Papers

Derivation algebras are natural structural invariants of Leibniz algebras. In a number of earlier papers, derivations were described for several classes of one-generated, nilpotent, non-nilpotent, and low-dimensional Leibniz algebras. In the present paper we complete the description for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are incorporated by reference, with changes of basis recorded when necessary. For the remaining cases we determine the derivations by direct coefficient calculations. Special attention is paid to characteristic two, where several derivation algebras increase in dimension. We also record simple decompositions of the resulting Lie algebras into natural ideals and subalgebras. A final table gives the derivation algebra of every type in the classification.

Automorphism groups are among the most natural structural invariants of an algebra, and their explicit determination is a basic problem in the structure theory of Leibniz algebras. In a series of earlier papers, automorphism groups were obtained for several classes of low-dimensional, one-generated, nilpotent, and non-nilpotent Leibniz algebras. In the present paper we complete this picture for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are not recomputed; instead, they are incorporated by reference, with changes of basis recorded when necessary. For the remaining types we determine the automorphism groups explicitly through a direct analysis of the automorphism conditions. Particular attention is paid to exceptional parameter values and to characteristic two. A final table summarizes the automorphism group of every type in the classification.

Matrices and tensors are ubiquitous throughout computer vision. The trifocal tensor $\caT$ is a $3\times 3\times 3$ tensor that plays a vital role in three-view geometry. However, this tensor, though displayed in a third-order form, is manipulated and operated primarily in a matrix style in standard literature. This article establishes the conformal tensor algebra by replacing ad-hoc matrix collections with well-defined multilinear operators. We treat all three views symmetrically by covariant subscript indexing ($T_{ijk}$). To make precise what is new beyond notation: the classical point, line, and mixed correspondence constraints are here re-derived, but uniformly, each as a single contraction in one shared algebra; the results that are genuinely new include the exact rank characterization of the trifocal tensor — all mode-$k$ unfoldings $\caT[k]$ in $\RR^{3 \times 9}$ satisfy $\rank(\caT[k])=3$ for non-degenerate setups, and rank deficiency of any unfolding certifies degeneracy of the camera configuration, which yields a near-zero-cost degeneracy alarm and a validation criterion for estimated tensors; the outer-product representation $\caT = A^{\top}\times \bfb_{4} - B^{\top}\times_{2} \bfa_{4}$; and direct epipole extraction via double contractive traces. The coordinate-free multilinear operators seamlessly map to tensor auto-differentiation frameworks (e.g., PyTorch, TensorFlow); a concrete PyTorch experiment demonstrates that the induced $24$-parameter bilinear parametrization acts as a structural prior that eliminates the accuracy drift exhibited by an unstructured $27$-entry autodiff refinement.

Let $(V,q)$ be an anisotropic quadratic space of dimension $4$ over the rationals and let $p$ be a prime such that the local quadratic space $V\otimes \mathbb{Q}_p$ is the orthogonal direct sum of two hyperbolic planes. We show that the group of $p$-adic rigid meromorphic cocycles attached to $V$ has infinite rank. A key ingredient in the proof is the computation of the group of invertible rigid analytic functions on products of certain rigid analytic subspaces of the projective line such as the $p$-adic upper half-plane.

We consider a family of uniqueness problems in the setting of weighted $\ell^2$ Fourier spaces on the unit-circle, from the perspective of simultaneous approximation. We develop a general potential theoretical framework, which enables us to extend and unify several classical results by Ahlfors, Beurling, Frostman and Khrushchev, previously known in the more classical regimes. In addition, we prove a surprisingly sharp result that illustrates a deep discrepancy between unilateral and bilateral uniqueness sets.

We prove a decomposition theorem for canonical processes generated by independent symmetric random variables with log-concave tails, without the $\Delta_2$ condition. The proof uses the classical chaining method, based on admissible partitions and chopping maps. The argument also recovers the majorizing-measure characterization.

