Papers

We study quotients of free associative algebras by the two-sided ideals generated by positive-degree symmetric, quasisymmetric, or free quasisymmetric polynomials. Free quasisymmetric functions give a word realization of the Malvenuto-Reutenauer Hopf algebra of permutations. For quotients on $n$ variables, finiteness in the symmetric and quasisymmetric cases follows from a theorem of Bergman–Isaacs when $n!$ is invertible in the field. Our main result is a nilpotence criterion for families of relations compatible with restriction to subalphabets. The criterion proves finiteness over every field for quasisymmetric, $r$-quasisymmetric, and free quasisymmetric coinvariants, with explicit degree bounds exponential in $n$ for fixed $r$. In the quasisymmetric and $r$-quasisymmetric cases, these improve the bounds inherited from Bergman–Isaacs. A principal application is to the free quasisymmetric case, where, for $n\geq3$, the group-invariant argument used for the symmetric case does not apply. For the symmetric quotient $C_n$, we prove that invertibility of $n!$ is also necessary for finiteness. In characteristic zero, an auxiliary presentation and the Golod–Shafarevich inequality give $\log\dim C_n\geq(1-o(1))n^5(\log n)^2$ as $n\to\infty$. The same inequality shows that quasisymmetric and free quasisymmetric coinvariant dimensions grow faster than every fixed power of $n!$. For all three families on several freely noncommuting alphabets, we determine all multigraded Hilbert series and, for symmetric coinvariants in characteristic zero, Frobenius characteristics from one-alphabet data. In characteristic zero, projection gives stable Schur multiplicities for compatible quotients in mixed commuting, anticommuting, and free alphabets.

This paper presents GpFlash, a Gaussian process library that is focused on fast training and posterior evaluations, minimal memory usage, and scalability to high-dimensional problems. GpFlash was compared to the popular GP libraries scikit-learn, GPy, GPflow, GPyTorch, and SMT on a MacBook Pro and the supercomputer Trillium. GpFlash was found to use less memory and to be faster for training and posterior evaluations relative to the other GP libraries for all cases considered. For example, when using $\nx = 10^4$ training points in a 10-dimensional space GpFlash required 9.6 to 35 times less memory and was 2.6 to 54 times faster than the other GP libraries for the evaluation of the marginal log likelihood and its gradient. The memory reduction was achieved by only using one matrix of size $\nx \times \nx$ to store different symmetric matrices in its lower and upper triangular entries. The computational speedup was the result of designing algorithms to access memory efficiently, using BLAS and LAPACK methods extensively, and factoring kernel equations so that expensive exponential calculations could be reused. GpFlash was more than 440 times faster than all of the other GP libraries considered when evaluating the posterior mean together with its gradient for the case with $10^4$ training points and 10 dimensions. GpFlash is available at https://github.com/marchildon/gpflash.

We prove that every Lawson surface $\xi_{k,l}\subset\mathbb S^3$ of genus $kl>1$ is non-algebraic: any real polynomial vanishing on $\xi_{k,l}$ vanishes identically on $\mathbb S^3$. This answers a question posed by Hsiang in 1967 and proves Lawson's 1970 conjecture for $\xi_{2,2}$. The proof uses a reflection principle for the monodromy of the associated family of flat connections.

We derive explicit formulas for the dual Moore–Penrose generalized inverse (DMPGI) of dual complex matrix sums under some conditions. We first express the DMPGI of a matrix partitioned into column blocks in terms of these blocks and associated projectors, and then derive two formulas for matrix sums. Two examples illustrate the results. We further apply the formulas to rigid-body motion estimation with two observation points subject to a first-order perturbation of their positions that preserves the rank-deficient observation geometry. The DMPGI recovers the observable motion components from the velocity data.

