Papers

A coloring of the vertices of a hypergraph is called balanced if the sizes of the color classes differ by at most one. We say that a hyperedge is rainbow if its elements have pairwise distinct colors. In this paper, we provide a general upper bound on the balanced upper chromatic number of arbitrary linear hypergraphs, that is, the largest integer $k$ such that there exists a balanced $k$-coloring of the vertices of the hypergraph without rainbow hyperedges. We focus on the cube $C_t^n$, defined as the linear hypergraph whose vertices are the lattice points in $[0,t-1]^n$, and whose hyperedges are the sets of $t$ collinear points. We determine the exact balanced upper chromatic number of $C_t^n$ for $t\geq 4n-2$. For smaller values of $t$, we present bounds and determine this parameter (with few exceptions) in dimensions $2$ and $3$.

The number of non-Clifford gates needed to perform a task, or simply magic, is a resource for fault-tolerant quantum computation. Von Neumann algebras provide a formal mathematical structure to describe infinite-dimensional quantum systems, such as those in quantum field theory or quantum statistical mechanics. A particularly important class of von Neumann algebras is called Type III algebras, for which the standard notions of finite-dimensional systems such as density matrices and traces break down. In this work, we argue that Type III von Neumann algebras fundamentally require an infinite amount of magic. Specifically, we consider the thermodynamic limit of finite-dimensional quantum systems, such as lattice systems, and show that if the states in the thermodynamic limit possess only a bounded amount of magic, the resulting local von Neumann algebra cannot be of Type III. Our result has direct implications for the quantum simulations of quantum field theories, for which the algebra of a local subregion is known to be of Type III.

We study descendant integrals on moduli stacks of simplicially stable curves. For each finite collision complex, we factor the reduction morphism into elementary wall crossings and obtain a reconstruction formula in terms of ordinary Witten-Kontsevich correlators. We use the formula to derive corrected Virasoro recursions. We then package all finite-support theories in a completed disjoint-support Fock space, where the extended potential is a cumulant translate of the Witten-Kontsevich potential.

We find a generating function for the number of ordered commuting pairs of elements in a symplectic Lie algebra over a finite field of characteristic $2$, following the analogous work done in odd characteristic in Fulman and Guralnick (arXiv:1611.05499).

We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic $2$, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component containing all nonidentity unipotent class sizes and that every other component is either an isolated vertex or an explicitly described two-vertex component. We also determine the isolated vertices outside the distinguished component. Additionally, we prove in Appendix A that the divisibility graph of a Frobenius group has exactly two components.

Tornehave and Yalçin proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any $p$-group $S$. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on $S$. As a corollary, we close the main question posed by Mazza–Miller in arXiv:2508.07404 by showing that given a field $k$ of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of $p$-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if $k = \mathbb{F}_2$.

We prove that every countable left uniformly discrete subset of a Hausdorff functionally balanced topological group is right thin. As applications, we answer two questions of Bouziad and Troallic: every left precompact subset of a Hausdorff functionally balanced group is right precompact, and every Hausdorff $\omega$-narrow functionally balanced group is a SIN group.

We study weak limits of multiple Wiener integrals together with the limits of their kernel contractions. After passage to a subsequence, bounded kernel sequences admit a Gaussian Hermite representation whose coordinates carry weights determined by the original chaos orders. In the limits of kernel contractions, only pairings of whole Gaussian coordinates of equal weight contribute. We construct kernel projections with explicit bounds for Gaussian approximation and asymptotic independence. For homogeneous inputs, spectral extraction converges to the decomposable and primitive projections in the limiting kernel spaces. Conversely, prescribed weighted Hermite expansions are realized by smooth compactly supported kernels away from the diagonals, with exact inner products for finite expansions and asymptotically prescribed contractions. We then classify fixed-chaos limits with an absolute exponential moment. For a standard Gaussian vector $X$ and a symmetric matrix $\mathsf A$, a quadratic target $X^{\mathsf T}\mathsf A X-Tr\mathsf A+b^{\mathsf T}X$ is such a limit if and only if $\mathsf A b=0$. We also give a finite cumulant criterion and a joint version. Finally, we identify limits under stationary Ornstein–Uhlenbeck noise: a coordinate of weight $s$ has relaxation rate $s$. Two fourth-chaos examples have identical one-time laws and covariance functions but different asymptotic variances for time averages of their squares. For kernels with nonnegative coordinates, contractions of the weight-one component along simple regular graphs are nonnegative; this excludes an explicit cubic representation that is attained by signed kernels.

