A permutation $π$ strongly avoids the pattern $τ$ if both $π$ and $π^2$ avoid $τ$. In this paper, we enumerate permutations of size $n$ that strongly avoid the pattern 132. This enumeration allows us to prove a conjecture that the growth rate of such permutations is 2.

Boundary controllers have been recently proposed in the literature, via modal decomposition, to achieve $H^1$ stabilization of linear parabolic equations in two and three dimensions. In one dimension ($1$-D), $H^1$ exponential stability is known to imply boundedness and asymptotic convergence of the state to zero in the sense of the max norm. However, in two ($2$-D) and three dimensions ($3$-D), this implication does not systematically hold. In this paper, focusing on the full-state feedback case, our objective is to prove that the modal-decomposition based controller in \cite{Munteanu2017IJC} guarantees, not only $H^1$ exponential stability, but also $H^2$ exponential stability. This implies, in particular, boundedness and asymptotic convergence of the state to zero in the sense of the max norm. Our approach consists in rewriting the Laplacian of the state, required in the $H^2$ norm, as a linear combination of the state and its time derivative. The $L^2$ norm of the state being bounded by the $H^1$ norm, we only analyze the $L^2$ norm of the time derivative of the state.

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.

We prove a sharp stable $2$-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If $M$ is diffeomorphic to $\mathbb{C}\mathrm{P}^n$ and $\mathrm{scal}_g\ge 4n(n+1)$, then $\mathrm{sys}_2^{\mathrm{st}}(M,g)\le π$. Moreover, equality holds only for the Fubini-Study metric, up to biholomorphism after choosing the corresponding complex structure. The proof uses Spin$^c$ Dirac operators, a comass estimate for the curvature term in the Lichnerowicz formula, and stable norm-comass duality.

We prove a structural result for sets of integers with doubling at most $4 + δ$, with $δ>0$ sufficiently small. This generalises earlier work of Eberhard--Green--Manners which dealt with sets of integers with doubling strictly less than $4$, and makes progress towards a question of Green.

We study certain dynamical and metric aspects of Kiselman's semigroup $K_n$. The level function $\mathcal{L}$ is introduced and shown to admit a simple description in terms of right multiplication by generators. We show that every sequence of partial products in $K_n$ is eventually constant. Using $\mathcal{L}$, we further study sequences of random partial products in $K_n$ and show that, in the independent and identically distributed setting where every generator is chosen with positive probability, the hitting time of the eventual constant value is distributed as a sum of $n$ independent geometric random variables. Finally, we define a natural ultrametric on $K_n$ arising from the level function and obtain some basic results on the associated metric balls and spheres.

Finite horizon Sinai billiard maps are examples of uniformly hyperbolic systems with singularities. These discontinuities make it more difficult to develop the classical theory of thermodynamic formalism. Nevertheless, Baladi and Demers established a variational principle for these systems, and proved that if the billiard table satisfies a certain sparse recurrence condition, then there is a unique measure of maximal entropy. We extend this existence and uniqueness result to all finite horizon Sinai billiard maps by giving a new proof that does not rely on the sparse recurrence condition. Our construction is very concrete: the unique MME is obtained as the product of the Hausdorff measures on the one-sided subshifts associated to the billiard map.

We study a natural analogue of Ulam's problem for random rooted trees distributed according to a Plancherel-type measure. This probability measure is closely related to the classical Plancherel measure on integer partitions. For a Plancherel random tree $T_n$ with $n$ vertices, we investigate the asymptotic behavior of its height $H_n$, defined as the maximal distance from the root to a leaf. We prove that this height grows logarithmically. More precisely, there is a one-parameter family of random trees $(T_n(θ))_{n \in \mathbb{N}}$ indexed by $θ>0$ such that $\frac{H_n}{\log n}$ converges in probability to $c_\star(θ)$, where $c_\star(θ)$ is an explicit constant depending on the parameter $θ$. The case of Plancherel trees corresponds to the parameter $θ=2$. The proof is based on the fact that the Plancherel random trees can be viewed as Ewens fragmentation trees, for which the height exhibits a sharp threshold phenomenon. An upper bound is obtained via $s$-mass functionals and contraction estimates, while the lower bound is derived by embedding the model into a branching random walk with logarithmic displacements governed by a Poisson--Dirichlet distribution. The constant $c_\star(θ)$ is characterized through a variational principle associated with this branching random walk.

