Papers

This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic form an ideal. Our hope is that the corresponding piecewise (strong) f-generic types can provide a substitute in arbitrary NIP groups for the (strong) f-generic types of definably amenable NIP groups, and in the rest of the paper we give several applications. Two of the applications deal with the Ellis group of an NIP group. Let $T$ be an NIP theory, $G$ a definable group, and $M$ a model. In our first result we show that the size of the Ellis group of $G(M)$ is bounded above by $2^{|T|}$, independent of the choice of $M$, giving a substantial step towards the question of whether the isomorphism type is independent of $M$. In our second result, inspired by a theorem of Hrushovski, we show that, if $T$ and $M$ are countable and the formulas of $T$ have uniformly bounded VC-codensity, then the Ellis group of $G(M)$ has `finite Archimedean rank', ie its connected component is profinite-by-Lie. A crucial tool for us in both results is the recent result of Chernikov-Gannon-Krupiński and Basso-Zucker that the $\tau$-topology on the Ellis group is Hausdorff. Finally, we use our techniques to obtain a `local' result valid in arbitrary NIP theories, without the assumption of uniformly bounded VC-codensity: for any `bi-invariant' formula $\phi(x,y)$, the group $G/G^{00}_\phi$ has finite Archimedean rank. More precisely, if the VC-codensity of $\phi(x,y)$ is at most $\delta$, then $G/G^{00}_\phi$ is an inverse limit of compact Lie groups of dimension at most $(4\delta)^2$. This connects to, though is different than, a question of Hrushovski.

Oscillatory inverse problems often exhibit highly nonconvex discrepancy landscapes due to signal misalignment and cycle-skipping phenomena, posing significant challenges for uncertainty quantification. While Gibbs posteriors provide a flexible framework for incorporating problem-specific discrepancy measures, their performance depends strongly on the empirical risk landscape induced by the chosen discrepancy measure. Optimal transport accounts for the spatial and temporal structure of the underlying feature domain when comparing oscillatory signals. However, the direct application of classical optimal transport is precluded because observed and predicted signals are typically signed and therefore do not satisfy the positivity requirements of classical transport formulations. We introduce a symmetrized Sinkhorn-Gibbs inference framework for oscillatory inverse problems. The proposed approach combines a normalization procedure for signed signals with a symmetrized Sinkhorn loss that exploits complementary transport information from the normalized signals and their normalized negations. The resulting loss is incorporated into the Gibbs posterior framework, yielding a Gibbs inference methodology tailored to oscillatory data. We establish smoothness properties of the proposed loss, prove well-definedness of the resulting Gibbs posterior, derive robustness guarantees, and develop an adaptive sampling strategy for posterior computation. Numerical experiments demonstrate less oscillatory empirical risk landscapes with fewer spurious local minima, more accurate posterior inference, greater robustness to observational noise, and improved population-level recovery than Gibbs posteriors based on Euclidean and trace-wise Wasserstein losses.

We introduce the multispecies version of the harmonic process on a one-dimensional chain in contact with boundary reservoirs. This process is a continuous-time Markov chain where each site can host an unbounded number of colored particles. The symmetric bulk dynamics is put in contact with reservoirs, which inject and remove particles driving the system out-of-equilibrium. The Markov generator of the process is identified with the integrable Hamiltonian of an open rational Heisenberg spin chain of higher rank. We derive the underlying R- and K-matrices in operator form and construct the double-row transfer matrix following Sklyanin. Similar to the monospecies case the R-matrix factorises into two R-operators, each factor corresponding to left and right moving particles. We further define three dual models: an absorbing particle model, a hidden parameter model and a heat conduction model.

