Papers

We present an elementary proof of the bivariate nested Artin-Popescu approximation theorem: it ensures the existence of a nested algebraic power series solution of a given polynomial (functional) equation. Such equations often appear in enumerative combinatorics, e.g., when counting lattice walks in the first quadrant. The ad hoc proofs of the algebraicity of the generating functions, as e.g. proposed by Bousquet-Mélou and Jehanne , can thus be replaced by applying the theorem without need to resort to the extremely difficult general case solved by Popescu. Our method of proof follows closely the arguments of Denef and Lipshitz who treat a more general case (namely, the Artin approximation for Weierstrass systems). This is combined with the techniques of Hauser and Woblistin for the description of the overall geometry of the infinite dimensional variety formed by all power series solutions. Putting both approaches together now provides combinatorialists with an accessible proof for the bivariate nested Artin-Popescu approximation theorem.

This paper presents the first systematic study of $[F]$-irregular graphs, a concept that parallels classical $F$-irregularity. For a fixed graph $F$, a graph $G$ is $[F]$-irregular if the numbers of its induced subgraphs isomorphic to $F$ containing a given vertex are pairwise distinct for all vertices of $G$. We prove that there exist infinitely many $[P_n]$-irregular graphs for any path $P_n$ of order $n \ge 3$. We establish that a non-trivial $[P_3]$-irregular graph of order $k$ exists if and only if $k \ge 7$. Finally, we propose the Strong Conjecture on $[F]$-irregular graphs.

This article extends Stein's functional method to establish explicit rates of convergence in the Wasserstein-1 distance for Donsker's invariance principle under weak dependence. Our approach first reduces the problem of bounding distances between distributions on a function space to estimating distances between finite-dimensional marginals. We then employ the block technique, which is standard in Stein's methodology, to handle dependence. While our analysis focuses on $\phi$-mixing sequences, the method extends to other forms of weak dependence admitting suitable covariance inequalities.

Consider a number field $K$, a polarized endomorphism $f:X\to X$ on a normal projective variety, and a non-exceptional point $\alpha$ for $f$. We show that the extension $K(f^{-\infty}(\alpha))/K$ generated by the backward orbit of $\alpha$ is virtually abelian if and only if $f$ is an exceptional map with CM and $\alpha$ is $f$-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over $\mathbb{C}$ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.

We study the Cauchy problem for the one-dimensional full compressible Navier–Stokes equations with viscosity $\mu(\theta)=\tilde\mu\theta^\alpha$ and heat conductivity $\kappa(\theta)=\tilde\kappa\theta^\beta$. For each fixed $\beta\ge0$, we prove the global stability of the combination of a viscous contact wave with rarefaction waves, provided that the viscosity exponent $\alpha$ and the total wave strength are sufficiently small. The initial perturbation may be large in $H^1$, and the specific volume and temperature are assumed to have positive initial lower bounds. The solution remains uniformly bounded in $H^1$ relative to the wave, and converges uniformly to that wave as time tends to infinity. We show that both the specific volume and the temperature admit time-uniform lower and upper bound. Estimates for the logarithmic volume derivative and the temperature gradient then close the $H^1$ estimates without requiring a second derivative of the initial specific volume.

Transcendence questions of periods and quasi-periods of Anderson modules, as well as the hyperderivatives thereof, are of major interest in number theory over function fields. In the case of Drinfeld modules, this question was answered by the second author. In this paper, we show that these results hold for general Anderson $t$-modules under certain assumptions on their Galois representations. The proofs use the explicit description of the $\mathfrak{p}$-adic Galois representation of the first author by means of a rigid analytic trivialization of the associated $t$-motive, as well as prolongations of $t$-modules and $t$-motives. The general result is applied to determine algebraic relations between periods and quasi-periods, and their hyperderivatives, of finitely many Drinfeld modules. Along the way, our results provide more evidence for the Mumford-Tate conjecture for Anderson modules.

Watanabe constructed many nontrivial family diffeomorphisms in $\pi_*\text{Diff}_\partial(D^4)$, which were obtained by doing clasper surgeries on trivalent graphs. In this paper, we generalize the barbell diffeomorphism discovered by Budney and Gabai and construct a series of nontrivial elements in $\pi_{k-1}\text{Diff}_\partial(M_{k+1}')$ with $M_{k+1}'=\natural_{k+1} S^2\times D^2$ the $(k+1)$-cuff barbell, which we call higher barbell diffeomorphisms. We show that Watanabe's constructions can be realized as implanted higher barbell diffeomorphisms in $D^4$ if the trivalent graph is homologically nonzero in the graph complex. Combining this realization with Watanabe's detection theorem, we establish the nontriviality of these implanted higher barbell diffeomorphisms.