A vertex cover of a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that every edge of $G$ is incident to at least one vertex in $C$. The vertex cover number of $G$ is the minimum cardinality of a vertex cover of $G$ and is denoted by $\tau(G)$. A snark is a connected, bridgeless, cubic graph that has an edge chromatic number of four, meaning its edges cannot be properly colored with only three colors. In this work, we investigate the problem of determining the value of a minimum vertex cover for classes of snark graphs. Given a positive integer $k$, we firstly prove that determining whether an arbitrary snark has a vertex cover $C$ with size $|C| \leq k$ is an NP-complete problem. Secondly, we determine the vertex cover number $\tau(G)$ for several subclasses of snark graphs, such as Flower snarks, Goldberg snarks, Generalized Blanuša snarks and Loupekine snarks.

Let $v_1,\ldots,v_T\in B_2^m$ be fixed in advance and revealed sequentially, and assume that $\|v_t\|_\infty\leqslant d^{-1/2}$ for some $d\geqslant 1$ and every $1\leqslant t\leqslant T$. There are absolute constants $L,C,c>0$ and a randomized online signing such that $$\mathbb{P}\left\{\max_{k\leqslant T}\left\|\sum_{t=1}^k\varepsilon_t v_t\right\|_\infty>6L\right\} \leqslant CT\exp\left(-\frac{cd}{\ln^2(ed)}\right).$$ Consequently, constant prefix discrepancy holds with probability at least $1-\varepsilon$ once $d$ is at least $C\ln\frac{3T}{\varepsilon}\left[\ln\left(e+\ln\frac{3T}{\varepsilon}\right)\right]^2$. In particular, every fixed sequence of vectors $a_t\in[-1,1]^m$ with at most $d$ nonzero coordinates admits an online signing with prefix discrepancy $O(\sqrt d)$ and failure probability at most $CT\exp[-cd/\ln^2(ed)]$. We also prove a nonuniform version in which the failure probability depends on the individual parameters $d_t=\|v_t\|_\infty^{-2}$, and a lower bound showing that a universal constant prefix discrepancy is impossible when $d=o(\ln T)$. We identify the corresponding $\ln^2 d$ barrier for the compact-potential method and extend the argument to general symmetric target bodies admitting a quadratic smoothness estimate.

The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.

We propose Sampled Local WeakIdent (SLW-Ident), a framework for identifying changing governing equations from a single set of given data. Different from a typical approach of using finite element based approximation to represent varying coefficients, this paper explores a local approach in identification of differential equations. First, we present the power of Local WeakIdent which gives good local identification and is also computationally efficient with a small patch size, yet it can be sensitive to local perturbations. We propose SLW-Ident which stabilizes the identification process and also incorporates global information: we first sample patches in the whole given domain, identify equations for each sampled patch, then use residual error of these equations to find the transitions between different equations. We refer to a region where the support of the identified equation does not change to be a region of one equation. Within each region of one equation, we pick the most frequently identified equation as the identified equation, and find constant as well as varying coefficient PDEs within each region of one equation. This is justified by an uncertainty quantification theory that gives the relation between statistical error of dominant support selection and the number of patches. We provide various numerical experiments showing that SLW-Ident accurately recovers the regions of one equation and the governing equations for changing PDEs with varying coefficients even with noisy given data.

Real-to-complex spectral transitions mark the onset of amplification in non-Hermitian systems, but their thresholds are often treated as model-specific quantities. Here we develop a general, non-perturbative activated-mode principle that governs the real-to-complex threshold across broad classes of non-Hermitian systems. A central insight is that only a small Hilbert subspace is “activated" at the transition onset, which can be variationally determined through the competition between spectral detuning and mode-level projected non-Hermitian couplings. The result is a closed-form exceptional-activation condition for arbitrarily large “disturbances", rather than a perturbative estimate. We apply our framework to three contrasting illustrative problems, establishing (i) a closed-form threshold for critical non-Hermitian skin amplification at all system sizes; (ii) a new link between impurity tunneling threshold and exceptional point switching; and (iii) activation channel switching without underlying topological phase transition. Overall, our findings recast real-to-complex transitions as generic mode-selection problems independent of any specific symmetry.

