Papers

Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(\theta^{-1})\!)$ spanned by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$ of weight $w$. We prove that $\sum_{w=1}^{\infty} \left(\dim_{\mathbb{F}_q} \mathcal{Z}_w^{(\mathbb{F}_q)}\right) x^w = \frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1})^2}$. Moreover, we construct an explicit $\mathbb{F}_q$-basis of $\mathcal{Z}_w^{(\mathbb{F}_q)}$, and prove that any $\mathbb{F}_q$-linear relation among Carlitz multiple polylogarithm values $\operatorname{Li}_A(\mathfrak{s})$ is an $\mathbb{F}_q$-linear combination of quadruple-carry relations. This result can be regarded as an $\mathbb{F}_q$-analogue of the corresponding $\mathbb{F}_q(\theta)$-theorem proved by Chang–Chen–Mishiba and independently by Im–Kim–Le–Ngo Dac–Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im–Kim–Ngo Dac. These relations may be viewed as $\mathbb{F}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta(\mathfrak{s})$.

Classical Turán theory shows that, without additional constraints, the extremal structure of $F$-free graphs is governed by the chromatic number of $F$ and is asymptotically the balanced Turán graph. We investigate how prescribing a large maximum degree changes this picture and leads to an unbalanced Turán problem. Let $F$ be a graph with $\chi(F)=r+1\ge3$, and let $\lceil(r-1)n/r\rceil\le\Delta\le n-1$. We study the maximum number of edges and adjacency spectral radius of an $n$-vertex $F$-free graph with maximum degree exactly $\Delta$. For $F=K_{r+1}$, the unbalanced $r$-partite graph $S_{n,\Delta}^{(r)}=(n-\Delta)K_1\vee T(\Delta,r-1)$ is the unique maximizer of both quantities. Let $a(F)$ be the minimum size of an independent set $I$ such that $\chi(F-I)\le r$. If $a(F)=1$, we prove edge and spectral stability with respect to $S_{n,\Delta}^{(r)}$. If $a(F)>1$, the extremal values are $t(n,r)+o(n^2)$ and $\rho(T(n,r))+o(n)$, respectively, and every extremal graph differs from $T(n,r)$ in $o(n^2)$ edges, uniformly all $\lceil(r-1)n/r\rceil\le\Delta\le n-1$. We further establish a general spectral transfer principle: for finite forbidden families, a decomposition-family bound of order $O(n^{1+s})$ yields a spectral bound with error $O(n^s)$ for $0\le s<1$.

We establish a new bridge connecting quantales, semigroups and the theory of formal concept analysis. By introducing residuated relations whose domains are semigroups, we show that every quantale arises as the formal concept lattice induced by such a relation. Furthermore, this construction yields an equivalence between the category of quantales and a category of such residuated relations whose morphisms are multiplicative bonds.

These open problems were presented in the Problem Sessions held during the Tianyuan Workshop on Definability and Computation, June 22-26, 2026. The problems are organized into sections named after their contributors, in the order of their presentations during the workshop. Notes were taken and compiled by Wei Dai, Xiangxi Hu, Yingying Jiang, Ruiwen Li, Tianhao Wang, Xu Wang, and Jie Zou.

We present a reduced-order modeling framework based on proper orthogonal decomposition for dynamic linear viscoelasticity governed by generalized Maxwell models. The hereditary constitutive law is reformulated using internal variables, and the resulting problem is discretized in space by a symmetric interior penalty discontinuous Galerkin method and in time by either the backward Euler or Crank–Nicolson scheme. The reduced model is obtained by projecting the fully discrete system onto low-dimensional spaces constructed from displacement snapshots. For the full-order discontinuous Galerkin formulation, we show well-posedness and derive an a priori stability bound without using Gronwall's inequality. Consequently, the bound grows at most linearly, rather than exponentially, with the final time. We further derive spatial error estimates and an error decomposition that separates the effects of spatial discretization, temporal discretization, and proper orthogonal decomposition truncation. The reduced formulation replaces the large full-order systems with low-dimensional problems, enabling efficient long-time simulations. Numerical experiments verify the predicted convergence behavior, assess different inner products for constructing the reduced basis, and demonstrate accurate recovery of transient oscillations and viscoelastic relaxation with substantial computational savings, even after accounting for snapshot generation and basis construction.

