Papers

We complete the classification of compact simply connected homogeneous Einstein manifolds in dimension six by resolving the remaining cases for left-invariant Einstein metrics on $G=\mathrm{SU}(2)\times\mathrm{SU}(2)\cong S^3\times S^3$. Previous work leaves two cases for the isotropy group $K$, namely $K=\{e\}$ and $K\cong\mathbb Z_2$. We first rule out $K=\{e\}$. Then we show that any $\mathbb Z_2$ symmetry generated by an inner involution $\sigma$ with $\tr\sigma=-2$ necessarily extends to a $\mathbb Z_2\times\mathbb Z_2$ symmetry. Together with the previously known classification results, these theorems imply that every left-invariant Einstein metric on $G$ is, up to homothety and isometry, either the standard product metric $g_{\rm can}$ or the Jensen nearly Kähler metric $g_{\rm NK}$.

All recent advances \cite{DRS}, \cite{S-Rita1}, \cite{S-Rita2}, \cite{Millet}, \cite{MaS}, \cite{MaS2} on the boundedness and decay for wave equations on subextremal Kerr spacetimes $\KK(a,m)$, with non-small angular rotation, rely in an essential way on Whiting's groundbreaking discovery of a powerful integro-differential transformation with the help of which he ruled out exponentially growing modes in \cite{W} (see also \cite{Yacov}, \cite{AMPW} and \cite{Rita} for further developments). We give a purely physical space, generalized version, of the transformation which takes general solutions of {spin-$\sk$} wave equations in $\KK=\KK(a,m)$ to solutions of a secondary wave equation on a new metric manifold $\KKt=\KKt(a,m)$ with two stationary, asymptotically flat, ends. Moreover, for a certain subdomain $\DDt\subset\DDt_-$, which can be identified as the exterior of a black hole region, the new metric $\gt$ of $\KKt(a, m)$ is Lorentzian and the time translation $\T $ is timelike. Using the geometric properties of the two ends one can then deduce, by classical asymptotic Fourier analysis techniques, the desired bound for the flux of $\psi$ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

We propose a symmetry-preserving Boson-informed neural network (BINN) for the variational Monte Carlo computation of interacting-boson ground states in continuous space. The framework explicitly separates the two dominant physical scales of the system. At the microscopic interaction scale, a cell Jastrow factor absorbs the leading near-contact stiffness, while at the macroscopic scale, a potential-aware envelope encodes the external confinement, leaving the neural network to capture a smoother many-body residual. Across both two- and three-dimensional tests, the BINN approaches the Hartree and Gross-Pitaevskii limits with an empirical $\mathcal{O}(N^{-1})$ finite-size trend in both energy and density. The energies agree with diffusion Monte Carlo to within $0.06\%$ for the particle number tested up to $350$ in the three-dimensional shrinking-range case. Applications are made under various trapping scenarios, justifying robust performance of BINN.

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

We propose a method for closing moment chains for the Vlasov-Poisson kinetic system (the Landau fluid model), which yields a hyperbolic system of equations at each subsequent step. This system is interpreted as a multicomponent medium, where each component has its own density, velocity, and pressure. The coupling between the components is mediated by an electric field. The principle of closing moment chains is that the final component of the fluid is assumed to be pressureless.

The dimer spectral transform gives algebro-geometric action-angle coordinates for cluster integrable systems. Fock gave an explicit formula for its inverse. We study Fock's inverse map in the tropical limit where Harnack spectral curves degenerate to nodal curves. We determine the leading order asymptotics of the classical objects appearing in Fock's formula and show that they are governed by their tropical counterparts. This yields a tropical inverse spectral transform on tropical spectral data. We show that Fock's inverse spectral transform commutes with tropicalization, and also derive a tropical version of Fay's trisecant identity.

This paper deals with two kinds of Bäcklund transformations for scalar hyperbolic partial differential equations. We prove that both these types of transformations map solutions of a Darboux integrable equation into solutions of, generally speaking, another but also Darboux integrable equation. The latter fact can be used to roughly check the completeness of a list of Darboux integrable equations. To illustrate this, we apply the above transformations to several equations from a well-known list of Darboux integrable equations and, as a result, obtain a Darboux integrable equation which is absent in this list, but is already known at present. As a generalization of the last equation, we construct a family of Darboux integrable equations that is parametrized by three arbitrary functions, each of which depends on two arguments. This family is probably new.