We study a spin-$\frac12$ impurity coupled to the boundary of a strongly correlated spin-$s$ Takhtajan–Babujian chain, an integrable model whose low-energy physics is described by a perturbed $SU(2)_{2s}$ Wess–Zumino–Witten conformal field theory. While boundary conformal field theory determines the low-energy universality class of the weak-coupling regime, exact Bethe Ansatz methods reveal a sequence of boundary quantum phase transitions in which impurity-bound states emerge and reorganize the Hilbert space into multiple excitation towers built on distinct boundary configurations. This tower restructuring provides the organizing principle for a rich boundary phase diagram extending beyond the conventional Kondo regime. Weak antiferromagnetic coupling realizes the overscreened $2s$-channel Kondo universality class, whereas stronger couplings generate localized boundary modes and qualitatively new screening mechanisms. To describe the resulting thermodynamics, we develop a generalized thermodynamic Bethe Ansatz framework that captures the multi-tower structure across all regimes. The impurity entropy reproduces the boundary conformal field theory prediction in the overscreened Kondo regime but develops pronounced nonmonotonic temperature dependence once boundary-bound states appear, in quantitative agreement with large-scale finite-temperature matrix-product-operator simulations. Complementary dynamical calculations reveal sharp threshold features in the impurity spectral function that directly track the underlying tower structure. Together, boundary conformal field theory, exact Bethe Ansatz, generalized thermodynamic Bethe Ansatz, and tensor-network simulations provide a unified description of impurity screening, boundary-bound-state formation, and excitation-tower reconstruction in a correlated spin-$s$ chain.

In this paper, we establish Poisson limit theorems on Poisson and Rademacher chaoses. Our principal result is a total-variation bound, valid in both settings, for an integer-valued functional whose highest-order chaos is dominant. The approximation error is the sum of the pure-chaos four-moment terms and an explicit remainder controlled by the $L^4$-size of the lower-order chaoses. On Poisson space the pure-chaos term is controlled solely by the moment defect, whereas on Rademacher space an additional maximal-influence correction is required. When the lower-order remainder vanishes, we recover exactly the corresponding total-variation bound for a shifted pure chaos. As consequences, for nonnegative integer-valued shifts of random variables in a fixed Poisson chaos, convergence of the first four moments is equivalent to convergence in total variation to a Poisson law, together with uniform integrability of the fourth powers; on a fixed Rademacher chaos, the analogous conclusion holds under a vanishing maximal-influence condition. The proof reveals a unit-jump rigidity phenomenon: the four-moment defect simultaneously suppresses unwanted spectral components and rules out non-unit jumps. We show, through an explicit quadratic counterexample, that the maximal-influence condition in the Rademacher setting is necessary for a general Poisson limit theorem. Finally, for every order $q\geq2$, we construct pure Poisson-chaos sequences with vanishing moment defect that converge weakly to a centered Poisson law. These examples show that the exact higher-order rigidity is not uniform once the lattice condition is removed. In both the Poisson and Rademacher settings, our proofs follow a unified strategy combining the Chen–Stein method, exchangeable pairs, and Ledoux's spectral viewpoint.

A classical problem in graph theory known as the Italian domination number(also called Roman 2-domination number), involves assigning labels of 0, 1,or 2 to each node v. The goal is to ensure that every node with a label of 0 has a sum of labels of the nodes in its closed neighborhood that is 2 or greater. In computer systems, it is coined encompassing a robust cyber security strategy that will protect networks from potential threats, such as hacking, malware, and unauthorized access, by deploying security measures to provide the highest level of protection while reducing the misuse of resources. Toeplitz graphs are a special kind of graphs built over Toeplitz matrices from linear algebra, which are matrices with constant straight diagonal members. In this paper, we provide a detailed analysis regarding the Italian domination numbers for every Toeplitz graph family. We provide comprehensive results on Italian domination numbers across multiple graph families and identify the specific values at which the Italian domination number alters with increasing generator values.

Neural combinatorial optimization (NCO) relies on parallel solution sampling for training, yet existing methods fail to fully exploit the rich information latent in a co-sampled solution group. Preference-optimization methods anchor on the single best solution and discard fine-grained quality and structural signal from all other peers-a failure we term gradient signal polarization. Mean-based baselines instead weight peers uniformly, so structurally near-identical peers flood the baseline with redundant information and keep gradient variance high-a failure we term baseline redundancy. We propose SSPO (Structure-Aware Similarity-Weighted Preference Optimization), which scores all $B$ sampled solutions jointly through a dissimilarity-weighted leave-one-out baseline: structurally distinct peers receive higher weight, resolving both failures in a single mechanism. The baseline uses zero-parameter, problem-adaptive solution embeddings built from the encoder's existing node representations. Experiments on TSP, EFL, and JSP benchmarks show consistent gains over prior best-anchor and uniform-weight baselines. A direct comparison against uniform RLOO on TSP and EFL confirms that structure-aware weighting is the primary driver of improvement. The SSPO-trained EFL policy has been deployed in a production facility-location system at JD$\mathord{.}$com, confirming practical viability at scale.