The line graph $L(G)$ of a graph $G$ is defined to be the simple graph whose vertices are the edges of $G$, where two vertices in $L(G)$ are adjacent if and only if the corresponding edges in $G$ are incident with a common vertex, and define $L^2(G)=L(L(G))$. For positive integers $d$ and $k$, the function $\kappa_{L^2}(d,k) = \inf\{\kappa(L^2(G)): \kappa'(G) \ge k \mbox{ and } \delta(G) \ge d\}$ has been investigated. Niepel and Knor proved that $\kappa_{L^2}(d,1)\geq d-1$, for any integer $d \ge 3$. In this research, it is proved that if $d\geq 3$ and $k\geq 1$, then $\kappa_{L^2}(d,k)= \min\{f(d,k), 4d-6\}$, where \begin{equation} f(d,k) = \left\{ \begin{array}{ll} k(d-k), & \mbox{ if $1\leq k\leq \lfloor\frac{d}{2}\rfloor$, } \\ kd-k^2+2k(\lceil \frac{d}{2}\rceil)-(\lceil\frac{d}{2}\rceil)d, & \mbox{ if $\lfloor\frac{d}{2}\rfloor< k <\frac{3d-1}{4}$, } \\ kd-k^2+2k\lfloor \frac{d}{2}\rfloor-2(\lfloor \frac{d}{2}\rfloor)^2, & \mbox{ if $\frac{3d-1}{4}\leq k<d$, }\\ d(\lceil \frac{d}{2}\rceil), & \mbox{ if } d=k. \end{array} \right.\nonumber \end{equation}

We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow $(M^2, \partial M^2, g(t))$ on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.

For fixed $k\ge2$, Garvan's modified $k$-rank moments are encoded by odd elliptic Taylor coefficients of an odd-level Appell function. The completed coefficient of order $2n+1$ has weight $2n+3/2$, so the family cannot be realized as a fixed-weight vector-valued modular form. Nevertheless, all non-holomorphic parts arise from scalar contractions of canonical higher Serre derivatives of a single weight-$3/2$ vector-valued harmonic Maass form. Among lifts with the prescribed unary-theta shadow, the first $d=k-1$ Appell corrections are coordinates on the weakly holomorphic ambiguity and select a unique lift; the next correction is nonzero. The initial Taylor data recover the principal part without using positive Fourier coefficients of the modified moment series, and the principal part determines the shadow. The positive coefficients of the corresponding Maass–Poincaré lift have convergent Rademacher expansions; the canonical lift differs from it by a unique cusp form. This yields exact recursions for all even modified moments. For $k=2$, the cusp-form adjustment vanishes, giving an exact formula for the numbers of $2$-marked Durfee symbols involving a convergent Kloosterman–Bessel series.

We introduce finite-dimensional approximations to the space of convex bodies based on polytopes in vertex representation. For a prescribed set of directions, the admissible point configurations form a polyhedral convex subcone of the Euclidean vector space, described by a finite system of linear inequalities. The interior of this cone contains only nondegenerate representations, in which all parameter points are distinct vertices of the represented polytope. We study the geometry of the parameter cone, redundancies in its defining inequalities, and natural projections of convex bodies onto the resulting polytope spaces. We derive quantitative approximation estimates in terms of how densely the prescribed directions cover the unit sphere and construct nested Galerkin sequences whose approximation error converges locally uniformly to zero. Finally, we use these approximation properties to establish the convergence of finite-dimensional approximations of constrained global optimization problems in the space of convex bodies.

In this paper, we study the problem of augmenting a tiny target sample with a massive auxiliary sample. Utilizing Tukey's factorization, there are two popular approaches: the inverse probability weight (IPW) and the full-likelihood (FL) methods. We show that the IPW approach suffers from the limited target sample problem while the FL method may estimate some model parameters at the rate of the massive auxiliary sample size, a phenomenon we call full efficiency gain. We study the theory behind the full efficiency gain for exponential families and mixtures of exponential families. We also study the efficiency gain for the IPW method under a nonparametric procedure and show how it can achieve a parametric rate of the target sample size. As a side note, we also discuss how one may use FL to train neural network models simultaneously for both the target distribution and the odds model.