Two linear error-correcting codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$ are called linearly equivalent if there is a linear isometry mapping $\cal{C}_1$ to $\cal{C}_2$. In this work, we generalize the notion of linear equivalence and study the minimum distortion $\cal{D}(\cal{C}_1, \cal{C}_2)$ of a linear mapping between codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$, which quantifies how similar $\cal{C}_1$ and $\cal{C}_2$ are. We introduce and study the Code Distortion Problem (CDP), which asks to find a minimum distortion mapping between two input codes $\cal{C}_1$ and $\cal{C}_2$. CDP generalizes the Linear Code Equivalence Problem (LCE), which is essentially the special case of CDP where $\cal{D}(\cal{C}_1, C_2) = 1$ and which is well-studied because of its role in cryptography. We prove that (decisional) CDP is $\mathsf{NP}$-hard to approximate to within any constant factor, and that it is in $\Sigma_2^P$. We also give a single-exponential-time $k^2$-approximation algorithm for CDP, where $k$ is the dimension of the input codes. Furthermore, we give a single-exponential-time $\big(\frac{2k + 1}{3})^2$-approximation algorithm for a natural special case of CDP, and we show that our analysis is tight in this case. We use techniques from analogous work on the Lattice Distortion Problem (LDP) by Bennett, Dadush, and Stephens-Davidowitz (ESA, 2016). We also introduce or study a number of additional concepts that might be of independent interest. These include an adaptation of the celebrated reduction of Goldreich, Micciancio, Safra, and Seifert (IPL, 1999) from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP) on lattices to the analogous problems on codes; successive minima bases for codes; and the matrix $0 \to 0$ "norm" on subspaces.

In this paper, we derive a conditional copula representation for expectations of the form $\mathbb{E}[g(\boldsymbol{X})]$, where $\boldsymbol{X}$ is a random vector with arbitrary marginal distributions and $g$ is a measurable function satisfying suitable integrability conditions. The proposed representation explicitly separates the contributions of the marginal distributions and the dependence structure through conditional copula distributions, yielding a unified quantile–copula framework for a broad class of statistical functionals. This framework encompasses numerous quantities of practical interest, including moments, probabilities, dependence measures, inequality indices, entropy measures, and multivariate functionals. We further establish extremal bounds under fixed marginals by exploiting the concordance order on copulas and characterize the classes of functions for which these bounds apply through the notion of $\Delta$-antitonicity. Finally, several illustrative examples illustrate the versatility of the proposed framework through applications to risk measures, stochastic superiority probabilities, information measures, and option pricing under dependence uncertainty.

We propose a stochastic model of sea spray as a shot noise process attached to a homogeneous Poisson process whose intensity measure is the sea spray generation function, that is to say the average number of droplets of a given radius ejected into the atmosphere per unit area per unit time. We assume the impulse response function defining the shot noise is determined by the trajectory of the center of a spherical droplet whose radius may change with time. We also assume the droplet trajectory is vertical and droplets don't interact with one another. Under these assumptions we show the total mass of airborne sea spray droplets is a stationary stochastic process whose mean and covariance can be written explicitly as functions of the physical parameters. We illustrate the model through examples of increasing physical relevance the most general of which includes gravity, Stokes drag forces, and Langmuir's D-squared law of evaporation. A consequence of this particular example is an estimate of the height of the marine boundary layer at sufficiently large wind speeds.

The imposition of boundary velocities in finite element models of port-Hamiltonian elastodynamics typically relies on Lagrange multipliers, yielding Differential-Algebraic Equations (DAEs). Alternatively, weak imposition methods that maintain an Ordinary Differential Equation (ODE) structure often exhibit poor accuracy at Dirichlet boundaries. To address these limitations, this paper introduces an additive kinematic decomposition at the continuous level, splitting the displacement and velocity fields into a relative dynamic component that vanishes on the boundary and a prescribed lifting function extending into the interior domain. This decomposition induces a distributed port that maps the effects of the boundary actuation inside the domain. By incorporating this mapping into suitable virtual power principles, we derive lifted port-Hamiltonian system (PHS) models that, upon finite element discretization, reduce to ODE systems in which Dirichlet boundary velocities are strongly imposed. The framework is applied to derive 2-field and 4-field formulations suited to distinct PHS geometric representations. Furthermore, we show that under specific shape functions, standard FEM schemes are recovered, demonstrating that the lifting framework in the discrete models is equivalent to the classic algebraic matrix partitioning in computational mechanics practice. The energy-balance properties and computational performance of the proposed methodology are verified through numerical simulations.