We study the amplification and concentration of wave energy in smooth, periodic space-time media through the dynamics of the characteristic curves of the associated wave equation. In an impedance-matched setting, we show that stable limit cycles attract neighbouring characteristics, focusing the wave energy density. The exponential attraction rate is estimated analytically from the system's Lyapunov exponent. We further show that the attraction of characteristics and energy accumulation persist under a range of nonperiodic perturbations. Finally, we present a Floquet-Fourier numerical method for computing stable and unstable limit cycles and their attraction rates. Throughout, we exemplify the results using the example of modulation given by a superposition of travelling waves, revealing saddle–node bifurcations where limit cycles may appear depending on the relative wave speeds. Full-wave simulations confirm that the characteristic mechanism continues to describe energy accumulation beyond the impedance-matched case.

This paper studies fundamental solutions of uniformly parabolic systems with stationary random coefficients depending on space and time. Under ergodicity and local regularity, we establish quenched Gaussian bounds for the kernel and its spatial derivatives, with stationary random prefactors. A spectral gap assumption gives moments of every finite order for these prefactors and hence annealed Gaussian bounds. We also prove quantitative two-scale expansions for the kernel, its spatial gradient, and its mixed spatial derivative. The error estimates hold in every finite probability moment and retain Gaussian decay. Their dimension-dependent growth is described by a corrector weight, with a single additional logarithmic factor in the derivative estimates. The proofs combine large-scale regularity, extended correctors, weighted estimates, and a parabolic weak-norm estimate. The latter supplies the passage from deterministic pairings to local annealed estimates without interchanging expectation and a supremum over test functions. The mixed-derivative estimate builds on the gradient expansion and does not introduce a further logarithmic loss.

Standardizing a rare feature can help stochastic gradient descent learn its effect within a finite budget, but it also changes the noise cost of learning. We characterize the lowest rare-subgroup risks attainable by standardized and unscaled training when each chooses a nonnegative constant learning rate and a deterministic stopping time under the same observation cap. For two independent binary features, we derive both optimized risks as prevalence vanishes while the expected number of rare observations remains finite. Uniform control over rates and stopping times reduces each limiting oracle to a scalar minimization. With positive noise, the preferred scaling changes exactly once as we move signal to the rare feature while the limiting risk at initialization stays fixed and positive. The formulas also explain how noise and budget affect the comparison. Separate finite-prevalence bounds favor standardization on a rare-dominant signal family and unscaled training on a common-dominant family. The results concern final iterates with population preprocessing and fresh observations. Exact-moment searches assess the approximation at finite prevalence. In a controlled study, both orderings persist under observation reuse, estimated preprocessing, and intercept fitting when tuning choices are fixed before evaluation. Selecting a predictor after observing training can reverse an ordering, even when selection uses exact population risk.

We construct a meromorphic Wilf–Zeilberger pair for the hypergeometric series for \(3\ln 2\), completing the WZ proof left open by Guillera and Rogers. The same pair yields a one-parameter identity and, after a half-integer shift and differentiation, an inverse-binomial series for \(\zeta(3)\). We also study the finite truncations of the logarithmic series. For every prime \(p\ge 5\), we prove that the lower truncation is congruent to \(q_p(2)-1\) modulo \(p\), while the upper truncation is congruent to \(-\frac{7}{16}p^2B_{p-3}\) modulo \(p^3\). The proofs combine the WZ method with terminating hypergeometric identities, the Pfaff–Saalschütz identity, and classical congruences for harmonic sums.

