Papers

We introduce a generator-matrix-based geometric approach to the covering radius of $\mathbb F_{q^m}$-linear rank-metric codes. Starting from the $q$-system associated with the code, we define the notion of covering lift and show that the covering radius can be recovered from the hyperplane weights of such lifts. This yields an exact criterion, in terms of the length and effective length of the code, for the covering radius to attain its largest possible value, together with the upper bound $\rho(\mathcal{C})\le \min\{m,n\}-1$ in all remaining cases. We then specialize our approach to $1$-dimensional codes, for which the effective length coincides with the minimum rank distance. In this setting, codes attaining $\rho(\mathcal{C})= \min\{m,n\}-1$ are related to covering lifts that are clubs. More generally, we characterize this case through linear maps into suitable quotient spaces, obtaining equivalent interpretations in terms of generalised evasive subspaces and auxiliary matrix rank-metric codes. These results determine the covering radius for several boundary values of the minimum distance and for all $1$-dimensional codes with extension degree $m\in\{3,4,5\}$. For $m=6$ and every $q$ we provide constructions and, for $q\in\{2,3\}$, computational results showing that $1$-dimensional rank-metric codes with the same length and minimum distance can have different covering radii.

In this paper we consider the $n-3$ dimensional linear section $X_\Pi$ of the Grassmannian $\mathbb G(1,n)$ of lines in $\mathbb P^n$ with a general linear space $\Pi$ of codimension $n+1$, with $n=2k+1$ odd and $n\geq 5$, that is a Calabi-Yau variety. Related to $X_\Pi$ there is a degree $k+1$ Pfaffian hypersurface $\mathcal C_\Pi$ that is the intersection of the dual of $\mathbb G(1,n)$ with $\Pi^\perp$. We prove that $\mathcal C_\Pi$ is rational and we provide a birational parametrization of it. Moreover we study in some detail the congruence of lines of $\mathbb P^n$ given by the centers of the linear complexes parametrised by the points of $\mathcal C_\Pi$.

In this paper we consider random walks moving in a dynamic random environment that is completely resampled with probability $p$ after each step of the random walker. This resampling yields a straightforward regeneration structure and limit theorems, such as a law of large numbers with a limiting speed $v(p)$ that depends on $p$. We investigate several properties of $v(p)$ as a function of $p$, such as differentiability and monotonicity. Similarily, we study the limiting diffusivity $\sigma^2(p)$ in a central limit theorem.

Expectiles are the only law-invariant risk measures that are both coherent and elicitable. Unlike Conditional Value-at-Risk (CVaR), however, they do not admit a Rockafellar–Uryasev representation that admits tractable Wasserstein reformulations. We address this difficulty by developing an envelope theorem for worst-case expectiles that characterizes the worst-case expectile over a Wasserstein ambiguity set as the unique root of a worst-case expectation with a two-piece affine integrand. This representation permits direct application of standard Wasserstein duality. Using this result, we reformulate a multi-period tri-level mean–expectile portfolio problem as four parametric linear programs with constraints. We establish four structural properties of the proposed model: an endogenously damped price of robustness, a decision-dependent critical radius beyond which the expectile tail component becomes inactive, exact recovery of the nominal model at zero ambiguity, and a characterization of how the Wasserstein ground metric determines whether the limiting portfolio becomes more concentrated or more diversified. Numerical experiments on 90 FTSE constituents over 3,341 out-of-sample trading days show that the expectile model outperforms a CVaR model matched on ambiguity set, radius, ground metric, and trade-off weight in all nine parameter cells—significantly so whenever the radius is non-trivial. The experiments further confirm the predicted degeneracy under the ground metric.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a piecewise linear function with exactly \(m\) jump points and no fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(2m\). The conjecture was confirmed for \(m=1\) in [J. Lond. Math. Soc. 113 (2026)], and a lower bound of \(2m\) for arbitrary \(m\) was established by Chen et al. [arXiv:2608.19542]. In this paper, we construct systems with at least \(4m-2\) hyperbolic crossing limit cycles for every positive integer \(m\), thereby disproving Tonnelier's conjecture for \(m\geq2\). We also establish the uniform upper bound \(2^{224(m+1)^2}\) for the total number of crossing, grazing, sliding, and composite limit cycles.