The Word Problem has been a subject of intensive mathematical study for over a century, initially driving advances in combinatorial group theory and more recently emerging as a foundational hardness assumption in post-quantum cryptography (PQC). While generally undecidable, several families of infinite non-abelian groups exhibit solvable or algorithmically fast word problems, making them attractive platforms for cryptographic design. This paper introduces WPNet, a novel Graph Neural Network architecture capable of solving the Word Problem heuristically, which is demonstrated on the Baumslag-Solitar group $BS(1,2)$ and on an Artin group. By mapping unreduced words to dynamic graph structures, the model learns to cluster algebraically equivalent elements in a continuous embedding space, effectively identifying the geodesic representative of a word without executing discrete reduction steps. As an application, a model variant is developed that can predict the geodesic length of an unreduced word in both groups. To demonstrate the cryptographic severity of this structural leakage, WPNet is successfully deployed against the Wagner-Magyarik public-key cryptosystem.

In this paper, we study Ricci solitons on the three-dimensional solvable Lie group $\mathrm{Sol}^{3}_{m,n}$ equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical $\mathrm{Sol}^{3}$ geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from $\mathrm{Sol}^{3}_{m,n}$ into Euclidean spaces.

Featuring Hessian-driven damping, two inertial primal dual dynamical systems are proposed for solving smooth saddle point problems with bilinear coupling. For convex-concave functions, we establish a convergence rate $\mathcal{O}\left( \frac{1}{t^2} \right)$ for the primal dual gap; for strongly convex-strongly concave functions, we obtain an asymptotic rate $\mathcal{O}\left( \frac{1}{t^{\alpha-1}} \right)$ ($\alpha\ge 3$ is the damping parameter) without knowledge of the strong convexity parameters, and an accelerated linear convergence rate when the strong convexity parameters are known. As an application of the proposed inertial systems, we also consider the affinely constrained convex optimization problem, and develop an inertial system with Hessian-driven damping, which complements existing results.

In the theory of abelian Anderson $t$-modules, pure Anderson $t$-modules play an important role. Namoijam and Papanikolas also introduced the notions of strictly pure and almost strictly pure $t$-modules which are special classes of pure $t$-modules. Whereas it is well known that not all pure $t$-modules are isomorphic to strictly pure ones (e.g.~the tensor powers of the Carlitz module), the same question for almost strictly pure $t$-modules was not answered yet. In this article, we show that every pure Anderson $t$-module defined over a field $K$ is indeed isomorphic to an almost strictly pure Anderson $t$-module after base change to a suitable finite algebraic extension of $K$. We also give a necessary and sufficient criterion when such an isomorphism to a strictly pure Anderson $t$-module is possible.

Number fields, which generalize the rational numbers, are fundamental objects in number theory. Many of their key arithmetic properties are captured by invariants whose computation is among the central tasks of computational algebraic number theory and a focus of several computer algebra systems and databases. In this paper, we describe a Lean 4 formalization for certifying several of these number field invariants. Building on previous work on certifying rings of integers, we extend this certification approach to further invariants including the signature, the unit group modulo $p$-th powers, and, ultimately, the class group. We also improve discriminant certification, allowing verifications for higher-degree number fields infeasible in previous work. We introduce structures based on representations of algebraic objects suited to computation, including reusable ones for certifying ideal arithmetic. Along the way, we formalize several underlying mathematical results, for instance on real closed fields and pseudo-remainder sequences, which are of independent interest. We apply our framework to verify hundreds of entries of the $\textit{L-functions and modular forms database}$ (LMFDB) concerning the discriminant, signature, class number, and class group structure of various number fields. To this end, we wrote a SageMath script that computes the certificates and outputs Lean proofs of the corresponding statements.

Given a sequence of orthogonal polynomials $\{L_n\}_{n=0}^\infty$, orthogonal with respect to a positive Borel $\nu$ measure supported on $\mathbb{R}_+$, let $\{Q_n\}_{n=0}^\infty$ be the sequence of orthogonal polynomials with respect to the modified measure $r(x)d\nu(x)$, where $r$ is certain rational function, and {$L_n(-1) = Q_n(-1)= (-1)^n$}. This work is devoted to the proof of the relative asymptotic $$ \frac{Q_n^{(d)}(z)}{L_n^{(d)}(z)} \unifn \prod_{k=1}^{N_1}\left(\frac{\sqrt{a_k}+i}{\sqrt{z}+\sqrt{a_k}}\right)^{A_k}\prod_{j=1}^{N_2} \left(\frac{\sqrt{z}+\sqrt{b_j}}{\sqrt{b_j}+i}\right)^{B_j},$$ on compact subsets of $\mathbb{C}\setminus\mathbb{R}_+$, where $a_k$ and $b_j$ are the zeros and poles of $r$, and the $A_k$, $B_j$ are their respective multiplicities.

