Papers

Let $\Gamma$ be a group acting properly and cocompactly on the product of two trees $T_1$ and $T_2$. An anti-torus is a non-periodic flat plane in $T_1 \times T_2$ that is the convex hull of two secant periodic lines. That notion was introduced by Dani Wise as a tool to show that $\Gamma$ is irreducible. We establish a new criterion ensuring the existence of anti-tori, and use it to prove that if $\Gamma$ is an $S$-arithmetic lattice in a product of simple algebraic groups of rank one, then $T_1\times T_2$ contains anti-tori. As a byproduct, we obtain a sufficient condition ensuring that a group defined by a bi-reversible automaton contains non-abelian free sub-semigroups. We also introduce a new class of irreducible lattices acting regularly on the vertex set of a product of two trees, and containing anti-tori that are fractal aperiodic tilings of the plane. This establishes a connection between lattices in products of trees and substitution tilings.

For integers $n \geq k \geq 2$ and $0 \leq a,b \leq k-1$, let $m_{k,n}(a,b)$ denote the maximum size of an $(a,b)$-town (mod $k$) family of an $n$-element set, a collection of subsets of whose cardinalities are congruent to $a$ modulo $k$ and whose pairwise intersections are congruent to $b$ modulo $k$. This notion generalizes the classical Oddtown and Eventown problems. We prove that $m_{k,n}(a,b)\leq n$ whenever $a\not\equiv b\pmod{k}$, thereby resolving a conjecture of Veselinov and Marinov. We also disprove another conjecture of theirs by showing that $m_{3,11}(2,2)>m_{3,11}(1,1)$. For the diagonal case $a\equiv b\pmod{k}$, we establish the general bound $m_{k,n}(a,a)\leq 2^{\lfloor n/2\rfloor}$ and completely determine when equality holds. We further obtain improved bounds and exact values in several special cases. The proofs combine characteristic-zero linear algebra with methods from coding theory and finite geometry.

We develop a hierarchical multi-fidelity (MF) framework for efficient uncertainty quantification of porous-medium equation (PME) tumor growth models with moving free boundaries. The proposed approach combines coarse-grid PME solvers, level-set approximations of the Hele–Shaw limit, and fine-grid asymptotic-preserving PME discretizations, thereby integrating both discretization-based and asymptotic-model-based fidelity reduction. To guide the selection of high-fidelity samples, we introduce a residual-based farthest-point sampling (RFPS) criterion that combines projection residual information with a distance-based separation term in the low-fidelity snapshot space. Based on this criterion, we construct both bi-fidelity and tri-fidelity approximations, together with empirical error indicators for adaptive refinement. Numerical experiments are conducted in both bi-fidelity and tri-fidelity settings under several uncertainty scenarios, showing that the proposed multi-fidelity approximations achieve accurate results with reduced high-fidelity sampling cost in the reported tests.

Band-structure calculations for three-dimensional anisotropic phononic crystals require the repeated solution of large elastic generalized eigenvalue problems along Bloch paths. In standard staggered-grid discretizations, anisotropic coupling may involve derivative components located at incompatible grid positions, so additional interpolation or averaging closures are often introduced. This paper proposes a parity-blocked rotated staggered discretization based on four Bloch-periodic body-diagonal differences. The directional derivatives are reconstructed from these diagonal differences, leading to a Hermitian $B_hC_hB_h^H$ generalized eigenvalue formulation that incorporates anisotropic derivative coupling without separate interpolation closures. On even grids, when the stiffness and mass matrices are nodewise local multiplication matrices, the body-diagonal shifts preserve two independent parity invariants. The discrete velocity space is then decomposed exactly into four mutually independent block subspaces, and the full discrete spectrum can be recovered by solving the four smaller eigenvalue problems and merging their spectra. The full and block formulations are further organized in a unified Fourier SVD framework, which supports $\Gamma$-point zero-mode treatment, shift-invert Krylov iteration, inner PCG solves, and GPU matrix-vector products. Numerical experiments for a three-dimensional two-phase anisotropic phononic crystal show that the block implementation preserves the full-space spectrum while substantially reducing the wall-clock time. The results demonstrate that the proposed method provides a structured and efficient solver for large-scale band-structure computations of three-dimensional anisotropic phononic crystals.

