Papers

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.

Consider $n$ independent random vectors sampled from uniform distribution on $(p-1)$-dimensional unit sphere. This paper investigates the limiting joint distribution for the minimum and the maximum values of their pairwise angles. It proves that the minimum and the maximum angles are asymptotically independent when both $n$ and $p$ tend to infinity, which solves an open problem raised in Cai, Fan and Jiang (2013) [Journal of Machine Learning Research 14, 1837-1864]. Cai, Fan and Jiang (2013) obtained the limiting marginal distributions for both the minimum and the maximum angles under assumption $\lim_{n\to\infty}{\ln n}/p=\beta$ according to whether $\beta=0$, $\beta\in (0,\infty)$, or $\beta=\infty$. This paper presents unified limits for both joint distributions and marginal distributions regardless of the relative divergence rate of $n$ and $p$. The paper also derives the limiting distributions for some statistics based on the minimum and the maximum angles,

We study finite-step replica symmetry breaking (RSB) in mixed $p$-spin Ising spin glasses at positive temperature and zero external field. For every integer $k\geq1$, we construct polynomial mixtures whose Parisi measures are exactly $k$-RSB, with each such phase persisting on an open set of coefficients. More generally, for a broad class of polynomial covariances with an $r$-RSB Parisi measure, adding the interaction term $\lambda q^p$ for sufficiently large even $p$ produces a unique $r$-RSB-to-$(r+1)$-RSB transition as $\lambda$ varies near a critical value. Since such covariances exist for every $r\geq0$, this yields transitions of arbitrary depth. The main idea of the proof is that, for large $p$, the perturbation $\lambda q^p$ is negligible for overlaps bounded away from $1$, while its effect is concentrated at overlaps very close to $1$.

We develop a new framework, inspired by Schmidt's games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension $d\geq 2$. We show the Hausdorff dimension of the set of points in $\mathbb{R}^{d}$ whose best approximation constant lies in $[\varepsilon,\varepsilon(1+\delta\varepsilon^{d})]$, for some constant $\delta>0$, is positive, and for a slightly larger set approaches full dimension as $\varepsilon \to 0$. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. We also use an elementary observation to show that the naturally defined Lagrange spectrum for systems of linear forms is uncountable in the case of square matrices.

Bayesian pseudo-posteriors based on moment conditions, such as Bayesian empirical likelihood (EL) and Bayesian exponentially tilted empirical likelihood (ETEL), provide a robust route to Bayesian inference when the model is specified only through moment restrictions, but their computation is often prohibitive. In this work, we address two barriers to Bayesian pseudo-inference under moment restrictions. First, scalable sampling is challenging for tall data because the pseudo-likelihoods are not log-additive across observations, so standard likelihood-subsampling methods do not apply. Second, in simulation-based and latent-variable problems, the moment function itself may be defined through an intractable expectation, making full-data pseudo-posterior inference infeasible. We develop a new family of subsampled pseudo-posteriors that replaces the full-data pseudo-posterior with an aggregate of mini-batch pseudo-posteriors. We show that naive mini-batching induces a shifted-mixture distortion, yielding aggregates that are correctly centered but overly diffuse. To remove this distortion, we introduce moment-function-level control variates that align mini-batch moment equations around a reference estimator and recover the full-data posterior shape. For Bayesian ETEL, we establish finite-sample total variation bounds to the full-data pseudo-posterior, allowing discontinuous and intractable moment functions. Numerical experiments show that the method applies broadly across moment-condition-based Bayesian pseudo-inference, closely approximates full-data pseudo-posteriors, and substantially reduces computational cost.

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.

Ferraguti and Micheli conjectured that non-trivial $F$-sets of every prescribed width exist over each finite field. We prove the finite-width part of their conjecture over $\mathbb{F}_q$ for every $q\neq2,3$. Fix a suitable prime $\ell$. For a degree bound $D$, we consider the saturated family of all irreducible polynomials $g(X^{\ell^j})$ whose core $g$ has degree at most $D$. A factor-descent lemma shows that every irreducible factor arising from a shifted difference belongs to the same family and has strictly smaller core degree. The saturated family is therefore an $F$-set of finite width. Dirichlet's theorem for $\mathbb{F}_q[X]$ produces core ladders of arbitrary length, and Kummer lifting reproduces each ladder at infinitely many scales. These parallel ladders force the width to be sufficiently large; an appropriate tail of the nullity filtration then has the prescribed width.