We show stability of persistent path homology of path complexes. As a consequence, we deduce the stability of persistent path homology of hypergraphs and of sequence hypergraphs, and recover the known stability result for digraphs, originally due to Chowdhury and Mémoli.

We establish a connection between quantum resource theory and probability theory, under repeated application of free operations. We show that strong monotonicity of a resource measure implies that the resource is a supermartingale. We then use the optional stopping and martingale convergence theorems to derive bounds on the efficiency of post-selection, and other free adaptive strategies. We also describe the asymptotic dynamics of the conditional state, where resource fluctuations are necessarily absent, showing one of two distinct phenomena occurs: resource vanishing or resource freezing.

Can a bounded planar domain have a simple Dirichlet spectrum with uniformly separated consecutive eigenvalues? Dimension two is critical: Weyl's law permits both uniform separation and arbitrarily small gaps. We prove that uniform separation is nevertheless exceptional in a natural rough-domain setting, even after multiplicities are removed. Let $D\subset\mathbb{R}^2$ be a bounded domain and, for $\ell\geq 1$, let $\mathcal{C}_\ell(D)$ be the space of nonempty connected open sets $\Omega\subset D$ such that $\overline{D}\setminus\Omega$ has at most $\ell$ connected components, endowed with the complementary-Hausdorff topology. We prove that $\mathcal{C}_\ell(D)$ is completely metrizable and Baire, and that smooth domains are dense in it. If $0<\nu_1(\Omega)<\nu_2(\Omega)<\cdots$ are the distinct Dirichlet eigenvalues, our main result states that \[ \left\{\Omega\in\mathcal{C}_\ell(D):\inf_{m\geq 1}\bigl(\nu_{m+1}(\Omega)-\nu_m(\Omega)\bigr)=0\right\} \] is residual. This statement requires no simplicity assumption. Combining it with our transfer of Micheletti's classical generic-simplicity theorem to $\mathcal{C}_\ell(D)$ shows that a generic domain has simple spectrum and consecutive gaps with zero lower limit. The proof uses Šverák's planar spectral continuity theorem and a local surgery that implants an arbitrarily high pair of close consecutive distinct eigenvalues.

We derive an explicit formula for the pair correlation function of the Sine$_6$ process in terms of Bessel functions of the first kind. This provides the first single-variable special function representation of the pair correlation function for the bulk limit of a beta-ensemble beyond the classical values of $\beta=1,2,$ and $4$.

Let $S=\sum_{i=1}^r A_i\otimes s_i$ be a matrix semicircular element, with Hermitian coefficients $A_i\in M_n(\mathbb{C})$ and free standard semicircular generators $s_i$. Its scalar spectral density $f$ is governed, through Speicher's equation (a matrix Dyson equation), by the completely positive covariance map $\eta_S(X)=\sum_i A_iXA_i$. We treat the singular regime, where the pencil $\sum_i A_i x_i$ is full but not semisimple and $f$ is unbounded at the origin, in contrast to the bounded real-analytic density of the regular case. We prove three results. (i) The leading singularity exponent at $0$ is invariant under congruence $A_i\mapsto bA_ib^{*}$ of the pencil ($b$ invertible), and more generally under symmetric scaling of the covariance map. (ii) For binary elements ($r=2$) we obtain a complete classification: in Lancaster-Rodman canonical form every indecomposable cell is of one of three types, and $f(x)\sim c|x|^{-(n^*-1)/(n^*+1)}$ as $x\to0$ with an explicit constant $c$, where the exponent depends only on the size of the largest Jordan block (the effective chain length $n^*$) and not on the coupling. With (i) and the direct-sum behaviour, this classifies all full binary Hermitian pencils. (iii) The spectral classification is strictly coarser than the algebraic one: a Type III cell of size $2m$ with non-real $\beta$ and the direct sum of two Type II cells of size $m$ with parameter $|\beta|$ have identical scalar densities, yet their covariance maps are not symmetrically scalable; the scalar spectrum cannot detect the phase of $\beta$. Each type calls for a different method: a reduction of Speicher's equation to an autonomous discrete Painlevé I (McMillan) map (Type I), a Lyapunov-Schmidt reduction at the branch point (Type II), and a gauge reduction by a diagonal unitary (Type III).