For $T>0$, we consider the Benjamin–Ono-type Dirac equation $$ i\partial_t\psi(x,t) + D(-\Delta)^{\frac{1}{2}}\psi(x,t) = D\,G(\psi(x,t)), \qquad (x,t)\in\mathbb{R}^n\times[0,T], $$ where $(-\Delta)^{\frac{1}{2}}$ is the fractional Riesz operator, $D$ is the Dirac operator associated with the Clifford algebra of signature $(0,n)$, and $G(\cdot)$ a scalar-valued quadratic nonlinearity. Under a pointwise anti-commutation condition on $\psi$ and a uniform dyadic regularity assumption on its spatial Fourier profile, we construct a unit-modulus gauge map $u$ by $ u=\exp\bigl(-iw\bigr),$ where $w$ is a solution of $Dw=\psi$. Assuming, in addition, the stated regularity and commutation conditions on $w$, we show that the Plemelj components $u_{\pm}=P_{\pm}u$ solve a coupled Schrödinger system with gauge nonlinearities $P_{\pm}N(\psi,u)$, where $$ N(\psi(x,t),u(x,t)) = -2P_{-}\!\left[ G(\psi(x,t)) \right]u(x,t). $$ We also obtain a priori estimates for the homogeneous Sobolev seminorms $\dot H^\sigma$ of $\psi(\cdot,t)$, with at most exponential growth on $[0,T]$, together with estimates for $P_{\pm}N(\psi(\cdot,t),u(\cdot,t))$. Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for $n\geq 3$ and every sharp Schrödinger admissible pair $(q,p)$, an a priori Strichartz-type estimate for $\psi$ in $L_T^q\dot W_x^{\sigma,p}$, expressed in terms of the initial datum $\psi(\cdot,0)$ and the a priori controls of $\psi$ in $L_T^\infty H_x^{\sigma+1}$ and $L_T^\infty L_x^\infty$.

The total variation profile $\mathrm{TV}_n$ of a random walk measures how far its distribution at time $n$ is from being invariant under the generators. Three lower bounds are given: an elementary bound $\mathrm{TV}_n \geq 1/\big(44\,L(n)\big)$ in terms of the speed $L(n)$; a bound by the entropy increment $H(P^{n+1}) - H(P^n)$, which is a reverse Pinsker inequality; and a bound in terms of the entropy and the isoperimetric profile. Deciding whether the entropy-increment bound always improves on the isoperimetric one is related to conjectures on the growth of the entropy and of groups. The motivation comes from the question of whether the distributions of a random walk can be transported back onto their translates at bounded cost.

The purpose of this note is twofold. First, we revisit the classical regularity theory of harmonic maps in the setting of maps defined on a lattice $\mathbb{Z}^d$ instead of $\mathbb{R}^d$ or some other manifold. The target space of our harmonic maps will be spheres $\mathbb{S}^{N-1}, N\geq 2$. A harmonic map $\sigma : \mathbb{Z}^d \to \mathbb{S}^{N-1}$ is a map which in any finite subgraph $\Lambda$ minimizes the internal Dirichlet energy $\frac 1 2 \sum_{i \sim j \in \Lambda} \|\sigma_i -\sigma_j\|^2$ given the boundary condition $\sigma_{|\partial \Lambda}$. One key difference between our setting and the continuum is that our main Hölder regularity statement in a large domain $\Lambda\subset \mathbb{Z}^{d=2}$ does not require any regularity assumption on the map $\sigma$ restricted to the boundary of the domain. For example, when $d=2$ and $N=3$, the Hölder regularity induced in the bulk of the domain holds uniformly in all boundary configurations $\sigma \in (\mathbb{S}^{2})^{\partial \Lambda}$ (this is in contrast with the continuum which requires extensions in suitable Sobolev spaces). The fact we obtain results uniformly in the boundary conditions allows us to extend our results for positive temperatures. Namely, as $T \searrow 0$, we obtain a local regularity theory for all infinite volume Spin $O(N)$ Gibbs measures, at least when $d=2$ and $N\geq 3$. We also discuss implications of our analysis in dimensions $d\geq 3$.

