Papers

Careless or insufficient-effort responding threatens the validity of self-report measurement. Mixture models for careless responding specify its form, and screening rules and robust methods depend on a threshold or tuning constant with no direct meaning in terms of careless responses. We propose ExHuber, a Bayesian method for inference on the engaged population under Huber's contamination model, in which an unknown proportion of responses comes from a contaminating distribution of unspecified form instead of the structural model. With a finite support of responses, such as the values of a scale score, a Dirichlet prior and a Beta prior on the contaminating distribution and its proportion yield a closed-form exact marginal likelihood of the structural parameters. Under a prior favoring a contaminating distribution on a few values, we show that the posterior of the structural parameters is consistent when careless responses occupy few values of the support, without knowing which values. Dynamic programming evaluates the marginal likelihood and its gradient at a cost quadratic in the sample size, which enables gradient-based sampling. Simulations from graded response models show accurate inference under straightlining and repeated response patterns, and an application to Big Five scales estimates the engaged mean of each scale.

This paper investigates a multidimensional non-homogeneous stochastic linear-quadratic optimal control problem featuring random coefficients and a terminal mean-field term in the cost functional, enabling its direct application to mean-variance models in financial engineering. Employing the Lagrangian duality method together with a decomposition approach for linear backward stochastic differential equations, we provide two types of sufficient conditions for solvability and derive the corresponding optimal controls. In particular, in the deterministic-coefficient case, our condition is weaker than the standard condition found in the existing literature on mean-field stochastic LQ problems. Finally, a numerical example drawn from optimal portfolio selection with multiple assets under mean-variance utility demonstrates the applicability of our results.

The increasing densification of small-cell networks substantially expands cable-based backhaul infrastructure, creating heightened vulnerability to cable link failures. This paper proposes a reconfigurable intelligent surface (RIS)-assisted backup framework that exploits a key insight: during backhaul cable failures, base station (BS) radio components remain functional, enabling wireless backhaul traffic redistribution. Our framework maintains network connectivity by redistributing disconnected BS backhaul traffic to neighboring BSs through RIS-assisted wireless links. To maximize survivability across varying traffic conditions, we formulate a joint optimization problem that maximizes total resolvable backhaul traffic by jointly deciding BS selection, RIS phase shifts, and precoding vectors. The inherent non-convexity arising from coupling and quadratic fractional term is addressed through an alternating optimization algorithm that iteratively solves tractable convex subproblems via quadratic transformation. Comprehensive numerical evaluations demonstrate that the proposed RIS-enhanced framework significantly improves survivability from 58% to 72% under challenging high-intensity hotspot traffic conditions. Moreover, RIS provides the greatest gains for antenna-constrained systems by extending coverage to access more spare capacity of the distant BSs as well as enhancing the signal strength. Consequently, high survivability is achieved even with only two antennas per BS under moderate traffic intensity.

Pinching antenna systems (PASS) enable reconfigurable radiating elements and extended line-of-sight communication, mitigating path loss effects. However, existing designs lack fully controllable radiation weights, as they are governed by structural parameters rather than explicitly assigned variables. In this paper, we introduce a new degree of freedom (DoF) for PASS by enabling radiation weight control through phase-mismatch manipulation of guided waves under single-mode excitation within a coupled-mode framework. By tuning the propagation constants of pinching antennas, independent complex-weight control of individual elements is achieved, transforming PASS into a weight-adaptive analog beamforming architecture. Based on this principle, we present a physics-based hardware model that provides a unified framework for both amplitude-tunable pinching beamforming and conventional equal-power radiation models, ensuring compatibility with existing PASS implementations, such as movable setups. To evaluate the proposed model, we formulate a sum-rate maximization problem for hybrid precoding in multiuser downlink systems and solve it using an alternating optimization framework that combines weighted minimum mean square error-based digital precoding with genetic algorithm-based optimization of PASS configurations, including various scenarios such as weight tuning, antenna movability, and discrete activation. Numerical results demonstrate that the amplitude-tunable PASS architecture achieves consistent performance gains over conventional arrays and existing PASS schemes, with pronounced improvements in interference-limited regimes under practical constraints.

We show that the Wirtinger criterion cannot be used to investigate the frame set conjecture for the first Hermite function. More generally, for odd functions, it cannot determine regions of the frame set with density less than 2.

