Papers

The technical goal of this paper is to construct and study profinite monoids DG_K for number fields K such that (1) the unit group of DG_K is isomorphic to the absolute Galois group G_K; (2) the maximal abelian quotient of DG_K is isomorphic to the Deligne-Ribet monoid DR_K; (3) the idempotents of DG_K bijectively correspond to subsets of the set P_K of (finite) primes of K; (4) the maximal Galois groups G_{K, S} with restricted ramifications S all appear as maximal closed subgroups of DG_K at idempotents; (5) all maximal closed subgroups of DG_K at idempotents are of this form. We also discuss the relationship between DG_K and the p-typical Witt vectors of Borger and de Smit, where we will describe the fundamental monoid of the semi-galois category of Lambda-rings of Borger and de Smit in terms of fields of norms of the local field K_p in particular.

Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.

We discuss the existence of a well-defined $\mathfrak{bms}_3$-integrable hierarchy, introduced in Fuentealba et al. (JHEP, 2018), using different structural methodologies. Specifically, from the variational complex on the ring of polynomial symbols, a $\mathfrak{bms}_3$ bi-Hamiltonian structure is constructed on which a Nijenhuis operator can be attached. A similar argument is made for the $\mathrm{AdS}_3$ case, where we check that the flat limit of the latter recovers the asymptotically flat situation. An alternative $\tau$-scheme description is also presented. Moreover, a Lie-Poisson description suggests that such a $\mathfrak{bms}_3$-hierarchy is not unique. For a subclass of so-called energy-dependent Schrödinger operators, it is shown that their Lax flows are described by the coadjoint orbits of $\mathfrak{bms}_3$.

Many physical phenomena are naturally modeled by dynamical systems subject to non-autonomous perturbations that decay in time, including the interaction of a laser pulse with a molecule, epidemiological models, nonlinear oscillatory systems, and celestial mechanics. In the present paper, we consider time-dependent perturbations decaying in time of Hamiltonian systems having a lower-dimensional isotropic normally elliptic (resp. hyperbolic) invariant torus with quasiperiodic solutions. Under suitable rates of decay in time of the perturbation, we prove the existence of invariant manifolds in the extended phase space, and determine the asymptotic behavior of the associated transverse dynamics. The above results are obtained for Hölder, smooth, and analytic Hamiltonian systems.

We answer positively a question of Aliaga and show that for any nonconstant real polynomial $p$, the Lipschitz-free space over $\{(p(n), p(m)):n, m\in \mathbb{N}\}$ is isomorphic to $\mathcal{F}(\mathbb{Z}^2)$. We in fact show more generally that if $d\in \mathbb{N}$, $q\in \mathbb{Z}_{\geq 0}$, and $((a_n^{(i)})_{n=1}^\infty)_{i=1}^d$, $((b_m^{(j)})_{m=1}^\infty)_{j=1}^q$ are sequences with $0<a_1^{(i)}<a_2^{(i)}<\cdots$, $0<b_1^{(j)}<b_2^{(j)}<\cdots$, $a_n^{(i)}\to \infty$ as $n\to \infty$, $\frac{a_{n+1}^{(i)}}{a_n^{(i)}}\to 1$ as $n\to \infty$ and $\underset{m\to \infty}{\liminf}{\frac{b_{m+1}^{(j)}}{b_m^{(j)}}}>1$ for all $i, j$, then the Lipschitz-free space over the product of these $d+q$ sequences is isomorphic to $\mathcal{F}(\mathbb{Z}^d)$.

We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(\Omega)$ for some $q\ge 1$. We prove that the unique solution $u$ is $\alpha$-Hölder continuous up to the boundary, for any $\alpha$ below $p'(s-N/pq)$ if $N/ps<q\le N/s$, and $\alpha=s$ if $q>N/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_\Omega^s$ admits a Hölder continuous extension to the closure of $\Omega$, where ${\rm d}_\Omega$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.