We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincaré-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$.

We investigate a natural subfamily of twisted linearized Reed--Solomon (TLRS) codes in the sum-rank metric, where the twist is applied only to the constant term. We establish a simple necessary and sufficient condition for these codes to be linear complementary dual (LCD): the twisting parameter \(η\) must satisfy \(η^2 \neq -1\) in the underlying field. This criterion is independent of the evaluation subgroup, the dimension parameter, and the twisting exponent (subject only to a mild restriction on the code length). Furthermore, we construct infinite families of additive twisted linearized Reed--Solomon codes that are simultaneously additive complementary dual (ACD) and maximum distance separable (MDS) over quadratic extensions \(\mathbb{F}_{q^2}\), with respect to the trace-Hermitian inner product. These codes are explicit and achieve optimal parameters for all admissible lengths.

This paper is focused on decoding delay guarantees in wireless networks, where messages have a given signal-to-interference-plus-noise ratio threshold $η_0$ to meet in order to be successfully decoded, and where this should occur within some strict time constraints. Its main contribution consists in quantifying the worst-case transmissions decoding delays in the uplink of a cell-free network using successive interference cancellation. We show how such decoding delay guarantees can be obtained using spatial network calculus, a new tool introduced recently, and in particular spatial regulation.

This article extends weak convergence bounds of Markov transition kernels to convergence bounds on the variance of the Markov kernel applied to Lipschitz functions. In the reversible case, weak convergence rates of the transition kernels imply chi-squared divergence convergence bounds if the density of the initialization measure is Lipschitz. These results provide new tools to establish central limit theorems for Lipschitz functions used in Markov chain Monte Carlo simulations. Applications are explored to the stability of Metropolis-Hastings algorithms in high dimensions, stochastic gradient descent, and solutions to stochastic delay equations.

The Smoluchowski diffusion equation describes diffusion in the presence of external forces. Studying the mechanical response of soft materials to linear forces, such as shear, results in a boundary value problem involving an Ornstein-Uhlenbeck operator in an exterior domain with non-constant, unbounded coefficients. In this article, we present efficient and highly accurate boundary element methods in the frequency domain, motivated by applications in soft matter physics. Our key contributions concern the accurate assembly of the Galerkin matrix, combining the approximation of the fundamental solution as a Fourier integral with the resolution of near-field singularities. Numerical experiments demonstrate the accuracy and efficiency of the proposed methods and show their relevance for the computation of rheological quantities.

We derive a general scheme to construct infinitely many probabilistic counterparts for solutions to nonlinear PDEs by recasting the latter as different nonlinear Fokker--Planck equations and by constructing, for each of these equations, a solution to the associated McKean--Vlasov SDE with one-dimensional time marginal densities given by the PDE solution. We utilize this scheme to prove that nonlinear Markov processes in the sense of McKean as introduced by Rehmeier--Röckner (J.\,Theor.\,Probab. 38, 60 (2025)) are not uniquely determined by their one-dimensional time marginals. This is in sharp contrast to the case of classical Markov processes, which are uniquely determined by their one-dimensional time marginals. We demonstrate our results by constructing a continuum of nonlinear Markov processes with one-dimensional time marginal densities given by the Barenblatt solutions to the porous medium and $p$-Laplace equations, as well as by the fundamental solution to the heat equation. This includes a novel martingale representation for the $p$-Laplace Barenblatt solutions. We also prove that a nonlinear Markov process is uniquely determined by its two-dimensional time marginals. Moreover, for the porous medium equation, we show that the different McKean--Vlasov SDEs we investigate are consistent with corresponding gradient flow interpretations of the equation in the sense of Otto calculus.