In this article, we prove the existence of a sectorial transformation to a polynomial normal form a a non-degenerate 1-resonant biholomorphism in ${\mathbb C}^n$ . We also give their holomorphic classification.

The cotangent space to any subvariety of a Grassmannian is naturally identified with a space of linear homomorphisms. We use this to define a new statistic on a subvariety of a Grassmannian, called conormal rank. We show that Chow–Lam forms and their natural generalizations have low conormal rank, and moreover that hypersurfaces of sufficiently low conormal rank are all of this form, generalizing a result of Gelfand–Kapranov–Zelevinsky. We also compute the conormal rank of many families of varieties, including Schubert varieties, torus orbit closures, and positroid varieties.

Discrete gradient methods provide a structure-preserving approach to energy-consistent time integration of dissipative Hamiltonian systems. In their conventional formulation, however, they generally attain at most second-order convergence. Extending the perturbation principle previously developed for Poisson systems to the dissipative setting, we derive fourth-order average vector field discrete gradient schemes. A congruence-based completion preserves the positive semi-definiteness of the dissipation matrix, thereby guaranteeing unconditional energy dissipativity for all positive step sizes. This completion concept provides a systematic route to higher-order energy-consistent schemes. For purely dissipative systems, an alternative exponential completion is derived, together with structure-preserving rational approximations. In particular, we characterize Padé approximants that preserve positive semi-definiteness and can be incorporated into energy-associated splitting schemes. Numerical experiments on a damped physical pendulum and a dissipative Fermi–Pasta–Ulam system demonstrate the importance of the positive semi-definite completion for avoiding nonphysical numerical instabilities and capturing the correct qualitative dynamics. They further show that the proposed methods achieve the desired accuracy and energy consistency at a computational cost competitive with existing Galerkin-type approaches. Moreover, the fourth-order schemes outperform the conventional second-order discrete gradient method in computational efficiency over the considered accuracy range, despite their higher per-step cost.

Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions. These inequalities have been established primarily for the uniform measure and for biased product measures, often through Fourier-analytic or inductive arguments. We develop a curvature-based semigroup framework to establish isoperimetric inequalities for measures with weakly dependent coordinates. In particular, we show that a Dobrushin-type condition, together with marginal boundedness of the distribution, implies the local Bobkov inequality, Talagrand's $L^1$–$L^2$ and variance–surface-area inequalities, the Kahn–Kalai–Linial inequality, and the Eldan–Gross inequality. Our results apply to zero-field Ising models with interaction matrix $J$ throughout the Dobrushin uniqueness regime $\|J\|_1<1$. The framework builds on discrete Bakry–Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.

We establish a strongly sublinear counterpart of a recent result of Chudnovsky, E S, and Lokshtanov (arXiv 2025) on treewidth and tree-independence number. Namely, we prove that a hereditary graph class has strongly sublinear tree-independence number if and only if, for every fixed clique bound, its graphs of bounded clique number have strongly sublinear treewidth. In fact, this is part of a broader equivalence theorem. For hereditary classes, these conditions are also equivalent to having clique-dependent polynomial expansion, to admitting balanced separators whose size is bounded by $K\omega(G)^s |V(G)|^{1-\beta}$ for fixed $K,s,\beta>0$, and to admitting balanced clique-based separators of strongly sublinear size (equivalently, weight). Thus, we show that all these properties, which arose independently in the study of subexponential-time exact algorithms and polynomial-time approximation schemes, in fact describe the same hereditary graph classes. As a consequence of our equivalence theorem, we also show that every hereditary class $\mathcal C$ with strongly sublinear tree-independence number admits a subexponential-time algorithm that, given $G\in\mathcal C$, computes a tree decomposition of $G$ with strongly sublinear independence number.

We propose a kernel-based estimation framework for Multivariate Generalized Gamma convolutions (MGGC), a class of probability distributions on R d + . MGGC are useful for risk modeling e.g. Their densities are in general not explicit. The construction relies on a new Stein operator for the MGGC class, derived from a partial differential equation satisfied by the characteristic function and expressed as an integral against the underlying Thorin measure. This leads to construct a new kernel and an estimation procedure in the derived reproducing kernel Hilbert space (RKHS); with consistency results. Numerical experiments illustrate the practical behaviour of the proposed estimator.

