianxi Mao
Abstract:
The transformation of the $h$-vector of a finite simplicial complex under an $\mathcal{F}$-uniform subdivision is encoded by a transformation matrix. Mu and Welker conjectured that the transformation matrix of the barycentric subdivision is totally positive. In this paper, we give a new combinatorial proof of this conjecture. We also prove the total positivity of the transformation matrix of the i…
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The transformation of the $h$-vector of a finite simplicial complex under an $\mathcal{F}$-uniform subdivision is encoded by a transformation matrix. Mu and Welker conjectured that the transformation matrix of the barycentric subdivision is totally positive. In this paper, we give a new combinatorial proof of this conjecture. We also prove the total positivity of the transformation matrix of the interval subdivision. In addition, we establish a sufficient condition for the transformation matrix of a uniform subdivision to be totally positive of order $2$ (TP$_2$), thereby partially answering a question of Mu and Welker. As an application, we show that the transformation matrix of the $r$-colored barycentric subdivision is TP$_2$.
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Submitted 1 July, 2026;
originally announced July 2026.