The computation of the vertices of a polyhedron described by a system of linear inequalities is a central problem in polyhedral computation. It is a fundamental step in the conversion between H-representations, by linear inequalities, and V-representations, by vertices and extreme rays. This operation plays an important role both in the study of polyhedra and their combinatorics in mathematics and in applications to software and system verification. We present a certificate-based approach for formally verifying the computation of the vertices of a polyhedron. Given an informally computed list of vertices, our method allows to certify in the proof assistant Rocq that the list is complete, or even exact. The cornerstone of the method is a new completeness criterion based on an abstract simplicial complex that generalizes a triangulation of the normal fan of the polyhedron. A significant advantage over previous approaches is that the usually expensive numerical computations are essentially reduced to membership tests to the polyhedron, while the other steps are cheap combinatorial tests. We implement the certification method and prove its correctness in the proof assistant Rocq. We experiment with it on a variety of polyhedra, including Birkhoff polytopes, cross-polytopes, cubes, permutahedra, hypersimplices, and high-dimensional polytopes involved in the disproof of the Hirsch conjecture. Our experiments show that certification with the Rocq-to-OCaml extracted checker is typically 1.5x to over 5x faster than vertex enumeration by the state-of-the-art informal C implementation lrslib of the reverse search method.

In this paper, we propose a new approach called preflattening to speed up the resolution of nonlinear algebraic systems by Newton-type methods. Preflattening is meant to be for nonlinear systems what preconditioning is for linear ones. The idea is to transform the system into an equivalent one that is more favorable to Newton's method. To this end, we introduce the notion of flatness, which is intended to play for nonlinear systems a role similar to that of the condition number for linear systems. For a physical model of interest in the geosciences, namely the chemical equilibrium problem, we combine this strategy with the nonlinear elimination technique and demonstrate the value of the resulting approach through numerical simulations.

We prove the Hawking and Penrose singularity theorems for Lorentz–Finsler structures of mixed horizontal/vertical regularity $C^{(1,2)}$ in the $C^1$ weighted case and $C^{(1,3)}$ in the unweighted case. At this regularity, geodesics need not be uniquely determined by their initial conditions, and the (weighted) Ricci curvature must be interpreted distributionally. Our approach relies on a two-stage approximation procedure: first, a homogeneity-preserving convolution, which is of independent interest in the broader semi-Riemann–Finsler setting, and second, a causality-adapted approximation à la Chruściel–Grant. In the special case of $C^1$ Lorentzian metrics, our results extend the $C^1$ singularity theorems of Graf to the $C^1$ weighted case.

We consider the problem of learning linear dynamical systems under adversarial contamination from a single trajectory of length $T$. While identification of linear dynamical systems itself is well-studied, the problem of robust system identification under adversarial contamination is relatively less explored. In this work, we study the setting where a fraction of the $T$ observations are contaminated by adversarial outliers. We propose different estimators based on relaxations of least-trimmed squares along with an alternating minimization algorithm. Furthermore, we also propose two estimators which exploit the group-sparsity (through penalization/hard-constraints) of the outliers. For the estimator with group-sparse penalty, we derive non-asymptotic error bounds which establish its robustness to outliers. We also show empirically that the proposed estimators work well in practice.

We prove uniform height gaps for subvarieties of abelian varieties in arbitrary characteristic, extending the new gap principle of Gao-Ge-Kühne. Our proof goes through the theory of adelic curves and globally valued fields, and proves the Bogomolov conjecture for an arbitrary globally valued field. This gives a different way to obtain uniform Bogomolov-type results, differing from the approaches of Dimitrov-Gao-Habegger-Kühne and Yuan. First, we prove the Bogomolov conjecture over globally valued fields for divisors. We then reduce the case of general subvarieties to the case of divisors by an induction argument. Specializing to the case of global function fields, we obtain a new proof of the geometric Bogomolov conjecture following the strategy of Gubler and Yamaki.