Regression estimates from observational data can depend on specification under multicollinearity, while sequential sums of squares (SS) depend on term order. We introduce Retrospective Orthogonal Design (ROD), which reconstructs conditional mean surfaces on a probability-balanced lattice. ROD preserves observed cell means, completes unsupported cells, applies weighted tensor-product contrasts, and evaluates the reconstructed surface through piecewise-affine interpolation over Freudenthal polyhedra. Resolution and completion are selected jointly by validation among rank-admissible candidates, followed by refitting and evaluation on an untouched test set. For an admissible lattice, $\mathbf{X}^{\top}\mathbf{W}\mathbf{X}=c\mathbf{I}$, yielding specification-invariant contrast effects and unique, order-independent SS within the retained contrast space. Response-free projection calibration maps the fixed reconstruction onto a declared scientific basis and corrects finite-resolution recovery loss. Across 6,480 simulation conditions spanning nine data-generating processes, ROD matched or exceeded polynomial regression in five processes and performed strongest on threshold, sign-interaction, and localized surfaces. For the quadratic-interaction process, mean out-of-sample $R^2$ differed by only $0.0001$, while calibrated coefficient bias remained small across prespecified targets. A Rao-based information adjustment provides dependence-aware sample-size guidance for ROD planning. In a weighted Mincer application, ROD produced the highest out-of-sample $R^2$ point estimate, with substantial interval overlap with polynomial regression, and provided exhaustive SS allocations invariant to term-entry order.

The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\F_{q^2}$ over $\F_q$, is such an alphabet. Quantum error correction, however, only ever sees self-orthogonal additive codes, and whether that rigidity rescues the theorem—equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation—was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematically and at the smallest possible scales. For every prime power $q$ we construct a pair of $[[q+1,q-1]]_q$ stabilizer codes and a weight-preserving isomorphism between them extending to no monomial transformation; the codespaces are inequivalent even under arbitrary local unitaries combined with permutations, though they share Shor–Laflamme enumerators. Self-orthogonality is automatic here, by two elementary lemmas which also show that Dyshko's threshold-length counterexamples were already self-orthogonal, unremarked. For qubits we prove by exhaustive search that length $3$ is minimal and the counterexample essentially unique. Dropping the “idle qudit” invariant that detects these, we find the minimal full-support lengths: $4$ for a non-extendable isometry, $5$ for a weight-isometric pair that is not monomially equivalent, realized by explicit $[[5,2]]$ codes; at length $6$ all nontrivial stabilizer elements can have weight $\ge 4$. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU–LC circle of questions.

In six-dimensional $\mathcal{N}=(1,0)$ supergravity, the structure of tensor moduli space is governed by primitive BPS string charges known as BPS generators and their intersection pairing. We derive bounds on the intersection numbers of these generators from a purely effective field theory (EFT) perspective. Although gauge anomaly cancellation constrains intersections between generators supporting gauge algebras, bounds for E-strings intersecting generators with self-intersection numbers $-2$ and $-3$ have previously remained incomplete. We show that the Zariski decomposition, interpreted as the charge lattice counterpart of the attractor mechanism, together with current algebra embeddings on the E-string worldsheet theory, yields strong universal bounds on these intersection numbers. These results establish the finiteness of tensor charge intersection numbers up to duality. The underlying structure was identified through AI-guided investigation and is proven here analytically using EFT arguments.

