Papers

Chimera states, characterized by the coexistence of coherent and incoherent dynamics in networks of coupled oscillators, are among the most intriguing collective phenomena in nonlinear systems and are strongly shaped by the underlying topology and initial conditions. Their analysis in large-scale networks remains computationally demanding and motivates the development of coarse-graining strategies that reduce network size while retaining the essential dynamical features of chimera behavior. In this Letter, we investigate whether equitable-partition-based quotient graphs can preserve chimera patterns under substantial reductions of network size. Using nonlocally coupled FitzHugh-Nagumo (FHN) oscillators, we exploit the fact that, for an equitable partition, the full-network dynamics restricted to the associated cluster-synchronous subspace coincide exactly with the quotient dynamics. We ask whether this structural exactness is sufficient to preserve chimera signatures after the network size reduction. Our results show that this is not necessarily the case: agreement between quotient and full-network chimera dynamics depends on the equitable partition, and exact quotient dynamics do not necessarily guarantee the transverse stability of the corresponding full-network trajectory. Importantly, when the cluster-synchronous solution becomes transversely unstable, the full network may depart from the exact quotient trajectory while remaining in a macroscopic chimera regime. Therefore, structural exactness and dynamical preservation are distinct requirements, and transverse stability provides a complementary criterion for assessing quotient-based coarse-graining of chimera states.

In recent years a number of finite element methods have been developed for the surface Stokes equations. Constructing standard conforming surface finite element methods for the velocity-pressure formulation of this problem would require simultaneously enforcing $H^1$ conformity and tangentiality of the velocity field, which is not possible on the generalized polyhedral approximating surfaces on which surface FEM are typically posed. Thus methods for this problem weakly enforce either tangentiality or $H^1$ conformity of the velocity field. In this paper we provide an error analysis for an $H({\rm div})$-conforming method based on classical Brezzi-Douglas-Marini elements. In this method continuity of the velocity field is weakly enforced using an interior penalty-type formulation. Advantages of this formulation include exact enforcement of the incompressibility constraint and its construction based on a standard mixed finite element pair. Error analysis of this method in previous work was only valid for lowest-order elements and did not account for “geometric errors”, i.e., variational crimes due to approximation of the continuous surface on which the PDE is posed by an approximating surface on which the finite element method is posed. Here we provide a full error analysis in a natural discontinuous Galerkin energy norm, with geometric errors accounted for. Optimal estimates for the $L_2$ velocity error are also presented. In the case of lowest-order (polyhedral) surface approximations, our proof of optimal $L_2$ estimates requires a stricter than usual condition on the placement of mesh nodes relative to the surface. Numerical experiments are used to illustrate our results.

We study Bayesian linear bandits with an isotropic Gaussian parameter, independent Gaussian candidate arms, and Gaussian reward noise when the horizon is proportional to the dimension. The normalized posterior uncertainty has an explicit limit that is uniform over all causal policies. Gaussian posterior identities then determine the limiting parameter overlaps without an assumed closure of the adaptive recursion. These results yield exact regret curves for Thompson sampling, posterior-mean greedy selection, and a family of policies that scale the posterior sampling covariance. The normalized realized cumulative regret converges in L1, uniformly on compact proportional-time intervals. A policy-uniform lower bound identifies the limiting optimal Bayes regret and proves that posterior-mean greedy selection attains it. Thompson sampling incurs a strictly larger leading regret; its instantaneous regret ratio relative to greedy selection lies between one and two and approaches two at long proportional horizons. Closed-form cumulative curves also identify a different comparison in the vanishing-noise limit. Finally, the instantaneous regret converges to a nondegenerate Gaussian decision-loss distribution, rather than to its mean. The analysis separates the amount of information acquired by a bandit policy from the quality of the decisions made using that information.

This paper proves generic structural stability for Riemann solutions to $n \times n$ systems of hyperbolic conservation laws in one spatial dimension. Under assumptions of strict hyperbolicity, genuine nonlinearity, and a regular manifold hypothesis on the Rankine-Hugoniot map, we show that for almost every pair of left and right states, any $n$-wave Riemann solution consisting of Lax-admissible shocks and rarefactions is structurally stable under perturbations of the left state, the right state, and the flux function in the $C^2$ topology. The central new idea is sequential transversality, which chains the $n$ waves through intermediate states and transports their tangent contributions to a common reference point via pushforward maps, reducing the structural stability condition to the invertibility of an $n \times n$ transversality matrix. We apply the results to the $p$-system, polydisperse particle-laden thin films, and machine-learned flux approximations.

