Papers

Let $k=\overline{\mathbb{F}}_p$ and let $f\in k[X]$ be a polynomial of degree at least $2$ such that $f-f(0)$ is non-additive. For $\alpha,\beta$ in an algebraically closed extension $L$ of $k$, we prove that there are infinitely many $\lambda\in L$ for which both points are preperiodic under $f(X)+\lambda$ if and only if $\alpha,\beta\in k$ or $f(\beta)-f(\alpha)\in E_f$. Here $E_f=\{c\in G_f:(f-f(0))^n(c)\in G_f\text{ for every }n\ge0\}$, where $G_f=\{c\in k : f(X+c)-f(X)\in k\}$. We also provide a necessary and sufficient criterion for the corresponding colliding orbits set to be infinite.

We give an explicit local description of ray class refinements in cyclotomic Iwasawa theory. For distinct primes $p\geq5$ and $\ell$, and even $2\leq k\leq p-3$, we prove that the $\omega^{1-k}$-component of the $p$-primary ray class group of $\mathbb{Q}(\zeta_p)$, with squarefree modulus supported above $\ell$, is nonzero if and only if $p\mid\ell(1-\ell^k)B_k$. In the cyclotomic tower, we determine the local modules and their correction modulo the diagonal global Tate module, obtaining an explicit determinant formula. Their Frobenius Euler factors, combined with the Mazur–Wiles theorem, yield a ray class form of the Iwasawa main conjecture for nonexceptional odd characters. We also obtain exact class number valuation comparisons at every layer, including those before stabilization, recover Finat's formula for $\ell=2$, and deduce refined $\mu$-vanishing from Ferrero–Washington. The cyclotomic character explains the Wieferich exception, while existing Eisenstein congruence theorems connect the divisibility criteria with prime-level Ramanujan-style congruences.

By equipping the category $\mathcal{M}$ of modules over a finite dimensional algebra with the action of a finite tensor category $\mathcal{C}$, we introduce a subcategory $\mathcal{PM}$ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander–Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, $\mathcal{PM}$ is identified with the $\mathcal{C}$-module functors that are projective, and we also prove a $\mathcal{C}$-equivariant version of the Auslander(–Solberg) correspondence. As examples, we show that $\mathcal{PM}$ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga–Gorenstein algebras introduced by Geiss–Leclerc–Schröer, we show that $\mathcal{PM}$ is exactly the subcategory of $\tau$-locally free modules.

We explore properties of the Berezin range $B(C_\phi)$ of a composition operator $C_\phi$ on the Hardy space $H^2$, including constraints on its location in the complex plane $\mathbb{C}$, topological properties, shape, symmetry, and convexity. For example, we show that $B(C_\phi) \subseteq \{z\in \mathbb{C}: \Re(z) > 0 \ \text{and}\ |z| \le 1 + |\phi(0)|\}$, that $B(C_\phi)$ is closed if and only if $C_\phi$ is the identity operator, and that $B(C_\phi)$ containa all its boundary points except for $0$ if $C_\phi$ is compact. Addressing questions raised by Cowen and Felder, we show that $C_\phi$ can be a (nondegenerate) disk and characterize when $B(C_\phi)$ is convex if $\phi$ is linear.

A complete mapping of a finite group $G$ is a permutation $\theta$ of $G$ such that $x\mapsto x\theta(x)$ is also a permutation of $G$. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the question of which finite groups admit complete mappings of both parities. A useful approach to this question is to construct complete mappings stabilizing suitable subgroups, in particular Sylow $2$-subgroups. We show that, for every $n\geq2$, every prime power $q$, and every $Z\leq Z(\mathrm{SL}_n(q))$, the group $\mathrm{SL}_n(q)/Z$ admits a complete mapping stabilizing a Sylow $2$-subgroup if and only if $(n,q)\neq(2,2)$.