This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar [On the sum of k largest eigenvalues of graphs and symmetric matrices, J. Combin. Theory Ser. B 99 (2009) 306–313]. Furthermore, in the case of the Laplacian matrix, we prove that the well-known Brouwer's conjecture [Spectra of Graphs, Springer, New York, 2012] holds for small values of $k$ for almost all graphs, thereby taking a significant step toward its complete resolution.

Let $X$ be a complex space of pure-dimension $n$. For a pseudoconvex relatively compact domain in $X$ with $\mathscr{C}^3$-smooth boundary and embedded in a domain of the complex number space, we prove that the $L^2$- and $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-cohomology groups are vanishing for $q>0$. Thereby, we include the case that the forms have values in a Nakano semi-positive holomorphic vector bundle. Using this local vanishing theorem, we also prove the equivalence of the $L^2$- and $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-cohomology groups of relatively compact domains $\Omega=\{\rho<0\}$ in $X$ which are defined by a $\mathscr{C}^3$-smooth function $\rho$ which is strictly plurisubharmonic on a neighbourhood of $\partial\Omega$ except of finitely many points.

We propose a degree-preserving variant of the Builder–Chooser clique game of Pettie, Tardos, and Walczak. In each round, Builder chooses a matching, performs a degree-preserving growth (DPG) step by replacing the chosen edges with edges incident to a new vertex. Then partitions the entire edge set into two parts, and Chooser keeps one part. We begin the study of this game with the first nontrivial target, namely forcing a triangle. For triangle-free initial graphs we prove an exact one-round criterion, derive an exact one-round threshold on paths and exact forcing times on cycles, and identify the $5$-cycle as the first genuine two-round example. We then formulate a one-round criterion for larger cliques, prove a sharp exact result for forcing $K_4$ from triangle-free seeds. We establish general lower bounds on clique-forcing times from clique-free seeds, and isolate a conjectural template-amplifier lemma which, if proved, would imply that every clique is forceable from some triangle-free seed.

We prove the open question posed by Zhuang and Hu in Remark 3.1. More generally, we consider symmetric joint probability mass functions and joint densities whose associated quadratic form is non-negative. In this class, for every \(r>0\), the inequality \(\E\abs{X+Y}^{r}\ge \E\abs{X-Y}^{r}\) holds for all distributions with finite \(r\)-th absolute moment if and only if \(0<r\le2\).

Convex integration has revealed that the Euler system of gas dynamics is ill-posed in the class of weak solutions even if the entropy inequality is imposed as an additional constraint. A natural question arises, namely, if a physically relevant solution can be selected by maximizing the entropy production rate. Firstly, we present an example of Riemann initial data in 2-D, for which the standard self-similar solution fails to satisfy the maximal entropy production principle. Hence, maximizing the entropy production rate rules out the 1-D self-similar solution which intuitively seems to be the physically relevant solution in this context. Secondly, we show for a large class of initial data that there exist entropy admissible weak solutions with an arbitrary (non-decreasing) total entropy profile.

We study odometer maps $W_L$ on vector-valued full Fock spaces arising from Fock representations of the odometer semigroup. We obtain a canonical upper triangular block decomposition \[ W_L= \begin{pmatrix} W_{11} & W_{12}\\ 0 & W_{22} \end{pmatrix}, \] where $W_{11}$ is unitary and $W_{22}$ admits a Hardy space realization as an analytic Toeplitz operator $M_\Theta$. The associated symbol $\Theta\in H^\infty_{\mathcal{B}(\mathcal{E})}(\mathbb{D})$ is used to characterize the isometric, unitary, and invertible cases, as well as norm identities and Douglas-type factorization properties of $W_L$. We also derive an explicit formula for $W_L^*$ for arbitrary bounded symbols $L$. In the isometric case, this identifies $\ker W_L^*$ with $\mathcal{E}_L\ominus L\mathcal{E}$, and hence $\operatorname{mult}(W_L)=\dim(\mathcal{E}_L\ominus L\mathcal{E})=\operatorname{mult}(M_\Theta)$. In the same setting, the condition $\dim(\mathcal{E}_L\ominus L\mathcal{E})<\infty$ is equivalent both to Fredholmness and to essential normality of $W_L$, with $\operatorname{ind}(W_L)=-\dim(\mathcal{E}_L\ominus L\mathcal{E})$. We further obtain Coburn-type spectral consequences and a necessary condition for hyponormality.

