Papers

Weighted right-angled Artin groups generalize right-angled Artin groups by adding weights to the edges of the defining graphs. We classify the weighted right-angled Artin groups that occur as fundamental groups of smooth complex quasi-projective varieties. They are precisely the finite direct products of finitely generated free groups and groups of the form $\langle a,b\mid [a,b]^m=1\rangle$.

For a connected graph $G$ with at least one edge, the eternal connected vertex cover number $ecvc(G)$ is the minimum number of guards that can maintain a connected vertex cover after every response to an arbitrary sequence of edge attacks. Denote the minimum size of a connected vertex cover by $cvc(G)$. It is known that $cvc(G)\leq ecvc(G)\leq cvc(G)+1$. A necessary condition for $ecvc(G)=cvc(G)$ is that every vertex belongs to some minimum connected vertex cover. We show that this condition is not sufficient: there exists a $32$-vertex graph $G$ that has $cvc(G)=19$ and $ecvc(G)=20$, although every vertex of $G$ belongs to some minimum connected vertex cover. For connected graphs with minimum degree at least two, we establish the sharp bound $cvc(G)\geq 2|V(G)|-|E(G)|-1$ and $\mathcal F$ denotes the class attaining equality. We prove that $G\in\mathcal F$ if and only if the vertices of degree at least three induce a forest. Within $\mathcal F$, the conditions $ecvc(G)=cvc(G)$, membership of every vertex in some minimum connected vertex cover, and the presence of at least two degree-two vertices on every cycle are equivalent. As applications, we obtain $ecvc(G)=cvc(G)$ for full subdivisions of connected graphs of minimum degree at least two and for minimally $2$-connected graphs. In both the families, every minimum connected vertex cover is an eternally winning configuration. This is not true in general for graphs outside $\mathcal{F}$, we show one example of such a graph.

We establish an explicit formula for the Kostka–Foulkes polynomial of type $C_n$ with unequal parameters at fundamental weights. As an application, we deduce an explicit Lusztig–Kato formula for the Hecke algebra of the special (but not hyperspecial) maximal compact subgroup of a quasi-split but non-split special orthogonal group over a $p$-adic field.

We spent five weeks using an LLM coding agent on open problems in coding theory: finding large sets of four-letter words, such as DNA barcodes, that stay far apart in edit distance. The agent wrote the verifiers and search code; a human chose the problem and set the verification protocol. Restricting the search to codes with a prescribed symmetry, a classical technique, shrank the problem about fourfold and raised the best known code of length 6 and minimum edit distance 3 from 114 to 120 words ($E_4(6,3) \geq 120$). The same pipeline improved twelve further lower bounds at lengths 6 to 9 and distances 3 to 6. We give the failures equal space. Our own search stopped at 116 and recorded the last symmetry class as topping out at 112; a second agent session, running the same search with a better operator, found the 120. A later verdict that the method did not carry over to length 7 was wrong for the same reason, and an earlier instance cost three weeks. Each time, an intermediate result was written down, never rechecked, and treated as a fact that ruled out further search. Checking final outputs, as our protocol required, does not catch such errors.

We study the singularities of orbit closures in loop symmetric varieties, extending our previous work https://arxiv.org/abs/2310.20006 to the twisted setting. We prove that the IC complexes of orbit closures are pointwise pure and satisfy a parity-vanishing property. We apply these geometric results to the study of twisted affine Lusztig–Vogan modules, establishing foundational results, including positivity properties of the twisted affine Kazhdan–Lusztig–Vogan polynomials. Along the way, we construct conical symplectic transversal slices in loop symmetric spaces. We provide applications to Langlands duality for real groups.

We give a new proof of the fact that every 2-coloring of the positive integers contains a monochromatic Pythagorean triple. This result was originally proven by Heule, Kullmann, and Marek with SAT solvers in 2016, answering a decades-old question of Graham. This proof uses elementary techniques and requires neither lengthy computation nor difficult analytic machinery.