We consider stochastic nonconvex optimization with sample-wise Lipschitz gradients and Hessians, and reusable opaque sample handles. Gradient and complete-Hessian evaluations are charged separately, including both endpoints of a same-sample difference. For fixed positive problem parameters and in the high-dimensional worst case, we prove that the minimax total derivative-query complexity of finding an $(\varepsilon,\gamma)$-second-order stationary point is \begin{equation*} \widetilde\Theta(\varepsilon^{-3}+\gamma^{-3}). \end{equation*} The upper bound combines curvature-first randomly signed negative-curvature steps with binary-ancestor derivative estimators; a truncated increment estimate charges predictable drift linearly and martingale motion quadratically. The curvature lower bound uses a deterministic smooth bounded-gradient chain, a radial barrier, and a random rotation. The new first-order lower bound is specifically robust to persistent sample reuse and to complete-Hessian access. It uses a slowly parameterized conservative sign chain, a third-order reverse stick-breaking localization inequality, and stateful relative-entropy accounting for reusable handles. Thus the total rate is near-optimally matched for every relation between $\varepsilon$ and $\gamma$.

Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to $Z(\mathbb{C}[G])$, the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.

Polyakov predicted in 1975 ([Pol75) that the spin $O(3)$-model (or classical Heisenberg model) should have exponential decay of correlations at all positive temperatures on $\mathbb{Z}^2$. Our main result is as follows: if we replace the Euclidean scalar product \begin{align*} \langle \sigma_i , \sigma_j\rangle = \sigma_i^x \sigma_j^x + \sigma_i^y \sigma_j^y + \sigma_i^z \sigma_j^z \end{align*} by the $\lambda$-modified scalar product \begin{align*} \langle \sigma_i , \sigma_j\rangle_\lambda := \sigma_i^x \sigma_j^x + \sigma_i^y \sigma_j^y + \lambda \sigma_i^z \sigma_j^z \end{align*} we show that for all $\lambda\in(-1,1)$, the $\mathbb{S}^2$ valued spin system with Gibbs measure $\exp(\beta \sum_{i\sim j} \langle \sigma_i,\sigma_j\rangle_\lambda)$ has power-law decay of correlations if $\beta$ is chosen large enough. This is the first evidence that arbitrary small smooth modifications of the classical Heisenberg model produce a BKT phase in $2d$. It was proved in the 1970s and 80s that there exist arbitrary small smooth modifications of the classical Heisenberg model which produce instead an Ising phase. In these work, they considered the regime $\lambda>1$ while in this paper, we consider the regime $\lambda<1$ which cannot be analyzed via reflection positivity and chessboard estimate. The two main ingredients used in our proof are the stability of the BKT phase under percolation type disorder ([DG25]) and weak regularity type properties of the groundstates or $\mathbb{S}^2$-valued harmonic maps associated to our $\lambda$-energy functional. This analytic step is interesting on its own. Finally, the exponential and power-law decay regime should be separated by a curve $\lambda \mapsto \beta_c(\lambda)$. We prove a rigorous lower bound in the case of the $\lambda$-anisotropic hemisphere model $\mathbb{S}^2_+$ which matches with RG flow predictions.

This paper introduces and systematically investigates several classes of numerical ranges associated with real-linear operators on Hilbert spaces, including the essential numerical range, maximal numerical range, essential maximal numerical range, $q$-numerical range, and trace numerical range. We establish fundamental structural properties of these sets and characterize their geometric shape. In particular, we analyze how the real-linear nature of the operators influences the geometry and spectral behaviour of the corresponding numerical ranges, thereby extending and enriching the theory of numerical ranges from complex-linear to real-linear operator settings.

Let $G$ be a finite group and let $\pi(G)$ be the set of prime divisors of $|G|$. For each $p\in \pi(G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $\Psi(G)= \prod_{p\in \pi(G)} \Phi(G_p)_G$ is called the local Frattini subgroup of $G$. Here, for $J \leqslant G$, the symbol $J_G$ denotes the core of $J$ in $G$. The subgroup $\Psi(G)$ is a characteristic subgroup of $G$ that is contained in $\Phi(G)$, the Frattini subgroup of $G$. We prove that $\Psi(G)$ satisfies many properties similar to $\Phi(G)$. Let $J$ be a normal $p$-subgroup of $G$ with $J \leqslant \Phi(G)$, and let $S$ be a Sylow $p$-subgroup of $G$. John S. Rose proved that if $J \leqslant \mathbf{Z}(S)$, then $J \leqslant \Phi(S)$. So, in fact, $J \leqslant \Phi(S)_G \leqslant \Psi(G)$. Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure $J$ is contained in $\Phi(S)$. In this article, we add a few more conditions to this list.

