Papers

Based on a recent generalization of the Reidemeister spectrum of a finite group [1], in this short note we extend the main result of our previous paper [2].

Let $A$ be a finite-dimensional local algebra over a splitting field. We prove that an irreducible tensor factorization $A \cong P_1 \otimes \dotsb \otimes P_n$ of the algebra $A$ is unique up to isomorphism and order of $P_1, \dotsc, P_n$.

Matchgate circuits describe classically tractable dynamics in a physically relevant setting while nonetheless exhibiting features characteristic of complex quantum systems, such as high amounts of magic. This invites the broader question: to what extent are typical complexity measures saturated under trivial dynamics? Here, we address this by studying the behaviour of different Heisenberg-picture complexity measures in random matchgate ensembles. Our main technical contribution is a tractable and generalisible free-fermionic unitary Weingarten calculus formulated entirely in the Majorana basis. This method is tailored to computing moments of operators, and simplifies to a combinatorics problem which depends on the boundary conditions prescribed by a given operator functional. Using this framework, we derive closed-form expressions for the typical local-operator entanglement, operator stabiliser entropy, and higher-order out-of-time-ordered correlators. For extensive initial Majorana strings, both operator entanglement and stabiliser entropy obey volume laws, even though the underlying circuits remain classically simulable. Out-of-time-ordered correlators can likewise become exponentially small, showing that their late-time saturation value alone does not witness computational hardness and constituting a classically tractable example of emergent free independence. Finally, we relate these separations to operator resource theories, showing how operator entanglement beyond the Gaussian baseline lower-bounds the number of non-Gaussian gates in doped matchgate circuits. Our results present a practical toolbox for computing exact averages over random matchgates, while clarifying the distinction between quantum resources versus simulation complexity, and their dependence on the representation being exploited.

We prove a dimension-free Poincaré inequality for isotropic unconditional log-concave probability measures, establishing the Kannan-Lovász-Simonovits (KLS) conjecture in the unconditional setting. The proof combines a new application of Dunkl operator theory combined with an appropriate transformation of the log-concave measure. Specifically, the proof follows by a direct spectral analysis of a new Dunkl-Langevin operator adapted to the transformed measure. The core ideas of the proof were generated by AI generative tools through interactions with the authors over several weeks. The authors refined and formalized the argument.

In this article I revisit the frequency interpretation of probability of von Mises, and argue that it contains the key elements of any coherent interpretation of scientific probability - notwithstanding the heavy criticisms von Mises collected from mathematicians and philosophers alike. I propose a modified definition that is not based on von Mises' "infinite collectives" but retains an essential ingredient of his interpretation, namely that probability can only quantitatively be defined for events that are, or can be, repeated in similar conditions and that exhibit frequency stabilisation. New is that the mentioned "conditions" should be "partitioned". Thus, I will partition probabilistic systems into object and environment, or in object, initiating, and probing subsystem, and show that such partitioning solves a series of classic problems. Other consistent interpretations can be unified with frequentism, e.g. Bayesianism à la Jaynes. I explore in detail the tantalising subjective or operationalist touch of probability, and argue that it can be put on objective grounds. One further upshot of the model is that several quantum puzzles can be seen as deriving from an imprecise interpretation of probability.

Recent advances in the classification of certain classes of Yang-Baxter operators through a holistic categorical approach has some of us wondering about ways to identify solutions. In particular the classification of charge conserving solutions in all dimensions inspires the question: \emph{what fraction of solutions are close to charge conserving solutions, under any reasonable notion of equivalence?} Of course reasonable and close to are open to interpretation. Here we suggest some precise notions of both, and prove that in dimension $2$, 10 out of 11 local families of solutions are close to charge conserving, and the one that misses is a singleton.

For a graph $H$, let $f(n,e,H)$ be the least number of colors in an edge-coloring of some $n$-vertex graph with at least $e$ edges in which every copy of $H$ is rainbow. Burr, Erdős, Graham, and Sós conjectured that $f(n,\lfloor n^2/4\rfloor+1,C_{2k+1})=(1/8+o(1))n^2$ for every fixed $k\ge3$, and Bucić, Chen, and Ma recently proved this for all $k\ge4$. We prove the remaining case $k=3$: \[ f\left(n,\left\lfloor n^2/4\right\rfloor+1,C_7\right) =\left(\frac18+o(1)\right)n^2. \] The lower bound rests on a weighted palette inequality, which we prove with an exact rational certificate on five sampled vertices. Its main ingredients are a fractional matching of compatible triangular edges and private resources attached to nontriangular edges. A stable form of the inequality, combined with regularity, triangle removal, and a direct argument for graphs close to bipartite, transfers the bound to arbitrary edge-colorings. We also describe a Lean 4 formalization of the conjecture for every fixed $k\ge3$, which combines the new seven-cycle proof with a formalization of the Bucić-Chen-Ma argument for $k\ge4$.