We record five improved lower bounds for Zarankiewicz numbers with s = t = 3: z(13,19;3,3) >= 118, z(14,19;3,3) >= 126, z(16,18;3,3) >= 136, z(14,20;3,3) >= 126, z(16,19;3,3) >= 136. The first three are certified by explicit K_{3,3}-free 0/1 matrices; the last two follow from the second and third by monotone padding. Compared with the lower bounds compiled in Figure 2 of arXiv:2605.01120, namely 114, 121, 130, 125 and 132, the improvements are +4, +5, +6, +1 and +4 respectively. The three matrices were produced by a kicked-greedy local search operated autonomously by a discovery system and were verified exactly, by exhaustive inspection of every 3x3 row/column triple, in several mutually independent implementations; they were also re-verified independently by the authors of arXiv:2605.01120 using their own verifier in July 2026. All three witnesses are printed in full in Appendix A and accompany this note as machine-readable ancillary files together with standalone, dependency-free verifiers.

The permanent of an $n\times n$ $0/1$ matrix $A$ equals the number of perfect matchings in the bipartite graph with edges defined by $A$. Jerrum, Sinclair, and Vigoda (2004) presented an FPRAS for approximating the permanent of any nonnegative matrix using a novel simulated-annealing algorithm. The running time was improved by Bezáková, Štefankovič, Vazirani, and Vigoda (2008) to $O(n^7\log^4 n)$ for $0/1$ matrices, for any fixed approximation and success parameters. We present the first asymptotic improvement over this running time bound, obtaining an $O(n^6\log^5 n)$-time algorithm. As in the previous works, our algorithm extends to arbitrary nonnegative matrices. The analysis of Bezáková et al. yields an $O(n^4)$ relaxation time bound for the JSV Markov chain on perfect and near-perfect matchings with ideal hole weights, under which each hole pattern (the unmatched vertices, if any) is equally likely in the stationary distribution. We introduce a restricted Poincaré inequality for the partition into hole patterns and prove an $O(n^3)$ bound on the corresponding restricted relaxation time. Our proof uses a coupled multicommodity flow argument inspired by a recent transport-flow argument of Chen et al.~(2025) for the Jerrum-Sinclair chain on all matchings.

We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE $\frac{\mathrm{d}}{\mathrm{d}t}R=Q(R)$ in dimensions $n=7,8$, thereby extending the pinching estimate established by Brendle for $n\geq 12$ and by Chen for $9\leq n\leq 11$. In dimension $n=8$, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at $n=8$ by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension $n=7$, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension $n=8$. The $n=8$ pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible $(n-1)$-dimensional space forms, extending a theorem of Brendle from $n\geq 12$. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact $\kappa$-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from $n=4$ or $n\geq 12$.

Let $X$ be an $\epsilon$-lc projective variety of dimension $n\geq 2$ such that $-K_X$ is nef and $0<\epsilon \leq 1$ is a real number. Then for any nef divisors $D_1,\dots,D_{n-2}$, \[ c_1(X)^{2}\cdot D_1\cdots D_{n-2}\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot D_1\cdots D_{n-2}. \] In particular, there exists a Kawamata–Miyaoka type inequality \[ c_1(X)^n\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot c_1(X)^{n-2}. \]

This article studies identifying codes in oriented graphs from a computational complexity perspective. We investigate the $\mathcal{F}$-Id Code problem, where given a simple graph $G$ and a vertex subset $C$, which induces a subgraph in the family $\mathcal{F}$, as inputs and ask whether it is possible to orient $G$ in such a way that $C$ becomes its oriented identifying code. Focusing on the family $\mathcal{F}_d$ of $d$-regular graphs, we establish a complete dichotomy by proving that the problem is polynomial-time solvable for $d\leq1$ and NP-complete for all $d\geq2$.

Let $I$ be a squarefree monomial ideal, and let $d$ denote the maximum degree of a minimal generator of $I$. We prove that \[ v(I^t)\le v(I^{(t)})+(t-1)(d-1) \] for all $t\ge1$, where $I^{(t)}$ denotes the $t$-th symbolic power of $I$. In particular, this bound does not depend on the number of variables in the ambient polynomial ring. On the other hand, for every fixed exponent $t\ge2$, the difference \[ v(I^{(t)})-v(I^t) \] can be arbitrarily large. Finally, we determine the $v$-numbers of the ordinary and symbolic powers of edge ideals of paths and cycles.

We prove that for any given frequency resonance exponent and phase resonance exponent, there exist a frequency and a phase exactly realizing these values such that the Almost Mathieu operator exhibits Anderson localization for $\ln|\lambda|>\max\{\beta(\alpha),\delta(\alpha,\theta)\}$. This resolves the conjecture in \cite{MR4686650}.