Reduced-order models (ROMs) are widely used to accelerate the solution of parametrized partial differential equations, with projection-based methods relying on a linear representation of the governing equations in a low-dimensional space. Their efficiency, however, is fundamentally limited by the Kolmogorov N-width of the solution manifold: when the N-width decays slowly, accurate approximations require a large number of linear modes. This limitation has motivated nonlinear ROMs, which typically replace the linear representation of the solution field with nonlinear latent representations learned through autoencoder architectures. In this work, we propose a different perspective by introducing nonlinearity directly into the representation of the governing operators. Rather than compressing only the solution field, we learn nonlinear low-dimensional representations of the full-order operators while retaining an explicit reduced system of equations. Convolutional architectures are employed to exploit the locality and spatial structure of differential operators, and the resulting compressed operators are coupled with a nonlinear representation of the solution through a training strategy that accounts for both reconstruction accuracy and the error of the reduced-order solution. To address large sparse operators arising from fine and unstructured discretizations, continuous convolutions are used to operate directly on their non-zero entries. Finally, radial basis function interpolation enables the prediction of the compressed operators at unseen parameter configurations, providing an efficient online stage without assembling the corresponding full-order system. Numerical experiments on three parametrized problems demonstrate the potential of nonlinear operator compression as an alternative strategy for equation-based ROMs.

We consider a generalisation of the probabilistic reformulation of the supercooled Stefan problem from arXiv:1902.05174 allowing for non-integrable initial data. From an application of Ito's formula, we obtain a relationship between the Laplace transform of the initial data and a transform of the solution, together with its possibly discontinuous free boundary. Using this relationship we give a sharp existence result for global solutions, we extend previous results on the uniqueness of solutions, and we determine the asymptotic behaviour of solutions for a certain class of initial data.

Landau solution is a special family of solutions to the stationary Navier Stokes equations, which is important for the study of stationary problems such as asymptotic behavior and regularity. In this paper we prove that the Landau solution is rigid in rotated self-similar class when rotation parameter $\alpha$ sufficiently small or large. The arguments in the two parts rely on different approaches. For small rotation, we use the compactness argument and establish a classification lemma of the self-similar kernel of the linearization around any Landau solution, this method can also be extended to DSS and RDSS cases. For large rotation, the strong angular dissipation produces additional coercivity on the non-axisymmetric component, which forces the solution to reduce into Landau solution.

We study the problem of adversarially robust PAC learning. In this framework, the learner observes independent samples from an unknown distribution over $\mathcal{X} \times \{0,1\}$, as in classical PAC learning. However, given a perturbation map $\mathcal{U} : \mathcal{X} \to 2^{\mathcal{X}}$ known to the learner, the goal is to output, with high probability, a predictor that correctly classifies every perturbation $z \in \mathcal{U}(x)$ of most future examples $(x,y)$ drawn from the same underlying distribution. We determine the optimal $\mathcal{U}$-independent sample complexity of this problem in both the realizable and agnostic settings. More specifically, for every concept class $\mathcal{H}$ of $\operatorname{VC}$ dimension $d$, we prove upper bounds of $\mathcal{O} \big( d/\epsilon + \log(1/\delta)/\epsilon \big)$ in the realizable setting and $\mathcal{O} \big( d/\epsilon^2 + \log(1/\delta)/\epsilon^2 \big)$ in the agnostic setting, together with an optimal first-order refinement of the latter. These bounds match the corresponding lower bounds for classical PAC learning. Consequently, and perhaps surprisingly, adversarial robustness incurs no additional distribution-free statistical cost, uniformly over all perturbation maps. Our bounds improve exponentially on those of [Montasser, Hanneke, and Srebro; COLT '19]. On the technical side, we present short and elementary proofs based on a new algorithmic principle that we call binomial-bagging. We believe that binomial-bagging and its analysis may be of independent interest.