Regression analysis for stochastic processes has been an important topic over the last decades. Typically, regressing a dependent process $Y$ on an explanatory $Z$ results in an integral relationship of the form $dY_t = \beta_t dZ_t + dX_t$ with some residual process $X$, and often with constant or piecewise constant $\beta$. In the case of jump processes such a relationship essentially boils down to $\Delta Y_t = \beta \Delta Z_t + \Delta X_t$ on the level of the jumps. Here the residual process $X$ is such that it never jumps together with $Z$, and so any jump in $Y$ is either exactly proportional to $Z$ or purely idiosyncratic. In this paper we discuss a related setup which appears more realistic and is closer to classical regression, namely $\Delta Y_t = (\beta + \eta_t) \Delta Z_t$ for i.i.d. $(\eta_t)_{t \ge 0}$. We propose an estimator for $\beta$ based on high-frequency observations of $(Y,Z)$ and discuss consistency and associated central limit theorems in two different asymptotic regimes.

We prove sharp volume bounds (up to dimensional constants) for the $\delta$-neighbourhoods of a general family of Besicovitch-Rado-Kinney (BRK) type sets associated with surfaces of revolution in $\mathbb R^n$, $n\ge 3$, which, in particular, include spheres, elliptic paraboloids, and cones. As a result, we obtain sharp $L^{p}$-$L^q$ bounds for their associated Wolff-type maximal operators for all $1\le p,q\le \infty$. In $\mathbb R^3$, it is worth noting that our result necessitates a variant of Sogge's cinematic curvature condition.

Accurate long-time prediction of Cahn-Hilliard dynamics, particularly in the late-stage coarsening regime, is computationally demanding because it requires simulations over extended physical time. Various machine learning approaches have been developed to speed up the simulations through efficient surrogate evaluations. While the learned map can be accurate over a single learned interval, small prediction error will accumulate when the map is repeatedly applied in a long-time prediction, leading to large numerical errors. In this work, we identify the structure function for the coarsening dynamics of the Cahn-Hilliard equation and propose a two-stage structure-preserving Fourier Neural Operator (FNO) learning framework. Stage~I introduces a bound-conforming FNO (bcFNO) with a soft, magnitude-dependent amplitude penalty. Stage~II retains the bound-conforming objective and supplements it with structure-function regularization, yielding a structure-preserving FNO (spFNO) that corrects late-stage coarsening statistics. Numerical experiments on the various spatial resolutions and reference-solver time steps suggest that the bcFNO stabilizes the long-time prediction by suppressing large amplitude excursions, without changing the low online cost of the FNO. Furthermore, the structure-function term consistently reduces discrepancies in the normalized structure function and provides further correction in associated late-stage coarsening quantities, including the $L^3(t)$ growth trend and the scaling collapse. These results indicate that our multi-stage implementation of physical principles in operator learning could be applied to other multiscale systems with statistical scaling, such as the functionalized Cahn–Hilliard equations and turbulence, provided that the state constraints and statistical observables are adapted to the governing dynamics.

Broadening the study of quantum optimization algorithms from binary to $k$-element alphabets has been shown to open new avenues for potential quantum advantage. A quantum advantage claim for approximate optimization requires showing that, under the same assumptions, an efficient quantum algorithm provably achieves a better performance than can be proven for the best known efficient classical algorithms. We study local classical and quantum algorithms for Max-$k$-Cut on $d$-regular graphs of girth $g$. We advance classical algorithms for this problem by developing a local vector algorithm based on the explicit vector construction of Thompson, Parekh, and Marwaha (TPM). Our algorithm gives the best provable cut fraction guarantee among known efficient classical algorithms on regular graphs for $k\geq3$. To evaluate the performance of QAOA under identical assumptions of girth and regularity, we develop tensor network techniques for general $d$ and $k \geq 2$. In addition, using an equivalence to a coupled qudit–boson system, we compute the QAOA performance in the infinite-degree limit. Together, these techniques give provable guarantees on QAOA performance on large graphs. Despite the improvements we introduce to the classical algorithm, QAOA achieves a better cut fraction guarantee for depths $p\geq 9$, corresponding to girth $g\geq 20$, for both finite- and infinite-degree regimes. Thus, we obtain an apparent quantum advantage from applying QAOA to the Max-$k$-Cut problem.