We prove, modulo almost one-to-one extensions, that the topological rational Kronecker factors of a minimal $\mathbb{Z}^k$-system are the topological characteristic factors for commuting transformations along polynomials with distinct degrees. As an application, we obtain necessary and sufficient conditions for the joint transitivity of these polynomial iterates.

We study the relation between $\NilBohr{d}$ sets and $\SG_d^*$ sets. We prove that every $\NilBohr{d}$ set is an $\SG_d^*$ set, answering a question of Host and Kra. More generally, for a compact Hausdorff distal system whose Ellis group admits a closed filtration \[ E=E_1\supseteq E_2\supseteq\cdots\supseteq E_{d+1}=\{e\}, \ [E_j,E]\subseteq E_{j+1}, \] we prove that every return time set is an $\SG_d^*$ set.

Let $S\subset\mathbb R^3$ be a properly embedded mean-convex planar surface with finitely many ends. Designate one end as asymptotically flat, assume that $H_S$ is integrable there, and denote its extrinsic mass by $m_+(S)$. Let $A_S$ be the infimum of the areas of compact surfaces separating the distinguished end from all the others. We prove \[ m_+(S)\geq\sqrt{\frac{A_S}{\pi}}. \] No outermost free-boundary minimal surface is assumed, and no asymptotic or integrability condition is imposed on the other ends. If $A_S=0$, equality holds precisely for the Euclidean half-space. The complete catenoid realizes equality with $A_S>0$. Conversely, if equality holds with $A_S>0$, then $A_S$ is attained by a flat free-boundary disk that is outermost toward the distinguished end, and the corresponding component of $S$ is a half-catenoid.

We study real ideal points of $SL_2(\mathbb C)$-character varieties of knot exteriors containing essential Conway spheres. Under explicit deformation-theoretic hypotheses on the two complementary tangles, we show that essential Conway spheres are detected via Culler-Shalen theory with real ideal points; such ideal points are then deformed into arcs of $SL_2(\mathbb R)$ representations on both sides. We then compute the asymptotic behavior of the representations on these arcs and compute the translation numbers of their lifts to $\widetilde{PSL}_2(\mathbb R)$ representations. Combining this with a result of Gao, we conclude that for certain knots with essential Conway spheres, all sufficiently large positive and negative rational fillings have left-orderable fundamental group. This verifies the $L$-space conjecture for large-slope Dehn fillings of knots which were previously unknown in the literature. We verify the criteria for three infinite families of knots assembled from torus-trivial and twist-trivial tangles, then count the resulting real ideal points, determine the longitudinal translation numbers of all constructed branches, and identify exactly the zero-translation arcs. Computations for the $28$ distinct verified census exteriors listed in Table \ref{tab:instances} agree with the certified branches.

We show that there exists a suspension flow defined over the full shift map which does not have local product structure. This gives a negative answer to an open problem posed by Gelfert-Kwietniak-Lima recently.

Poincaré duality is a classical theorem relating the homology and cohomology of closed oriented manifolds. This theorem has been extended to more general settings and to generalized (co)homology theories, including $K$-theory. In this paper, we construct an infinite-dimensional analogue of the $K$-theoretic Poincaré duality homomorphism. More precisely, for an infinite-dimensional Hilbert manifold $\mathcal{M}$, we construct a homomorphism $$K^{geo}_*(\mathcal{M})\to K_*(\mathcal{A}(\mathcal{M})),$$ where $K^{geo}_*(\mathcal{M})$ denotes the geometric $K$-homology of Baum and Douglas, and $\mathcal{A}(\mathcal{M})$ is a $C^*$-algebra associated to $\mathcal{M}$, based on a construction of Gong, Wu, and Yu. We also prove that the constructed homomorphism is non-trivial in certain cases.

While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel backpropagation-free framework for both forward and inverse problems in differential equations. The core mathematical contribution lies in the analytical evaluation of local Jacobians for nonlinear differential operators, which facilitates a linearized representation of the physical loss and projects it into a unified, linearized algebraic subspace. This reformulation allows for the explicit determination of optimal network weights via a sequence of generalized linear least squares solvers, effectively bypassing the iterative traps of traditional nonlinear optimizers. We develop a progressive algorithmic suite comprising localized construction (PI-SC-I), sliding-window updating (PI-SC-II), and global updating (PI-SC-III), and rigorously establish their universal approximation properties. Extensive experiments demonstrate that PI-SCM achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs. Our work provides a highly efficient and scalable foundation for next-generation, real-time Scientific Machine Learning applications.