We study the parameter dependence of weighted Bergman kernels on fixed bounded domains in $\mathbb C^n$. Our main result establishes real-analytic dependence on $t\in(-1,\infty)$ for the kernels associated with $\delta^t\,dV$ on bounded pseudoconvex domains with $C^2$ boundary, where $\delta$ is the Euclidean distance to the boundary. The parameter derivatives satisfy factorial estimates in $C^\ell(S\times S)$ for every $S\Subset\Omega$ and $\ell\ge0$, uniformly on compact parameter intervals. The proof combines weighted $L^2$ estimates for $\bar\partial$ with a holomorphic family of bounded operators and gives a local holomorphic extension with values in a fixed weighted Bergman space. For weight families smooth jointly in space and parameter up to the boundary, we also express all parameter derivatives in terms of iterated weighted Bergman projections and complete exponential Bell polynomials. This formula implies preservation of the Gevrey class $G^s$, $s\ge1$, under uniform parameter estimates up to the boundary. We show that the same derivative formula holds for the weights $-t\log\delta$.

This paper proposes a novel discretely entropy conserving/stable flux reconstruction (FR) scheme for nonlinear conservation laws. The scheme combines a hybridized split-form differential operator with a generalized entropy projection, enabling discrete entropy conservation for arbitrary nodal distributions and arbitrary correction functions without modifying the original quadrature weights. The formulation is built directly upon the classical FR framework and is shown to preserve conservation properties while achieving entropy conservation in the semi-discrete sense. Numerical experiments using the isentropic vortex problem confirm the theoretically predicted accuracy and demonstrate robust entropy behavior with entropy-dissipative interface fluxes.

Let \(\Kl(1;q)\) denote the normalized Kloosterman sum modulo \(q\). We prove that, for each sign, there are \(\gg X/\log X\) square-free moduli \(q\in(X,2X]\) with at most four prime factors for which \(\Kl(1;q)\) has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for Kloosterman sums; the uniform estimate for shifted sieve weights established in Part~I controls the contribution of moduli having small prime factors.

We prove that a complete, two-sided, stable capillary minimal hypersurface in $\R^4_+$ is flat provided its contact angle $\theta$ satisfies $|\cot\theta|<\kappa_*$, where $\kappa_*=4\Gamma(3/4)^2/\Gamma(1/4)^2\approx 0.45$.

We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by \(s,u\in[0,1/2]\). The error is \(O_g(Y/(\log Y)^2)\), uniformly for \(Y^{1/5}\leq R\leq Y^{1/3}\), including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.

In this paper, we present three new constructions of optimal FDRM codes, all derived from subcodes of maximum rank-distance (MRD) codes. The first construction (Theorem~\ref{theo5}) is based on a new family of generator matrices for systematic MRD codes and yields several previously unknown optimal FDRM codes. In particular, for $q\geq 7$, it establishes the optimality of $[\mathcal{F},7]_q$ FDRM codes with $ \mathcal{F}=[1,2,3,4,8,8,8,8,8]$, thereby resolving an open problem posed by Zhang et al. (Des. Codes Cryptogr., 87(1):107–121, 2019). Our second construction exploits structural properties of generator matrices of a family of systematic MRD codes to obtain new optimal FDRM codes whenever each of the rightmost $\delta-2$ columns of the Ferrers diagram $\mathcal{F}$ contains at least $n-1$ dots. Building upon this approach, we further develop a third construction by substantially relaxing this requirement: it is sufficient to assume that each of the rightmost $\delta-2$ columns of $\mathcal{F}$ contains at least $n-r$ dots, where $r<\kappa$ and $\kappa=n-\delta+1$. Furthermore, by exploiting the notion of proper combinations of Ferrers diagrams, we develop several recursive constructions that produce large FDRM codes from smaller building blocks, yielding a number of new optimal families. In particular, for an $n\times n$ Ferrers diagram $\mathcal{F}$ with prescribed parameters, one of these constructions establishes the optimality of $[\mathcal{F},\frac{n}{2}-1]_q$ FDRM codes whenever $n$ is even, thereby settling an open problem posed by Etzion et al. (IEEE Trans. Inf. Theory, 62(4):1616–1630, 2016).