Let $d(G)$ denote the least size of a generating set of a finite group $G$. We prove that if $G$ has a family $\mathcal H$ of subgroups such that $d(H)\leq d$ for every $H\in\mathcal H$ and $\gcd\{\lvert G:H\rvert:H\in\mathcal H\}=1$, then $d(G)\leq d+1$. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow $2$-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow $2$-subgroups of finite simple groups.

In phylogenetics, Metropolis-Hastings methods are commonly used to sample phylogenetic trees or networks, for example from Bayesian posteriors. These methods generally use transitions that distinguish all nodes involved, and thus require fully labelled representations of phylogenetic networks. We argue that sampling leaf-labelled phylogenetic networks demands a correction for the number of fully labelled representatives of a leaf-labelled network, or, equivalently, for its internal symmetry. Without correction, there is a danger of undersampling networks with internal symmetries. We show that this correction can be realized by a quotient construction on the Metropolis-Hastings Markov chain, which, in practice, requires the calculation of the size of the network's automorphism group. Using $\mu$-vectors, we show that the automorphism group is trivial for orchard networks, and thus also for tree-child networks and trees. This implies that a correction for symmetry is not needed when sampling only from such network classes. More generally, using our Python implementation of the algorithms in this paper, we show that using $\mu$-vectors can significantly speed up calculations of automorphism group sizes and thus of Metropolis-Hastings sampling of leaf-labelled networks.

Memory is a core component of AI agents, enabling them to accumulate experience, maintain personalization, and adapt over long-term interactions. However, existing memory systems often remain fixed after development, limiting their ability to adapt their memory models, organization strategies, and procedural knowledge through continued use. We present MindMemOS, a portable and self-evolving memory operating layer that organizes open-world information using a unified entity property timestructure. MindMemOS supports scenario-adaptive memory modeling, higher-order pattern discovery, autonomous memory refinement, and continuous skill evolution. Its MindMemEvolve algorithm employs validation-driven evolutionary search to optimize memory schemas for target scenarios, whiledreaming consolidates accumulated memories by merging redundant records and resolving conflicts. In addition, implicit corrective feedback serves as a human-in-the-loop signal for identifying and revising potentially inaccurate or misaligned memories. Its MindSkillEvolve algorithm further transforms agent execution trajectories into reusable and progressively refined skills. MindMemOS achieves 94.03% accuracy on LOCOMO and 70.63% on PersonaMem. MindSkillEvolve improves SpreadsheetBench success by 9.2 percentage points over the initial-skill baseline.

The degree/diameter problem asks for the largest order of a graph with a given diameter and maximum vertex degree. This has been widely studied and given rise to a recent variation for mixed graphs (graphs with both undirected edges and directed arcs), where an additional bound is placed on the maximum directed out-degree of any vertex. Both problems have applications to network design. Counting the possible number of vertices at each distance from a given vertex gives a bound on the order of a mixed graph satisfying the degree and diameter constraints (the mixed Moore bound). In this paper, we settle an open problem posed by Tuite and Erskine concerning the total regularity of mixed graphs whose order is one less than the mixed Moore bound (almost mixed Moore graphs). We use this result to show that the three known almost mixed Moore graphs of diameter at least three are the only such mixed graphs.

An analog of the classical Kronecker-Weyl theorem for weak uniform distribution is obtained for an arbitrary countably infinite independent subset of a discrete abelian group using a type of interpolation sets called epsilon-Kronecker sets. Specifically, the topological size of the set of homomorphisms inducing weakly uniformly distributed sequences is established by showing that it forms a dense G-delta set.