This paper considers a two-color, single-draw urn model with two types of balls, denoted type $1$ and type $2$, with initial counts $Y^1_0\in N^+$ and $Y^2_0\in N^+$, respectively. At each discrete time step, a ball is drawn uniformly at random, its type observed, and then it is returned to the urn. The urn is subsequently updated according to a mixed replacement matrix: with fixed probability $p\in(0,1)$, the Friedman replacement matrix is applied, adding $a$ balls of the drawn type and $b$ balls of the opposite type; with fixed probability $1-p\in (0,1)$, the Pólya replacement matrix is applied, adding $c$ balls of the drawn type. We establish the central limit theorem for the proportion of type $1$ balls after $n$ draws. Furthermore, we provide corollaries that yield large deviation inequalities and the law of the iterated logarithm related to the proportion of type $1$ balls after $n$ draws.

We prove that the continuous-time, single-flip Glauber dynamics for lozenge tilings of the size-$N$ hexagon mix in time $N^{2+o(1)}$. This was predicted to hold on fairly general domains of diameter $N$ (on the basis of the “Lifshitz law” heuristic) but had previously only been established in domains such that the associated limit shape has no frozen facets. To access the hexagon, we introduce a multi-scale comparison argument between the height function of the random tiling evolving under the Glauber dynamics and the limit shape of a volume-tilted tiling (whose tilting parameter varies suitably in time).

A linear-complexity algorithm for computing the Wasserstein-1 distance on non-uniform meshes is proposed. This work extends the fast Sinkhorn algorithms from [Q. Liao et al., Commun. Math. Sci., 20(2022)] and [Q. Liao et al., J. Sci. Comput., 98 (2024)] to non-uniform meshes. In those prior works, a distinctive collinear structure of the kernel matrix on uniform meshes was identified, enabling \(O(N)\) acceleration via dynamic programming. While non-uniform meshes are prevalent in practical applications like computational fluid dynamics and finance, their lack of collinearity has hindered direct acceleration. In this paper, we introduce the concept of a “dividing index”, which partitions the kernel matrix into two blocks. We demonstrate that each block exhibits a quasi-collinear property, a generalization of the structure found in uniform meshes. Leveraging this insight, we develop Fast SInkhorn algorithm on NOn-uniform Meshes (FINOM), a dynamic programming approach that reduces the per-iteration complexity of the Sinkhorn algorithm from \(O(N^2)\) to \(O(N)\). Extensive numerical experiments on 1D and 2D problems confirm these improvements, achieving speed-ups of several orders of magnitude while maintaining accuracy.

Reinforcement learning (RL) often has a hierarchical structure, where an upper-level (UL) learner selects model parameters and a lower-level (LL) decision-making process responds, naturally leading to a bilevel optimization problem. Most existing bilevel RL methods assume a single-policy LL Markov decision process (MDP), and therefore fail to capture competitive structures arising in applications such as incentive design, where multiple policies interact. We study bilevel optimization problems in which the LL problem is a regularized min-max zero-sum Markov game and the UL objective is optimized through the saddle-point equilibrium induced by the LL game. In this work, we propose penalty-augmented Nikaido-Isoda descent-ascent (PANDA), a penalty-based first-order policy-gradient method based on the Nikaido-Isoda function. By exploiting the min-max game structure, PANDA avoids computing UL hypergradients and does not require second-order information. We prove that PANDA converges to stationary points without convexity assumptions on either the UL or LL objectives. Moreover, PANDA reaches an $\epsilon$-stationary point in $\tilde{\mathcal{O}}(\epsilon^{-1})$ iterations with sample complexity $\tilde{\mathcal{O}}(\epsilon^{-3})$, matching the best-known rates for bilevel RL with single-policy LL MDPs. Experiments demonstrate the superior performance of PANDA over closely related baselines.

We study dynamic large deviations for the Kipnis–Marchioro–Presutti process on the discrete torus $\mathbb{T}_N^d$. By recasting the candidate rate function in terms of a linear hyperbolic-parabolic equation with rough drift, we show that pathological trajectories appear in the large deviations with finite rate in any dimension $d\ge 2$. Along the way, we rigorously validate the lower bound derived by Bertini-Gabrielli-Lebowitz in any dimension.