We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an existing simplex is uniform among simplices of dimension $<m$, the number $S_d$ of simplices of any admissible dimension $d\leq m$ is an asymptotically self-averaging random variable. This feature allows us to determine the asymptotic growth law of the average of $S_d$ when the number of vertices diverges. We also probe the degree distribution, examine the probabilities of various extreme outcomes, and analyze the characteristics of the first vertex.

We prove local Calderón-Zygmund estimates for distributional solutions to non-uniformly elliptic equations in divergence form modeled on the energy density $a(x)G(|z|)+b(x)H(|z|)$, where $G$ and $H$ are Young functions and $a(\cdot)$, $b(\cdot)$ are nonnegative coefficients, Hölder continuous with exponents $\alpha$ and $\beta$. Either coefficient may vanish, and only their sum is bounded away from zero, so that each phase can dominate or disappear. Under a gap condition that reduces to $q/p\le1+\min\{\alpha,\beta\}/n$ for $G(t)=t^p$ and $H(t)=t^q$, including the borderline case of equality, we show that the energy density of the gradient inherits the Orlicz integrability of the energy density of the datum. The proof combines a fractional differentiability estimate, which gives higher integrability up to the borderline growth, with comparison arguments that freeze the two coefficients in the appropriate order.

This paper studies finitistic dimensions in triangulated categories with a compact silting generator. We unify several notions of (big) finitistic dimension appearing in the literature, show that they are essentially equivalent, and relate them to the abelian heart when the generator is tilting. This yields a new categorical perspective on the finitistic dimension conjecture. Furthermore, we establish explicit inequalities for (big) finitistic and global dimensions under recollements, demonstrating that finiteness in the middle category is equivalent to finiteness in the outer categories. These results generalize classical ring-theoretic theorems and apply to triangular matrix rings, exact contexts and trivial extensions.

We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle–Matérn process is assigned to the unknown initial condition. We prove that the solution has \(\dot H^3\)-regularity when the nonlinearity satisfies a Lipschitz condition in the Sobolev norm. We then establish posterior contraction rates for the prediction error in the \(L^2\)-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback–Leibler divergences.

We establish high-probability bounds for mixed input derivatives of wide random neural networks whose activation derivatives satisfy a factorial growth bound. Our main result specializes these estimates to $\tanh$ networks with Xavier initialization. A direct deterministic analysis based on Euclidean operator norms of the weight matrices yields derivative bounds that generally grow exponentially with the depth. We show that this growth can be substantially improved for sufficiently wide Gaussian networks by isolating the term that is linear in the highest-order derivative and controlling the corresponding tangent directions by measurable finite nets. For scalar-output $\tanh$ networks with Gaussian weights and Xavier initialization, we prove that there exist constants $C,C_0,C_1>0$ such that, whenever the common hidden width satisfies $n \geq C\left(L^3n_0^2(1+\log n_0)+L^2\left(1+\log(L/\eta)\right)\right)$, then, with probability at least $1-\eta$, the estimate $\left|D^u\mathcal{R}_{\Phi^{(L)}}(x)\right| \leq C_0 |u|! (C_1L)^{|u|-1}\prod_{j\in u}\beta_j(\eta,n_0)$ holds simultaneously for every non-empty $u\subseteq[n_0]$ and every $x\in[0,1]^{n_0}$. Thus, the first-order derivative bound is independent of the depth, while a square-free mixed derivative of order $|u|$ grows at most polynomially as $L^{|u|-1}$, apart from the coordinate factors. As consequences, we obtain high-probability bounds for the Euclidean Lipschitz constant and for weighted Sobolev norms of the network realization. The latter connect the derivative estimates to quasi-Monte Carlo integration and indicate how such regularity can enter the analysis of QMC-based training.

We develop a novel method of inference for network-dependent high-dimensional random vectors. Dependence is characterized via a functional dependence measure based on graph distance, allowing the approximation theory to capture the interaction between the decay of dependence and the growth of network neighborhoods. We establish Gaussian approximation results for the maximum norm under finite-moment and sub-Weibull conditions, providing explicit conditions under which the dimension may increase with the network size. We also propose a high-dimensional network HAC covariance estimator and establish its convergence properties, yielding a feasible procedure for simultaneous inference. Simulation studies demonstrate favorable finite-sample performance of the proposed method. We apply the procedure to study how spillover effects vary with an index of network homophily by constructing confidence bands for the conditional spillover-effect function. The application reveals heterogeneity and local significance that would be obscured by conventional low-dimensional inference.