We study the two-parameter exponential family associated with the complete-monotonicity problem of Alzer and Berg. Strict necessary parameter bounds place every possible Bernstein function in the domain of a regularized Laplace representation. A quantitative positivity-transfer inequality then yields a linear sufficient condition with the largest possible universal coefficient, expressed in terms of the exact one-parameter critical exponent. To describe the whole admissible region, we derive convolution identities for parameter derivatives and prove that increasing either normalized parameter destroys positivity at every zero of a nonnegative density. Combined with uniform tail estimates, this excludes gaps in the admissible parameter intervals and gives a continuous, strictly monotone optimal boundary. Global nonnegativity of the density and contact with zero characterize that boundary; its inverse determines the complete admissible interval for the second parameter. The characterization is implicit and requires neither uniqueness nor nondegeneracy of the contact points. Published numerical approximations of the one-parameter exponent are distinguished from the exact results and from the finite rational certificate used in the proofs. We also derive a variational formula and a rigorous framework for validated numerical enclosure of the boundary, and formulate a boundary-contact conjecture asserting uniqueness and quadratic contact at every interior boundary point.

Let $\rho(n,d)$ denote the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $d$. A graph attaining this maximum is called an extremal graph. Liu [J. Combin. Theory Ser. B, 2024] determined the extremal graphs for $d=3,4$ and formulated two conjectures for general $d$. For each fixed odd integer $d\ge3$, the conjectures assert that: (1) $\displaystyle\lim_{n\to\infty}n^2\bigl(d-\rho(n,d)\bigr) =(d-1)\pi^2/4$. (2) For all sufficiently large $n$, the degree sequence of every extremal graph is $(d,\ldots,d,d-1)$ for odd $n$ and $(d,\ldots,d,1)$ for even $n$. We prove the first conjecture for every fixed odd $d\ge3$ and, more precisely, obtain the asymptotic expansion \[ \rho(n,d) =d-\frac{(d-1)\pi^2}{4n^2} +\frac{(d-1)^2\pi^2}{4n^3} +O_d(n^{-4}) \qquad(n\to\infty). \] We further prove the second conjecture for every fixed odd $d\ge3$.

We study square-complementary graphs $G$ (satisfying $G^2 \cong \overline{G}$). We show that in such graphs, no two vertices have comparable closed neighborhoods. This implies that nontrivial square-complementary graphs have no simplicial vertices and are not chordal, thus solving two open problems posed in Discrete Mathematics 327 (2014) 62-75. We also show that no nontrivial square-complementary graph is distance-hereditary.

Equivalence groups associated with the field equations of arbitrary motions of inhomogeneous hyperelastic solids are investigated. The explicit forms of the infinitesimal generators, namely the components of the isovector field associated with the equivalence groups are derived. Additional constraints on the isovector components are established to ensure that the transformed equations are supported by a strain energy function. Possible mappings between the equations of materials with different properties are examined. Not only the transformations between homogeneous and inhomogeneous hyperelasticity equations, but also the transformations between the equations and strain energy functions of isotropic and anisotropic materials are investigated. The results are illustrated with several examples.

The random hopping Hamiltonian is a toy model for a disorder-driven topological phase transitions in one-dimensional chiral Hamiltonians. It has a vanishing Lyapunov exponent and a Dyson peak in the density of states at zero energy. This work shows how the quantum dynamics leads to a logarithmic growth of the moments of the position operator, similar as in classical Sinai diffusion. The mechanism at the origin of this phenomenon is that the eigenfunction of the smallest eigenvalue is well-approximated by the exponential of a classical random walk, leading to two essentially independent subexponential localization centers.