This paper compares three different multirate splitting approaches for the application on field-circuit coupled magnetoquasistatic simulations. For these methods, again three different variants for exchanging values between the field and circuit are tested, namely voltages, currents and flux correction terms. All scenarios are applied on two different benchmark problems, i.e. a coil inductor and transformer model coupled to different circuits. The convergence behavior of different time steppers (Implicit Euler and Trapezoidal Rule) is determined for all possible settings, and guidelines for practical applications are derived.

A class of Kuramoto models with a general coupling function that can be expressed in terms of a finite number of harmonics, each comprising sinusoidal terms, is studied. We propose a novel approach for certifying local phase synchronization in this class for all initial conditions lying on an arc. The trace parametrization property and Gram matrix representation of a trigonometric polynomial are utilized along with Putinar's Positivstellensatz to obtain semidefinite programming certificates for the stability of the phase-difference system, which in turn implies synchronization of the original system. The results can be extended to any system of coupled oscillators where the forward-invariance on arcs can be established.

In this technical note, we show that the relativized bounded arithmetic $T^{1}_{2}(R)$ equipped with $UCP(\Delta^{b}_{1}(R))$ cannot prove $injPHP^{n+1}_{n}(R)$. The result partially supports [K, 2024] Conjecture 1.

In this paper we prove that an $(n+1)$-manifold, compactly $n$-enlargeable, where $3\leq (n+1)\leq 7$, has connected, immersed Morse index one, closed minimal hypersurfaces with unbounded volumes for bumpy metrics. We prove that in the three-dimensional case the hypersurfaces are geometrically distinct using cyclic coverings of manifolds with boundary. The proof extends to $(n+1)$-fiberings. We prove a scalar curvature rigidity theorem for area-nonincreasing maps of three-dimensional manifolds. The case of stable surfaces is also discussed by using cohomology classes and incompressible surfaces.

We study selection problems for the vanishing discount limit of first-order stationary mean field games with local coupling. The associated ergodic problem may admit several value functions for the same density and ergodic constant. We decompose the state space into the positive-density region, the essential zero-density interior, and a residual set, and show that possible non-uniqueness of selected value functions can occur only on the gap part of the residual set. If the gradients of selected value functions coincide on this gap residual set, then the selected value function is unique up to additive constants; under compactness and stability assumptions, this yields convergence of the whole normalized discounted family. We show that the gap residual set is null for one-dimensional problems and for a specific Hamiltonian in the multidimensional setting, and hence obtain convergence results in these cases.

For a possibly ramified $\mathbb{Z}_p^d$-covering of connected graphs, we establish an Iwasawa-type asymptotic formula for the growth of the $p$-adic valuations of the complexities. The formula is expressed as a polynomial in $n$ and $p^n$ with explicit leading coefficients $\lambda$ and $\mu$; in particular, we eliminate the error term of the form $O(p^{(d-1)n})$ appearing in earlier work. We then establish a Kida-type formula describing the behavior of $\lambda$ under a $p$-covering between $\mathbb{Z}_p^d$-coverings, assuming $\mu = 0$. Finally, for any fixed $p$ and integer $d \geq 2$, we construct an unramified $\mathbb{Z}_p^d$-covering of a bouquet with prescribed $\lambda$- and $\mu$-invariants.

Let $G$ be a nonempty finite simple graph. We study when the Stanley-Reisner ideal of its neighborhood complex has a $2$-linear resolution. Combining Fröberg's theorem with the classical hypertree criterion, we obtain the following equivalent description in graph terms: $G$ is bipartite, its indexed open neighborhoods are Helly, and every induced cycle of length at least eight has a filling from each color class. This class properly contains the chordal bipartite graphs without isolated vertices. Hochster's formula gives all squarefree multigraded Betti numbers, while face counts determine the complete graded Betti table. If $G$ has $n$ vertices and $c$ connected components, then its Stanley-Reisner ring has terminal Betti number $2c-1$, projective dimension $n-1$, and depth one. We also determine the multiplicity and the initially Cohen-Macaulay and Cohen-Macaulay cases. A second formula separates degree data from overlaps caused by repeated common neighbors and yields closed expressions for bipartite graphs without $K_{2,3}$, cactus graphs, pseudoforests, and forests. For square cactus graphs, the Betti table recovers every degree multiplicity at least three; for forests, it recovers the complete degree sequence. Finally, the dominance complex has a $2$-linear Stanley-Reisner ideal precisely for nontrivial stars.