We study the standard nonuniform arithmetic quotient of the affine Bruhat–Tits building attached to $\operatorname{PGL}_d(\mathbb F_q(\!(t^{-1})\!))$, with Haar measure normalized so that a maximal compact subgroup has volume one. We first compute its vertex volume in closed product form. The proof is entirely building-theoretic: vertices are parametrized by a dominant sector, their stabilizers are counted exactly, and the resulting sum over block compositions is evaluated by a cut-set recursion. On the same quotient, we introduce a homothety-invariant normalized lattice-minima height $\alpha$. We determine its exact integrability threshold, proving that $\alpha$ belongs to $L^r$ precisely for $0<r<d$, and establish a sharp cusp-tail estimate of order $T^{-d}$. The associated positive-moment height zeta function, equivalently the Mellin transform of the cusp-height distribution, converges exactly in the half-plane $\operatorname{Re}(s)<d$. It admits a meromorphic continuation as a rational function of $q^{s/d}$ and has a simple pole at $s=d$, with an explicit critical coefficient. We also compute the resulting rational functions explicitly for $d=3,4,5$. Thus the same dominant-sector coordinates simultaneously control volume, cusp decay, and the analytic structure of the height zeta function.

For an $n$-dimensional convex body $K$, let $\theta_L(K)$ denote its lattice covering density, and let $\Theta_L^{\mathrm{conv}}(n)$ and $\Theta_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantities were known only to lie between a lower bound of order $n$ and an upper bound of order $n^2$, so even their polynomial order was undetermined. We prove that there are absolute constants $c,C>0$ such that \[ c n\log n \le \Theta_L^{\mathrm{sym}}(n) \le \Theta_L^{\mathrm{conv}}(n) \le Cn\log n\,(\log\log n)^{10/3+o(1)}. \] Thus both worst-case quantities are $n\log n\,(\log n)^{o(1)}$, and the upper and lower bounds differ by a factor at most $(\log\log n)^{10/3+o(1)}$. For the upper bound, a vertical–horizontal amplification based on weighted Boolean cubes combines covering estimates for low-codimensional sections into an exact lattice covering of an arbitrary convex body. For the lower bound, a random-slab construction and Poisson witnesses on flat tori show, with positive probability, that the resulting body admits no lattice covering of density below $c n\log n$.

We propose Kohn-Sham Spectral Embedding (KSSE), an energy-based model replacing the top-layer classifier of convolutional networks with a sparse-graph spectral embedding at the Nishimori temperature of an associated Random-Bond Ising Model the spectral detectability threshold where class structure becomes marginally distinguishable from disorder. Mapping pre-trained features onto quasi-cyclic low-density parity-check graphs, we construct a regularized Laplacian (Bethe-Hessian) as an effective Kohn-Sham Hamiltonian, yielding D independent spectral problems-one per feature channel-solvable in $O(N log N + k_{mode}^{2} N)$ time by FFT on circulant blocks (Pontryagin self-duality), with low-mode Rayleigh-Ritz refinement ($k_{mode}=5$). Physically, this is a k.p effective-mass reduction on a one-dimensional ring crystal: the circulant support is the perfect crystal, the data weights a slowly varying impurity potential, and the Nishimori crossing a Fermi level at the band edge. Star-domain surgery optimizes the graph: instead of eliminating all frustrated cycles impossible without destroying the codewords-edge shifts create certified convexity around codewords with bounded residual frustration, with multi-scale fractal certification (basins $D_{2}<1$ vs rough landscapes $D_{2}>3$). The theory includes a generalized Ihara-Bass identity with a sharp spectral threshold, a non-backtracking growth trichotomy with frustration as a gauge-invariant $Z_{2}$ flux, a trapping-set spectral test, exact channel separability with a cup-product obstruction, plus loop-series, convexity, surgery, and quasi-stationarity bounds. On ImageNet-1000 with frozen EfficientNet-B4 features (D=1792) under a transductive protocol, KSSE achieves 88.93% Top-1 accuracy with ~21.24M parameters-beating Swin-L (197M, 86.4-87.3%) and matching the lower end of ViT-H/14 (632M, 88.0-89.5%) with 10x and 30x fewer parameters.

Let $\mu(d x)\propto e^{-U(x)} d x$ on $\R^d$, where $U$ is $m$-strongly convex and $L$-smooth, and denote by $\kappa=L/m$ the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error $\varepsilon$, the expected query counts are $O(\kappa^{1/2}d\,(d\log\kappa+\log\frac1\varepsilon))$ gradient queries for the bouncy particle sampler and $O(\kappa d^{1/4}(d\log\kappa+\log\frac1\varepsilon))$ full-gradient equivalents for Zigzag, where $d$ coordinate-partial queries count as one equivalent.