In adaptive quantitative group testing with exactly two defective items, each test reports how many defectives lie in a chosen subset. We study configurations in which the possible defective pairs form the edges of a star forest. Aigner proved a necessary majorization condition on the ordered star sizes for identifying the defective pair within a prescribed number of tests, and conjectured that this condition was sufficient. We disprove the conjecture by an explicit six-test counterexample and give an analytic family of counterexamples for every test budget $k\ge6$. The obstruction uses two tight prefix sums to force incompatible demands on the outcomes of the first test. We also prove, by exhaustive computation combined with analytic reductions, that the converse holds for $0\le k\le5$. Thus six is the first test budget at which majorization alone fails. The counterexamples and their infinite extension do not depend on the exhaustive computation.

In the nearly unstable heavy-tailed regime of Jaisson and Rosenbaum, where the intensity of a linear Hawkes process converges to a rough square-root Volterra process, we ask what a record of length $T$ reveals about the relative distance to criticality $1-\rho$ and the exogenous rate $\mu$. For the first, the information is governed by $\mathcal I_T=T\mu(1-\rho)$, not by the number of events; $\mathcal I_T$ stays bounded although the event count diverges like $T^{2\gamma}$, so no estimator of the relative distance to criticality is uniformly locally consistent. For the second, the Fisher information diverges because the rough limit has an atom at zero: the asymptotic information geometry is singular in the exogenous direction and, unlike the light-tailed case, has no Feller-type threshold. We prove that the joint maximum likelihood estimator of $(\mu,\rho)$, whose likelihood is jointly concave, recovers $\mu$ consistently at the random rate $\{Q_T/\log\log Q_T\}^{1/2}$ given by the observed information $Q_T$, while its estimate of the relative margin is only tight: the exogenous rate is learned although the endogenous parameter it is coupled to is not. For empty-start records $Q_T$ diverges because the rough limit started at zero spends positive time at zero; on windows observed long after the start, still with the branching ratio unknown, learnability holds with probability tending to one as the window grows, through the ergodicity of the rough limit and the atom of its stationary law. We also prove finite-dimensional convergence of the stationary intensity to the stationary rough Volterra process.

The branching ratio $\rho$ of a self-exciting point process determines the branching margin $1-\rho$ to the critical boundary. We study inference on $\rho$ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback–Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order $T^{-\gamma/(2\gamma+1)}$ for a power-law envelope with tail exponent $\gamma$. A block-dispersion estimator whose block length grows as $T^{1/(2\gamma+1)}$ attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown $\gamma$, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.

Feature selection in neural networks remains a challenging problem, particularly in the presence of noisy or contaminated data. LassoNet is a recent approach that addresses this issue by combining neural networks with hierarchical sparsity constraints, enabling simultaneous prediction and variable selection. However, its standard formulation relies on the mean squared error (MSE) loss, which is known to be highly sensitive to outliers. In this paper, we present Robust LassoNet, an extension of LassoNet that incorporates robust loss functions, such as Huber, Cauchy, Tukey's bisquare, and Nonnegative Garrote, to mitigate the effect of extreme observations. The proposed approach preserves the original optimization framework while improving stability under data contamination. Through experiments on synthetic and real datasets, we show that robust LassoNet significantly improves both predictive performance and feature selection accuracy in the presence of outliers or heavy tailed noise, while maintaining comparable performance in clean settings. These results highlight the importance of robustness in neural network based feature selection and suggest practical guidelines for choosing appropriate loss functions.

We construct systematic $q$-ary codes correcting two deletions with three redundant symbols over polynomial-size alphabets. For every information length $n\ge2$, our first construction has length $n+3$ over an alphabet of size $q=2^s<2^{32}n^7$ for some integer $s$, with polynomial-time encoding and decoding. A cardinality bound of Liu and Xing implies that three redundant symbols are necessary, so the construction is optimal in this measure. We also establish the existence of systematic codes with the same redundancy over alphabets of size $O(n^4)$, leaving efficient encoding and decoding at that alphabet scale open. For a single burst of at most two adjacent deletions, we give a polynomial-time systematic construction with three redundant symbols over a power-of-two alphabet of size $q<(n+2)^4$. A converse bound for adjacent deletions shows that three redundant symbols are again optimal for $n\ge2$.