In this paper, we obtain an exact sequence connecting $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ to $H_2(\rm{SL}_2(\mathcal{O}_{K,T}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ with the second $K$-group $K_2(\mathcal{O}_{K,S})$, for $S$ large enough. This leads to a description of the rank and size of the torsion of $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$. As an application, we propose a homological version of the Birch-Tate formula (conjecture) under these assumptions.

Longitudinal spacing models describe the speed–spacing states that vehicles maintain, while longitudinal controllers regulate vehicle motion through spacing and speed errors. Less explicit is how one spacing state should be connected to another through a finite, physically meaningful vehicle transition. This paper develops such a connection through r-Safety and Adaptive Kinematic Smoothing (AKS). The r-Safety relation represents speed-dependent spacing through a braking-normalized reserve, whose change, together with leader motion, determines the follower displacement required during a transition. AKS then converts this displacement into a low-dimensional analytical trajectory with consistent position, speed, spacing, and acceleration. Experiments using controlled platoons and NGSIM trajectories identify r-Safety values of 0.307 and 0.249–0.281, respectively, and show that AKS can both represent observed transitions and generate prospective references under prescribed endpoint states, transition horizons, and leader motion. In same-controller ablations, adding AKS reduces integrated acceleration effort by 21–47% and integrated squared jerk by 73–81% across the controlled and naturalistic datasets. FHWA ACC/CACC trajectories further provide an empirical illustration of finite-horizon execution under automated following, showing how spacing adjustment can continue beyond the prescribed speed transition. The framework therefore provides a direct path from a speed–spacing relation to an analytical finite transition and its use as a longitudinal control reference.

Let $\pi=\pi_1\cdots \pi_n$ be a permutation. We say that $\pi$ is $run$-$sorted$ if $\pi_1=1$ and the entries immediately following the descent positions of $\pi$ form an increasing sequence. Let $\mathcal{R}_n$ denote the set of run-sorted permutations of length $n$, which has cardinality given by the Bell number $B_{n-1}$ for all $n \geq 1$. In this paper, we consider the joint distribution $A_n(q,u)$ on $\mathcal{R}_n$ for the parameters tracking the numbers of inversions and runs leading to a new polynomial generalization of the Bell numbers. Among our results, we find a general recurrence for $A_n(q,u)$, from which one may derive explicit formulas for the total numbers of inversions or runs in all the members of $\mathcal{R}_n$ as well as for the sign-balance on $\mathcal{R}_n$ of either parameter. A simple expression for the Eulerian generating function for $A_n(q,1)$ may be found upon making use of Gessel's $q$-exponential formula which can be extended to general $u$. Finally, a formula is found by a direct argument for the maximum number of inversions within a member of $\mathcal{R}_n$.

We study combinatorial properties of translations of meager and closed sets of measure zero. We discuss constellations of Borel conjectures for related classes of small sets and show that there are no uncountable null-additive sets in the Miller model. We also study cardinal invariants of those classes. In particular, we investigate the cardinal invariants of $\sigma$-ideals $\mathcal{H}_{F}$ related to meager-additive sets. We obtain a new characterization of additivity of meager ideal and answer some questions of Cardona. We also show that $\mathrm{non}(\mathcal{E}^{*})$, that is equal to the translation version of the covering number of the ideal $\mathcal{E}$, is close to the cardinal invariant related to the Laver property. This strengthens a result of Bartoszyński and Judah and a result of Elekes and Steprāns. Finally, we show that every $\mathcal{E}$-Luzin set is in $\mathcal{E}^{*}$.

We study magnetic flows on closed Riemannian surfaces of genus at least two. Let $(M,g)$ be a surface without focal points and let $\Omega$ be a magnetic field. We prove that, above the Mañé critical value, the absence of conjugate points imposes geometric restrictions on the magnetic flow. First, we obtain a bound for the integral of the Lorentz force in terms of the geodesic curvature of the horocycles of the underlying Riemannian metric. As a consequence, a lower bound on the Gaussian curvature yields an explicit bound in terms of the area of the surface. The proofs combine the global geometry of magnetic geodesics with Liouville's formula for the geodesic curvature in the orthogonal coordinates given by the level sets and geodesics of a Busemann function on the universal cover.