We determine the exact second-order asymptotics of covert communication over binary-input discrete memoryless channels when covertness is measured by variational distance. Previous work by Tahmasbi and Bloch [IEEE Trans. Inf. Theory, Apr. 2019] characterized the first-order asymptotics and derived achievability and converse bounds on the second-order term, but these bounds do not match. The gap arises from an additional penalty of order n^{1/4} in the achievability bound. We show that this penalty can be removed through a sharper analysis of the distribution of the warden's output induced by pulse-position modulation. Specifically, we express the variational distance through the Bhattacharyya coefficient of two distributions and the expectation of a continuous function of the log-likelihood ratio. Because the resulting expectation involves a continuous function rather than the probability of a likelihood-ratio event, an analysis of the characteristic function combined with a Gaussian smoothing argument reduces the approximation error from O(n^{-1/4}) (derived from the Berry–Esseen bound in prior work) to O(n^{-1/2}). With this better controlled approximation error, we manage to derive a matching achievability result to the existing converse result, thus establishing the exact second-order asymptotics.

A key motivation for Heyting's intuitionistic logic was to gain a formal notion of Brouwer's idea of mathematics as a "construction of the mind". One might thus argue that Heyting's Calculus should also correspond to a notion of provability. Inspired by this idea, Gödel formalised this connection via an embedding into a modal calculus, which is now known as the modal logic S4. While in his original publication, he only proved soundness for the embedding from intuitionistic propositional logic into S4, the converse was proved fifteen years later by McKinsey and Tarski. Although, it was later discovered that Gödel also had obtained a proof of the faithfulness of his embedding in unpublished notes in 1941. Rasioa and Sikorski later extended Gödel's embedding to first-order intuitionistic logic. In 1954, Maehara independently obtained the same results using proof-theoretic methods, even extending the embedding to one from intuitionistic first-order logic into intuitionistic first-order modal logic. This document presents a faithful English translation of Maehara's 1954 paper.

For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into $\mathrm{PU}_{2,n+1}$ with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension $10(g-1)$ or $10(g-1)+2(n-1)$ in the character variety depending on the Toledo value. Our construction uses $4$-cyclic Higgs bundles and their correspondence with $\partial$-alternating surfaces in complex pseudo-hyperbolic spaces. The associated equivariant minimal maps into the symmetric space are bi-Lipschitz embeddings. We further construct cocompact domains of discontinuity in the Shilov boundary. Their quotients are smooth fiber bundles over the surface with fiber homeomorphic to $S^{2n-1}\times S^{2n-1}$.

We report on frequency bursting oscillations in a system of phase oscillators with adaptive and delayed coupling. Adaptation of the coupling strengths is considered slow and depends on the phase shift between the oscillators. We find due to the combined chain of adaptation, collective dynamics, and time delays, the system robustly achieves a state in which the oscillator's frequencies are nearly synchronized but detuned by an integer number of small adaptation frequencies. We demonstrate that this quantization of the detuning is caused by alternating slow and fast transitions. Moreover, the observed motions take the form of bursts of instantaneous frequency, and the number of spikes in each burst corresponds to the quantization level of the detuning. We provide a fast-slow analysis of this phenomenon and explain the mechanisms behind the emergence of bursts. Our findings indicate that these frequency bursting oscillations are robust and exist stably within finite parameter regions.

We compute an explicit contribution of supercuspidal representations arising from Yu's construction to the middle étale cohomology of local Shimura varieties. For the proof, we construct special affinoids in local Shimura varieties whose reductions are isomorphic to Heisenberg Deligne-Lusztig varieties and then apply positive-depth Deligne-Lusztig theory. Along the way, we propose a stacky analogue of special affinoids using the stack of local shtukas and observe that the `general canonical subgroup problem' can be solved purely on the stack of local shtukas, which yields the corresponding claim on Shimura varieties via the Hodge-Tate period map.

We study arithmetic operations and maps which preserve normality. Revisiting a recent argument of Dayan, Ganguly and Weiss, we give a streamlined, self-contained presentation of the probabilistic proof that the non-zero rationals are exactly the multiplicative normality preserving numbers in a fixed integer base. We also include the extension to all bases simultaneously, using the theorem of Hochman and Shmerkin. In the additive case, we extend Rauzy's characterization of deterministic numbers in a fixed integer base to absolutely normal numbers. Namely, a translation preserves absolute normality if and only if its parameter is deterministic in every integer base. We prove this extension by a sparse random perturbation argument. Finally, we show that all locally $C^2$ normality preserving maps need to be affine-linear. The regularity assumption cannot be weakened to $C^{1,1}$: we construct a non-affine $C^{1,1}$ diffeomorphism which preserves normality in every integer base, and hence absolute normality.