We develop a rigorous renormalization group (RG) formalism on Wiener space. We introduce an RG operator $\mathcal R$ acting on probability measures over continuous paths, whose fixed point is the Wiener measure. We linearize $\mathcal{R}$ around the Brownian fixed point and analyze the spectral structure of the resulting operator $\mathscr{L}$. Its eigenvectors are too singular to be realized as honest measures on path space, and we therefore develop a generalized spectral theory within the framework of white noise analysis (Hida calculus). The eigenvectors of $\mathscr{L}$ are realized as Hida distributions, satisfying $\mathscr{L} U=\lambda U$ in the weak sense. This analysis yields structural results such as a spectral gap and a diagonal-concentration property of the eigenvectors. Moreover, we identify a family $\{\mathfrak{U}_n\}_{n \geq 0}$ of eigenvectors with eigenvalues $\lambda_n = 2^{1-n/2}$, consisting of Wick polynomials of white noise formally expressed as $\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\, dt$, and rigorously constructed as Hida distributions. We then use this spectral structure to uncover a finer, second layer of universality in the Donsker invariance principle: not only is the convergence of random walks towards the Brownian scaling limit universal, but the entire hierarchy of leading corrections to the Brownian limit is universal as well, governed by the eigenpairs $(\lambda_n, \mathfrak{U}_n)$. We also show, at a formal level, that the framework developed here extends to Gibbs-type perturbations of the Brownian fixed point, in the spirit of self-interacting quantum field theories. In particular, we recover the standard irrelevant/marginal/relevant classification, usually obtained in the physics literature by power-counting, solely from the spectral analysis of $\mathscr{L}$.

Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from $4^{100}$ terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the $D_4$ code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.

The Vlasov-Hartree system models a mixture of bosons and fermions interacting via Coulomb forces in which the bosons are described quantum-mechanically, while the fermions are described classically. For small initial data, we prove that the system exhibits modified scattering, i.e. the bosonic and fermionic subsystems each converge to solutions of the free Schrödinger and Vlasov equations respectively, but with coupled logarithmic corrections owing to the long-range nature of Coulomb interactions. We use a mixed Lagrangian and Fourier-based approach to derive explicit descriptions of the asymptotic dynamics of the solution and associated fields and Lagrangian trajectories. Notably, due to our purely Lagrangian analysis on the Vlasov side, we require no derivatives on the initial fermion density, the asymptotic fermionic profile is as regular as the initial data, and convergence occurs in almost the same space as the initial data.

This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence–contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement–estimation–feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.

We generalize the one-dimensional infinite-order linear differential equation satisfied by the $p$-adic Hurwitz-type Euler zeta function (Abh. Math. Semin. Univ. Hambg. 91: 117–135, 2021) to $n$ variables. By encoding the differential operator as a convolution with a distribution kernel, we introduce partial zeta functions and partial operators indexed by subsets of $\{1,\dots,n\}$. A tensor product expansion of the distribution kernel is established, whose Möbius inversion via the binomial theorem yields the higher-dimensional analogue of the original equation in the form of an alternating-sum identity, reducing to the one-dimensional case when $n=1$. Convergence of the resulting series is confirmed by non-Archimedean estimates. As a second main contribution, we prove that, under the condition $|a|_p > 2p^{1/(p-1)}$, within the Banach space of bounded analytic functions on $\mathbb Z_p$, the shifted $p$-adic Hurwitz-type Euler zeta function is the unique solution to the one-dimensional equation. This uniqueness result establishes that the infinite-order $p$-adic differential equation admits at most one bounded analytic solution, and together with the existence theorem, it characterizes the shifted $p$-adic Hurwitz-type Euler zeta function uniquely. This sharply distinguishes it from the complex setting, where the operator series diverges on the Hurwitz zeta function, and the formal kernel fails to converge in the usual spaces of analytic test functions.