We continue research on the burning game on graphs. Given a graph $G$, two players, Burner and Staller, take turns in selecting vertices of $G$ to burn. All burned vertices spread fire to unburned neighboring vertices, as in the burning process. The goal of Burner is to burn the graph as quickly as possible, while Staller wants the process to last as long as possible. If both players play optimally, then the number of time steps needed to burn the whole graph $G$ is the game burning number $b_{\rm g}(G)$ if Burner makes the first move, and the Staller-start game burning number $b_{\rm g}'(G)$ if Staller starts. In this paper, we study this game further, establishing Nordhaus-Gaddum bounds on the game burning number, as well as bounds for four different types of graph products: strong, Cartesian, lexicographic and corona products.

The Thurston spine of the genus 2 Teichmüller space of closed compact surfaces is obtained. In particular, we identify all mapping class group-orbits of filling sets of systoles in genus 2. Thereby, surprisingly rich cases already appear in this simplest case which is genus 2. This is contrasted with a cell decomposition of a bordification of Teichmüller space obtained computationally using Rivin's angle coordinates, on which the cells are labelled by Delaunay graphs closely related to our systolic graphs.

A skew-morphism of a finite group $G$ is a permutation $\sigma$ of $G$ fixing the identity element such that there exists a function $\pi\colon G\to\mathbb{Z}$ satisfying $\sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y)$ for all $x,y\in G$. For a given skew-morphism $\sigma$ of $G$, the product of the left regular representation of $G$ and the cyclic group $\langle\sigma\rangle$ forms a permutation group on $G$, called a skew-product group of $G$. Skew-product groups of finite elementary abelian $p$-groups were investigated in \cite{DLYZ}. In the present paper, we extend this study to all finite abelian $p$-groups.

Murphy and Petridis proved that, for subsets $A$ of an arbitrary finite field $\F_q$, the condition $\abs A>q^{2/3-\delta}$ implies $\abs{(A-A)(A-A)}>q/2$ whenever $\delta<1/13542$ and $q$ is sufficiently large. We improve the explicit saving to $1/1689$. The proof replaces the main physical-space extraction in their large-energy branch by a common Fourier spectrum, followed by a popular-ratio Balog–Szemerédi–Gowers step. The resulting additive energy is treated level by level, retaining difference multiplicities and combining the Reiher–Schoen relation lemma with Mohammadi's difference–ratio estimate. This gives the structural exponent $188$ and hence \[ \abs A\ge Cq^{375/563}=Cq^{2/3-1/1689} \quad\Longrightarrow\quad \abs{(A-A)(A-A)}>\frac q2 \] for an absolute constant $C>0$.

The problem of the uniform law being a sum of two independent distributions has been well studied. Here, we study the approximation of the uniform law to a sum of two distributions with fixed support, under the following discrepancies: the Manhattan distance, the Euclidean distance, the forward Kullback–Leibler divergence and the Wasserstein-one distance based on the line metric. The problem of membership of the uniform law in this model has been well studied. Explicit results are obtained in two models, one where one of the distributions is a Bernoulli and the other when both distributions have the same support, including one conjecture. Finally, we show an application of our reconstruction method to a recent conjecture about the coding capacity of an additive noise channel.

We prove that every commutative Gaussian ring has finitistic weak dimension at most two. The proof works with a fixed module of finite flat dimension and does not assume that the weak global dimension of the ring is finite. For a local Gaussian ring $A$, with nilradical $N$, the kernel $I$ of $A\to A_N$ satisfies $NI=I^2=0$. A hypothetical module $M$ of flat dimension three gives a nonzero torsion module $H=\Tor_2^A(A/N,M)$ over the valuation domain $A/N$, with $H\cong\Tor_1^A(I,M)$ and $I\otimes_{A/N}H=0$. Two carefully controlled localizations, a propagation argument in the chain ring $A/D$, where $D=\{x:x^2=0\}$, and an annihilator factorization in a filtered system of cyclic modules yield a contradiction.

A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.