We study deterministic relative approximation of the total variation distance between high-dimensional distributions given by succinct descriptions. We develop an abstract deterministic approximation framework based on representing the total variation distance as a support function of a low-dimensional zonotope. As applications, we obtain FPTASs for several models. Given two mixtures of product distributions over $[q]^n$ with a total of $K$ component distributions, our algorithm approximates their TV-distance within a factor of $1+\varepsilon$ in time $\widetilde O_K(nq(n/\varepsilon)^{2K})$. We also give an FPTAS for mixtures of $n$-step Markov chains over $[q]^n$ with a total of $K$ component distributions, with running time $\widetilde O_K(nq^2(n/\varepsilon)^{2K})$. Finally, for two latent-tree Ising models with the same underlying tree topology, we give an FPTAS for the TV-distance between their leaf marginals in time $O(|V|^{13}\varepsilon^{-12})$.

This paper develops a staggered discontinuous Galerkin (SDG) scheme based on Cartesian grids for Stokes equations that is simple to implement, intrinsically pressure-robust, and superconvergent for all variables. Instead of the composite meshes used in standard SDG, we construct staggered quadrilateral meshes from Cartesian grids. The scheme takes velocity, pressure, and velocity gradient as unknowns, and employs piecewise-constant spaces with carefully designed staggered continuity. The additional gradient unknowns are locally eliminated via static condensation and can be further removed by mass lumping without loss of accuracy. An explicit pointwise formulation of the scheme is derived, which facilitates implementation and enables a detailed pointwise analysis. We rigorously prove pressure robustness and second-order superconvergence, which holds on general non-uniform Cartesian grids. The scheme is further extended to Navier-Stokes equations by introducing a novel discrete convection term with second-order consistency. Combined with the scalar auxiliary variable (SAV) approach and the Crank-Nicolson (CN) scheme, this yields an unconditionally energy-stable and second-order accurate scheme. Numerical experiments validate the theory and demonstrate accuracy and robustness.

Covert navigation in an uncertain flow couples motion planning with information acquisition. A vehicle must estimate the local flow to navigate, while estimation error induces corrective maneuvers that increase statistical distinguishability, or leakage, and hence the vehicle's detectability. We model this coupling by augmenting position with the estimation-error variance, whose evolution depends on the route through the local information rate. A small-error expansion shows that estimation uncertainty contributes a leading-order correction to the expected detectability rate. The resulting route-planning problem is characterized by a first-order Hamilton–Jacobi–Bellman (HJB) equation. The deadline-indexed value determines the covert-time frontier, while an exact sensitivity formula with respect to sensing quality shows that information acquired earlier along the route can reduce leakage incurred later, whereas information acquired later cannot reduce leakage already incurred. We then formulate a zero-sum sensing-allocation game, show that the leakage rate is convex in the sensing allocation, and prove that the defender may restrict attention to randomization over extreme single-site allocations. The resulting equilibrium is computed by column generation, with vehicle best responses obtained from the HJB equation. Numerical experiments illustrate the learning detour, the deadline–leakage tradeoff, the spatial-ordering effect, and the benefit of randomized sensing allocations.

Shah and Peters showed that a conditional independence test with finite-sample or uniformly-controlled level has only trivial power against any alternative. Most practical tests, however, only claim pointwise asymptotic level. There have been incorrect claims in the literature of conditional independence tests with pointwise asymptotic level and consistency against any alternative; whether such a test actually exists has remained open. We resolve this question. Even restricting to hypotheses with a bounded density on compact subsets of Euclidean spaces, for any sequence of (possibly randomized) tests with pointwise asymptotic level $\alpha$, for every $\varepsilon > 0$ there is a conditionally dependent distribution where the test's limsup power is at most $\alpha + \varepsilon$. This remains true when the level control is required only over conditionally independent distributions with a continuous density and a uniformly continuous conditional law, and the conditionally dependent distributions have smooth densities. On the other hand, the extreme limitation of trivial power against any alternative for uniform-level tests does not apply to tests with pointwise level. We exhibit a test that, without regularity assumptions, has pointwise asymptotic level $\alpha$, strictly higher power for all alternatives, and power tending to one for all distributions whose dependence exceeds a chosen scalar threshold.