We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full $L^p$-operator algebras for every $p\in[1,\infty]$. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for $1<p<\infty$, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For $1\leq p<\infty$, we also compare the concrete $p$-linking algebra of the reduced Morita equivalence with the reduced $L^p$-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for $p=2$. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an $L^p$ version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding $K$-theory.

We study how well deep neural networks approximate and learn spectral Barron functions. Recent studies have shown that these function classes can be efficiently approximated by shallow neural networks without suffering from the curse of dimensionality. We complement these results by providing new approximation bounds for deep networks with ReLU activation and establishing the minimax rates for learning these function classes. Specifically, we show that $d$-dimensional spectral Barron functions with smoothness index $s>0$ can be approximated by deep ReLU neural networks with approximation rate $\widetilde{\mathcal{O}} (S^{-\frac{1}{2}-\frac{s}{d}})$, where $S$ denotes the number of nonzero parameters in the network. Using this approximation result, we further show that deep ReLU neural networks can learn spectral Barron functions in a fast rate $n^{-\frac{d+2s}{2d+2s}}$ with $n$ training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.

We show that the associahedron $\mathcal{A}_n$ has chromatic number $O(\log\log n)$, improving on the previously best known upper bound of $O(\log n)$.

In this paper, we consider certain uniformity conditions on the radii of balls. Under these conditions, we study the semicontinuity of the Hausdorff dimension of measures with respect to measured Gromov–Hausdorff convergence and box convergence. We also examine the semicontinuity of other dimensions of measures, which are similar to the Hausdorff dimension of measures.

Let $\mathcal{E}$ be an essentially small idempotent-complete exact category with enough projective objects. We develop derived delooping levels in $\mathcal{E}$. We apply this framework to a basic support $\tau$-tilting module $T$. We then define the derived delooping level relative to $T$ and compare it with the ordinary relative delooping level and the derived delooping level of $B=\operatorname{End}_A(T)$. We show that the big finitistic dimension of $B^{\mathrm{op}}$ is bounded above by the derived delooping level of $\operatorname{Fac}T$ relative to $T$. Moreover, we prove that a self-orthogonal $\tau$-tilting module $T$ is a $1$-tilting module whenever $2\text{-}\ddell_B(DT)$ is finite.

In this paper our aim is to prove a conjecture of Á. Baricz on Bessel functions of the first kind, which improves the classical Turán type inequality for Bessel functions of the first kind proved by O. Szász. The idea is to consider the normalized Turán expression for Bessel functions of the first kind and to show that between two consecutive zeros of the Bessel functions of the first kind the branch of this normalized Turán expression has a local minimum, and the minimum values form a strictly increasing convergent sequence, whose terms are strictly decreasing with respect to the order. Moreover, we extend this result to general Bessel functions and regular Coulomb wave functions, and we use a similar approach to show sharp Turán type inequalities for modified Bessel functions of purely imaginary order and parabolic cylinder functions. In addition, we prove a complex analogue of the result on Bessel functions of the first kind about the strictly decreasing property of the successive minimum values: an inclusion property in the complex plane of the Turán expression for Bessel functions of the first kind by using the subordinating factor sequence technique in the sense of H.S. Wilf. The techniques employed in the paper may be useful to treat similar problems where Turánians or normalized Turánians of other oscillatory special functions appear.

Let \(\cB\) be an abelian category with enough projective and injective objects, and let $ \cA=\cB\ltimes_\eta\sF $ be an \(\eta\)-extension induced by a right exact endofunctor \(\sF\). We introduce uniform image-nilpotence for \(\eta\)-extensions and characterize it in terms of the nilpotence of the associative natural transformation \(\eta:\sF^2\to\sF\). Using the associated \(\eta\)-Loewy filtration, we prove a lifting theorem for right \(n\)-cotorsion pairs over image-nilpotent \(\eta\)-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.