We study a one-dimensional self-consistent transfer operator generated by a repulsive mean-field coupling. The interaction admits a natural interpretation as a myopic relocation dynamics balancing congestion and uniformly distributed resources. Using the quantile representation of probability measures, combined with self-consistent transfer operators methods, we prove the existence and uniqueness of a physically relevant invariant measure for the system and characterize it explicitly through a nonlinear fixed-point equation. We then show that the corresponding finite-particle equilibrium converges to the continuum equilibrium with optimal order in Wasserstein distance and establish a quantitative thermodynamic limit.

Pattern avoidance is a central topic in the study of permutations. Parking functions provide a natural setting in which to ask analogous questions. In this paper, we study parking functions whose parking permutations avoid the patterns $1234$ and $2134$, obtaining an analog of the Schröder numbers for parking functions. To enumerate these objects, we develop an enumeration scheme that runs in polynomial time and adapt the prefix-based framework introduced by Zeilberger in 1998, and later improved by Vatter, by instead using “suffixes.”

For nonsquare $d\ge 2$, we construct fixed straight-line programs for the least non-trivial solution $(X_1,Y_1)$ of $x^2-dy^2=1$, using addition, truncated subtraction, multiplication, integer division, exponentiation, and remainder. A geometric-sum identity recovers $(X_1,Y_1)$ from the coordinate sums of an initial segment of Pell solutions, without knowing how many solutions were summed. Weighted binary encodings make these sums accessible to arithmetic computation. Counting operations with reuse of computed values, one program uses $94$ operations, with intermediate bit lengths bounded by $2^{2^{2^{O(d)}}}$. Four additional operations give a $98$-operation program with the bound $2^{2^{O(d)}}$. Hua's bound gives corresponding programs with $103$ and $107$ operations and replaces $O(d)$ by $O(\sqrt d\log d)$ in these size bounds. Once the fundamental solution is known, generating-function formulas compute the $n$-th positive solution in $21$ further operations by extracting its coordinates as base-$b$ digits. We prove that $2X_1(X_1+1)-1$ is the least integer base for which both formulas are defined and correct for every $n\ge1$.

We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero–one testing law, and the subchannel asymptotic equipartition property.

The ${L}_\Lambda w_{1+\infty}$ algebra is the cosmological-constant deformation of the celestial $Lw_{1+\infty}$ symmetry algebra of gravity, providing an infinite-dimensional extension of the three-dimensional conformal algebra at the AdS$_4$ boundary. We construct the Hamiltonian charges associated with these symmetries from boundary ambitwistor-space data, using LeBrun's Heaven on Earth correspondence, and translate them to spacetime via the Penrose transform. We relate the resulting expressions to light-ray operators in the boundary CFT$_3$, including ANEC operators, their conformal descendants, and their commutators.

In this time of uncertainty and anxiety about artificial intelligence (AI) in mathematics, it is more important than ever to reflect on how we can prepare the next generation of professional research mathematicians for success. In this article, we report, from the front lines, on how AI is affecting the long-running summer math research program at the University of Minnesota Duluth. We do not have answers to the big questions about how AI will shape the future of professional mathematics. What we can offer instead is a collection of practical observations about the role of AI in the 2026 Duluth program: how it was useful and where it presented potential pitfalls; what surprised us and how we adapted; and what went well and what we wished we had done differently.

We study global convergence guarantees of third-order Langevin dynamics for non-convex optimization via simulated annealing with fixed friction and decreasing noise. An explicit three-block distorted entropy transfers dissipation from the noisy auxiliary variable to the full state. Under dissipativity, regularity, and low-temperature functional-inequality assumptions, logarithmic cooling drives the objective values to the global minimum in probability at the barrier-controlled kinetic rate. For the exact-force-integral and midpoint three-stage discretizations, polynomially decreasing steps preserve this rate on the physical time scale. The cubic local endpoint estimate gives a less restrictive sufficient step-size condition than the available frozen-force kinetic result. A comparison with the one-gradient UBU integrator shows how its centered stochastic local error leads, under the same strong-coupling analysis, to a smaller sufficient iteration exponent. Numerical experiments are conducted to illustrate our theory. For a double well objective, third-order Langevin terminal-success point estimates are higher than UBU at both a common horizon and an equal gradient budget. For a high-dimensional nonconvex neural-network objective using synthetic data, independently tuned UBU and third-order Langevin schemes both outperform overdamped Langevin dynamics; the third-order Langevin point estimate is higher. For the same neural-network objective on real data, we show the same point-estimate ordering for best-basin probability and post-quench test accuracy. Numerical code and associated experiment results are publicly available at https://github.com/gagawjbytw/simulated-annealing-third-order-langevin.