We study the sample complexity of policy gradient for log-growth control – the problem of learning, from observed state transitions, a feedback gain that optimally stabilizes a scalar linear system driven through a multiplicative-noise actuation channel. The objective $J(K) = \mathbb{E}[\log|1+BK|]$ is the top Lyapunov exponent of the closed loop. This problem carries a structural difficulty we call the cusp obstruction: the optimal gain $K^*$ always places the noise singularity $b_{\rm sing}(K) = -1/K$ in the interior of the support. At this singular optimum the policy gradient exists only as a Cauchy principal value, not as a Lebesgue integral, and the natural single-sample gradient estimator has infinite variance. Standard first-order stochastic-optimization analysis is thus inapplicable at the optimum, and merely smoothing the objective does not resolve the difficulty. The obstruction, however, has an exploitable symmetry: the Cauchy kernel is an odd function of the displacement from the moving pole, so pairing each observation with its reflection through the pole cancels the divergent part. This one cancellation simultaneously controls the population curvature, the gradient-estimator variance, and the bias incurred when the noise density is estimated. Combining these bounds with a closed-form single-transition gradient oracle, we prove that projected mini-batch policy gradient, initialized in any compact subset of the stabilizing region, attains total sample complexity $\tilde{O}(1/\eta)$ when the noise density is known and $\tilde{O}(\eta^{-(2s+1)/(2s)})$ when it must be estimated, for $C^s$ noise densities with $s \geq 2$.

Practical ISAC is constrained by static clutter and NLoS multipath, which obscure target-coupled echoes and induce spurious peaks for beam alignment. Existing receiver-side methods largely model targets as passive scatterers, limiting the structural separability of target echoes from the environment. This paper establishes a structural correspondence between these limitations and target-side Backscatter-MIMO responses: reflection modulation enables waveform-domain separation from unmodulated clutter, while retro-directional passive beamforming concentrates the tagged echo toward the BS-facing direction and suppresses NLoS-induced false-peak locking. To operationalize this correspondence, dual-end spatial locking is required to overcome cascaded backscatter loss and provide beam-domain angular information. We propose a downlink-triggered blind dual-end alignment protocol that jointly selects the BS and Backscatter-MIMO codeword indices from the tagged echo observed at the BS, without pilots, CSI feedback, or target synchronization. We further derive a clutter-aware remodulation waveform robust to fractional timing offsets and construct adjustable-width BS/Backscatter-MIMO codebooks via quadratic phase spoiling. For reliability characterization, we derive closed-form expressions for the coherence-averaged end-to-end success probability. The analysis shows that beam narrowing is not universally beneficial: in NLoS-dominated regimes, enlarging the array aperture may degrade alignment reliability. The optimal beamwidth is instead governed by cross-phase competition between discovery and alignment, yielding a nontrivial feasible region with an analytically characterized boundary. Simulations validate the analysis and demonstrate improved reliability-gated locked-link performance under strong clutter, severe NLoS multipath, and finite coherence time.

This paper proves a conjecture by Solomon about Steiner shallow-light trees (SLT) in Euclidean $d$-space: It is shown that for any finite point set $\mathbb{R}^d$, any root, and any $\epsilon>0$, there is a Euclidean Steiner $(1+\epsilon,O(\sqrt{1/\epsilon}))$-SLT without any dependence on dimension. We also revisit the core example, designed by Solomon, in the plane and its generalization to $d$-space.

The Allen-Cahn equation with Flory-Huggins potential is a fundamental and crucial model in phase field simulation for describing phase separation phenomena, which serves as a core tool in diverse branches of natural sciences. The numerical simulation of the Allen-Cahn equation is of great importance but poses significant challenges due to the strong nonlinearity and the presence of logarithmic singularities at $u=0,1$ in the Flory-Huggins potential. In this paper, we consider convex splitting schemes to %preserve this bound and guarantee unconditional unique solvability, which reduces the numerical simulation to solving a singular nonlinear system arising from spatial discretization at each time step. We propose an iterative solver that is specifically designed for such systems based on the alternating direction method of multipliers (ADMM) approach. The scheme possesses properties such as bound preserving and discrete energy stability. Building upon the recent unconditionally convergent ADMM framework for the Cahn-Hilliard equation (Li et al., 2026), our key theoretical contributions are twofold: (a) a proof of unconditional convergence when the multiplier update step size $\alpha \in (0,\frac{\sqrt{5}+1}{2})$; (b) a rigorous establishment of the linear convergence for the embedded ADMM solver. This effectively liberates the solver from time-step constraints or strict separation conditions. Comprehensive numerical experiments validate our proposed ADMM framework, where its theoretical predictions are fully substantiated in practice, showcasing efficiency and robustness.