In a deviation event, the empirical measure of $n$ independent draws lands in a set of distributions. The exponent of its probability splits, at every $n$, into two nonnegative terms. The first is the rate: $n$ times the relative entropy, from the population, of the distribution of a typical draw under the event. The second is the dependence that conditioning induces among the draws, measured by their total correlation. Their relative sizes depend on the geometry of the set. This paper focuses on this exact split and on the geometry that the interplay of its two terms reveals. The dependence vanishes exactly when the event confines every draw to a single set, and the classical rate of that constraint is exact. On a set invariant under a compact group that fixes the population and leaves no invariant set of intermediate probability, the rate vanishes; uniform convergence over a hypothesis class that such a group permutes is one. On a half-space whose threshold sits a fixed number of standard errors above the mean, the fraction of the exponent that is dependence tends to a function of the event's probability alone. On a set split into pieces, the dependence is the pieces' average plus $n$ times the Jensen-Shannon divergence of their marginals, minus the entropy of their weights. The split extends to a general reference law. When the information projection is such a law, the split gives the exact value of the expectation in the exponential change of measure to the projection.

We propose a general goodness-of-fit framework for distributions on separable metric spaces. Under suitable identifiability conditions, probability distributions are characterized by distance profiles, which motivates their use in goodness-of-fit testing, for simple and composite null hypotheses. For composite null hypotheses, parameter estimation is incorporated via a Bahadur-type expansion, and the asymptotic distribution of the empirical process for distance profiles is obtained under the null. We define test statistics based on this empirical process and derive their asymptotic null distributions. We further study the behavior of the proposed tests under fixed and local alternatives, establishing consistency results. Multiplier bootstrap procedures are developed, and their conditional asymptotic validity is established under both simple and composite null hypotheses. The methodology is illustrated with simulation studies and real-data applications for data on the sphere, hyperboloid, and simplex.

We determine the exact strong converse exponent for secret-key transmission and generation over general finite-dimensional quantum wiretap channels, measuring reliability and secrecy jointly by squared fidelity to an ideal secret key. We introduce a novel Renyi private information whose regularization characterizes this exponent. An exact integral representation in terms of ordinary private information gives a uniform continuity bound, showing that the regularized quantity converges to private capacity as the Renyi order approaches one. This establishes for any wiretap channel an exponential fidelity decay at every rate above private capacity.

The Maker Breaker Triangle Game involves two players, Maker and Breaker, who alternately claim 1 and $q$ edges of $K_n$, respectively. Maker's goal is to claim all three edges of any triangle, whereas Breaker's goal is to prevent this. The threshold bias, i.e. the minimum $q$ such that Breaker wins, is known to lie between $\big\lceil\sqrt{2n-2}-\frac{3}{2}\big\rceil$ and $\big(\sqrt{8/3}+o(1)\big)\sqrt{n}$. Determining its exact value is a longstanding open problem. In this paper, we introduce a novel version of the game in which Breaker may claim fewer than $q$ edges per round to build up a budget that he can spend in later rounds. With this additional power for Breaker, we determine the threshold bias to be precisely $\big\lceil\sqrt{2n-2}-\frac{3}{2}\big\rceil$ for all $n$, matching the known lower bound. This is the first version of the Maker Breaker Triangle Game for which the exact threshold bias is known. Even if Breaker is not allowed to use the budget for threats, we prove that the threshold bias is still $\big(\sqrt{2}+o(1)\big)\sqrt{n}$. Furthermore, we study the budget version of other Maker Breaker Games. Specifically, for the $K_4$-Game, we prove the first explicit lower and upper bounds on the threshold bias in both the original and the budget version of the game.

The celebrated conjecture of Lovász from 1969 asks whether every connected vertex-transitive graph has a Hamiltonian path. Bucić, Christoph, Pokrovskiy and Steiner recently proved that every such graph on $n$ vertices contains a cycle of length $n^{2/3-o(1)}$. In this paper, we improve this bound to $n^{1-o(1)}$. Our proof uses a structure theorem of Tessera and Tointon to first obtain a partition of the vertex set into sets of small diameter in the original graph. When the parts are large, we repeatedly traverse a spanning tree of maximum degree at most three in the quotient graph, and use the Lovász local lemma to join random short paths along this traversal and extract a long path in the original graph. When the parts are small, we apply Babai's contraction lemma to reduce the problem to finding a long path in a connected Cayley graph of a nilpotent group with boundedly many generators and bounded nilpotency class, and then show that such a Cayley graph on $m$ vertices contains a path on $m^{1-o(1)}$ vertices.