We consider discrete-time dynamical systems generated by the iteration of asymptotically autonomous maps, whose asymptotic behavior may be either conservative or dissipative. These systems are of particular interest in the modeling of complex phenomena with time-dependent parameter variation. With an appropriate time compactification, the past and future infinity become normally hyperbolic invariant hyperplanes, and thus the plausible asymptotic states of the system can be determined by considering the appropriate fibers of the stable and unstable foliations or suitable sections of the asymptotic manifolds to their compact invariant subsets. In this context, we explore how tipping induced by a change of phase–namely, how a phase shift affects the evolution of the system–is encoded in the unstable manifolds of compact invariant subsets of the past equation. We further connect the tipping phenomenon to the splitting of invariant manifolds and identify certain system reversibilities able to prevent tipping. The theoretical framework we use clarifies the relationships among the various notions of tipping induced by phase-change introduced in the literature.

In this paper, we study the Cesàro-type operator $$ \Ces_{\mu,\beta}f(z)=\int_{[0,1)}\frac{f(tz)}{(1-tz)^\beta}\,d\mu(t), $$ where \(\mu\) is a positive Borel measure on \([0,1)\) and \(\beta>0\). For \(0<p,q<\infty\) and radial doubling weights \(\omega_1,\omega_2\), we prove that \(\Ces_{\mu,\beta}:\AL_p^q(\omega_1)\to\AL_p^q(\omega_2)\) is bounded if and only if $$ \mu([0,1))+\sup_{1/2\le r<1}\frac{\mu([r,1))}{(1-r)^\beta}\left(\frac{\what\omega_2(r)}{\what\omega_1(r)}\right)^{1/p}<\infty, $$ and its norm is comparable to this quantity. For standard weights, our result extends \cite{BlascoMas2026,GalanopoulosSiskakisZhao2025} to the full range $0<p=q<\infty$. We further characterize compactness and the essential norm on average radial integrability spaces induced by doubling weights.

We uniformly generalize Goodstein's theorem (Goodstein 1944) and Cichon's independence proof (Cichon 1983) via the concept of maximality quotients.

Radio frequency fingerprint identification (RFFI) is a promising approach to distinguish physical wireless devices using hardware-induced signal imperfections. Conventional RFFI methods only provide discrete device labels and no human-interpretable representation of the relations among received signals. We propose supervised device charting, which maps location-insensitive channel-state information (CSI) fingerprints to a low-dimensional chart that visualizes cluster compactness, overlap, and outliers. We evaluate the method with real-world 5G New Radio (5G NR) measurements from six commercial smartphones and introduce the neighbor label error rate (NLER) to quantify class-separation accuracy. Our results demonstrate that two- and three-dimensional device charts provide an interpretable visualization of the learned RFFI representation. For three-dimensional device charts, the NLER is 0.22% for same-day measurements and 7.38% for measurements from the next day. The device charts reveal a cross-day distribution shift and map the held-out device close to the known device of the same model. Increasing the device chart dimensions further improves cluster separation at the expense of interpretability.

Let \(R\) be any unital ring equipped with a Sylvester matrix rank function, and let \(Q\) be the rank completion of the matrix system defined by block-diagonal embeddings along a factor sequence. We prove that both \(\GL(Q)\) and \((Q,+)\) are exotic: every strongly continuous unitary representation is trivial. We also prove that \(\GL(Q)\) is Bourbaki bounded and admits no nonzero escape function. For every \(\varepsilon>0\), let \(B_\varepsilon\) be the open ball centered at the identity. The least positive integer \(N\) such that \(\GL(Q)=B_\varepsilon^N\) is \(\lfloor\varepsilon^{-1}\rfloor+1\). Neither regularity nor irreducibility is required. Our proof of exoticness uses a uniform Kazhdan estimate for additive integer matrix groups together with rank-one matrix perturbations, rather than the character-theoretic and continuous ring methods used previously. The geometric conclusions follow from factorizations using diagonal corner subgroups.