Recently, the study of the number of $k$-colored generalized Frobenius partitions, denoted by $c\phi_k(n)$, has witnessed renewed interest. In this paper, we investigate congruence properties of $c\phi_{16}(n)$ and $c\phi_{18}(n)$. Our main result is a proof of the conjecture of Cui, Gu, and Tang \cite{CGT25} that, for all $n\ge0$, $c\phi_{18}(3n+2)\equiv0\pmod{2187}$. The proof uses a $(p,k)$-parametrization together with $q$-series identities and dissections. We also establish congruences for $c\phi_{16}(n)$ modulo $1024$ and $2048$, and for $c\phi_{18}(n)$ modulo $8$ and $81$.

Let $(R,\mathfrak m)$ be a Noetherian local ring and $J$ be an arbitrary ideal of $R$. Suppose that $f_1,\ldots,f_r$ is a $J$-filter regular sequence in $R$ and $I=(f_1,\ldots,f_r)$. In this paper, we establish an improved upper bound for the perturbation index of the associated graded ring $\mathrm{gr}_J(R/I)$, refining the bound obtained by Quy and N. V. Trung.

We consider the adjacency matrix of a random directed $d$-regular graph on $N$ vertices. For fixed $d\geq 2$, we prove that the empirical eigenvalue density converges in probability to the oriented Kesten–McKay law as $N\to \infty$. The key technical input is the small-ball probability estimate for the smallest singular value. The proof combines a fixed-rank transposition argument with finite-field anticoncentration for shifted inverse compressions. We also prove a polynomial hard-edge estimate, which allows us to deduce the global law from the vanishing small-ball probability.

We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time — even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.

Let $C$ be an $[n,k]_q$ maximum distance separable (MDS) code with $4\le k\le q-3$, and suppose that it has a $2$-transitive permutation automorphism group. In this paper we show that $n\le q+1$, so the MDS conjecture holds for this class of codes.

Let an infinite sequence of independent and identically distributed random variables over a countable alphabet generate a graph by joining consecutive symbols and suppressing repeated edges. We determine the exact tail and local asymptotics of the limiting degree distributions of this range-renewal graph. If the ordered sampling probabilities satisfy \(\pi_k\in\mathrm{RV}_{-1/\gamma}\) with \(0<\gamma<1\), then the directed and undirected degree tails are asymptotic to \(\pi_k^\gamma\) and \(2^\gamma\pi_k^\gamma\), respectively, while the corresponding local masses are asymptotic to \(\pi_k^\gamma/k\) and \(2^\gamma\pi_k^\gamma/k\). Consequently, both limiting laws are regularly varying with index \(-2\), independent of \(\gamma\); forgetting edge orientation affects only the leading amplitude. The proof combines infinite-occupancy estimates for discovery times and residual unseen mass with a conditional geometric representation of inter-discovery gaps. Uniform integrability yields the tail asymptotics, whereas a geometric-smoothing argument obtains the local masses without differentiating a regularly varying tail. We also prove that deleting self-loops leaves the limiting laws unchanged. Finite-sample simulations for normalized Zipf frequencies illustrate the asymptotic result.

Suppose that the eventual use of data is not known when the data are reduced or collected. This note considers two simple boundary cases. In a finite statistical experiment, a statistic preserves the Bayes risk for every finite later decision problem if and only if it is sufficient. Hence, when the minimal sufficient statistic is one-to-one, exact preservation of all later decision problems permits no nontrivial reduction. We then consider adaptive sampling from $m$ independent Gaussian streams when an external query specifies the coordinate to be classified only after sampling stops. Under coordinatewise error control, the optimal symmetric average sample size is exactly $m$ times the one-coordinate optimum. A change-of-measure argument gives the corresponding pointwise lower bound in terms of binary relative entropy.

Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance $k_\varepsilon > 0$ on a “small” subset $\omega_\varepsilon$ of the boundary of the ambient domain, vanishing at the limit $\varepsilon \to 0$. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset $\omega_\varepsilon$ and the parameter $k_\varepsilon$. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called “Robin-Dirichlet” capacity or a “Robin-Neumann” capacity, which depend on the geometry of $\omega_\varepsilon$ and on the value of the parameter $k_\varepsilon$. We analyze how these quantities compare to more classical measures of the “smallness” of $\omega_\varepsilon$, such as the capacity or the Neumann capacity of $\omega_\varepsilon$, according to the behavior of $k_\varepsilon$ as $\varepsilon \to 0$.

We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in neighbourhoods of points where the action of the projective Lie symmetry algebra fails to be regular, including all singular behaviours that can occur. Together with the known classification in the regular case, this completes the solution of the problem posed by Sophus Lie in 1882.

We compute the limit of the eigenvalue of the $k$-Hessian operator on the unit ball in $\mathbb{R}^n$ when $k\rightarrow \infty$, assuming that the ratio $\frac{n}{k}$ remains fixed. When $n< 2ke$, we moreover identify the limit of the corresponding eigenfunctions. We also derive monotonicity results and consider when the ratio varies in $n$.

Relative ($\tau$) is equivalent to a statement that the sequence of Cayley graphs associated to group quotients $\Gamma_q=\Gamma/N_q$, $q\in\mathbb{N}$, form an expander family. There is a philosophy that expander graphs give rise to good mixing; for instance, one has exponential mixing for the geodesic flow uniformly the along a family of congruence covers of the modular surface, stemming from the symmetry in the $\mathrm{SL}(2,\mathbb{R})$ action and the uniform spectral gap for the Laplacian. Do we see similar phenomena in less structured settings? We investigate this question for a tower of finite sheeted covers of certain expanding dynamical systems. We use transfer operator machinery that applies in particular in the cases of subshifts of finite type and expanding interval maps. We make fruitful connections with KMS states of Cuntz–Krieger algebras.

We study a natural extension of Ramsey theory relative to the classes of maximally planar and maximally outerplanar graphs. This can be seen as a continuation of the study of `Planar Ramsey theory', introduced by Axenovich et al. The question we ask is the following: For a fixed family $\mathcal{K}$ of graphs and a pair of graphs $\{H,F\}$, does there exist an integer $r_{\mathcal{K}} (H, F)$ such that for every graph $G \in \mathcal{K}$ with $|G| \geq r_{\mathcal{K}}(H, F)$, every red/blue edge-colouring of $G$ admits a red copy of $H$ or a blue copy of $F$? When such an integer exists, we say $\{H,F\}$ is unavoidable in $\mathcal{K}$,, and otherwise $\{H,F\}$ is avoidable in $\mathcal{K}$. Our work focuses on this problem where $\mathcal{K} = \mathcal{K}_{\mathrm{MOP}}$ and $\mathcal{K} = \mathcal{K}_{\mathrm{MP}}$, which denote the families of maximal outerplanar (MOP) graphs and maximal planar (MP) graphs, respectively. This framework generalises the classical Ramsey problem relative to these classes, as the case with $\mathcal{K} = \{K_n \colon n \geq 2\}$ corresponds to classical Ramsey. We also study the corresponding Ramsey numbers for MOP and MP, which we denote as $r_{\mathrm{MOP}}(H, F)$ and $r_{\mathrm{MP}}(H, F)$. In the case when $\mathcal{K} = \mathcal{K}_{\mathrm{MOP}}$, we completely determine all unavoidable pairs $\{H, F\}$ with $|E(F)| \geq 2$, together with upper bounds and sometimes exact values of $r_{\mathrm{MOP}}(H, F)$. When $\mathcal{K} = \mathcal{K}_{\mathrm{MP}}$, we completely determine all unavoidable pairs in the diagonal case $\{H, H\}$ when $H$ is connected, showing that $H$ must be one of the graphs $P_3$, $P_4$, $P_5$, $K_{1, 3}$ or the fork graph $S_{2,1,1}$. This work opens up further possibilities in the study of Ramsey theory relative to a class, and we offer several open problems in this vein.