We construct a compressible isotropic energy on $\mathrm{GL}^+(3)$ combining polyconvexity and global true-stress-true-strain monotonicity (TSTS-M++), answering a question of Wollner, Holzapfel and Neff. For every $\mu>0$ and $k>0$, the energy $W_k(F)=\frac{\mu}{2k}[\exp(k(\lVert F\rVert^2+3J^{-1}+J-7))-1]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge1/(8\sqrt3)$. The reference state is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the smaller sharp threshold $k_{\rm B}\approx0.0083$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it a global smooth diffeomorphism. Thus, for $k_{\rm B}\le k<1/(8\sqrt3)$, the Cauchy-stress map remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, each prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and the inner pressure increases smoothly and strictly from zero to infinity. Comparisons with other energies distinguish the roles of polyconvexity, TSTS-M++, reduced convexity and endpoint growth. Under incompressible spherical inflation, global pressure inversion holds for the Demiray family with $b\ge b_*$, including $W_k|_{J=1}$ for $k\ge k_*$ and the stress-free $I_1$-exponential energy, and for the exponentiated Hencky energy when $k_{\mathrm{eH}}>3/8$. Incompressible neo-Hooke instead has a limit point. A further compressible $I_1$ energy retains the global constitutive properties and reference moduli of $W_k$ but exhibits two pressure turning points under incompressible spherical inflation, showing that these properties allow a descending pressure branch.

Let $X$ be a toric Fano surface. Let $\pi: \widehat{X} \rightarrow X$ be a toric blow up at a point. We use the Topological Vertex [AKMV] to prove a higher genus, higher winding, open-closed duality between the toric Calabi-Yau 3-folds $K_X, K_{\widehat{X}}$. For $g \geq 0, w \geq 1$, we show the equality $n_g(K_{\widehat{X}}, \pi^*\beta - wC) = (-1)^g N_{g, (w)}^{LMOV}(K_X/L, \beta)$, where $n_g(K_{\widehat{X}}, \pi^*\beta - wC)$ is the genus-$g$, Gopakumar-Vafa invariant of $K_{\widehat{X}}$ in curve class $\pi^*\beta-wC$, where $\beta \in H_2(X, \mathbb{Z})$ and $C$ is the exceptional curve, and $N_{g, (w)}^{LMOV}(K_X/L, \beta)$ is the genus-$g$, LMOV invariant of an outer Aganagic-Vafa brane $L \subset K_X$ in class $\beta$ and representation $(w)$, or the Young Tableau of a single row with $w$ boxes.

The Normalized Energy Density (NED) was introduced in Goto et al.(2011) as a quantity that, unlike the conventional energy, is conserved across material interfaces in a layered elastic medium. Its conservation was proved analytically only for a two-layer structure; for three or more layers it was supported by Monte Carlo simulation. I supply the missing proof for a general multi-layer (n-layer) structure in the 2D SH problem. The key step is to stop treating the angular frequency as a single variable and instead lift the layer phases to independent coordinates on the n-torus. The obstruction identified in the original paper—that the ratio of travel-time combinations is generically irrational, so that the periodicity argument used for two layers fails and disappears, and the proof reduces to Weyl's equidistribution theorem combined with n repetitions of the same elementary integral used in the two-layer case. I prove conservation in two independent settings that together cover all cases of practical interest: travel times that are linearly independent over Q, and commensurable travel times, including the equal-travel-time (Goupillaud) discretization. I also show that an arbitrary layered structure can be subdivided by reflectionless interfaces into one whose travel times take only d distinct, Q-independent values, where d is the dimension of the Q-span of the original travel times; this places the two proved cases at the endpoints d=1 and d=n and isolates the remaining open case 1<d<n in a canonical form.

The Lorentz integral transform (LIT) method is a powerful tool for calculating quantum response functions; however, it requires an ill-posed inversion. Here we show that solutions of the LIT equation can also determine two additional integral transforms: a complex Stieltjes transform and a double-pole transform, at little additional computational cost. A Fourier analysis of the associated deconvolution problem shows that both alternative kernels are better conditioned than the Lorentzian kernel at the same width parameter $\Gamma $. We benchmark the resulting inversions for deuteron photodisintegration by adding controlled noise to the LIT solutions. Relative to the standard LIT, the alternative kernels yield response functions with reduced noise-induced scatter: by a factor of 2-3 for the complex Stieltjes kernel and 3-4 for the double-pole kernel.

This paper provides an algebraic proof of the gauge-fixing independence of perturbative asymptotic expansions in a finite-dimensional model of non-Abelian gauge theory. We develop an algebraic framework using the BRST formalism and perturbation theory. Our main theorem shows that under appropriate transversality conditions, normalized expectation values of observables remain unchanged under smooth deformations of the gauge-fixing conditions that preserve these assumptions.