We investigate the differentiation and integration operator defined on resp. between weighted spaces of analytic functions in the special situation when the weight is given in terms of a weight sequence. More precisely, in this case the weight is defined via the associated weight function which is a crucial object in the ultradifferentiable setting and the information can be then expressed purely in terms of the underlying weight sequence. Due to the additional structure this setting is exploited in order to construct (counter-)examples, to compare conditions on weights in the different weighted frameworks, and to study and to understand in a more precise qualitative way required and admissible growth restrictions on the weights.

This paper considers the reach-stabilize prob- lem for a class of nonlinear control-affine systems with unknown parametric uncertainties, where the system states must remain in a safe set at all times, or enter a safe set in finite time and then remain in it, and converge to a goal point. An estimator is designed that generates a parameter estimate and a computable, nonincreasing bound on the estimation error from a known initial error bound, without requiring persistence of excitation. The bound defines margins that are added to a control barrier and a control Lyapunov function condition, which are then enforced in a quadratic program for efficient control design. It is shown that, for the closed-loop system, the safe set is forward invariant when the system starts within the safe set and is finite-time reachable from outside the safe set with a recovery-time bound that does not depend on the unknown parameter, and that the goal point is exponentially stable. In two numerical examples, with the initial state outside and inside the safe set, the proposed control policy, a baseline oracle control policy that uses the true parameter, both recover or remain in the safe set and converge to the goal point, while another baseline control policy that uses a fixed, incorrect, parameter does not remain

Let $X_1,\ldots,X_n$ be almost surely distinct independent real-valued random variables. Let $\sigma$ be the random permutation of $\{1,\ldots,n\}$ such that $X_{\sigma(1)}<X_{\sigma(2)}<\cdots< X_{\sigma(n)}$. We show that the set of laws of $\sigma$, as the laws of $X_1,\ldots,X_n$ vary, has semialgebraic dimension \[ \sum_{k=2}^n {n \choose k}(k-1)! \] as a subset of the $(n!-1)$-simplex. This establishes a conjecture of Babson, Duchin, Iseli, Poggi-Corradini, Thurston, and Tucker-Foltz who proved that the dimension is upper bounded by the above expression. We prove their conjecture by constructing a family of finitely supported laws for $X_1,\ldots,X_n$ that provides the correct dimension. We also give an alternative proof of the upper bound using a theorem of Radford. Furthermore, we show that the Mallows law on permutations cannot arise as a law of $\sigma$ for $n\geq 4$, answering a question of Ed Crane.

We study common weighted sampling designs for operator systems that are exactly observable at nonnegative integer times. For positive operators with original frame condition number at most $\kappa$, we construct a common integer design with arbitrarily small relative Gramian error and $O(\log R)$ distinct times up to $R$. The optimal logarithmic counting coefficient for preserving the frame property satisfies $C^*(\kappa)\sim\pi^{-2}\log\kappa$ as $\kappa\to\infty$. Real and integer times give the same infimum, and we determine its asymptotics as $\kappa\downarrow1$. A finite bound on the sampled condition number strictly increases this coefficient. For tight frames, we determine the exact optimum under this constraint and construct an integer design attaining it. Without this constraint, the infimum is never attained. For positive operators, a design valid for every finite $\kappa$ requires counting faster than $\log R$, with arbitrarily slow divergence of the ratio. For unrestricted normal operators and fixed $\kappa>1$, at least linear counting is necessary. Under the logarithmic sector condition $|\arg\lambda|\le M(-\log|\lambda|)$ with fixed $M\ge0$, logarithmic sampling is possible with a sharp leading coefficient. For fixed Borel spectral sets satisfying a radial density condition, we characterize logarithmic sampling and determine the minimum upper counting exponent, attained by an integer design. Counterexamples show that the radial condition cannot be omitted.