We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $\rho$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{\alpha,\rho\}$ for target functions of mixed smoothness $\alpha$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{\alpha,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{\alpha,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $\alpha$ for ELU and cosine activations, again up to logarithmic~factors.

A total coloring of a graph is an assignment of colors to its vertices and edges so that adjacent or incident elements receive distinct colors, and it is equitable when the cardinalities of any two color classes differ by at most one. Stemock conjectured that every $4$-total coloring of a cubic graph of order less than $20$ is equitable. In this paper, we disprove this conjecture: the circular ladder $L_{12}$ admits a non-equitable $4$-total coloring and, moreover, no smaller counterexample exists: order $4$ is vacuous, and every $4$-total coloring of a cubic graph of order $6$, $8$, or $10$ is equitable. We also prove that the same property holds at order $14$. Our proofs rely on a decomposition lemma, which states that, in any $4$-total coloring of a cubic graph $G$, each color class consists of an independent set $S$ together with a perfect matching of $G-S$. We use the lemma to determine all possible color class configurations for orders $12$, $16$, and $18$, and we show that every listed configuration is attained. Finally, we provide a splicing construction showing that, for every even $n\geq16$, some connected cubic graph of order $n$ admits a non-equitable $4$-total coloring. We may conclude that $14$ is the largest order for which every $4$-total coloring of every cubic graph is equitable.

Large language models (LLMs) are increasingly used to formulate optimization models from natural-language problem descriptions, yet realistic operations research (OR) requests are often incomplete: missing objectives, constraints, or business rules can change the resulting mathematical program. Existing evaluations largely assume a complete specification and therefore overlook whether an agent knows when clarification is needed before modeling. We introduce OR-Clarify, a benchmark for pre-formulation clarification. Each task presents a partial public problem description, withholds structured hidden slots, and evaluates agents through bounded interaction with a simulated user. The benchmark supports both openended and choice-based clarification, and measures slot recovery, stopping behavior, silent assumptions, and interaction cost. We further propose Interactive Optimization (InterOPT), a two-stage framework that identifies unresolved formulation-critical gaps and uses them to guide whether to ask the next question or to stop. In our choice-based experiments, InterOPT substantially outperforms all baselines in exact slot recovery; in the open-ended setting, it remains competitive with strong prior methods. Together, OR-Clarify and InterOPT reframe OR assistance as a selective completeness decision: clarify when needed, stop when ready, and quantify what remains missing.

This paper proves a generalisation of Coleman's isomorphism (between norm compatible cyclotomic units and a group of invertible power series) to various cohomology theories evaluated on proper regular $p$-adic formal schemes, for future applications to Iwasawa theory. This generalisation is an immediate consequence of a description, as a cyclotomic synthetic spectrum, of the limit over transfer maps as one goes up the cyclotomic tower of motivically filtered $\mathrm{THH}$. This crucially uses a calculation of $\mathrm{THH}(\mathbb{Z}_p[\zeta_{p^n}])$ due to Devalapurkar–Raksit as well as a calculation of the free loop transfer due to Schlichtkrull. Along the way, we construct transfer maps for various cohomology theories along finite locally free regular maps, using some elements of $\mathbb{P}^1$-stable motivic homotopy theory.