We extend the construction of stability conditions on projective schemes over a field to projective families over an arbitrary base, and prove that they admit proper relative moduli spaces of semistable objects. We also prove a number of complementary results: the existence of mass-Hom bounds for these stability conditions, as conjectured by Halpern-Leistner and Robotis; a comparison with tilt-stability on surfaces and threefolds; a construction of stability conditions on the supported derived category of total spaces of certain vector bundles, including all local Calabi–Yau varieties; and a simple new proof of Bondal and Orlov's reconstruction theorem.

We investigate a numerical method for the Vlasov-Poisson (VP) system utilizing asymmetrically-weighted (AW) Hermite bases in velocity space, which is an hyperbolic system. In particular, we concentrate on spectral methods in velocity. For the Hermite spectral form of the VP system, we analyze the resaon that the form with AW Hermite bases can be instable. To obtain L2 stability properties, we consider a Galerkin method intead of the classical Petrov-Galerkin method, which naturally ensures stability with respect to the L2 norm. We also present an equivalent form of the method that maintains a computational cost modest compared to that of the Petrov-Galerkin method. Finally, we present numerical simulations based on the proposed Hermite spectral method, showcasing its stability.

This thesis studies policy learning in interactive systems where an agent observes a context, selects an action from a very large set, and receives partial feedback. The main framework is contextual bandits, with two paradigms: on-policy learning, where the agent interacts sequentially with the environment and minimizes regret, and off-policy learning, where it learns from logged data collected by a logging policy. In large action spaces, both settings face major challenges: inefficient exploration, sparse data coverage, high-variance importance weights, extrapolation bias, and difficult optimization landscapes. The first part develops structured Bayesian methods for on-policy learning. We introduce meTS, a mixed-effect extension of Thompson sampling, and dTS, which leverages diffusion-inspired priors to model dependencies between actions. These methods share information across actions and yield regret guarantees depending on an effective number of actions. The second part addresses off-policy learning. We propose sDM, a structured direct method based on latent variables, show that optimization error can dominate estimation error in large action spaces, and introduce concave, efficiently optimizable policy-weighted log-likelihood objectives. Finally, we develop differentiable pessimistic methods based on exponential smoothing and PAC-Bayesian bounds to control the bias-variance trade-off of regularized importance-sampling estimators.

Thompson's groups $F \subset T \subset V$ were introduced in 1965 and have since found widespread application in fields as diverse as logic, group theory, homotopy theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of $F$, factoring through a surjection from $F$ to isotopy classes of links in $S^3$. The second author extended Jones' surjection to $T$, thereby constructing all isotopy classes of checkerboard colorable (CC) links in the thickened annulus. We complete this program for $V$, defining a surjection $\mathcal{L}_{V}$ from $V$ to virtual equivalence classes of CC links in thickened compact oriented surfaces. This yields a new oriented subgroup $\vec{V} \subset V$ containing Jones' oriented subgroups $\vec{F} \subset F$ and $\vec{T}\subset T$. We prove $\vec{V}$ realizes all oriented almost classical virtual links. We then construct unitary representations of $V$ and $\vec{V}$ from kei and operator quandle coloring invariants, respectively.

In this paper, we prove a conjecture of D. Ginzburg on the unramified computation of an Eulerian global integral associated with a cuspidal automorphic representation of $\mathrm{GL}_{2r}$ and the generalised Speh representation $\Delta(\tau_2, r)$ attached to a cuspidal automorphic representation $\tau_2$ of $\mathrm{GL}_2$. We also study an analogous Eulerian global integral involving degenerate Eisenstein series and compute its unramified local factors.

This paper investigates various classes of cotorsion modules relative to the hereditary torsion theory $\tau_w$ induced by the so-called $w$-operation. To achieve this, we first introduce the Tor and cotorsion pairs relative to $\tau_w$, which serve as our main tool. Additionally, we also introduce and study the strongly flat modules relative to $\tau_w$. As applications, several known rings (including perfect rings, almost perfect domains, Prüfer $v$-multiplication domains, Dedekind domains, Krull domains, and DW-rings) are characterized in terms of these classes of modules.

We prove that every positive rational number has eventually greedy best Egyptian underapproximations, both when repetitions of the denominators are allowed and when the denominators are required to be distinct. This answers affirmatively a problem originating with Erdős and Graham and later revisited by Nathanson, and yields an application concerning the maximal asymptotic growth of denominators in unit fraction series converging to a given rational number. We reformulate the question as an optimal control problem for a dynamical system, construct an appropriate payoff function, and study properties of the associated Bellman function. We also answer another question of Nathanson by constructing an irrational number with unique and greedy best Egyptian underapproximations.