We investigate the relation between fundamental complements and regional operator algebras in global de Sitter space. For a finite union of open arcs on the time-symmetric circle of global dS$_2$, we show that the fundamental complement is determined by the largest complementary gap: a gap contributes precisely when its angular length is at least $\pi$, and at most one gap can do so. In higher-dimensional global de Sitter space, the corresponding criterion is containment of an open hemisphere. We then study the operator-algebraic consequences in an abstract Möbius-covariant chiral conformal net. Each regional algebra is crossed with its vacuum modular flow using a shared auxiliary clock. We derive an exact criterion for inclusions between the resulting continuous cores and show that vacuum correlations obstruct this criterion when the larger region contains an additional spacelike arc. A compatible family can nevertheless be obtained on the finite Boolean algebra generated by a fixed collection of separated arcs by using a split product state. Motivated by the holograms prescription of Bousso and Penington, we introduce a representation-dependent commutant model for regional algebras. When the protected gap overlaps a fixed reference arc, the assigned algebra is a proper von Neumann subalgebra of the reference Type-II$_1$ factor. For an explicit two-arc family, adding an arbitrarily short antipodal component removes the fundamental complement and changes the assigned algebra discontinuously to the full reference algebra.

Given a pair of complex algebraic varieties, we can ask: does every analytic map admit an algebraic representative? Such questions are most interesting when the source and target are non-compact varieties, in which case the problems are closely linked with Hodge-type conjectures. Concrete avatars of this kind of problem, which we discuss from several points of view, include: which holomorphic vector bundles on an affine variety admit an algebraic structure, and which homotopy classes of maps admit algebraic representatives?

We construct two families of commutative Frobenius rings: discrete and continuous Frobenius quotients $K_f$, resp. $A_g$, which are defined as the quotients of shift, resp. differential operator algebras by the annihilator of a polynomial. These constructions model the numerical $K$-rings and Chow rings of smooth complete varieties. We show that there is a naturally defined isomorphism $\mathbb{Q} K_f \cong \mathbb{Q} A_g$ playing the role of the Chern character, precisely when the polynomials satisfy a combinatorial analogue of the Hirzebruch-Riemann-Roch theorem, which states that there exists an invertible element $\mathrm{td} \in \mathbb{Q} A_g^\times$, called the Todd class, such that $f = \mathrm{td} \cdot g$. In this case we show that $K_f$ carries all the structure needed to make it a suitable model for $K$-rings of complete smooth varieties: $K_f$ is a $\lambda$-ring, has well-defined Chern classes, determinants, and satisfies a combinatorial analogue of Serre duality. We further show that the construction is functorial and obtain a combinatorial analogue of the Grothendieck-Riemann-Roch theorem. Lastly, we investigate the existence of larger families of isomorphisms between Frobenius quotients defined by the action of a power series on a generating set, such as the truncated Chern character which yields an integral isomorphism $K_f \cong A_g$. We provide numerous examples and applications: we give a new formula for computing Snapper polynomials of matroids, show that the dualizing class of $K_f$ coincides with dualizing classes of matroids and linear families of polytopes, realize $K$-rings of toric variety bundles as Frobenius quotients, and study $K$-rings of Ehrhart fans as well as exceptional isomorphisms in this setting.

In this paper, we establish a one-to-one correspondence between a natural monoid and the endomorphisms of the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. As a consequence, we classify endomorphisms of $\mathcal{Q}_2$ with prescribed images and derive several criteria for the uniqueness of extensions of endomorphisms of a canonical copy of the Cuntz algebra $\mathcal{O}_2$ inside $\mathcal{Q}_2$ to endomorphisms of $\mathcal{Q}_2$. Moreover, we construct an example of an extendable automorphism of $\mathcal{O}_2$ that is not a composition of the flip-flop automorphism, the gauge automorphisms, and inner automorphisms, thereby providing negative answers to certain open questions. Using this explicit construction, we also show that the canonical image of $\operatorname{Aut}(\mathcal{Q}_2,\mathcal{O}_2)$ in the outer automorphism group $\operatorname{Out}(\mathcal{Q}_2)$ is non-abelian. Finally, we characterize the automorphisms of $\mathcal{Q}_2$ that globally preserve $C^*(u)$, and completely describe the Weyl group $\mathcal{W}(\mathcal{Q}_2, C^*(u))$. Consequently, several related open questions are answered affirmatively.