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb U$. Due to the universality of $\mathbb U$, the collection $\mathbb X_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) $\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$ of the space $\mathscr P(\mathbb U)$ of Borel probability measures on $\mathbb U$, where $\mu\sim\nu$ if $\nu$ is the pushforward of $\mu$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb U$, we show that, under the above identification, Gromov's box topology on $\mathbb X_1$ coincides with the quotient topology induced by the weak topology of $\mathscr P(\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathscr P(\mathbb U)$ induces a complete and separable distance ${\sf d}_{\rm mG}$ on $\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$, which metrises the quotient topology of $\mathscr P_\sim(\mathbb U)$ and is Hölder equivalent to the box distance $\square$.

Recently, the existence of regular solutions $(u,p)$ to the stationary incompressible Navier–Stokes equations on $\mathbb R^n$ with bounded compactly supported forces $f$ for $5\le n\le15$ was established by Li–Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for dimensions $n>15$. Here we answer this question and obtain the existence of regular solutions to the stationary Navier–Stokes equations in arbitrarily high dimensions. The key innovation is to combine an averaged signed pressure-potential identity with $L^2$-BMO estimates for the Bernoulli source flux, yielding a Bernoulli supremum bound independent of the size of the skew coefficient representing the drift. Together with the energy estimate for the comparison solution, this yields a sublinear bound for the Newtonian potential of the positive Bernoulli function, removing the upper dimension restriction in the earlier existence argument. We also construct periodic solutions and whole-space solutions for forces in $L^\infty\cap L^{2n/(n+2)}$. For compactly supported forces, we record two far-field profiles, their velocity, gradient and pressure remainders, and the corresponding cancellation criteria.

In this work, we propose a novel design of high-rate Calderbank–Shor–Steane (CSS) quantum low-density parity-check (QLDPC) codes based on quasi-dyadic matrices. The Tanner graphs associated with the two component classical codes used to construct the proposed CSS codes are characterized by a girth of at least $6$. We also derive an exact count of length-$4$ cycles in the quaternary representation of the parity-check matrix. Monte Carlo simulations under code-capacity and phenomenological noise models show that the proposed codes have competitive finite-length logical error rate performance compared with state-of-the-art codes in both these settings.

Many spatial (3D) vehicles, including AUVs and fixed-wing aircraft, are effectively nonholonomic and subject to limited actuation, such that continuous time-invariant stabilization is fundamentally obstructed by Brockett's necessary condition. To overcome this obstruction, we exploit the geometric singularity of spherical coordinates to design a backstepping continuous, time-invariant feedback law that exponentially stabilizes a 3D nonholonomic vehicle actuated solely by forward surge velocity, pitch rate, and yaw rate. The resulting closed-loop region of attraction excludes only the coordinate singularity, codimension-two, measure-zero set of initial conditions in which the vehicle lies on the line through the target orthogonal to the target plane, thereby covering the largest possible domain. We further construct a strict control Lyapunov function to prove global exponential stability of the origin on this domain with a user-specified decay rate, while simultaneously preventing the system from approaching the singular set. Finally, we show that the closed-loop system is exponentially attractive to the origin in Cartesian coordinates. Numerical simulation examples in both spherical and Cartesian coordinates illustrate the effectiveness of the control law.

This paper studies general leader-follower linear-quadratic stochastic graphon games with indefinite control weights. The model consists of one leader and a continuum of followers, where the followers interact through graphon aggregate terms and are coupled with the leader through their dynamics and cost functionals. Compared with existing related models, the state equations considered here allow more general diffusion terms, including diffusion coefficients depending on the states, controls, leader variables, and graphon aggregate terms. Using the stochastic maximum principle, we characterize the followers' Nash response and the leader's optimal control through coupled forward-backward stochastic systems with graphon aggregate terms. Under suitable Riccati solvability and convexity conditions, we construct a Stackelberg-Nash equilibrium for the limiting graphon game. We further design decentralized strategies for the associated finite-player network game and prove that they form an approximate Stackelberg-Nash equilibrium, which yields a propagation of chaos result. The framework covers, as special cases, indefinite linear-quadratic stochastic graphon games, Stackelberg mean field games, and stochastic graphon games with random effects.