We prove that the category $\mathsf{SKB}$ is not algebraically coherent. As a consequence, it is not locally algebraically cartesian closed. Then the same can be deduced for cocommutative Hopf braces.

The local behavior of zeros of orthogonal polynomials is determined by the local scaling behavior of Christoffel-Darboux (CD) kernels. All previously studied behaviors are described by scaling limits, and different values of the limit kernel correspond to different universality classes. In this paper, we describe new universality classes in which instead of a single limit kernel, there is a limit cycle. These are naturally suited to Cantor spectra and to fractal behaviors of the measure. We show that these new phenomena occur for two canonical models with singular measures: the middle third Cantor measure and the balanced/equilibrium measure on a real Julia set of an expanding polynomial. In particular, this is the first result on the local behavior of CD kernels for an almost periodic operator with singular spectrum. As a complementary result, we describe the asymptotics of the scaling function, and the Christoffel function, at a fixed point of a quadratic iteration. This is the first such analysis for an almost periodic model with singular spectrum. It allows us to conclude that, at the fixed point, the local scaling of zeros of the polynomial of degree $n$ is precisely of order $n^{-1/\alpha}$, where $\alpha$ is the local dimension of the measure. We also study the limit chain in this case, and prove that it can be parametrized by its asymptotics with respect to a Martin function (so-called $M$-type); this is the first result of this kind for a chain with a singular measure.

We show that, among all measurable sets $\Omega \subset \mathbb{C}$ with finite area $s$, the disc of area $s$ maximizes the Ky Fan norm, which is defined as the sum of the first $N$ eigenvalues of the Toeplitz operator with symbol $\mathbf{1}_{\Omega }$ on the Fock space. For $N=1$ this reduces to Nicola-Tilli's celebrated Faber–Krahn inequality and for general $N$ the result was conjectured by Nicola, Riccardi and Tilli, who proved it for radial sets. The proof combines Ky Fan's maximum principle with methods from quantum information theory based on Fock rearrangements of density operators. As a by-product, we obtain isoperimetric inequalities for the Schatten sums in the range $0<p\leq \infty $.

Operators are called twisted commuting if the deformation from commutativity is determined by a unitary operator, including multiplication by a unimodular constant. The aim of the paper is to investigate constraints on the diversity of twisted commuting pairs depending on the deforming unitary called the twist. It turns out that some operators, like projections, are never twisted commuting. For many twists, twisted commutativity is possible only if at least one of the operators has a nontrivial kernel. In particular, bounded below operators, like isometries, are never twisted commuting by such twists. Twisted commuting unitaries are modeled by pairs of unitarily equivalent and commuting unitaries.

Constructing log canonical compactifications of open varieties is a central problem in birational geometry. Finding a natural coordinate system and obtaining the equations of these models is difficult in general. We show that for a large class of varieties explicit coordinates for the log canonical model are provided by canonical forms of positive geometries, and use this to compute the equations of these models. Our theory applies, for instance, to complements of hyperplane arrangements, cubic surfaces with lines removed, and the moduli space of marked cubic del Pezzo surfaces.

In this paper, we obtain non-testing characterizations, in terms of dyadic capacity gauges, of the boundedness and compactness of the differentiation operator $$ \frac{d}{dz}:Q_K\longrightarrow L^q(W\,dA), \qquad 0<q<\infty. $$ We also characterize the limiting case as $q\to0^+$, formulated in terms of a logarithmic geometric mean, while the endpoint $q=\infty$ is treated separately using a standard testing argument. These results greatly extend the previous work on ${\mathcal Q}_p$-spaces to the general setting of $Q_K$-spaces. As applications, we characterize composition operators and Volterra-type integral operators between different $Q_K$-spaces. In particular, the off-diagonal characterization established here, together with the previously established diagonal case, completely resolves Zhao's 2009 open question on composition operators between ${\mathcal Q}_p$-spaces.