Channel charting enables location-based services (LBSs) without requiring explicit position information by using pseudo-locations from the channel chart. While this property implies inherent privacy advantages, it does not provide formal privacy guarantees. In this work, we address location privacy in channel charting referred to as chart location indistinguishability (CLI), which extends geo-indistinguishability (GI) to channel charting representations. In order to achieve CLI, a standard planar Laplace mechanism is investigated and a geometry-aware Mahalanobis norm planar Laplace (MNPL) mechanism is devised. The proposed MNPL mechanism perturbs the channel chart by injecting noise aligned with the local structure of the chart. In the CLI framework with MNPL, privacy is defined in latent channel chart manifolds using locally adaptive covariance derived from chart neighborhoods, while preserving manifold topology under privacy constraints. In addition, differential privacy is considered as a privacy baseline. The proposed approach is evaluated across multiple channel charting schemes. The performance is assessed using utility metrics such as quality loss (QL) and range query error (RQE), as well as geometry-aware metrics including trustworthiness (TW) and continuity (CT). Numerical results demonstrate that the proposed privacy mechanism provides strong privacy guarantees while preserving the channel chart for LBSs tasks.

Burrows-Wheeler-transform-based compressors rely on local context regularity, but structured text also contains dates, counters, identifiers, coordinates, and other digit runs whose values vary differently from their surrounding tokens. STC is presented as a new algorithm found by the authors through the self-evolving AI system zeelin. It is a practical BWT-family compressor that separates this source of variation before the component BWT stage. It replaces digit runs in the main stream with an unambiguous placeholder and stores the removed digits in length- and context-conditioned side streams. The side streams use stable bucket ordering and compact digit packing, so the decoder can reconstruct the original run order from the normalized main stream without storing a separate permutation. The resulting components are encoded by a fixed internal BWT/M03-style component coder. On enwik9, STC produces a 157,388,188-byte archive with a 183,174-byte decoder source package, giving a local LTCB-style total of 157,571,362 bytes. A full-enwik9 same-coder ablation shows that the digit-context decomposition reduces the archive by 2,629,561 bytes relative to the no-split control. The result is locally verified by full decode and SHA-256 matching; official benchmark status requires independent maintainer-side verification.

For nonconvex optimization problems whose objective is the prediction function of a trained Support Vector Regression (SVR) model with the Gaussian radial basis function (RBF) kernel (RBF-SVR), we present a framework that applies the difference of convex functions (DC) algorithm (DCA) by exploiting the analytical structure of the RBF kernel to construct an explicit DC decomposition. Specifically, we derive in closed form both the lower bound $\mu$ of the strong convexity parameter of the DC components and the upper bound $L$ of the gradient Lipschitz constant of the subproblem. Both $\mu$ and $L$ are determined solely by the post-training dual-coefficient sum $C_{\alpha}$ and the RBF kernel parameter $\gamma$, together with the DC decomposition parameter $\rho$, and they share a common leading term $C_{\alpha}\rho$. Through numerical experiments on six benchmark functions, we show that $C_{\alpha}\rho$ is the primary single quantity characterizing both the convergence properties and the initial-point dependence of DCA, and further demonstrate that it decomposes into two independent pathways, $C \to C_{\alpha}$ and $\gamma \to \rho$, with its primary variation governed by the SVR hyperparameters $(C, \gamma)$. Together, these results allow the convergence properties of DCA on RBF-SVR to be assessed in advance through the single scalar quantity $C_{\alpha}\rho$: approximately from $(C, \gamma)$ before training, and exactly in closed form after training.

While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.

Hyperparameter optimization (HPO) for Random Forest faces a specific difficulty in tuning the number of trees: the predictive score typically improves monotonically with ensemble size, so standard methods such as Tree-structured Parzen Estimator (TPE) and Hyperband require a predefined search range and often drive the estimate toward its right boundary. Early-stopping strategies avoid fixing such a range, but can be sensitive to score noise and prone to premature stopping. To address this, we propose an integrated triplet-based plateau-search algorithm that removes the number of trees from the direct TPE search space and still exploits information accumulated across HPO trials. The method adaptively tracks a near-minimal sufficient ensemble size by monitoring relative changes in the out-of-bag (OOB) score across a triplet of forest sizes and shifting this triplet accordingly. This yields an automated and user-interpretable procedure based on a tolerance parameter. We also provide a theoretical analysis: we relate the proposed relative OOB-score criterion to the gap between the current and limiting scores, and derive an asymptotic variance estimate for the corresponding OOB-based absolute relative difference. Experiments show that the selected number of trees can differ substantially from the common heuristic: for most classical benchmark datasets it is smaller, whereas for some high-dimensional bioinformatics datasets, such as Arcene and Dorothea, it is larger. The source code and reproducible experiments are available at https://github.com/lange-am/rf_plateau_hpo.