Detecting hidden convexity is one of the tools to address nonconvex minimization problems, and find global minimizers. We introduce the notion of conditional infimum, develop the theory, and establish a tower property, relevant for minimization problems. Then, we illustrate how the conditional infimum is instrumental in revealing hidden convexity. Thus equipped, we provide a new sufficient condition for hidden convexity in nonconvex quadratic minimization problems, that encompasses and goes beyond known results (with the notion of block-signed pair matrix-vector). We also show how the conditional infimum is especially adapted to tackle the so-called S-procedure.

We consider relatively prime integer numbers $m$ and $n$ such that each group of order $mn$ has a normal subgroup of order $m$. We prove that any skew brace of size $mn$ has an ideal of size $m$ and, if the subgroup of order $m$ of its additive group is abelian, a subbrace of size $n$. We give a method to classify skew braces of size $mn$, such that the subgroup of order $m$ of their additive group is abelian, from the classification of braces of size $m$ and the classification of skew braces of size $n$. We apply this result to determine all skew braces of size $p^2q^2$ of non-abelian type, for $p$ and $q$ odd primes, $p<q$, such that their ideal of order $q^2$ is cyclic.

In this paper, we prove that there exist a symmetric quantum Latin square with maximal cardinality of order $n$ for all $n\geq 6$, and a symmetric idempotent quantum Latin square with maximal cardinality of order $4k+2$ for all $k\geq 1$.

We study randomized stopping times for a Markov process, described by a progressively measurable intensity $\alpha_t(\omega)$. We prove that every admissible intensity has a Markovian representative $\lambda(t,X_t)$, explicitly obtained from the observed measure and the surviving occupation measure. This representative preserves both the family of surviving mass measures and the joint distribution of the stopping time and the state at stopping. Whenever the relative entropy with respect to a reference intensity $r(t,x)$ is finite, we prove an exact decomposition showing that Markovization does not increase the entropy. As a consequence, variational problems in which a randomized stopping time enters only through its observed measure and an entropy penalty can be reduced to optimization over Markovian intensities $\lambda(t,x)$. For every observed measure admitting a finite-entropy representative, there is a unique minimum-entropy randomized stopping time, and it is Markovian. We discuss the connection with optimal Skorokhod embedding and prove a finite-constraint realization result for Brownian stopping. We also approximate arbitrary observed measures by those generated by bounded Markovian intensities.

We compare the performance of the RPA and CHIRRUP decoding algorithms for second-order Reed–Muller codes. The RPA algorithm recovers a binary quadratic form associated with a skew-symmetric matrix $M$, whereas CHIRRUP recovers a $\mathbb{Z}_4$-valued quadratic form associated with a symmetric matrix $P$. We describe how the Gray map connects the evaluation vectors of these forms, inducing a rank-preserving correspondence between $M$ and $P$. We analyze how Euclidean distance between evaluation vectors is governed by matrix rank differences using Delsarte–Goethals sets $DG(m,r)$. We demonstrate that CHIRRUP uses repeated measurements more effectively than RPA on these ensembles. While a kernel-aware variant of RPA improves performance on singular quadratic forms by aggregating projection directions across cosets of $\ker M$, CHIRRUP's tree search combines multiple row estimates more efficiently on $DG(m,r)$ when $r < \frac{m-1}{2}$, outperforming vanilla RPA.

We continue to develop a theory of Besov and Triebel-Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, specializing to the case of groups with polynomial volume growth. We consider the full scale of spaces and consider interpolation, localization, and pointwise multiplication.