We give an exact translation between fat-point interpolation in complex projective space and a dark-state problem for fixed-particle-number bosons. Under the unitary Fischer–Bargmann realization, homogeneous degree becomes particle number, the Veronese lift becomes a bosonic symmetric product state, and order-$m$ vanishing becomes orthogonality to all coherent jet excitations of orders below $m$. The initial degree of a symbolic power is therefore the first particle number admitting a dark state of a positive constraint Hamiltonian, and the Waldschmidt constant is the asymptotic particle cost per suppression order. After developing both sides of this dictionary from first principles, we prove a finite-scale no-dark-state amplification theorem. Frobenius amplifies the associated full-rank certificate in one good positive-characteristic fiber; a nonzero maximal minor returns it to characteristic zero. Its asymptotic consequence is Demailly's conjectured inequality. The paper is written so that the projective statement, the quantum reformulation, and the arithmetic part of the solution can be read independently and then identified exactly.

Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch–Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.

Quantum many-body scars are rare nonthermal states embedded within otherwise thermalizing spectra, making them promising candidates for storing and manipulating quantum information. However, a general framework that can both identify such states without prior knowledge of their microscopic scarring mechanism and determine when they enable a quantum error correcting code has remained elusive. Here, we develop an end-to-end framework connecting nonthermal-state discovery to certifiable quantum error correction using a many-body spectral localizer. By searching for states jointly localized in energy and a diagnostic observable in systems with a many-body grading symmetry, the spectral localizer identifies candidate scars within dense many-body spectra in tandem with a topological index whose shifts determine their graded subspace. For suitable pairs of same-graded scars, the grading renders grading-odd errors exactly detectable, while their localization can suppress the logical action of sufficiently local grading-even error products, yielding an approximate Knill–Laflamme condition and an operator-algebra quantum error-correction structure. We demonstrate the scar-identification framework in one- and two-dimensional density-difference-dependent bosonic models and in the PXP model, and explicitly construct the resulting approximate quantum code in the one-dimensional bosonic system. Our results establish a systematic route from identifying nonthermal many-body states to exploiting them for protecting quantum information, and position pseudospectral methods as a broadly applicable tool for discovering useful behaviors in interacting quantum systems.

We construct bounded Bell observables for a free scalar field in complementary spacetime wedges using measurable involutions of the spectral variable of a local canonical quadrature. These involutions yield wedge-local Hermitian unitary operators that anticommute with a chosen bounded observable. For a cosine observable, we characterize the admissible translations and reflections and derive exact correlations at finite modular bandwidth. The affine-reflection construction yields a violation of the Bell-CHSH inequality of approximately 2.06.

Let \(T_\Lambda=\mathbb R^d/\Lambda\), \(d\geq2\), be a flat torus with quotient metric \(\rho_\Lambda\), and consider the distance energies \(I_\alpha(\mu)=\iint \rho_\Lambda(x,y)^\alpha\,d\mu(x)\,d\mu(y)\) of Borel probability measures \(\mu\). We prove a quantitative Fourier signature of the cut locus: for every Voronoi facet and every \(\alpha>0\), there is a sequence of dual-lattice frequencies approaching the facet normal along which the Fourier coefficients of \(\rho_\Lambda^\alpha\) are positive, with an explicit leading asymptotic determined by the facet. As direct consequences, Haar measure is not a local maximizer for any positive distance power, even among smooth densities, and every flat torus of dimension at least two has supremal negative type and generalized roundness zero. We then solve the global maximization problem for two classes of flat tori. On an orthogonal rectangular torus, the maximizers undergo a transition at \(\alpha=2\): for \(1\leq\alpha<2\) they are the translated uniform measures on the two-torsion subgroup; at \(\alpha=2\) all balanced couplings on translates of that subgroup are extremal; and for \(\alpha>2\) only equally weighted diametral pairs remain. On the regular hexagonal torus, the maximizers are precisely the uniform measures on translates of a distinguished cyclic subgroup of order three for every \(\alpha\geq1\).