We study a broad class of stochastic transport problems under controlled McKean-Vlasov dynamics, allowing control of both drift and volatility, with weak terminal constraints. In contrast to the classical Schrödinger problem, the reference dynamics depend on the evolving law of the process, leading to nonlinear optimization problems on path space, while the terminal constraint is formulated through weak optimal transport, permitting a flexible treatment of distributional targets. This framework encompasses several problems of current interest, including the mean field Schrödinger problem, martingale optimal transport, and Schrödinger-Bass bridges. We establish three main results. First, we prove the existence of optimal controls under general coercivity, continuity, and convexity assumptions. Second, we derive a convex dual representation that extends weak Kantorovich duality and stochastic optimal transport duality to controlled McKean-Vlasov dynamics. Finally, for the mean field Schrödinger problem, we prove convergence, at the level of value functions, of the associated finite-particle Schrödinger bridge problems to the mean field limit, thereby identifying the nonlinear problem as the exact infinite-particle limit of finite-particle Schrödinger bridges. As a byproduct, we establish the existence of an optimal mean field Schrödinger potential, providing a variational foundation for duality in nonlinear stochastic transport.

We prove that the fundamental group of every compact graph $3$-manifold is linear over the complex numbers. As a corollary, all fundamental groups of compact $3$-manifolds are linear over the complex numbers.

We develop analytical sufficient conditions for certifying the local asymptotic stability of nonlinear autonomous systems based on Lyapunov functions derived from their finite-order Taylor polynomial approximations. This approach translates the stability analysis problem from the original nonlinear dynamics to a more computationally tractable polynomial domain. The derived conditions allow for the explicit calculation of a guaranteed inner approximation of the region of attraction ($\mathrm{ROA}$). We show how sum-of-squares (SOS) programming and grid search can be used to systematically search for polynomial Lyapunov functions that yield a large certified threshold radius and extract a certified invariant set inner approximation of the $\mathrm{ROA}$ of the nonpolynomial original system. The framework is then applied to a control tuning problem: finding reduced-effort Linear Quadratic Regulator (LQR) controllers that ensure that a given initial state of interest belongs to the certified $\mathrm{ROA}$ of the nonpolynomial closed loop system. The effectiveness of the overall methodology is demonstrated using a pendulum with nonlinear damping and a nonlinear control system.

After a basketball game, each of the $n$ players on Team A must shake hands with each of the $n$ players on Team B. What is the minimum number of (reversible, ambidextrous) gloves required so that zero germ transmission occurs? We show that the answer is $3n/2+o(n)$ and develop a general theory for prescribed handshake sets.

In this paper, we research an anisotropic elliptic equation in the entire Euclidean space R^{N} . Under the assumption that the nonlinearity of the equation has exponential or critical growth, we prove, by means of the mountain-pass theorem, that the equation admits a nontrivial positive weak solution. Moreover, applying the mountain-pass theorem and Ekeland's variational principle, we show that a perturbation of the equation possesses two distinct positive weak solutions.

We establish a strong convergence rate for the implicit midpoint method applied to the periodic stochastic Korteweg-de Vries equation with additive noise, addressing a question that has remained open since the work of Debussche and Printems. To control nonlinear error growth, we obtain exponential integrability estimates for the exact and comparison solutions using the coercivity of a classical KdV invariant. We then derive local error bounds from estimates for the comparison solutions and control error accumulation through an $L^2$ error propagation identity that isolates a martingale increment. Under $H_0^{9/2}$ regularity assumptions on the noise and the deterministic initial data, we prove that, on every fixed finite time interval, the maximum $L^2(\mathbb{T})$ error at the time points is $O(h)$ in $L^p(\Omega)$ for every $1\le p<\infty$. Notably, the exponential integrability and the convergence rate hold for every damping coefficient $\lambda\ge0$, including the undamped case $\lambda=0$. Numerical experiments confirm the predicted first-order convergence.