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 develop an energy-optimal generalized scalar auxiliary variable (EOP-GSAV) framework for nonlinear index-one port-Hamiltonian differential-algebraic equations (pH-DAEs). Exploiting the port-Hamiltonian structure, we separate the nonlinear effort, interconnection, and dissipation terms from a constant implicit core. The resulting BDF-1 and BDF-2 schemes require one linear solve per time step with a reusable factorization, while retaining discrete passivity and accurate tracking of the Hamiltonian. The schemes are compared with the implicit midpoint method equipped with full, modified, and frozen-Jacobian Newton iterations. Numerical experiments ranging from a strongly state-dependent nonlinear stress test to large-scale benchmarks demonstrate robust and competitive performance, with substantial efficiency gains in matched-accuracy regimes. A comparison with SUNDIALS IDA shows comparable work-precision behavior at equal order despite a non-specialized Python/SciPy implementation, while unrestricted variable-order adaptive IDA is faster in the high-accuracy regime.

We consider a rectangular random matrix formed by adding a rank-one term to a matrix with independent entries. The direction on the left side of that term is estimated by the leading left singular vector, and the error is multiplied by the square of the size of the added term. For entries with mean zero and variance one, we identify the exact tail condition for the following statement to hold for every deterministic sequence of directions: the largest error over all sizes of the added term converges to the same fixed value determined by the limiting ratio of rows to columns. This condition (weaker than the existence of fourth moments) requires that the tail probability, multiplied by the fourth power of the threshold, to tend to zero. If it fails, convergence to this value already fails when both directions are coordinate vectors, even at a size fixed before the matrix is drawn. For regularly varying tails of order between two and four, the largest error tends to infinity.

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.

Cryptocurrency markets exhibit extreme volatility and non-stationary dynamics that challenge conventional forecasting methods. Although Large Language Models (LLMs) have shown promise for time series forecasting, the combined effects of adaptation choices remain largely unexplored in financial settings. This study introduces PRICE, a structured approach for adapting LLMs to short-term Bitcoin price forecasting. Built on a 4-bit quantized LLaMA-3 8B model, PRICE investigates how fine-tuning, numerical representation, prompting, inference, and decoding jointly influence forecasting performance. PRICE integrates Parameter-efficient fine-tuning with Low-Rank Adaptation (LoRA), Recursive multi-step inference, Integer-rounded numerical representation, Context-Task-Format (CTF) prompting, and Exact zero-temperature decoding. Ablation studies show that each component contributes to forecasting accuracy and reliability. LoRA enables efficient training on limited hardware, recursive inference improves accuracy, integer-rounded values reduce errors, CTF prompting outperforms Chain-of-Thought, Implicit Chain-of-Thought (iCoT), and few-shot prompting, and zero-temperature decoding improves stability during recursive forecasting. Comparative evaluation against eight transformer-based and time-series foundation models shows that PRICE achieves the lowest forecasting errors on both validation and test sets while maintaining robust performance across evaluation periods. Despite being based on a model primarily pretrained on text rather than time-series data, PRICE achieves competitive or superior performance relative to specialized foundation models. These findings demonstrate that adaptation choices critically determine the accuracy and robustness of LLMs for numerical time-series forecasting.

We study latency-optimal file assignment in geo-distributed storage systems modeled as weighted graphs, where edge weights represent communication delays and each node stores one (possibly coded) file. Our goal is to minimize the average time required to retrieve an original file, taken uniformly over all nodes and files. We show that for every fixed number of files $k \geq 3$, computing a latency-minimizing assignment is NP-hard via a reduction from the domatic number problem. On the positive side, we identify natural network topologies that admit uncoded, structured optimal assignments in which, for every node, one can choose its $k$ closest nodes, including itself, so that they store distinct original files. We prove that every weighted tree, certain weighted cycles, and unit-weight graphs with sufficiently large minimum degree admit such assignments. For these graph classes, we provide efficient algorithms to construct latency-optimal file assignments.

One-shot information theory measures often require an optimization over states, but the form of these optimizers can be complicated or depend on the initial problem in nonlinear ways. In this note, we show that in many instances using the marginal instead of the optimal state is sufficiently good and only changes the result by a small factor. We prove that for the Petz–Rényi and sandwiched–Rényi divergences of order $\alpha\in(0,1)$, replacing the optimizing state on $B$ by the marginal $\rho_B$ results in a multiplicative overhead of at most $1/\alpha$.