Optimization modeling underpins real-world decision-making in logistics, manufacturing, energy, and public services, but reliably solving such problems from natural-language requirements remains challenging for current large language models (LLMs). In this paper, we propose \emph{Agora-Opt}, a modular agentic framework for optimization modeling that combines decentralized debate with a read-write memory bank. Agora-Opt allows multiple agent teams to independently produce end-to-end solutions and reconcile them through an outcome-grounded debate protocol, while memory stores solver-verified artifacts and past disagreement resolutions to support training-free improvement over time. This design is flexible across both backbones and methods: it reduces base-model lock-in, transfers across different LLM families, and can be layered onto existing pipelines with minimal coupling. Across public benchmarks, Agora-Opt achieves the strongest overall performance among all compared methods, outperforming strong zero-shot LLMs, training-centric approaches, and prior agentic baselines. Further analyses show robust gains across backbone choices and component variants, and demonstrate that decentralized debate offers a structural advantage over centralized selection by enabling agents to refine candidate solutions through interaction and even recover correct formulations when all initial candidates are flawed. These results suggest that reliable optimization modeling benefits from combining collaborative cross-checking with reusable experience, and position Agora-Opt as a practical and extensible foundation for trustworthy optimization modeling assistance. Our code and data are available at https://github.com/CHIANGEL/Agora-Opt.

Label noise presents a fundamental challenge in modern machine learning, especially when large-scale datasets are generated via automated processes. An increasingly common and important data paradigm, particularly in domains like medical imaging, involves learning from a large dataset with coarse, noisy labels supplemented by a small, expert-verified, clean dataset. This setting constitutes a typical information transfer and fusion problem. However, the significant distribution shift between the noisy and clean data violates the core overall parametric similarity assumptions of existing statistical transfer learning methods, while their reliance on parametric models is ill-suited for complex data like images. To address these limitations, this paper develops a generic model-agnostic nonparametric framework for classification with label noise, which applies to a broad class of classifiers. Our approach leverages the small clean dataset to ``purify'' the large noisy one and carefully manages the remaining ambiguous samples. This framework is underpinned by a rigorous statistical theory. Its empirical performance is demonstrated through simulations and a real-world application to medical image analysis for pneumonia diagnosis.

Metric-preserving functions (here, metric aggregation functions) offer a natural method for constructing metrics on Cartesian products of metric spaces or for aggregating multiple metrics defined on a common set. Strongly metric-preserving functions represent a more specialized subset of these functions, ensuring that the new metric aligns with the product topology, in the Cartesian product case. However, these strong functions have not been previously explored for quasi-pseudometrics. Furthermore, in the case where all metrics are defined on the same set, the problem has not been addressed previously. In this paper, we investigate the class of strongly (quasi-)(pseudo)metric aggregation functions, extending the classical concept. We begin by examining the case where the aggregation function produces (quasi-)(pseudo)metrics on Cartesian products, characterizing these functions through continuity at zero and a minimal zero preimage condition. In addition, we will examine the scenario where the aggregation function produces a (quasi-)(pseudo)metric defined on a fixed set. Within this context, we will demonstrate that the appropriate topology to consider is the supremum topology. We will also provide both necessary and sufficient conditions for an (quasi-)(pseudo)metric aggregation function on sets to qualify as a strongly one, thereby addressing a gap in the existing literature.

The planar Turá number of a graph $F$ is the maximum number of edges an $n$-vertex $F$-free planar graph can have. We study the case where $F$ is forbidden as an induced subgraph, thereby introducing the induced planar Turá numbers. We will determine a sharp upper bound when $F$ is $Θ_4$, a $4$-cycle with a diagonal edge, and obtain exact extremal values in case $F$ is a path $P_k$ on $k$ vertices, for $k=3,4$ and $5$.

Partial differential equations (PDEs) are fundamental across numerous scientific fields. As these problems scale to high dimensions, classical numerical schemes introduce severe computational bottlenecks, known as the curse of dimensionality. Attempts to solve this problem typically rely on either classical sparsity and low-rank decompositions, or neural network surrogate models. On the other hand, Quantum Computing offers a promising alternative, as it allows us to operate in significantly larger spaces while demanding far fewer resources. In this work, we present a quantum subroutine to solve second-order linear PDEs by exploiting the structural properties of the filter in Fourier space using Quantum Block Encoding (QBE) with quantum reversible arithmetic. This approach serves as a specialized alternative to standard quantum matrix inversion, which typically relies solely on Quantum Singular Value Transformation (QSVT) without exploiting the inherent structural properties of the matrix. We validate the proposed methodology against its classical counterpart to prove its correctness. This framework provides a foundation for extending these methods toward quantum group Fourier transforms, wavelet-based analysis, and equivariant quantum neural networks (EQNNs), offering a promising path toward solving broader classes of problems, including nonlinear PDEs.