When characterizing graphs defined by forbidden induced subgraphs, it is often desirable to understand how a graph $G$ can be generated incrementally from a given induced subgraph $H$. In this paper, we study a number of problems within this framework. First, for several notions of connectivity, we determine all minimal induced subgraphs of $G$ that contain $H$ and attain the connectivity of $G$. Second, we establish how to grow $H$ inside $G$ while preserving twin-freeness and certain forms of connectivity. Third, we establish finite test theorems for various graph classes, proving that agreement with the recursively generated graphs for a few initial iterations forces agreement at all subsequent iterations. Consequently, such an infinite graph family can be completely determined by finitely many of its initial generations. As an application, we characterize $\{\mathbb L_1,\mathbb L_3\}$-free graphs, where $\mathbb L_1$ is the claw and $\mathbb L_3$ is another forbidden graph for line graphs. Our structural results reduce this problem to a finite verification, which is completed through computer-assisted computation.

Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $\Phi\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circ\Phi=L\,{\mathfrak s}$. Pulling back Roth's theorem gives the optimal exponent $2L$ for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent $q_0+q_1$. We formulate this as a conjecture and prove an $S$-adic radical form under the $abc$ conjecture. For ${\mathfrak q}=(2,3)$ the resulting inequality at the target $[1:1]$ is the coprime Hall-type inequality, and Danilov–Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted $abc$ formulation is introduced; in signature $(3,2)$ finitely many coefficient pairs are equivalent to $abc$, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.

In this paper, we consider the Cauchy problem for the three-dimensional (3D) compressible micropolar equations with density-dependent viscosities. By classical $L^p$ estimates to elliptic system, defining suitable energy functions, after the elaborate index analysis, we establish the global existence of strong solutions to the system with all viscosities depending density as long as the initial density is large enough. By product, We also prove the existence of global large strong solutions with only large initial shear viscosity $\mu_1(\rho)$ and angular viscosity $\mu_2(\rho)$, which consists with the research work \cite{ZZ2014} of 3D compressible Navier-Stokes equations with constant viscosities and improve the recent result of \cite{HLZ} in some extent where they requiring both initial shear viscosity and bulk viscosity to be large.

Many systems in engineering applications may experience complex multi-dimensional degradation processes involving random jumps. Analyzing the reliability of such degradation processes presents significant challenges. Previously developed models have utilized multi-dimensional Lévy subordinators and Lévy copulas to describe these processes, with reliability functions expressed through characteristic functions and Laplace transforms (Shu et al., 2015, 2016, Shi et al., 2020, Shi and Feng, 2024). However, these models are restricted to stationary linear paths and homogeneous Lévy subordinators. In real-world scenarios, increments in a degradation process are often time-dependent and nonhomogeneous. To address this limitation of previous studies, this work extends existing degradation models by introducing nonhomogeneous Lévy subordinators, which are designed to capture time-varying complexities inherent in degradation processes. Unlike conventional degradation models that often assume a constant rate of jumps, nonhomogeneous Lévy subordinators aim to provide a more flexible representation of degradation behaviors that evolve under varying external conditions and internal system dynamics. Additionally, the joint behaviors of multiple degradation processes are characterized using Lévy copulas, enabling a more comprehensive analysis of system reliability. Numerical examples are provided to illustrate the applicability of these models in lifetime analysis.

Inference-time alignment steers diffusion models toward downstream objectives without costly fine-tuning, but existing methods typically optimize expected cost (or reward) without explicitly controlling tail risk. In this work, we propose a risk-averse framework for inference-time alignment of diffusion models that balances expected cost and Conditional Value-at-Risk (CVaR) under trajectory-level Kullback-Leibler regularization. We characterize the optimal path distribution and reverse transitions as functions of the CVaR threshold, reducing the alignment problem to a one-dimensional optimization problem over the threshold. We show that this one-dimensional optimization problem can be nonconvex in general, establish sufficient conditions for convexity, and introduce quadratic regularization to enable efficient derivative-free ternary search. Under suitable assumptions, we provide finite-sample guarantees for threshold optimization and value estimation, and combine them into a terminal total-variation bound relative to the regularized target under exact resampling. Experiments on synthetic data and image generation using Stable Diffusion demonstrate improved tail performance over risk-neutral baselines while maintaining competitive average cost.