Matching one distribution to many targets can require repeated evaluations of every target. CAL-$\alpha$-MODM aggregates a nonlinear predictor into one barycenter, calibrates its residual derivatives, and checks steps against the original $f$-divergence objective. Under smoothness assumptions, calibration controls parameter-step model error without requiring targets to be close to the model. Exact-oracle stationarity and local convergence guarantees are established. At common matching, canonical geometry cancels the leading quadratic iteration error, giving cubic convergence with second-order calibration and sufficiently accurate inner solves. High-precision experiments support this prediction. Sensitivity-based initialization also accelerates preference continuation along regular stationary branches. On 300-target Jensen–Shannon problems, CAL-p2 achieves median paired speedups of $3.15\times$ over L-BFGS and $1.10\times$ over second-order Direct-TR, across 23/25 and 24/25 objective-comparable stationary pairs. The savings arise when cheaper predictor queries offset calibration overhead.

We obtain very high genus asymptotics for double Hurwitz numbers when both the size $n$ and the genus $g$ go to infinity, under the constraint that $g$ grows much faster than $n$, more precisely that $\frac{2g}{n}-\ln n\to\infty$. Our proof relies on adapting arguments about the mixing time of the transposition random walk introduced by Diaconis and Shahshahani.

We prove that the classical Bloch constant is strictly smaller than the Ahlfors–Grunsky value, disproving the famous Bloch constant conjecture. The Lean 4 formalization of the main results are also provided.

Driven by the scientific value of Venusian atmospheric studies, numerous missions, such as Magellan and Venus Express, have relied on single-spacecraft radio occultation (RO) experiments. Despite their proven success, data throughput is inherently limited by the use of a single transmitter and its relative geometry with respect to Earth-based receivers. Therefore, the literature proposes leveraging a constellation of spacecraft to increase RO measurement opportunities and improve spatial and temporal coverage; however, current efforts impose strict simplifying assumptions that limit their applicability and responsiveness to complex mission narratives. Beyond initial constellation design, evolving mission objectives may require event-based constellation reconfiguration during operations. To address both design and operational needs, we propose two problems formulated as integer linear programs: the RO Constellation Configuration Design Problem and the RO Constellation Reconfiguration Problem. Further, we address the programs' solution-space explosion with a Benders decomposition approach. Numerical experiments show the applicability of our problems for constellation configuration design, tackle the full mission pipeline, transitioning from global atmospheric profiling to on-demand regional profiling above areas of volcanic interest, and show that the Benders decomposition approach efficiently obtains high-quality solutions to large-scale instances.

In this paper we develop several aspects of the algebraic theory of Girard algebras, regarded as the equivalent algebraic semantics of multiplicative additive Linear Logic $\mathsf{MALL}$. The main structural results concern the bottom of the subvariety lattice of the variety of Girard algebras: varieties generated by simple Girard algebras generated by the constants correspond exactly to the atoms of the lattice. Moreover the lattice itself is atomic, i.e. every nontrivial subvariety contains an atom. These atoms can be represented uniformly by the maximal filters of the free algebra generated by the constants. Moreover every congruence of this free algebra is realized by a subvariety, and the resulting atom-support topology records the behavior of arbitrary joins: the atoms below a join of atoms are precisely the points in the closure of the corresponding subset of the spectrum. The concrete constructions show that the bottom of the lattice is already quite rich. Besides determining the integral and degenerate boundary cases and exhibiting finite modular and nonmodular atoms and chains of covers, we constructed infinite families of atoms whose topology reflects genuinely infinitary behavior; thus finite and infinite joins behave essentially differently at the bottom of the subvariety lattice. On the logical side, these results yield $2^{\aleph_0}$ Post-complete extensions of $\mathsf{MALL}$.

Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a non-cylindrical graph of degree $D\ge3$ satisfies $|\mathcal{N}_S|\le p+1-\lceil p/(D-1)\rceil$ under the same hypothesis. This bound is attained when $p\equiv1\pmod{D-1}$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.

We study the information-theoretic limits of inference in rectangular rank-one spiked matrix models with additive rotationally invariant noise, thereby going beyond the standard assumption of independent noise entries. Using the replica method from statistical physics, we derive a variational formula for the mutual information between the signal and the data, together with a system of saddle point equations predicting the minimum mean squared error in estimating the rank-one spike. We further show that, in the Bayes-optimal setting, these saddle point equations coincide with the fixed point equations of the state evolution of a recently proposed orthogonal approximate message passing algorithm with Bayes-optimal denoisers. This equivalence provides an algorithmic interpretation of the replica prediction and supports the Bayes-optimality of orthogonal approximate message passing whenever the relevant fixed point is unique. Numerical evaluations of the saddle point equations for several rotationally invariant noise ensembles and signal priors, together with finite-size simulations of the algorithm, confirm the asymptotic predictions. They illustrate both regimes with a unique informative fixed point, where orthogonal approximate message passing is predicted to attain the minimum mean squared error, and regimes with multiple fixed points, where a statistical-to-computational gap exists.

Existing deep-learning-based channel state information (CSI) feedback methods reduce feedback overhead by compressing high-dimensional CSI into low-dimensional representations and/or exploiting historical CSI to predict future CSI. However, these methods may become unreliable when user equipment (UE) suddenly changes its motion. To address this issue, we propose a prediction-aided coding (PAC) framework for generative CSI feedback, which adaptively encodes CSI based on predicted channel/CSI distributions and the observed CSI in real time. Each CSI frame is represented by multiple CSI tokens obtained without using historical CSI, making token generation robust to abrupt UE motion. In the first stage, the conditional distribution, instead of a deterministic value, of the next-frame CSI tokens is predicted based on historical tokens. In the second stage, once the next frame arrives, the predicted distribution is used for variable-length entropy coding of its CSI tokens. To further reduce feedback overhead and inference latency, we propose efficient designs for both stages. For the first stage, a customized causal Transformer with a new positional embedding strategy accurately predicts next-frame token probabilities. For the second stage, a parallel causal decoding strategy reduces the inference latency caused by sequential token decoding. Extensive experiments show that, compared with existing history-aware baselines, PAC reduces feedback overhead by approximately 72% to 84% while achieving higher reconstruction accuracy. Additionally, PAC maintains CSI reconstruction accuracy under random-walk trajectories, whereas existing history-aware baselines suffer sharp degradation in reconstruction accuracy.

Selecting weights and projection directions from the observations used for testing can invalidate a reference distribution. We develop a framework for exact conditional inference under rotation symmetry. A maximal invariant permits adaptation while leaving a uniform common phase for calibration. This yields explicit Cauchy references for invariantly selected weights and projections. Under separate block rotations, the same principle gives exact normal-score combinations with coefficients selected from all blockwise invariants. The calibration results require neither densities nor moments. To illustrate the framework, we study direction learning and power in spherical Gaussian location models, identifying a regime in which consistent detection is possible without consistent direction recovery. For heterogeneous functional data, a matched mean–residual construction produces the required null symmetry under a conditional Gaussian model. Simulations compare Radius and Blend, two representative procedures, with sample splitting, and an electrocardiogram analysis illustrates their application.