In this note we prove an upper bound on the $\mathbb F_p$-rank of the incidence matrix of points and hyperplanes in $(\mathbb Z/p^k \mathbb Z)^n$, improving a recent bound of Laba and Trainer when $k$ is large.

We propose that the symmetry category associated to a 2D quantum field theory with 0-form $G$-symmetry with 't Hooft anomaly $k\in H^4(BG,\mathbb{Z})$ for a large class of Lie groups $G$ is the category of twisted measurable fields of Hilbert spaces over $G$ denoted by $\mathrm{Hilb}^k(G)$, which is equivalent to the category of unitary representations of $C_0(G)$ with convolution product twisted by a multiplicative bundle gerbe labeled by $k$ denoted by $\textbf{Rep}^k(C_0(G))$. We find that the Drinfeld center of the symmetry category $\mathcal{Z}(\mathrm{Hilb}^{k}(G))$ equivalent to the category of unitary representations of the groupoid $C^*$-algebra of the Fell line bundle $Σ_k$ over the conjugation action groupoid $G//_{\rm Ad} G$, denoted by $\textbf{Rep}(C^*(G//_{\rm Ad}G,Σ_k))$, where the twist is characterized by the transgression $τ(k)\in H^2(G//_{\rm Ad}G,U(1))$. To the full generality, our framework applies to a Lie group $G$ that is a direct product of a compact connected Lie group and a number of $\mathbb{R}$ or $GL(1,\mathbb{C})$ factors. We compute the braiding of anyon lines in the bulk 3D SymTFT from this formalism. Finally we provide physical examples for abelian and non-abelian $G$, and discuss the physical consequences of flat gauging continuous global symmetries.

We consider the one-dimensional quasilinear heat equation with state-dependent heat capacity and thermal conductivity, and design a boundary-output observer based on the backstepping design for a linear heat equation with constant coefficients. Viewing the quasilinear system as a perturbation of the linear one, we establish exponential stability of the origin for the observation error dynamics in $H^1$, with an explicit region of attraction depending on the system parameters, observer gains, and the mismatch between the nonlinear diffusivity and the constant design diffusivity. Importantly, the observation error converges to zero rather than merely to a neighborhood scaling with this mismatch, even though, in contrast to backstepping-based stabilization of nonlinear PDEs, the mismatch need not decay along trajectories and may remain bounded away from zero, acting as a persistent state-dependent multiplicative perturbation. A technical challenge was to perform a sufficiently-fine Lyapunov analysis that does not yield overly conservative results such as mere boundedness of the observation error. Interestingly, while in the linear case the relationship between one of the backstepping observer gains and the convergence rate is monotonic, we show that in the nonlinear setting this is no longer the case: there may exist an optimal value of that gain, beyond which further increases deteriorate the system's performance. Such behavior cannot be predicted without our analysis: one might expect a priori the decay rate to be freely tunable at the expense of a region of attraction that shrinks to zero as the prescribed rate tends to infinity. However, our Lyapunov analysis (supported by numerical experiments) reveals that this intuition is incorrect.

Defant and Kravitz introduced generalizations of West's stack-sorting map $s$ from permutations to finite words. This raises questions as to how such generalizations could be applied in the field of combinatorics on words. The Defant-Kravitz generalizations of $s$ depend on how repeated occurrences of the same character within a word may be repositioned, according to their $\textsf{tortoise}$ and $\textsf{hare}$ operations. As demonstrated in this paper, these operations provide a natural way of extending abelian complexity functions for infinite sequences, in a way that gives light to structural properties associated with infinite words. We apply these new ideas to two famous infinite words: the paperfolding word and the Thue-Morse word. In the case of the Thue-Morse word, we discover an interesting connection to the previous work of several authors, such as de Luca and Varricchio, on the ``special'' factors of the Thue-Morse word. This may be seen as providing a basis for a new and interdisciplinary area linking the combinatorics about the stack-sorting of permutations with the field of combinatorics on words.