A silver colouring is a proper colouring in which every colour appears in the closed neighbourhood of each vertex of a prescribed independent set. This local condition suffices when vertices are tested one at a time. We study a stronger requirement for simultaneous testing: whenever one or two vertices of the independent set are attacked together, each colour must supply distinct nearby defenders for them. We call this a pair-defensive silver colouring. For the hypercube $Q_d$, with one parity class as the attacked set, we prove that the maximum number of colours in a pair-defensive silver colouring is at most $\lfloor(d+3)/2\rfloor$, roughly half the ordinary silver-colouring target $d+1$. We construct colourings attaining this bound in four consecutive dimensions around every power of two, and we study the structure of the extremal, bound-attaining colourings. In each odd critical dimension, we characterize the extremal colourings by a partition of the defender parity into regular, triangle-free subgraphs of the halved cube. This characterization also has a local form in terms of coordinate matchings and a defect coordinate

We introduce a combinatorial criterion for the projectivity of standard modules over generalized Reedy categories, formulated in terms of a finite bipartite graph whose normalized weightings determine normalization idempotents and hence projective splittings of standard modules. We apply this framework to recover several classical normalization constructions and Dold–Kan–type equivalences, and to obtain new Dold–Kan–type equivalences for categories of finite-dimensional affine, semilinear, and semiaffine spaces over a finite field. These equivalences also yield explicit descriptions of uniformly continuous representations of the corresponding infinite transformation monoids. Together with our previous work on Hom-spaces between standard modules, the results provide a combinatorial machinery, based mainly on graph theory and linear algebra, for establishing Dold–Kan–type equivalences for generalized Reedy categories.

It has been found that utilizing the geometric properties of the graph dynamics can bring out crucial information of the data that a statistical analysis will not. The properties of Ricci flow bring out hidden dynamics of the data in the same way as decomposition of smooth manifolds. In this paper we explore an alternative to Ricci flow, where we find that we can bring out similar properties like it using a simpler and well defined Edge Laplacian to practice community detection in graphs. Moreover, the results obtained by the present approach show that a flow driven by the Edge Laplacian yields results similar to those of the Ricci flow. In contrast, computing the Forman-Ricci curvature requires looping over the adjacent edges, while computing Olivier-Ricci curvature involves the computation of Wasserstein distance and probability distribution associated with the graph nodes, making the Edge Laplacian approach computationally more efficient.

{Non-contingency logic $(\KWL)$ replaces the usual necessity operator of modal logic with an operator expressing that a proposition is necessarily true or necessarily false. Besides its intrinsic logical interest, it admits natural interpretations as knowing whether in epistemic logic and as decidability under the arithmetical interpretation of provability logic. Although the semantics of non-contingency logic have been extensively studied, its proof theory remains comparatively underdeveloped. In this paper, we develop a uniform proof-theoretic framework for $\KWL$ over a broad class of frame conditions. Our approach is based on generalized path conditions (GPCs), a grammar-theoretic formalism that uniformly captures many standard modal frame properties. For every finite set $\gpc$ of GPCs, we construct a corresponding labelled sequent calculus. All calculi share a common set of logical rules and differ only by a single structural rule generated from $\gpc$, which captures the underlying frame conditions. We prove that these calculi are sound and complete with respect to their corresponding frame classes, thereby providing a uniform proof theory for a large family of non-contingency logics.}

We prove that, for every irrational $\xi$, any $F_\sigma$ additive subgroup of $\R$ has an $F_\sigma$ $\Q$-vector space extension of the same Hausdorff dimension satisfying $A+\xi A=\R$. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in every prescribed dimension $d\in[0,1)$. We also construct in ZFC a zero-dimensional lightface $\Delta^0_3$ $\Q$-vector subspace $A$ for which $A+\xi A=\R$ holds for every irrational $\xi$, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two $F_\sigma$ sets, but is neither $F_\sigma$ nor $G_\delta$. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional $F_\sigma$ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand's projection theorem.

We develop a positive-conservative Fourier optimization (PCFO) method for computing ground states of Bose–Einstein condensates with higher-order interactions. The ground-state problem admits a convex density formulation, but the singular behavior near zero density poses difficulties for high-accuracy computation. We introduce convex regularizations of the density energy and establish their $\Gamma$-convergence on the computational domain as the regularization parameters vanish. The regularized problem is discretized by a Fourier pseudospectral method with mass conservation and nodal positivity. For the resulting constrained optimization problem, we use an accelerated first-order method for the main energy reduction, followed by a Newton refinement to reduce the remaining optimality residual. Numerical experiments show spectral-type convergence for regularized problems, quantify the effects of the regularization parameters on accuracy and spatial resolution, and demonstrate the effectiveness of the two-stage optimization method.