Let $(M,g)$ be a compact Riemann surface with smooth boundary $\partial M$. Let $K(x)>0$, $h(x)> 0$ or $h(x)\equiv 0$ be two smooth functions on $M$ and $\partial M$ respectively. In this paper, we consider the following double mean field type equation with a nonlinear Neumann boundary condition related to a prescribed curvature problem \begin{equation}\label{equation:01} \begin{cases} \displaystyle -\Delta_g u+\frac{4\pi}{|M|} =2\rho\frac{Ke^u}{\int_MKe^u\,dv_g}, & \text{in }M,\\[1.2em] \displaystyle \partial_\nu u =2\rho'\frac{he^{u/2}}{\int_{\partial M}he^{u/2}\,ds_g}, & \text{on }\partial M, \end{cases} \end{equation} where $\rho> 0,\ \rho'\geq 0$ and $\rho+\rho'=2\pi$, $\nu$ is the outward unit normal vector to $\partial M$. When $h\equiv 0$, the second equation changes to $\partial_\nu u=0$ on $\partial M$. By introducing a boundary integration constant, we obtain a sufficient condition under which \eqref{equation:01} admits a solution. As a geometric application, we solve the prescribed curvature equation for both the Gauss curvature and the geodesic curvature in the case that $M$ is topologically a disk. When $(M,g)$ is the standard hemisphere $\mathbb S^2_+$, let $p\in \partial \mathbb S^2_+$ be a maximum point of $$ \Phi(q)=h(q)+\sqrt{h(q)^2+K(q)}, \qquad q\in\partial \mathbb S^2_+. $$ Suppose \begin{equation*} K_\nu(p)+2\Phi(p)(-\Delta_{\partial \mathbb S^2_+})^{1/2}h(p)<0. \end{equation*} Then the prescribed curvature problem \begin{align*} -\Delta_g u+\frac{4\pi}{|M|}=2Ke^u, \text{in }\mathbb S^2_+,\ \ \partial_\nu u=2he^{u/2}, \text{on }\partial \mathbb S^2_+. \end{align*} admits a smooth solution.

Ferrers diagram rank-metric codes are essential components of the multilevel construction of constant dimension codes for random network coding. This paper aims to construct optimal Ferrers diagram rank-metric codes, focusing on the diagram \([3,3,3,5]\) with rank distance \(3\). We construct an optimal \([[3,3,3,5],6,3]_q\) Ferrers diagram rank-metric code for every prime power \(q\), thereby answering an open problem posed by Etzion et al. in 2016~\cite{Etzion2016}. The construction first builds a \([3\times4,4,3]_q\) MRD code and then proves that the Delsarte dual of its puncturing on the last column contains a \([3\times3,3,3]_q\) MRD code, which comes from semifield codes. We further construct optimal Ferrers diagram rank-metric codes for two infinite families of Ferrers diagrams: the first relies on the \([[3,3,3,5],6,3]_q\) construction and holds for every prime power \(q\), while the second is binary and obtained directly by the puncturing-inclusion construction.

In this paper, we obtain an $\varepsilon$-regularity criterion at one scale for 3D Navier-Stokes equations using a coarse-to-refined estimate argument based on mean-fluctuation decomposition. Making use of this $\varepsilon$-regularity criterion, we improve the upper bound on the box dimension of the interior singular set to $15/13$.

We construct a structure-preserving effective linear approximation of the quantum Boltzmann collision operator for Bose and Fermi gases in the normal phase. For variable hard-sphere kernels, the resulting Fourier spectral evaluation costs $O(N_v^4\log N_v)$, with $N_v$ velocity nodes per direction and no angular quadrature factor. The continuous operator conserves mass, momentum and energy and satisfies a linearized $H$-theorem. We also use the operator at local quantum equilibrium as an approximate collision Jacobian in a modified Newton solver that retains the full nonlinear residual. Numerical tests show smaller errors than the quantum BGK model in homogeneous relaxation and fewer full collision evaluations for steady Couette and Fourier flows than the same Newton solver using a quantum BGK Jacobian.