We construct a family of scalar-closed, non-additive maximum sum-rank distance codes (MSRD) by switching norm components of a $\sigma$-rational normal curve and lifting the resulting point set through a linearized Reed–Solomon syndrome map. We characterize the admissible switches by a multiplicative stability condition on the norm classes. In particular, for $1 \leq n_i \leq m$, $N=n_1+\ldots+n_{\ell}$ and $2\leq\delta\leq N-1$, every partition of $\mathbb{F}_q^*$ satisfying a specific condition, referred to as Condition $(\diamond)$, yields a code in $\mathbb{F}_{q^m}^{N}$ of size $q^{m(N-\delta+1)}$ and minimum sum-rank distance $\delta$; the code is non-additive whenever a curve component is retained. For $\ell\geq2$ these appear to be the first non-additive MSRD codes in the literature, all previously known families with more than one block being $\mathbb{F}_q$-linear. Moreover, every code of the family has the same sum-rank weight distribution as an $\mathbb{F}_{q^m}$-linear MSRD code with the same parameters, although the geometry of the switched set distinguishes it from linearized Reed–Solomon codes. In the single-block case, an explicit rank isometry identifies the construction with the cone codes of Durante, Grimaldi and Longobardi.

In this paper we study the geometrical entities as Gauss and mean curvatures attached to the ruled normal surfaces associated with rectifying curves. A particular and special example of rectifying curves is also investigated, via its attached ruled normal surface. Our main result is the proof of the non-existence of rectifying curves that are simultaneously Bertrand curves.

We extend the theory of De Giorgi-Nash-Moser to elliptic equations with symmetric coefficients $\mathbf{a}(x)$ which are possibly degenerate and unbounded but satisfy a coarse ellipticity condition. We prove local upper bounds for weak subsolutions, a weak Harnack inequality for nonnegative supersolutions, and a Harnack inequality for nonnegative solutions. The coarse ellipticity hypothesis requires spatial moments of coarse-grained matrices, suitably discounted and summed across scales, to be finite. In particular, it holds if, for some $\alpha,\beta\geq0$ and $1<p,q<\infty$, $$\mathbf{a}\in W^{-\alpha,p}\cap L^1, \quad \mathbf{a}^{-1}\in W^{-\beta,q}\cap L^1 \quad\text{and}\quad \frac{\alpha+\beta}{2}+\frac{d-1}{2}\Big(\frac1p+\frac1q\Big)<1.$$ We show that the coefficient range is sharp, including its boundary, in every dimension $d\geq3$, for $\alpha=\beta=0$ and for the corresponding Besov-type quasi-norms of negative order when $\alpha,\beta>0$. For $\alpha=\beta=0$, it corresponds to the results of Bella and Schäffner [8].

We study bond percolation on the one-dimensional lattice in which two sites at distance $r$ are connected with probability $p_r=C/r^{1+\sigma}$, where $0 < \sigma < 1$. An infinite cluster exists if $C$ exceeds a critical value $C_c(\sigma)$, which is not known exactly: the available results are the lower bound of Schulman, $C_c\ge1/[2\zeta(1+\sigma)]$, and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives $C_c\le\sigma+O(\sigma^2)$ as $\sigma\to0$ and is below $1$ for $\sigma < 0.667$. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: $C_c^{\rightarrow}\zeta(1+\sigma)\to1$ as $\sigma\to0$. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound $C_c < 1$ for every $\sigma < 1$. We compare the bounds with the known numerical values of $C_c$.

We consider the problem of learning structured linear dynamical systems over convex sets $\mathcal{K}$, where only a small subset of the observations are available at each time point. An estimator which minimizes a bias-corrected, potentially non-convex objective function is proposed. Non-asymptotic bounds are obtained for the statistical error, which depend on the local complexity of $\mathcal{K}$, the trajectory length $T$, and the sub-sampling probability $p$. Convergence of the projected gradient descent algorithm is also established. The general theory is applied to settings where (i) $\mathcal{K}$ is a subspace, (ii) $\mathcal{K}$ is the set of bi-isotonic matrices, and (iii) $\mathcal{K}$ is the set of matrices whose rows are formed by sampling Lipschitz functions. We show meaningful recovery of the transition matrix is possible for values of $T$ much smaller than what is required in the unconstrained case, and for $p = o(1)$.