In this paper, we study the incompressible, viscous and resistive Hall-magnetohydrodynamic (Hall-MHD) system. We first prove that every two-dimensional Leray-Hopf weak solution becomes smooth after a finite time. In three dimensions, where eventual smoothness for arbitrary Leray-Hopf weak solutions is not known, we construct Leray-Hopf weak solutions for which the magneto-vorticity field $B+\nabla\times u$ eventually gains additional regularity. Finally, under suitable low-frequency pseudomeasure assumptions on initial data, we establish decoupled algebraic decay rates for the velocity and magnetic fields by combining a generalized Fourier splitting method with the eventual smoothness in two dimensions and strong regularity in three dimensions.

We study a class of Sturm–Liouville problems of the form \[ -(p\,y')' + q\,y = \lambda\, w\, y, \] on a finite interval with complex-valued coefficients, where $p$ and $w$ are piecewise smooth, the ratio $w/p$ is real and positive, and $q$ is bounded. We prove that all eigenvalues in the open left half-plane are contained in a bounded set, which implies that only finitely many eigenvalues lie in this region. This stands in contrast to the known unboundedness, for real-valued coefficients, when $p$ or $w$ changes sign independently. A canonical instance of this class, with $w=p$, arises in transverse-magnetic (TM) diffraction by metallic lamellar gratings, a benchmark problem in computational photonics, central to the development of modal methods. In Fourier modal methods, in particular, the emergence of spurious modes with unbounded propagation constants (eigenvalues), rooted in discretization of sign-changing coefficients, leads to notorious convergence difficulties. These modes cannot be excluded a priori, since genuine eigenvalues are not constrained by conventional bounds in this regime. Nevertheless, our result shows that the physical eigenvalues remain bounded, providing a rigorous criterion for identifying spurious modes in low-loss metallic gratings.

Neurons receive information through their dendrites. During development, when synaptic connections are forming, dendrites grow, retract, and branch. The resulting dendritic tree shapes the structure of the broader neural network. Crucially, retraction and branching make it necessary to track whole dendritic paths rather than only their endpoints. While this is handled implicitly in some existing simulations, here we construct an explicitly path-valued stochastic process for dendrite growth. Combining this with a branching process, using ideas from measure-valued branching particle systems, we show that the model produces the typical tree structures of real dendrites. To complement this analytical work, we also outline several methods for numerical simulation, including time discretisations at different temporal scales and an approximation using a dynamic graph. This provides both a more rigorous mathematical framework and more structured simulation methods for modelling dendrite development.

In this paper, we introduce a new type of ring, called quasi-nil clean ring, where each element of the ring is the sum of a quasi-idempotent and a nilpotent element. We also investigate a particular class of ring, called strongly quasi-nil clean ring whose each element is the sum of a quasi-idempotent and a nilpotent element, where they commute. Our primary objective is to study the structural properties of these new class of rings, explore their relationships with existing classes of rings and establish some key characterizations. In the commutative setting, we provide a complete characterization in terms of quasi-Boolean quotients. Moreover we discuss about quasi-cleanness of amalgamated algebra and group ring.

We study finite palette extremal problems motivated by uniform Turán densities of $3$-uniform hypergraphs. Given a self-converse tournament $T$ with at least two vertices, we determine the largest possible number of admissible triples in an $m$-color palette that avoids the left and right palettes associated with $T$. The answer is the one-sided degree-square Turán number \[ \operatorname{pal}_T(m) = \operatorname{ex}_2^+(m,T) = \max\left\{ \sum_{v\in V(D)} d_D^+(v)^2: |V(D)|=m,\ D\text{ is }T\text{-free} \right\}. \] Thus this palette problem is reduced to an extremal problem for digraph out-degrees. We then prove a prefix-majorization lemma for convex out-degree moments and apply it to directed cycles. In particular, $\operatorname{ex}_2^+(m,\overrightarrow C_3)=\frac{m(m^2-1)}3$, which gives the sharp $m$-color palette endpoint $\frac13-\frac{1}{3m^2}$ for the directed triangle. Combining this endpoint with the palette characterization of uniform Turán density and the palette separation theorem, we show that for every $m\ge2$ there is a finite $3$-graph $H_m$ such that \[ \frac13-\frac{1}{3m^2}\le \pi_u(H_m)\le \frac13. \] Hence there is a sequence of individual finite $3$-graphs whose uniform Turán densities converge to $1/3$. We also describe the extremal palettes, prove a qualitative edit-distance stability theorem, and compute the Lagrangian of the endpoint palette $\mathcal P_m^\star$. As a consequence, for every $m\ge2$ there is a finite family $\mathcal F_m$ of $3$-graphs with $\pi_u(\mathcal F_m)=\frac13-\frac{1}{3m^2}$, so $1/3$ is an accumulation point of uniform Turán densities of finite forbidden families.