We study nonnegative solutions of the reciprocal cost law on the positive rationals and determine which of them admit regular extensions to the positive reals. Using the substitution $H=1+F$, we reduce the reciprocal composition law to d'Alembert's functional equation. We prove that every nonnegative solution on $\mathbb{Q}_{>0}$ is determined by one real weight $\alpha_p$ for each prime $p$, with only a global sign identification. Thus the rational solution space is infinite dimensional. We prove that every nonnegative rational solution has an algebraic extension to $\mathbb{R}_{>0}$, but a regular extension exists exactly when the prime weights satisfy $\alpha_p=\lambda\log p$ for some $\lambda\in\mathbb{R}$. In this case the extension is unique and belongs to the one-parameter family $F_\lambda(x)=\cosh(\lambda\log x)-1$. Otherwise the rational solution is unbounded on every nonempty open subset of $\mathbb{Q}_{>0}$, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of $\mathbb{R}_{>0}$, where the alternative is governed by rational rank. Finally, we show that unit logarithmic curvature selects the canonical reciprocal cost $J(x)=(x+x^{-1})/2-1$, while, for a carrier $G$ whose logarithm is dense in $\mathbb{R}$, the single asymptotic condition $F(e^s)\sim s^2/2$ implies both regularity and calibration.

We resolve the ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo, and Willsky up to a vanishing factor. Concretely, for $m$ independent Gaussian points in dimension $d$, we show that with high probability, for $m \leq (1-o_d(1)) \cdot d^2/4$, there exists a centered ellipsoid passing through all $m$ points; for $m\geq (1+o_d(1) )\cdot d^2/4$, no such ellipsoid exists. This confirms that the ellipsoid fitting problem has a sharp phase transition at $d^2/4$.

In this work, we present novel approaches for constructing linear codes over $\ZZ_4$ from the known ones. We succeeded in obtaining new linear codes, many of which are optimal. In particular, we found all optimal codes for $k_1=2,~k_2=0$ and many optimal codes for $k_1=3,~k_2=0.$

Quasicontinuous functions have found applications in many areas of mathematics. We study completeness properties of the space of quasicontinuous functions equipped with the topology of pointwise convergence. Let X be a Hausdorff topological space, Q(X) be the space of quasicontinuous real-valued functions and {\tau}_p be the topology of the pointwise convergence. For (Q(X), {\tau}_p) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), {\tau}_p) is completely metrizable, then X is countable and the set I(X) of isolated points of X is dense in X. If X is first countable, then (Q(X), {\tau}_p) is completely metrizable if and only if X is countable and I(X) is dense in X.

Let $X\subset\mathbb{R}$ be finite and let $\gamma(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(\gamma(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3\gamma(x_i)=\sum_{i=4}^6\gamma(x_i)\} \ll_{\epsilon}|X|^{3+\epsilon}. \] When specialized to the parabola, our result implies near-optimal estimates for the number of solutions to the diameter-free quadratic Vinogradov system. As a second application, we settle a conjecture from Krishnapur-Kurlberg-Wigman and Bombieri-Bourgain concerning lattice points on dilates of the unit circle. As a third application, we prove that $|A-A|\gg_\epsilon|A|^{5/3-\epsilon}$ and $|A+A|\gg_\epsilon|A|^{8/5-\epsilon}$ for any finite convex sequence $A\subset \mathbb{R}$.

We give a proof that the optimal constant in the weighted Hilbert inequality is strictly greater than $\pi$, answering in the negative a question asked by Montgomery and Vaughan. The proof proceeds via an explicit limiting counterexample.

Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157\varphi(n)/1440<\varphi(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.

We develop a probabilistic cavity-contiguity framework for proving replica-symmetric convergence of local marginals in mean-field Gibbs systems. The approach is based on cavity decompositions of the Hamiltonian together with direct comparison of probability measures through Radon-Nikodym derivatives and Hellinger-type estimates. At a conceptual level, the method separates the concentration of the relevant order parameters from the identification of the asymptotic cavity model and the comparison of the associated Gibbs measures. In contrast with interpolation-based approaches, the argument relies only weakly on the Gaussianity of the disorder and naturally accommodates concentration tools such as Poincaré and log-Sobolev inequalities. Rather than pursuing maximal generality, with the aim of making the method transparent, we implement the framework in a canonical example: the high-temperature Sherrington-Kirkpatrick model. In this setting, we prove that the marginal law of a fixed spin converges in total variation toward the effective one-dimensional cavity measure predicted by the replica method. Beyond the specific result for the Sherrington-Kirkpatrick model, the paper illustrates a broader cavity-contiguity methodology which is expected to extend naturally to other mean-field Gibbs systems, particularly Bayesian inference models with or without mismatch.