We introduce, motivated by noncommutative Carrollian geometry, the notion of a Carrollian–Poisson bundle understood as a Carrollian manifold with a compatible Poisson structure. Compatibility is expressed as the inclusion of the Carroll distribution in the image of the Poisson anchor. We explore the geometric consequences of this compatibility and present several simple examples of Carrollian–Poisson bundles, including the (outer) Kerr horizon, the BTZ horizon, and future null infinity. Furthermore, we establish that every Carrollian–Poisson bundle can be equipped with a Vaisman (contravariant) connection that is both metric-compatible and clock-compatible, and thus geometric kinematics can be defined and studied.

We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems. A loop (periodic orbit candidate) is parametrized as a Fourier series; a loss function is defined based on the deviation of the loop from the physical differential equations. Unlike previous work relying on hand-derived Jacobians, our method automates the process by leveraging automatic differentiation, a common machine learning technique. The continuation direction can be determined by the flat directions of the loss landscapes (directions with zero eigenvalues), making the search of periodic orbits efficient and guided. Our method is integrator-free, precisely initializes oscillations around unstable fixed points, and efficiently detects orbit family intersections and subharmonic bifurcations. As a demonstration, we present full continuations of periodic double pendulum oscillations from fixed points, showing bifurcations along orbit families and categorizing branches of periodic orbits. In particular, we find periodic orbits where both pendulum masses are never simultaneously at rest, which to our knowledge has been missing in the literature.

Chaotic evolution measures how rapidly nearby configurations separate, but the existence of such rapid evolution does not by itself determine what physical information the evolution actually carries. For fixed-coupling JIMWLK evolution with fresh transverse-local, color-isotropic noise, we show that this distinction becomes exact - leading to a no go theorem. At finite rapidity step and finite transverse regulator, the exact linearized update depends on the Wilson-line background only through local adjoint rotations of fresh color-isotropic noise. Once the past is conditioned upon, these rotations leave the complete local noise measure invariant, so the tangent process has the same probability law for any target ensemble evolved with the same regulated fixed-coupling dynamics, including dilute and saturated ensembles. After the exact global color-rotation mode is removed, the reduced tangent process found to have a positive leading Lyapunov exponent which indicate chaotic evolution. Fixed-coupling JIMWLK therefor can be dynamically chaotic while its tangent-only statistics carry no information about the target ensemble. Spatially nonlocal noise however evades the argument because correlations between distinct sites leave relative Wilson lines in the conditional covariance.

We investigate reflected diffusion in a bounded domain subject to two independent, competing stopping mechanisms: an exponentially distributed bulk lifetime of rate $p$ and a surface reaction triggered when the boundary local time exceeds an independent exponential threshold of rate $q$. Denoting by $T$ the stopping time and by $L$ the acquired boundary local time at stopping, we derive their marginal distributions, joint Laplace transform, and complete hierarchy of mixed moments. These statistics are determined by the splitting probability $\phi$ that surface reaction occurs before bulk decay. In particular, we establish the identity $p\expect{T}+q\expect{L}=1$ and show that the cumulative risk $pT+qL$ is exponentially distributed with unit rate. We further obtain equivalent representations of $\phi$ in terms of the Robin-Laplacian and the generalized Steklov spectra. Explicit results for a three-dimensional ball reveal how competing rates $p,q$ control $\phi$ and the $(T,L)$ statistics. Monte Carlo simulations test the universal cumulative-risk law.

We investigate the Weyl representation of the canonical commutation relations for a model describing a quantized massive scalar field under the influence of an external potential. The main problem is to determine whether the Weyl representation remains equivalent to or becomes inequivalent to the original one when the mass and/or potential are changed. This problem is reduced to the study of Schrödinger operators. It turns out that the Weyl representations are inequivalent when the masses differ. Moreover, when the masses are the same, the transition between equivalence and inequivalence occurs at the decay rate $-3/2$ of the difference between the potentials. This contrasts with the decay rate $-1$ that defines the short-range condition in scattering theory.

We explain a general procedure to compute characters of chiral algebras of 2d (S)CFTs geometrically from their coadjoint orbits, in analogy with the coadjoint orbit method for finite-dimensional Lie groups. We demonstrate that this method can be used to give concise rederivations of the characters of the Virasoro algebra, Kac-Moody algebras, as well as supersymmetric generalizations.

We present a simple algebraic proof of meromorphic non-integrability of a 'mysterious' case of a quadratic Hamiltonian studied by Sokolov and Wolf. The result is obtained by applying the Morales-Ramis method, based on differential Galois theory.