We introduce action integrals for partially integrable Hamiltonian systems that interpolate between the non-integrable and completely integrable theory. We show that under the assumption of ergodicity of the Hamiltonian system on the common level sets of its integrals, these actions become constants of motion for the averaged dynamics and, consequently, adiabatic invariants when the Hamiltonian function is allowed to slowly change with time.

This paper presents a scalable optimal-transport-based framework for terminal distribution matching in multi-agent systems. While optimal transport provides a natural way to measure distributional mismatch and assign agents to a desired spatial distribution, global discrete transport can become computationally expensive for large-scale systems. We address this bottleneck by partitioning agents and target samples into spatially corresponding blocks and solving smaller local transport problems. Under a mass-balance condition, the resulting restricted coupling remains feasible for the global problem and provides an upper bound on the Wasserstein cost. The local assignments generate target locations for finite-horizon agent control, applicable to both linear and nonlinear dynamics. By alternating local assignment and control, we establish a cycle-to-cycle descent guarantee for the resulting transport surrogate. The proposed framework therefore enables scalable terminal distribution matching while retaining a rigorous connection to the Wasserstein objective. The technical soundness of the proposed results is validated through simulations.

We prove two conjectures of Laugesen and Siudeja: the equilateral triangle uniquely maximizes the harmonic mean of the first two positive Neumann eigenvalues at fixed perimeter and their geometric mean at fixed area. The proof uses the two equilateral eigenfunctions and four projected polynomial corrections. Triangle symmetry reduces the associated variational matrices to two-by-two operations and two scalar invariants. Angular comparison and horizontal trial functions then yield the global inequalities. The same argument gives explicit quadratic stability bounds in terms of normalized side-length asymmetry.

We study the singular sets of real-valued solutions to periodic parabolic equations with rapidly oscillating coefficients. Under a nondegeneracy condition on the corrector matrix, we obtain a local parabolic $n$-dimensional Hausdorff measure bound and a codimension-two Minkowski estimate. The Minkowski estimate holds at every scale, including scales below the oscillation scale $\varepsilon$, with a constant independent of $\varepsilon$. The bounds depend on a coarse-scale ratio of space-time $L^2$ mass to terminal-slice $L^2$ mass. Independently of corrector nondegeneracy, we prove that this ratio controls doubling at all smaller scales uniformly in $\varepsilon$. The singular-set argument combines summable errors in truncated Gaussian approximations, comparison of caloric polynomials at different time centers, and a stopping cover with microscopic singular-set estimates.

We discuss a class of mixed characteristic rings defined analogously to Stanley-Reisner rings by replacing one variable with a uniformizing parameter for a discrete valuation ring. We adapt Hochster's formula for Tor and Ext modules, Hochster's formula for local cohomology, and Terai's criterion for Serre conditions to this new setting; one of the tools needed is cellular sheaf cohomology, for which we give a brief treatment.

We encode the vertices of every finite complete bipartite graph as binary strings at each scale. The information distances between these strings approximate the scaled graph distances with an additive error bounded by a constant independent of the scale. We also assign one fixed infinite binary sequence to each vertex; suitable prefixes satisfy the same distance estimate with logarithmic error. This extends Hutter's construction for $K_{3,3}$ and resolves his open problem on scale-embeddings of finite complete bipartite graphs. Combined with his complete bipartite obstruction, these embeddings show that no positive power of information distance can be represented exactly by distances in a real Hilbert space, resolving his question about such representations. For conditional prefix complexity, we further prove that at most $2^{\delta+C}$ strings lie between any two strings with total distance exceeding their mutual distance by at most $\delta$, where $C$ is independent of the endpoints. This bound rules out scale-embeddings of the integer line and every positive-dimensional integer lattice, resolving Hutter's lattice embedding problem for both string and sequence embeddings. Restriction to the integer lattice gives the same conclusion for $(\mathbb R^m,\|\cdot\|_1)$ for every $m\ge1$.

This is a continuation of our study of algebraic dymanics of birational automorphisms of complex Calabi-Yau manifolds in the view of primitivity and the existence of Zariski dense orbit. The universal Coxeter groups and their Coxeter elements also play important roles in our study.