Long-time simulations of evolution equations present substantial computational challenges due to the inherently sequential nature of conventional time-stepping schemes. The parareal method, a leading parallel-in-time (PinT) algorithm, offers a promising approach to overcome the challenge by introducing concurrency in the time domain. While its convergence theory is well-established for linear problems, extending the theory to nonlinear problems, particularly when the problem data have only limited regularity, remains a significant challenge. In this work, we provide the convergence analysis of the parareal algorithm for solving semilinear parabolic equations with an $H^2$ initial data. We employ stable rational approximations and first-order linearization as coarse propagators, establish the linear convergence of the parareal algorithm and provide a sharp estimate for the convergence factor. The analysis combines the error-splitting technique from the superlinear convergence analysis of the parareal method, a refined linear convergence theory for linear parabolic equations, and a priori error estimates that are optimal with respect to the regularity of the problem data. The analysis shows the close connection between the convergence behavior of nonlinear models and their linear counterparts. Numerical experiments fully support the theoretical findings.

Quantum computing presents a promising method to overcome the efficiency and memory constraints in large-scale mechanical problems, with numerous successful applications demonstrated in fluid mechanics. However, solid mechanics problems usually require irregular grids for spatial discretization, due to the Lagrange formulations and complex boundaries, which makes the quantum simulation of the system matrix, e.g., the mass or stiffness matrix which is often referred to as the Hamiltonian in quantum computing, difficult to be effectively conducted. This study proposes a voxel-based quantum computing method (VBQC) for the quantum simulation of Hamiltonians in solid mechanics. VBQC applies voxel grids to discretize the spatial domain, thereby enabling the system matrix to exhibit the tridiagonal fractal property. Based on this property, the system matrix can be decomposed into three groups of fundamental matrices, $\mathbf{k}_{n}$, $\mathbf{c}_{n}$, and $\mathbf{q}_{n}$. This decomposition process is referred to as the KCQ decomposition. By integrating the KCQ decomposition with the quantum Fourier transform and the quantum multiplexer, VBQC enables efficient quantum simulation of Hamiltonians in solid mechanics. Three specific solid problems with different dimensions and numbers of variables are applied to preliminarily verify the correctness of the proposed VBQC for solid mechanics problems.

This paper investigates the secrecy sum-rate (SSR) performance of optical intelligent reflecting surface (OIRS)-assisted multi-user visible light communication (VLC) systems under line-of-sight (LoS) blockages. To mitigate physical obstructions and internal eavesdropping, a joint optimization problem is formulated to maximize the SSR through the co-design of the transmission precoder and OIRS units assignment. Due to the binary constraints and coupled variables, the problem is highly non-convex. To solve it efficiently, an alternating optimization (AO) framework integrating the concave-convex procedure (CCCP) and first-order Taylor approximations is developed. Simulation results demonstrate the convergence of the proposed algorithm and show that increasing the number of OIRS reflecting units yields significant SSR gains.

Recall that a module is called weakly uniserial if its submodules are comparable regarding embedding. Weakly uniserial modules are a nontrivial generalization of uniserial modules. In this paper we define and study a new dimension, which measure how far a module deviates from being weakly uniserial. We call this dimension, weakly uniserial dimension. Also, we define and study monoartinian (mononoetherian) modules. We say that an $R$-module $M$ is monoartinian (mononoetherian) if in every descending (ascending) chain of submodules of $M$, except probably a finite number, each module in chain embedded in the next (previous) one. We show that a module has weakly uniserial dimension if and only if it is monoartonian.