We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: $$ \hom(H,G)\ge M(G)^e |V(G)|^{v-2e} \quad \text{ if and only if }\quad \hom(H,G)\ge \lambda(G)^{2e-v}M(G)^{v-e}.$$ We also introduce an operator-norm certificate which, via the Riesz–Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. The converse direction in the equivalence theorem is proved by a tensor-power spectral regularization lemma. Our main result provides a unified framework to prove sharp asymptotic edge-spectral supersaturation results for degenerate bipartite graphs with the Sidorenko property, including complete bipartite graphs and even cycles. Let $S_{t-1,m}$ be the split graph with $m$ edges obtained by joining a clique $K_{t-1}$ to an independent set. For every $m$-edge graph $G$ with $\lambda(G)>\lambda(S_{t-1,m})$, $$\texttt{#} K_{t,t}(G) \ge \Big(\frac{2^{-(t-1)^2}}{(t!)^2}-o(1)\Big)m^t \quad \text{and}\quad \texttt{#}C_{2t}(G) \ge \Big(\frac{(t-1)!}{2t^t}-o(1)\Big)m^t.$$ Both constants are best possible: the first is attained asymptotically by random graphs, while the second is attained by split graphs. The supersaturation proofs combine spectral Sidorenko inequalities with a heavy-edge pruning process, a Perron-vector localized/delocalized dichotomy, and incidence-matrix inequalities.

Cruz Chapital, Goto, Hayashi and the author showed that the game-theoretic variants $\mathfrak{s}_{\mathrm{game}^*}^\mathrm{I}$ and $\mathfrak{s}_{\mathrm{game}^{**}}^\mathrm{I}$ of the splitting number $\mathfrak{s}$ are consistently different, although the corresponding two games differ only in a minor case. This result suggests that even if two relational systems $\mathbf{R}=\langle X,Y,\sqsubset\rangle$, $\mathbf{R}^\prime=\langle X,Y,\sqsubset^\prime\rangle$ are the same modulo a countable set $C\subseteq X$, the associated cardinal invariants might be different. We study this phenomenon for the standard relational system of evasion and prediction and for a variation of it. We show that such a difference occurs for the standard one, but not for the variation.

Divide-and-conquer (D&C) is the natural parallel strategy for symmetric tridiagonal eigenvalue decomposition, yet eigenvalue-only solvers fall back to serial QR algorithm because the quadratic transform state each merge carries exhausts memory at scale. In this paper, we demonstrate that each merge stage in D&C needs only two boundary rows, the first and last rows of a node's eigenvector block, and these close into a recurrence whose parent rows follow from its children's and local merge data alone, so the state is carried up the tree with nothing replayed and persistent memory drops from quadratic to linear. Because this state keeps no replay workspace, adapting the schedule to each merge's post-deflation active rank is a scheduling choice, so both CPU and GPU spend finer-grained parallelism only where substantial secular work survives. We evaluate optimized CPU and GPU implementations, with complexity and correctness analysis, against QR, bisection, and D&C routines in standard libraries.

The purposes of this article are threefold. First, to determine numerically when an arbitrary blowup of a smooth surface is smooth. We show the surface is smooth if and only if certain rational parameters involving log discrepancy and multiplicity of the exceptional divisors form a generalised Farey sequence within the dual graph of divisors. Second, in doing the above we provide an exposition of the Berkovich projective line $\mathbb P^1_{\text{an}}(\mathbb K)$ over the Puiseux series as a universal dual graph for divisors on a ruled surface. Third, to explain how non-Archimedean skew products interact with this multiplicity structure of the tree.

We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary to itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the $C^*$-algebra techniques, plays an important role in our approach to the analysis of the problem.