We study the extent of universality in the scaling of the height $0$ probability of the stationary Abelian sandpile on lattice approximations of a region $U \subset \mathbb{C}$ with lattice spacing $\varepsilon$ and with open boundary conditions. We show that under certain symmetry assumptions on the lattice, in the scaling limit $\varepsilon \to 0$ this probability at a point $z \in U$ equals $p_{\mathcal{G}}(0) + \varepsilon^2 c_{\mathcal{G}} f_U(z) + O(\varepsilon^3)$, where $p_{\mathcal{G}}(0)$ is the height $0$ probability on the full lattice, $c_{\mathcal{G}} > 0$ is a constant depending only on the lattice, and $f_U$ is a conformally covariant positive real-valued function depending only on $U$. This generalises a result of Brankov, Ivashkevich and Priezzhev (1993), who considered the square lattice and the upper half plane. It also generalizes Example 4.10 in the recent study of the fermionic DGFF of Adame-Carillo and Ruszel (2025). We show via counterexamples that our symmetry assumption cannot be omitted, although conformal covariance may be recovered via an unusual scaling. In particular, the natural analogue of our result fails on general isoradial graphs. Our work is motivated by Jeng, Piroux and Ruelle (2006), who computed the correction terms for all height variables in the special case of the upper half plane in $\mathbb{Z}^2$. They conjectured that their results hold more generally and the present paper is a step in an attempt to make this conjecture rigorous.

Motivated by the recent deterministic approach of Fournier~\cite{F2025} to gelation for the continuous Smoluchowski coagulation equation, we adapt his method to the discrete Oort–Hulst–Safronov (OHS) coagulation system. We show that under a suitable condition on the coagulation kernel, every solution with finite initial mass loses mass in finite time, and we give an explicit bound on the gelation time. We also prove gelation in the critical logarithmic case and provide a sufficient condition for mass conservation. Finally, we study the positivity of solutions and show that, for any positive time, a cluster size has positive concentration if and only if it is at least as large as the smallest cluster present initially.

We consider resonances of (not necessarily self-adjoint) three-dimensional Dirac operators, defined as poles of the meromorphic continuation of the resolvent. We prove that a region of the resonance set can be characterized by the discrete spectrum of distorted Dirac operators, with preservation of multiplicities. As an application, we prove Kramer's degeneracy under time-reversal symmetry.

B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, $\frac{d\theta}{d\tau}=\frac1{\omega}(\cos\theta+B+A\cos\tau)$, which is known as the RSJ model. It depends on three parameters: $B$ called the abscissa, $A$ called the ordinate, and a fixed frequency $\omega$. We study its rotation number $\rho(B,A;\omega)$ as a function of $(B,A)$ and the phase-lock areas: those its level subsets that have non-empty interiors. They exist only for integer values of the rotation number (Buchstaber, Karpov, Tertychnyi). In this paper we study asymptotics of the phase-lock area portrait in a special slow-fast limit, as $\omega\to0$ and $(B,A)\to(0,1)$ so that $(B,A)-(0,1)=O(\omega)$. We show that in the rescaled parameters $\ell:=\frac B{\omega}$ and $u:=\frac{A-1}{\omega}$ the phase-lock area portrait converges to a parquet with boundary lines being parallel to the lines $\{ u\pm\ell=0\}$. Namely, the limit of phase-lock area with rotation number $r$ is the union of an infinite chain of squares going up, with integer vertices and diagonals of length two lying on the line $\{\ell=r\}$, and an infinite strip going down (sector in the case, when $r=0$). We state and prove a generalization of this result to a wide class of slow-fast systems on 2-torus.

Nonholonomic mechanics describes systems subject to non-integrable velocity constraints, such as rolling bodies and skating motion. These systems generally lack a canonical Hamiltonian formulation, obstructing standard quantization methods. Here we quantize nonholonomic systems as Markovian open quantum systems, with the nonholonomic constraint appearing in a large-dissipation limit. We find explicit Lindblad superoperators that reproduce the classical dynamics of the Chaplygin sleigh and the Suslov problem in the semiclassical limit. The master equation is numerically simulated, and the covariance is shown to satisfy a relation predicted by the theory of metastability in open quantum systems.