Universally robust dynamical decoupling (UR$n$) sequences were proposed to compensate pulse imperfections arising from arbitrary experimental parameters while achieving high-order error suppression with only a linear increase in the number of pulses. Although their performance was supported by analytical arguments, numerical simulations, and experiments, a complete mathematical proof of the claimed order of error compensation has been absent. In this work, we present a rigorous proof for UR$n$ DD sequences with even $n$. Using a series expansion of a quantity whose modulus is the fidelity $F$, we derive necessary and sufficient conditions for the cancellation of its coefficients up to, but not including, order $n$. The UR$n$ phase prescription satisfies these conditions, and therefore $1-F=O(ε^n)$. Our results establish the UR$n$ construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.

We give an explicit description (in component fields) of a holomorphic theory associated to a general supersymmetric background of $\mathcal N=1$ supergravity in ten dimensions. Conjecturally, this provides a sought-for holomorphic realisation of the supergravity twist in such backgrounds, generalising the minimal type I BCOV theory for Calabi-Yau manifolds. Our theory unpacks the recently introduced Courant contact model associated to a holomorphic Courant algebroid. We also show that a coisotropic reduction of this model reproduces the recent model of ref [1], which is formulated in terms of constrained fields.

As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.

We prove a criterion for the mildness of a finitely presented pro-$p$ group $G$. It implies as a special case a cohomological mildness criterion via Massey products, generalizing results due to Schmidt and Gärtner. It subsumes Labute's non-singular circuit criterion. We further show connections with the triangle condition for the mildness of pro-$p$ right-angled Artin groups, due to Quadrelli, Snopce and Vannacci.

We study the asymptotic distribution of level crossings for random matrix pencils A_n+λB_n in several ensembles, including complex and real i.i.d. matrices and Gaussian/Hermitian settings. We derive a representation of the normalized log-discriminant in terms of pairwise eigenvalue interactions and formulate conditions under which its limit is governed by a deterministic potential. Under assumptions combining a uniform circular law, logarithmic tail control, and small-spacing (repulsion) estimates, we prove convergence of the empirical measure of level crossings to an explicit deterministic limit. In the complex Gaussian case these assumptions are verified (modulo a uniformity step), while in the general i.i.d. setting the results are conditional and motivated by universality theory. We further analyze the real case, showing that any limiting measure does not concentrate on the real projective line under suitable hypotheses, and discuss analogous phenomena for elliptic/Hermitian ensembles. Our results highlight the role of logarithmic energy and universality in governing spectral degeneracies of random matrix pencils.

In this paper, we investigate arithmetical completeness with respect to finite Kripke models of quantified modal logic. We adapt the finite-model embedding techniques of Artemov and Japaridze to two settings involving finite Kripke models. First, for conversely well-founded finite Kripke models of quantified modal logic, we construct a $Σ_2$ Fefermanian provability predicate together with an arithmetical interpretation that embeds the model into arithmetic. Second, for finite constant domain Kripke models of quantified modal logic, we construct a $Σ_1$ provability predicate satisfying $\mathbf{D2^G}$ and an arithmetical interpretation yielding such an embedding.

We analyze a three-dimensional discontinuous piecewise linear system \(Z=(X,Y)\) whose switching manifold \(Σ\) contains visible-visible two-fold intersection lines. Assuming that the matrices \(DX\) and \(DY\) each have one nonzero real eigenvalue and one pair of complex conjugate eigenvalues, we reduce the system to a canonical form. Under a resonant condition, we use Darboux integrability theory to obtain a first integral common to \(X\) and \(Y\). Its restriction to \(Σ\) defines a hyperbola \(Γ\), which parametrizes the crossing points of symmetric periodic orbits. On this curve we construct the half-return maps, derive analytic expansions for the corresponding return times near infinity, and introduce a time-matching function given by their difference. By means of the Weierstrass Preparation Theorem, we prove the existence of a large-amplitude symmetric limit cycle for a suitable subfamily of systems. We then study stability through a saltation-corrected monodromy matrix and reduce the problem to Schur--Cohn inequalities for the two transverse Floquet multipliers.