This paper proposes a theoretical framework for establishing the energy dissipation of general implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, by constructing a dissipative modified energy consisting of the original energy and a non-negative quadratic modification. We first test IMEX-LMMs with a simple multiplier, the first-order time difference of numerical solutions. Then, it is shown that the associated non-negative quadratic modification can be constructed if and only if two generating polynomials (corresponding to the LMM) are positive on $[-1,1]$. Based on this, the modified energy is proved to decay over time under a mild time-step restriction depending on the lower bounds of the associated generating polynomials. As a consequence, the energy dissipation of the well-known backward differentiation formula methods up to fifth order can be obtained straightforwardly. Furthermore, we construct for the first time (to the best of our knowledge) a sixth-order energy-dissipative IMEX-LMM and also prove the sixth-order barrier of energy-dissipative IMEX-LMMs when testing the simple multiplier. Some numerical experiments are conducted to verify our theoretical results.

This paper develops a proof-theoretic framework for abstract interpretation by systematically associating logical systems with finite abstractions. Building on earlier work on the internal logics of abstractions, we propose a general procedure for generating a logic whose Lindenbaum-Tarski algebra is isomorphic to a given abstract lattice. The approach identifies logical connectives preserved by the concretization map and derives corresponding proof rules and axioms. The paper establishes soundness and completeness results under suitable conditions, extends the framework to Cartesian products and multi-variable settings, and investigates the logical structure of non-Cartesian abstractions such as octagons. These observations suggest new connections between abstract interpretation, proof theory, and algebraic logic, providing a foundation for a systematic logical analysis of program abstractions.

In shared-codebook unsourced random access (URA), a receiver must recover an unordered set of messages from simultaneous transmissions. Its outer decoder assembles ambiguous slot-wise evidence into distinct codewords while enforcing parity constraints. We propose CIDER, a learned outer decoder that combines masked-diffusion refinement with cross-row demixing and parity-aware propagation on the code's Tanner graph. At higher per-bin loads, quality-guided remasking selectively re-decodes low-confidence rows. On the evaluated GF(64) LDPC codes at $K=2$, CIDER has lower measured symbol and codeword error rates than SIC-BP and FFT-BP at all four code lengths. In same-GPU decoder-only measurements, the deployed mixed-precision implementation is $8.2\times$ to $167\times$ faster than our optimized FFT-BP implementation across those lengths. We also evaluate higher loads, estimated user counts, and detector mismatch. Code is available at https://github.com/jiyunyoung/CIDER/tree/neurips-2026.

Learning-based methods for synthesizing controllers have gained popularity due to their high expressiveness and strong empirical performance. However, in safety-critical scenarios such as autonomous driving, robotics, and power systems, empirical performance alone is insufficient, and formal verification of controller properties such as stability and safety is highly desirable. Unfortunately, many prior verification approaches are either tied to specific structural assumptions on the system or the certificate, making them difficult to transfer across settings, or suffer from poor scalability on higher-dimensional neural network systems. In this tutorial, we present a unified framework that aims to mitigate this gap via bridging control with the state-of-the-art neural network verifier $\alpha,\!\beta$-CROWN (alpha-beta-CROWN). At its core, $\alpha,\!\beta$-CROWN is a general-purpose bounding engine for nonlinear functions represented as computation graphs: given an input domain, it can produce certified bounds and explicit linear relaxation of the nonlinear function. These certified bounds are useful on their own for tasks such as reachability analysis, and they also provide the foundation for more complex routines that perform satisfiability checking and optimization. More specifically, many control problems reduce to verifying real-valued inequalities over a state domain (e.g., Lyapunov theory). Consequently, $\alpha,\!\beta$-CROWN enables scalable verification of such conditions by computing tight bounds and recursively partitioning and pruning subdomains based on the bounds. Thanks to GPU parallelization, this pipeline demonstrates superior scalability on verification and optimization problems that are challenging for traditional approaches. In this tutorial, we discuss the basics of $\alpha,\!\beta$-CROWN and introduce its application to various control-related tasks.

In this paper, we investigate the modulational stability of periodic traveling waves in a local model for shallow water waves, which is an extended version of the Hunter-Saxton equation. We construct a family of small-amplitude periodic traveling waves for this local model and provide a parameterization of these waves. Using Floquet-Bloch theory, perturbation theory, and spectral analysis, we then establish the modulational stability of these background periodic traveling wave solutions. Finally, we analyze the modulational instability of another extended Hunter-Saxton equation with cubic nonlinearities, following a similar approach.