Generative modeling is widely used for producing diverse objects from complex, multimodal distributions. However, its expressivity does not, in general, come with formal guarantees that the generated objects satisfy hard constraints or requirements. In multi-agent generation, this problem becomes more challenging because a hard requirement can depend on multiple agents, while each agent may need to determine its own guidance input without relying on the simultaneously computed guidance inputs of other agents. To this end, we introduce DeGG-Flow, a general framework for multi-agent flow matching with decoupled generative guidance. By representing the generative process as a control-affine dynamical system, we develop guidance conditions for two classes of coupled requirements: shared requirements whose satisfaction depends on multiple agents together, and private requirements associated with each individual agent dependent on its neighbors. For both classes, we establish feasibility conditions and finite-horizon convergence guarantees. We further derive a Wasserstein bound that characterizes the distributional deviation induced by the guidance. We demonstrate DeGG-Flow on multi-robot collaboration for crossing a spatial gap by reconfiguring the environment, and on multi-object scene generation with affordance requirements. Across both applications, DeGG-Flow directly generates objects that satisfy all corresponding hard requirements, including at team sizes unseen during training.

We study average-reward weakly-coupled Markov decision processes (WCMDPs), where a WCMDP consists of $N$ smaller MDPs, called arms, that share multiple per-step budget constraints. We consider the setting where the arms have identical model parameters, multiple actions, and state- and action-dependent costs. For restless bandits (RBs), a well-studied special case of WCMDPs, prior work has developed policies that achieve an $O(1/\sqrt{N})$ optimality gap under general conditions, and has further identified conditions under which policies can achieve a better-than-$1/\sqrt{N}$ optimality gap. However, for general WCMDPs, no prior result achieves an optimality gap better than $1/\sqrt{N}$. In this paper, we identify conditions analogous to those for RBs under which a better-than-$1/\sqrt{N}$ optimality gap is achievable, and design a policy that attains an $O(1/N)$ optimality gap. Notably, unlike prior approaches based on generalizing priority orderings, our policy is not priority-based but rather is designed to induce locally linear mean-field dynamics.

We study Ore extensions $R[t;\sigma,\delta]$ of rings graded by an arbitrary group $\Gamma$. The grading of $R$ extends to a grading with $t$ homogeneous of degree $\gamma$ if and only if $\sigma$ and $\delta$ are compatible with conjugation and translation by $\gamma$, and this grading is then unique. For nonabelian $\Gamma$ the degree of $t$ need not centralize the support of $R$; this twist can be removed by a change of variable when $R_\gamma$ contains a unit, but not in general. The graded forms of the classical necessary conditions for simplicity are not sufficient, as the quantized Weyl algebra shows. If $R$ is graded-simple and $\sigma$ is an automorphism, then $R[t;\sigma,\delta]$ is graded-simple if and only if it contains no homogeneous normal element that is monic of positive $t$-degree. For Ore extensions of group algebras, an affine action of the group on the line describes all graded ideals; in characteristic zero, graded simplicity is equivalent to the nonvanishing of a class in $H^1(G,k_\chi)$, and simplicity forces the group to be metabelian.

We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special $S^3$-$S^3$ bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He–Liu–Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser–Wraith gluing theorem gives the required positively curved metrics.

In this article, we analyze plane-symmetric solutions to the relativistic Euler equations on spatially flat spacetimes with decelerated expansion. We consider a linear equation of state $p(\rho)=K\rho$ and power law inflation $a(t)=t^{\alpha}$, where $0<K<\frac{1}{3}$ and $0\leq\alpha<1$. Previous results imply nonlinear stability of the homogeneous and isotropic solution above an expansion threshold $\alpha_\text{crit}=\frac{2}{3(1-K)}$. We prove that at and below this critical expansion rate there exist arbitrarily small data that exhibit finite-time singularity formation of the shock type, i.e., gradient blowup while the solution stays bounded. For this type of data, we establish lifespan estimates that are polynomial below and exponential at the critical threshold. In addition, we prove a theorem based on energy estimates that provides lower bounds on the lifespan of plane-symmetric solutions generated from small initial data and generalizes to solutions without symmetry in $3+1$-dimensions.

In [OS05] the authors proved an exact triangle for the Heegaard Floer homology of the double branched covers associated to a skein sequence of links, and constructed a spectral sequence from the reduced Khovanov homology to the Heegaard Floer homology of the double branched cover of a link. For $L \subset S^3$ a link and $\Sigma(L)$ its double branched cover, the invariant $HF(\Sigma(L))$ doesn't capture the information of the natural involution on $\Sigma(L)$. In this paper we study the equivariant version of $HF(\Sigma(L))$ and prove that similar exact triangle and spectral sequence hold for this equivariant version.