We confirm Vaught's conjecture for each partial order ${\mathbb X}=\sum _{{\mathbb I}}\prod _{j<m_i}{\mathbb X}_{i,j}$ from the closure $\langle \langle {\mathcal C}^{\rm rt}\rangle _\Pi\rangle _\Sigma$ of the class ${\mathcal C}^{\rm rt}$ of rooted trees under finite direct products and lexicographic sums. In addition, (a) ${\mathcal T}:=\mathop{\rm Th}\nolimits ({\mathbb X})$ is $\omega $-categorical iff all the theories ${\mathcal T}_{i,j}:=\mathop{\rm Th}\nolimits ({\mathbb X}_{i,j})$ are $\omega $-categorical; (b) ${\mathcal T}$ satisfies VC$^\sharp$ (that is, $I({\mathcal T})\in \{1,{\mathfrak{c}}\}$), if VC$^\sharp$ holds for all ${\mathcal T}_{i,j}$. If ${\mathbb X}\in \langle \langle {\mathcal C}^{\rm rt}_{\rm fa}\rangle _\Pi\rangle _\Sigma$, where ${\mathcal C}^{\rm rt}_{\rm fa}$ is the class of finitely axiomatizable rooted trees, then (c) ${\mathcal T}$ is finitely axiomatizable and (d) ${\mathbb Y}\in \mathop{\rm Mod}\nolimits ({\mathcal T})$ iff ${\mathbb Y} \cong \sum _{{\mathbb I}}\prod _{j<m_i}{\mathbb Y}_{i,j}$, where ${\mathbb Y}_{i,j}\in \mathop{\rm Mod}\nolimits ({\mathcal T}_{i,j})$, for all indices. As a by-product we prove that (e) For each $n$ the class ${\mathcal C} _n$ of partial orders isomorphic to a direct product of $n$ rooted trees of size $>1$ is first-order definable; (f) Each $\omega $-categorical partial order from the class $\langle \langle {\mathcal C}^{\rm rt}_{\rm fb}\rangle _\Pi\rangle _\Sigma$, where ${\mathcal C}^{\rm rt}_{\rm fb}$ is the class of finite-branching rooted trees, is finitely axiomatizable. Statement (f) is related to the results of Rosenstein (for the class of linear orders) and Schmerl (for the class of partial orders of finite width).

These are extended notes based on a series of lectures given by the author at the Institut de Mathematiques de Toulouse in June 2025. The goal is to present classical problems as well as the most recent developments in spectral geometry of the Laplace operator in Euclidean domains, subject to Dirichlet, Neumann and Robin boundary conditions. Among the topics covered, there are embedded eigenvalues in unbounded domains, spectral isoperimetric inequalities, nodal-line and hot-spots conjectures, inverse problems of hearing the shape of a drum and geometrically induced eigenvalues and Hardy-type inequalities in curved tubes.

We prove that topological isotopy implies smooth isotopy for neat surfaces with common nonempty boundary in a smooth orientable 4-manifold, in the presence of a geometric dual in the boundary. Together with our earlier work on surface smoothing, this shows that the natural map from smooth isotopy classes to topological isotopy classes of such surfaces with prescribed boundary is bijective.

Rooted phylogenetic networks are widely used to represent the evolution of species that have undergone reticulate processes. However, these networks can be highly non-planar, making them more difficult to visualise and interpret than evolutionary trees. In this paper, we investigate planarity properties of galled networks, an important subclass of phylogenetic networks. We show that all planar galled networks are necessarily upward planar. Furthermore, by leveraging recent results on planar phylogenetic networks, we provide three characterisations for each of the outerplanar and terminal planar galled network classes in terms of forbidden vertex configurations, forbidden directed subgraphs, and forbidden structures in their associated underlying undirected graphs. These results contribute to a deeper understanding of the structural properties of galled networks and may inform future methods for their construction and visualisation.