Submodular maximization arises in a wide range of combinatorial optimization problems and has been extensively studied from the perspective of approximation algorithms. For maximizing nonnegative, non-monotone submodular functions subject to a matroid constraint, randomized algorithms achieve a $0.401$-approximation, whereas obtaining comparable deterministic guarantees remains challenging. In this paper, we present a deterministic $0.401$-approximation algorithm in the exact value and independence oracle model, improving upon the previous deterministic guarantee of $(0.385-\varepsilon)$ \cite{ChenGLSZ2026}. Our algorithm builds on the extended multilinear extension and provides a deterministic implementation of the recursive guided framework underlying the randomized $0.401$-approximation algorithm. The main technical ingredient is a direction oracle supported on at most two partitioned matroid bases. This oracle simultaneously satisfies the growth condition required by the extended-multilinear update and an approximate stationarity condition needed to construct the recursive guides. The resulting finite-support representation enables exact evaluation of the relevant expectations and marginal values without random sampling. The algorithm uses polynomially many value- and independence-oracle queries.

This note proves that the Kuznetsov component of a quartic double solid or a special Gushel-Mukai threefold does not have non-trivial admissible subcategories.

We study several function-theoretic aspects of the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$. We introduce and investigate the notions of $\Gamma_{E(3;3;1,1,1)}$-inner functions and $\Gamma_{E(3;2;1,2)}$-inner functions, and establish their relationships with $\Gamma_{E(2;2;1,1)}$-inner functions and $\Gamma_{E(2;1;2)}$-inner functions. We provide explicit constructions and representations of rational $\Gamma_{E(3;3;1,1,1)}$-inner functions and rational $\Gamma_{E(3;2;1,2)}$-inner functions. As an application of these results, we obtain partial solutions to the associated interpolation problems and construct explicit rational inner interpolants for interpolation data lying in suitable subsets of $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, respectively.

We show that every proper variety of distributive $\ell$-pregroups is periodic; as a result, the subvariety lattice of the variety of distributive $\ell$-pregroups decomposes into a disjoint union of bounded intervals each consisting of all properly $n$-periodic varieties, for different $n \in \mathbb{Z}^+$. On the way, we study non-periodic (one-generated) subalgebras of $\mathbf{F}(\mathbb{Z})$ and show that they either contain a non-periodic idempotent or they contain idempotents of arbitrary big periodicity. The same results holds for non-periodic one-generated subalgebras of $\mathbf{F}(\mathbf{J} \overrightarrow{\times} \mathbb{Z})$, where $\mathbf{J}$ is a chain, generated by a composite element and the resulting idempotents can be taken to be local in the associated wreath product.

Since Brill's work of 1959, the conformal factor of finite-amplitude Brill waves, the axisymmetric and time-symmetric vacuum initial data on $\mathbb{R}^3$ that describe pure gravitational radiation, has been obtained only numerically, up to the most recent studies. Here an explicit one-parameter family of such data is constructed: the first explicit real-analytic, non-trivial, asymptotically flat vacuum initial data with trivial topology and a single end. The free metric function is a closed-form combination of elementary functions and a dilogarithm. The data have the toroidal conformal symmetry of Beig and Husa, and a change of variable of Natanzon type reduces every mode equation to the Gauss hypergeometric equation, so that the conformal factor is a series with explicit Gamma-function coefficients. The quantities that have so far required numerical work are thereby obtained exactly. A ground-state identity gives a sharp existence criterion: complete, asymptotically flat data exist precisely for $k<k_c=0.8048016772\ldots$, and the explicit series is identified with the unique variational solution. The ADM mass is an absolutely convergent digamma series, with $m/a=\pi k^2/10+O(k^3)$ for small amplitudes and an explicit residue at its critical divergence. Sufficiently strong waves contain trapped surfaces: it is proved that for $k$ close to $k_c$ the rescaled conformal factor converges to an explicit function and explicit tori and spheres become outer trapped, so that the data contain an apparent horizon. A numerical search, restricted to equatorially symmetric star-shaped surfaces, finds marginally trapped (minimal) spheres, which appear in a fold near $k=0.7670$. No thin-torus or weak-field approximation is used.