The paper discusses renormalization aspects of the quantum four-dimensional Yang-Mills theory with a cutoff regularization in the coordinate representation. The background field method is used to formulate a generating functional, and the regularization is introduced through quasi-local probabilistic averaging. Two main types of regularization are proposed: strong deformation, which consists in averaging fluctuation fields, and weak deformation, which is a covariant generalization of the first case with respect to gauge transformations of the background field. We study singular contributions for the first two quantum corrections in this paper and compare them in detail with the case of dimensional regularization. The consistency of the action and the equation of motion after introducing the regularization and making a renormalization procedure is analyzed. New counter-vertices are studied, in particular their locality properties and dependence on the regularization parameter.

We answer a question posed by Poggiolesi concerning a syntactic decidability proof for GL in the tree-hypersequent system CSGL, and resolve a challenge identified by Maggesi and Perini Brogi, who sought a PSPACE proof-search algorithm for GL in expressive sequent-based formalisms. We work with a notational variant of CSGL formulated in terms of (labeled) tree sequents. Our answer is complexity-optimal: we present a proof-search algorithm that decides the (in)validity of formulae and runs in PSPACE, matching the known PSPACE-completeness of GL. To achieve this, we introduce a "linearization method", which constructs only a single branch of a derivation and of a tree sequent at a time, avoiding the exponential blowup typical of naive proof-search in sequent formalisms. We show how to systematically combine fragments of tree sequents generated during proof-search to extract finite counter-models, which serves as a theoretical device for establishing the correctness of the algorithm when proof-search fails. Finally, we show that every valid formula admits a proof consisting solely of line sequents, which correspond to linear nested sequents. This establishes a connection between depth-first proof-search and linear nested sequent calculi. Our results not only answer the aforementioned questions, but also provide new insights into proof-search and correctness arguments in tree sequent systems for modal logics.

Regev and Stephens-Davidowitz conjectured that the integer lattice maximizes Gaussian mass among integral lattices of a given rank. We prove this, including the equality case, for all unimodular Construction-A lattices arising from binary self-dual codes. The proof reduces the theta-series inequality to a sharp majorization statement for codes: if $C$ is a binary self-dual $[2k,k]$ code, then the half-weight distribution of $C$ is dominated in convex order by $\operatorname{Bin}(k,1/2)$, which is the corresponding distribution for the repetition-code model of $\mathbb{Z}^{2k}$. Indeed, after putting $C$ in systematic form $[I\mid A]$, self-duality gives $AA^T=I$ over $\mathbb{F}_2$, so for a uniformly random message $a$ the two weights $\operatorname{wt}(a)$ and $\operatorname{wt}(aA)$ have the same binomial law. The half-weight of the resulting codeword is their average, and Jensen's inequality then gives convex-order domination. Applied to the convex test functions that build the theta series, this yields a sum-of-squares formula for the Gaussian-mass gap; applied to hinge functions, it gives coefficientwise nonnegativity of the reduced gap polynomial.

Let $Q=-x_1^2-x_2^2-x_3^2+x_4^2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\mathbb{Q}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $\mathrm{SL}_2(\mathbb{Z}[i])\backslash \mathbb{H}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between $\mathrm{SO}_{1,3}(\mathbb{R})$ and $\mathrm{SL}_2(\mathbb{C})$ gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.

Physics-informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with governing physical laws. However, most existing methods implicitly assume that the prescribed governing equation is accurate. This assumption may fail in practical applications, where model simplifications, missing physical effects, parameter drift, or incomplete constitutive relations can lead to model misspecification. In this work, we propose a physics-guided operator correction framework for learning solution operators under misspecified governing equations. At the operator level, the target mapping is decomposed into a prior operator induced by an approximate physical model and a learnable correction operator that accounts for the remaining discrepancy. We realize the operator model correction using a serial DeepONet architecture, where the first DeepONet provides a solution prediction and the second DeepONet learns an additive correction conditioned on both the input function and the solution prediction. The learned correction is incorporated into the physics residual and trained together with data-consistency constraints, allowing the model to retain useful physical structure while adapting to inaccurate governing equations. Numerical experiments on diffusion-reaction, Burgers, cavity flow, and hyperelastic problems show that the proposed method substantially reduces errors induced by misspecified physics. Additional tests under sparse and noisy observations further demonstrate the robustness of the framework and its ability to provide informative uncertainty estimates through deep ensembles.