Artificial dissipation improves the robustness of high-order methods for under-resolved flows, but can compromise accuracy and suppress physically relevant structures if its strength and spatial distribution are not appropriately controlled. We present a sensor-activated nonlinear artificial-dissipation framework, inspired by the residual correction method, for continuous and discontinuous summation-by-parts discretizations. In contrast to many existing spectral-element dissipation operators that either reduce the formal order of accuracy or target only the highest polynomial modes, the proposed framework permits design- and higher-order dissipation with explicit control over its magnitude, spatial distribution, activation, and convergence rate under mesh refinement. This modular construction also recovers several existing methods through appropriate choices of its components. The dissipation preserves discrete conservation and semidiscrete entropy stability whenever the base method satisfies these properties. It is dimensionally consistent, element local, and discretization agnostic, as the added term is constructed independently of the base spatial residual. For smooth solutions under suitable mesh assumptions, we establish parameter choices that retain the formal accuracy of the base discretization. Numerical experiments with summation-by-parts discretizations of the Burgers and Euler equations demonstrate substantially improved robustness for under-resolved flows and flows with shock waves while retaining high-order accuracy for smooth solutions, using the same parameter choices across several test problems. Overall, these results show that targeted dissipation can improve the robustness of high-order methods without excessive damping of resolved flow structures.

We develop an effective method for determining the real Waring rank and the real border rank of biharmonic binary forms. By means of real apolarity, we convert both of these problems into membership questions for regions bounded by a hypocycloid and an epicycloid. We further provide a computable algebraic criterion for the relevant memberships, which involves only the signs of certain real-valued resultants.

In this paper, we study the active vibration control of an elastic beam using reinforcement learning (RL). The physical system is modeled by a modified Timoshenko-Ehrenfest type linear partial differential equation with simply supported boundary conditions. The dynamics are approximated by a reduced-order model (ROM) obtained through eigenvalue expansion and truncated Galerkin projection. Control is applied through boundary actuation. The resulting finite-dimensional system is formulated as a continuous-time Markov decision process and solved numerically using the Proximal Policy Optimization (PPO) algorithm. All modeling steps, from physical formulation to temporal discretization and control design, are explicitly described. A systematic comparison is performed between the controller learned via PPO and the classical linear quadratic (LQR) regulator. The analysis highlights the strengths and limitations of each approach, both from a theoretical and practical point of view, with particular emphasis on energy dissipation mechanisms and actuator constraints.

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), its $\ell$-degree Turán density $\gamma^{(k)}_\ell(\mathcal{F})$ is the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all integers $k>\ell>2$ and every $\alpha\in[0,1)$, there exists an infinite family $\mathcal{F}$ of $k$-graphs such that $\gamma^{(k)}_\ell(\mathcal{F})=\alpha$, while every finite subfamily $\mathcal{F}'\subset \mathcal{F}$ satisfies $\gamma^{(k)}_\ell(\mathcal{F}')>\alpha$. This confirms the conjecture of Mubayi and Zhao on codegree Turán densities for every $k > 3$. As an immediate consequence, the set $\Gamma^{k,\ell}_{\infty}$ of $\ell$-degree Turán densities of nonempty $k$-graph families is $[0,1)$ whenever $k>\ell>2$, which answers a question of Lo and Markström from 2014.

An efficient shallow Ritz method was developed in [8, 9] for one-dimensional linear diffusion and diffusion-reaction problems. In this paper, the method is extended to a nonlinear diffusion-reaction equation with a cubic reaction term: (1) a basic error estimate of the shallow Ritz discretization is established and (2) a reduced block Newton (rBN) method for solving the resulting non-convex optimization problem is developed. Again, the computational cost of the rBN method is $\mathcal{O}(n)$ for one-dimensional problems, where $n$ is the number of neurons. Combining a convex splitting scheme for time discretization with the efficient shallow Ritz method, we develop a time-stepping neural network method for solving the Allen-Cahn equation. The unconditional energy stability of the time discretization and an error estimate in the $L^2$ norm are established. Numerical experiments demonstrate that the neural network approach yields accurate solutions for singularly perturbed nonlinear diffusion-reaction equations and the Allen-Cahn equation. The method effectively resolves interior and boundary layers without oscillations and significantly outperforms the finite element method on fixed uniform meshes.