The finite-block maximum-likelihood (ML) guarantee of soft-input GRAND requires querying noise-effect patterns in nonincreasing conditional-likelihood order. Under correlated Gaussian noise, additive reliability metrics and independent-block approximations need not preserve this order because the matched metric contains cross-coordinate interactions; the first codebook hit need not induce an ML codeword. We develop Low-Pathwidth GRAND (LP-GRAND) for binary phase-shift keying (BPSK) with precision matrix $Q$. The candidate-dependent part of the Gaussian negative log-likelihood is an observation-dependent quadratic pseudo-Boolean energy whose interaction graph has edge $\{i,j\}$ exactly when $Q_{ij}\neq0$. If $Q$ has half-bandwidth at most $\nu$, this energy admits a trellis with at most $2^\nu$ states per layer; a path decomposition of width $w$ yields at most $2^{w+1}$ bag assignments per layer. In real arithmetic, suffix dynamic programming and best-first complete-path enumeration enumerate patterns in nondecreasing energy. With complete enumeration and no abandonment, the first codebook hit induces an ML codeword for any nonempty binary codebook with equiprobable codewords. LP-GRAND agreed with exhaustive codeword ML in all $10{,}000$ frames for two $[20,12]$ codes. At nominal $E_b/N_0=2$ dB, its empirical BLER was lower than that of each block-based approximation for six $[64,52]$ codes.

In this work, we characterize the shadowing property for weighted shifts on a general class of Banach sequence spaces, extending a theorem proved by Bernardes et al. in ETDS (2020). We then apply this result to characterize this property for weighted composition operators acting on a class of Banach spaces of continuous functions. As an application, we present a characterization of shadowing for bilateral weighted translations on the spaces $C_b(\mathbb{R})$ and $C_0(\mathbb{R})$. In a more general setting, we provide sufficient conditions for weighted composition operators to have the shadowing property. These conditions allow us to obtain complete characterizations of this property for multiplication operators on the Hardy spaces $H^p(\mathbb{D})$, $p\in[1,\infty]$, as well as on spaces of the form $H^\infty(\Omega)$ or $C_b(\Theta)$.

We give a self-contained Hecke-algebraic derivation of the Casselman-Shalika formula and J.-S. Li's genericity criterion for irreducible spherical representations of unramified groups, without using the unramified principal series or intertwining operators. The argument centers on the spherical and sign idempotents $e_K$ and $e_{\mathrm{sgn}}$ of the Iwahori-Hecke algebra $\mathcal{H}$. Their left ideals $\mathcal{H}e_K=\mathcal{A}e_K$ and $\mathcal{H}e_{\mathrm{sgn}}=\mathcal{A}e_{\mathrm{sgn}}$ are free of rank one over the Bernstein subalgebra $\mathcal{A}$. Describing $e_K\mathcal{H}e_K$ inside $\mathcal{A}e_K$ recovers the Satake isomorphism. Describing $e_K\mathcal{H}e_{\mathrm{sgn}}$ inside $\mathcal{A}e_{\mathrm{sgn}}$ yields rank-one freeness of the $K$-invariants of the Gelfand-Graev representation and the Casselman-Shalika formula. Symmetrically, describing $e_{\mathrm{sgn}}\mathcal{H}e_K$ inside $\mathcal{A}e_K$ determines when the sign-isotypic part of a spherical module is non-zero, and hence yields Li's genericity criterion.

We introduce a new method for obtaining models of stationary reflection at the successor of a singular cardinal of any cofinality. In particular, starting from a cardinal $\kappa$ which is $\kappa^+$-supercompact, we get a model of countable simultaneous reflection at $\aleph_{\omega_1+1}$.

We study continuous-time quantum walks (CTQWs) on cycles. In particular, we prove that the cycle $C_9$ does not admit uniform mixing at any time via algebraic geometry and Gröbner basis techniques to rule out all cyclic 9-roots.

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

Let $\HH$ be a real Hilbert space and let $(f_i)_{i\in I}$ be a finite family of proper, lower semicontinuous, and convex functions on $\HH$. This study investigates the existence and uniqueness of exact solutions to systems involving proximity operators of the form: \ $(\forall i\in I)\ \pr{f_i}(x)=p_i,$ where $(p_i)_{i\in I}$ is a prescribed collection of proximal points in $\HH$, which naturally generalize classical projection problems. We establish necessary and sufficient conditions for the existence of approximate solutions to such systems. Moreover, we introduce and derive several characterizations of the inverse proximal property (IPP), as a generalization of the inverse best approximation property (IBAP). Applications of the obtained results are presented in the contexts of a feasibility problem and signal recovery problem, demonstrating the relevance of the proposed framework to optimization and inverse problems in Hilbert spaces.