This paper is devoted to studying permutation binomial $F_{r,a}(X)=X^r(X^{q-1}+a)\in\mathbb F_{q^e}[X]$ with $a\in\mathbb F_{q^e}^*$. We present a complete characterization for $F_{r,a}$ to be a permutation of $\mathbb{F}_{q^e}$. This yields a complete proof of the conjecture proposed by Masuda–Rubio–Santiago \cite{masuda2022permutation}. More precisely, we show that \(F_{r,a}\) permutes \(\mathbb F_{q^e}\) if and only if $(-a)^{(q^e-1)/(q-1)}\ne1, \gcd(r,q-1)=1$, and there exists \(1\le h<e\) with \(\gcd(h,e)=1\) such that $r(q^h-1)\equiv q-1\pmod{q^e-1}.$

The Erdős-Gyárfás conjecture posits that every graph with minimum degree at least three contains a cycle of length some power of two. We prove a few simple structural properties for any minimal counter-example to this conjecture. In particular, the fraction of its vertices of degree three must be greater than $2/3$, thus improving on the prior bound of $4/7$ (Carr, 2026). Furthermore, it is either biconnected or the $1$-clique-sum of two biconnected graphs. By exploiting some of these properties, we were able to verify the conjecture for every graph of order at most $40$, every bipartite graph of order at most $66$, and every cubic graph of order at most $48$.

Over three centuries ago our Enlightenment forefathers brought forth in Europe–and later on this continent–a new approach to knowledge production and dissemination, conceived in distrust of authority, and dedicated to the proposition that truth is accessible to human reason. Mathematics, the Enlightenment's firstborn, was its existence proof–knowledge that compels assent without appeal to authority–because the only way to make a theorem certain was to show it to a human mind. Now we are engaged in an existential struggle testing whether mathematics, or any discipline so conceived and so dedicated, can long endure. We are met in the wake of the OpenAI press release: 166 pages, 616,000 lines of Lean, produced in 88 hours by some 10,000 software agents, and read in full–at the time of this Siren Call–by no human being. A proof has historically been two things: a certificate that a theorem is true, and a demonstration that lets another person see why. OpenAI delivered the first without the second. But, in a larger sense, what matters is not what the Silicon Leviathan (OpenAI and its peer competitors) produces but rather what we accept. By accepting the certificate without the demonstration, we adopt the Hobbesian bargain: authority, not truth, makes the law. The age that this augurs is–in Panofsky's phrase–a Middle Ages in reverse–its authority a subhuman superintelligence. The great task before us is to frame a covenant under which what has become optional–understanding–remains required: that nothing counts as mathematics until a human being has understood it–and can show the next person why. Whether we will safeguard and keep our sovereign inheritance–decided at the moment of acceptance–which is not yet past–will determine if this Aufklärungsdämmerung spells the Enlightenment's end.

Massive Internet of Things (IoT) networks operate with short packets whose reliability is limited by finite block- length (FBL) effects, while remote deployments increasingly rely on Low Earth Orbit (LEO) satellites for backhaul connectivity that is sensitive to atmospheric attenuation. In this paper, we pro- pose a unified end-to-end framework for satellite-assisted massive IoT networks that jointly models uplink FBL random access, sensing assisted satellite backhaul, and worst user broadcast downlink transmission. Uplink reliability is characterized using stochastic geometry, while atmospheric sensing and conservative SNR margins enable FBL-safe backhaul adaptation. Numerical results reveal an optimal uplink access probability due to the tradeoff between spatial reuse and FBL reliability, and show that sensing assisted backhaul margins significantly improve robustness against attenuation uncertainty.

We prove that every character on a commutative complex complete Hausdorff locally multiplicatively convex algebra, not necessarily unital, is bounded on bounded sets. Consequently, every character on a commutative complex Fréchet–Arens–Michael algebra is continuous, giving an affirmative answer to Michael's problem. In the Fréchet case, each character is bounded by one seminorm from any increasing defining sequence of submultiplicative seminorms.