A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class are linearly ordered by inclusion. We study the generalized spectral determination of chain graphs of adjacency rank $4$. We first show that generalized cospectral chain graphs have the same number of isolated vertices, reducing the problem to the connected case. For connected rank-$4$ chain graphs, we derive four explicit equations in the cell parameters that are necessary and sufficient for generalized cospectrality. These equations reduce to a quadratic equation in a single parameter. As a consequence, every connected rank-$4$ chain graph has at most one generalized cospectral mate, and the possible mate can be reconstructed explicitly. We further obtain a simple sufficient condition for a rank-$4$ chain graph to be determined by its generalized spectrum (DGS) and, in particular, prove that every double star graph is DGS. Finally, we construct an explicit infinite family of non-double-star rank-$4$ chain graphs that are DGS but not determined by their adjacency spectra, with both the graphs and their complements being connected.

We investigate the relationships between fragments of SRP and diagonal reflection principles. It turns out that the subcomplete fragment of SRP, augmented with CH, implies that the nonstationary ideal on $\omega_1$ is not saturated, but that Larson's diagonal reflection principle $OSR_{\omega_2}$ holds. In contrast, it was known that the full principle SRP implies the saturation of the nonstationary ideal on $\omega_1$, but it does not imply $OSR_{\omega_2}$. We find several instances of this phenomenon: pairing (variations of) the subcomplete fragment of SRP with stronger failures of the nonstationary ideal on $\omega_1$ to be saturated allows us to conclude more instances of diagonal reflection principles, while boosting (the corresponding variations of) the full principle SRP with strong forms of the saturation of the nonstationary ideal allows us to produce models where these principles still hold, yet weaker versions of diagonal reflection fail.

We completely classify aperiodic automorphisms of the Razak–Jacelon algebra $\mathcal{W}$ up to outer conjugacy. As a technical ingredient, based on the works of Connes, Herman–Ocneanu, Kishimoto, and Sato, we introduce a notion of weak stability for automorphisms of simple separable monotracial C$^*$-algebras. We also consider automorphisms of UHF algebras of infinite type and the Jiang–Su algebra. In particular, we classify aperiodic “UHF-absorbing” automorphisms of UHF algebras of infinite type up to outer conjugacy.

Safety verification of a Markov decision process (MDP) asks whether the probabilities of reaching several failure states jointly lie in a convex safe region under every scheduler. Watanabe and Unno (2026) prove safety by refining an outer approximation of the failure probabilities, where each query bounds a nonnegative weighted sum of the probabilities. Their rule selects weights by the largest Euclidean violation. We present a unified framework to measure violation in an explicitly chosen error unit: directions are scored by the support-function gap between the outer approximation and the safe region, normalized by the support function of the unit. We show that the maximal score, when positive, equals the smallest enlargement of the safe region in multiples of that unit that contains the outer approximation. The Euclidean unit recovers the earlier rule; two new rules are generated by the unit cube and the safe region itself. We experimentally evaluate how the choice of measurement rule affects the performance of safety verification for MDPs.

We study the stability of the variable coefficient wave equation $u_{tt}-u_{xx}-(1+x^{2})u_{yy}+\chi_{\mathbb G}u_{t}=0$ on the rectangle $(-1,1)\times(0,\pi)$ with Dirichlet boundary conditions, where the viscous damping acts on a vertical strip $\mathbb G=(\eta-\delta,\eta+\delta)\times(0,\pi)$. We establish a precise relationship between the location of the damping region and the rate of energy decay. We prove that the associated semigroup is exponentially stable whenever the closure of the damping strip meets the line $x=0$, i.e., $|\eta|\leq\delta$, despite the existence of rays that never enter the damped region. When the damping region is strictly bounded away from $x=0$, the semigroup decays at the sharp rate $(\log(2+t))^{-1}$. The proofs combine separation of variables, one-dimensional resolvent estimates, a reflection identity for the harmonic oscillator, and Gaussian quasimodes. We then consider the spectral Galerkin approximation based on the Dirichlet eigenfunctions of the constant coefficient Laplacian and show that, although the sequence of semigroups resulting from the approximation converges to the continuous one, it fails to be uniformly exponentially stable.