We prove the existence and uniqueness of mild solutions to the ellipsoidal BGK (ES-BGK) model with the correct Prandtl number, under suitable weighted bounds and a free-transport non-vacuum condition. At the correct Prandtl number, the temperature tensor is no longer uniformly comparable to the scalar temperature, so classical scalar moment estimates do not directly yield the required Gaussian bounds. We derive anisotropic velocity moment estimates by splitting velocity space along the longest principal direction of the stress tensor and using the geometry of transverse elliptical slices. These estimates yield a linear weighted $L^{\infty}$ bound for the ellipsoidal Gaussian. Together with uniform ellipticity and weighted $L^1$-Lipschitz continuity under the stated assumptions, this bound allows us to construct unique mild solutions.

In deep learning, approximation theory motivates increasing representation size. We ask whether this benefit extends to dynamics learning through autoregressive prediction. We analyze the learned time evolution through the eigenstructure of Koopman operators, using relative residuals to detect spurious eigenpairs arising even as one-step error falls. For bounded Koopman operators, we show that minimal residuals over learned dictionary spaces converge pointwise to their full-space counterparts as these spaces approximate the observable space in $L^2$. Our hypothesis is that Koopman spectral reliability helps explain how consistently rollout error decreases with increasing dimension. We compare two models of a shared Koopman autoencoder trained alternately for reconstruction and latent evolution, using latent-prediction loss (one-step prediction errors in latent coordinates) or spectral-residual loss (relative residuals of candidate eigenpairs). Across six chaotic systems, both models reduced median windowed rollout error from smallest to largest dimension. The spectral-residual model achieved lower medians than the latent-prediction model for all systems and dimensions, and its median fell by a larger factor in every system. Its median decreased monotonically with dimension in four systems, against one for latent prediction. Against four baseline families, its mean valid prediction times were nearly always longer. At the largest dimension under two-stage training, we compared eigenvalue positions with each learned dictionary's residual contours. Spectral-residual eigenvalues concentrated in low-residual regions, whereas latent-prediction eigenvalues also appeared in high-residual regions, consistent with the hypothesis.

We show that the wild locus of an étale morphism of smooth adic spaces over an affinoid field is either empty or contains a divisorial point. As a consequence, we establish the equivalence of curve tameness and valuation tameness without using resolution of singularities. Moreover, given a finite étale map of regular schemes $f \colon Y \to X$ that is curve tame over a base $S$, we construct a compactification $\bar{f} \colon \overline{Y} \to \overline{X}$ over $S$ that is numerically tame.

Kernel Stein discrepancies (KSDs) provide a versatile tool for comparing distributions. One of their main applications is in quantifying the goodness-of-fit (GoF) between a data-generating distribution and a prescribed target distribution. In this work, we study the related problem of conditional GoF quantification: given only a (possibly non-normalized) conditional target model, without information on the distribution of its covariates, and samples from a joint distribution, the goal is to assess how well the conditional distribution of the samples matches the target. To tackle this setting, we present a framework that allows lifting unconditional KSDs to the conditional setting through an operator-valued kernel on the covariate space, going beyond the known Euclidean case. We establish that our suggested statistic vanishes if and only if the conditional model and the true conditional distribution agree for almost all covariates and deploy it to test conditional GoF on smooth manifolds and on discrete spaces. Our experiments on level, power, and runtime demonstrate the viability of testing on these domains using the proposed statistic.

We show that every triple of integers $(a,b,c)$ satisfying $1\leq a\leq b\leq c$ is realized as the smooth slice genus, smooth concordance genus, and Seifert genus of a knot, respectively. This proves a conjecture of Kearney.

We introduce an asymmetric variant of a classic problem by Roth about colourings of integers and midpoints. The new problem cannot be solved with the standard approach by Erdös-Sárközy-Sós which uses symmetry. We describe a Fourier Analytic approach which gives asymptotically tight bounds. In the $\mathbb{F}_3^n$ setting, the density increment we show is efficient and together with the Freiman-Ruzsa Theorem provides an affine subspace of near optimal codimension within the distinguished set. This approach adopted to the setting of the integers from $1$ to $N$ via Bohr sets and Bogolyubov-Ruzsa Lemma gives a progression of length being a power of $N$ which only depends on the number of colours. We then use Chang's Lemma and Balog-Szemerédi-Gowers Theorem to further improve the dependence on the number of colours.