Finite simple skew braces with non-abelian additive group remain largely unexplored. In particular, to the best of our knowledge, no finite simple skew brace with non-abelian nilpotent additive group had previously been constructed. We introduce an inverse Baer deformation which produces skew braces of additive nilpotency class at most two from braces of abelian type. Starting from a finite left brace $C=(A,\oplus,\circ)$ of odd order and a suitable biadditive alternating map $\kappa:A\times A\to A$, we define \[ x+y=x\oplus y\oplus\frac12\kappa(x,y). \] The resulting skew brace $\IB_\kappa(C)$ satisfies $\Br(\IB_\kappa(C))=C$ and $[x,y]_+=\kappa(x,y)$, and simplicity passes from $C$ to $\IB_\kappa(C)$. Applying this construction to simple left braces arising from asymmetric products, for every pair of odd primes $p,q$ with $p\mid(q-1)$ and every $m\ge1$ we construct a finite simple skew brace $X_{p,q}^{(m)}$ of order $p^{2m(q-1)+1}q$ such that \[ (X_{p,q}^{(m)},+)\cong E_{p,q}^{(m)}\times C_q, \] where $E_{p,q}^{(m)}$ is an extraspecial $p$-group of order $p^{2m(q-1)+1}$ and exponent $p$. Thus every fixed admissible pair $(p,q)$ yields infinitely many pairwise non-isomorphic finite simple skew braces with non-abelian nilpotent additive group of class two.

Katz centrality is a well-established measure to identify and rank the most important nodes in complex networks by means of a linear system solve. Recent works have developed notions of Katz centrality for temporal, i.e., time-evolving networks. Their drawback is that small changes in the network structure may drastically change centralities across large parts of the network. Moreover, an unproportional effort in the network science community has been devoted to the study of node-based quantities while edge-based measures are far less explored. In this manuscript, we introduce a novel failure-aware temporal multiplex network model for computing edge-based Katz centralities for spatial networks. This class of networks admits the use of the line graph as network representation. We use a block-triangular and -banded supra-adjacency matrix representation, modeling inter-layer connections as low-rank matrices, which assigns a spatially constrained emphasis on up- or downdated edges. These could represent, e.g., breakages of water/gas pipes or obstructions in street networks in real-time infrastructure monitoring applications. We analyze the structure of the of the block matrix inverse with respect to entry decay in the blocks above the block-diagonal. This gives rise to two truncation approaches that allow drastic computational runtime reductions at the cost of introducing a controlled truncation error. Numerical experiments on a range of real-world spatio-temporal networks of size up to $2\cdot 10^8$ illustrate accuracy and efficiency with runtime gains of up to a factor of $679$.

We study rough differential equations driven by controlled rough paths with drift. By means of the canonical lift of the controlled driver, we establish pathwise well-posedness and continuity. Under compatible time-shift conditions, we construct differentiable solution cocycles for the associated dynamical system generated by the equations considered here. Furthermore, assuming the corresponding regularity, moment, and spectral hypotheses, we prove the invariance of the local stable, unstable, and center manifolds.

We consider the parametrised family of sets $$ E(x,y)=\left\{\sum_{n=1}^{\infty}\frac{\varepsilon_n}{4^n}: (\varepsilon_n) \in \{0,x,y,x+y\}^{\mathbb{N}}\right\} $$ for $(x,y) \in \mathbb{N}^2$. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of $E(x,y)$ for $(x,y) \in \mathbb{N}^2$. Next, we collapse this topological classification to a binary one according to whether or not $E(x,y)$ has interior. When this binary classification is visualised, it reveals a two-colour tiling $T$ of the lattice $\mathbb N^2$, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family $\{E(x,y)\}_{(x,y) \in \mathbb{N}^2}$ is viewed through the theory of self-similar sets, $T$ can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that $T$ is a factor of a substitution tiling on four “hidden” arithmetically defined states.

We establish arbitrary-order a priori regularity estimates for the reverse heat flow in the Ornstein–Uhlenbeck setting, that is, for the probability-flow ODE of score-based diffusion models. For compactly supported, possibly singular targets satisfying a dimension-uniform doubling condition on supporting-cap masses, we bound the iterated material derivatives of the flow and their first spatial derivatives pointwise, with constants independent of the dimension $d$ and explicit in time and support radius. The proof rests on two ingredients: a spatially uniform bound on normal-direction posterior fluctuations, which is the only point where the geometry of the target enters, and a centered posterior-moment normal form closed under material differentiation. This algebraic structure, together with a weighted regularity calculus, yields quantitative bounds at every order, including the mixed space–time directional derivatives required by Runge–Kutta stages. As an application, for exact-score sampling stopped at forward time $\delta>0$, order-$p$ Taylor and explicit Runge–Kutta schemes reach accuracy $\varepsilon$ in $\widetilde O(d^{1/p}\varepsilon^{-1/p})$ steps in total variation, and the Taylor scheme in $\widetilde O(d^{1/(2p)}\varepsilon^{-1/p})$ steps in $W_2$.