Using notions from graded Poisson geometry, we develop a framework that allows for concise formulations of field theories such as Einstein-Cartan and Palatini-Cartan gravity. We show that Einstein-Cartan gravity is equivalent to a variant of Palatini-Cartan gravity in which the vielbein fields are constrained to satisfy appropriate symmetry relations, which we refer to as constrained vielbein gravity. To that end, we relate these theories via a diffeomorphism of graded manifolds. We then show how to obtain the full Palatini-Cartan theory by removing the symmetry constraint. Finally, we show how our graded geometric framework allows for a direct incorporation of the minimal BV extension of the graded constrained vielbein and Palatini-Cartan theories.

In this article, we show that the Sidon subsets of $[N]^d$ are Fourier uniform, and we prove a dense model lemma for Sidon sets. Using these results and a higher-dimensional version of Rado's theorem, we prove that given any partition regular linear equation in $s \geq 5$ variables with nonzero coefficients, and any finite partition of a sufficiently dense Sidon subset $S$ of $[N]^d$, the number of monochromatic solutions to this equation is $\gg |S|^s N^{-d}$ for all large $N$. As a corollary of the Fourier uniformity result, we show that dense Sidon subsets of $[N]^d$ are equidistributed in certain arithmetic and Bohr structures. Our proofs are motivated by the arguments of Ortega and Prendiville.

We present a boundary integral equation (BIE) framework for two-dimensional time-harmonic elastic transmission problems. Using the Helmholtz decomposition, the elastodynamic displacement fields are expressed in terms of scalar pressure and shear potentials, so that only the four classical Helmholtz boundary integral operators are needed. The key step is a rewriting of the Navier traction operator in the local normal–tangential frame of the interface, in which second-order normal derivatives are eliminated in favour of tangential derivatives and curvature terms. This leads to a one-parameter family of indirect combined field integral equations, depending on a coupling constant $\alpha\ne 0$, which we prove to be uniquely solvable. The principal part of the resulting operator is, however, defective, and the discretized systems are severely ill-conditioned. Using the pseudodifferential calculus of the Helmholtz BIOs, we construct explicit left and right analytical preconditioners and prove that the preconditioned operator is a compact perturbation of the identity. Numerical experiments based on high-order Nyström discretizations confirm that the preconditioner clusters the spectrum: the number of GMRES iterations becomes independent of the discretization and is reduced by one order of magnitude or more, also in the high-frequency regime and for non-convex scatterers.

We study the sample complexity of noisy one-bit compressed sensing for signals drawn from a prior distribution. By characterizing the effective distributional complexity of the prior via its approximate covering number, we prove that posterior sampling achieves accurate recovery with high probability when the number of measurements scales with the logarithm of the approximate covering number, up to a one-bit separation gap factor. This upper bound is robust to learned prior mismatch. Specifically, we show that posterior sampling with an approximate prior remains reliable, provided that the learned prior distribution is sufficiently close to the true signal distribution in Wasserstein distance. In addition, we establish a sample complexity lower bound for any reliable method of noisy one-bit compressed sensing, showing that our upper bound is nearly matched in its main prior dependent term. To approximate the ideal posterior sampling process for real world scenarios, we instantiate posterior sampling through a plug-and-play algorithm with diffusion priors. Experiments on the FFHQ and ImageNet datasets demonstrate the effectiveness of our proposed approach.

Let \(b(M)\), \(d(M)\), and \(r(M)\) denote the number of bases, the number of circuits, and the rank of a matroid \(M\), respectively. We prove that every nonempty simple binary matroid with no coloops satisfies \[ 2b(M)\ge(r(M)+1)d(M), \] with equality if and only if \(M\) is isomorphic to the Fano matroid. This confirms a conjecture recorded by Oxley in 1983. For the same class, we prove that deleting any element leaves at least as many bases as there are circuits in the original matroid: \(b(M\backslash e)\ge d(M)\) for every \(e\in E(M)\). We determine all equality cases and deduce a sharp linear lower bound for basis growth under successive series extensions. The main counting step is a joint estimate for the three largest possible circuit sizes, obtained from contraction-normalized fundamental-circuit counts and an exact folded-cube edge correspondence.