Papers

The construction of a $p$-adic height pairing for a geometric $p$-adic representation of the absolute Galois group of a number field depends on a global $p$-adic logarithm and on local splittings of the Hodge filtrations at the primes above $p$. We study the dependence on these splittings for suitable two-dimensional symplectic self-dual representations, including self-dual twists of representations attached to even-weight newforms at non-ordinary primes not dividing the level. We express the difference between the height pairings associated with the two splittings determined by Frobenius explicitly in terms of local Bloch–Kato logarithms. As an application over $\mathbb{Q}$, we prove that at least one of the two cyclotomic $p$-adic height pairings is non-trivial under the additional assumptions that the Frobenius eigenvalues at $p$ are distinct and the localization map at $p$ from the Bloch–Kato Selmer group is non-zero.

We compute the integral Bott–Chern cohomology of a class of generalized Calabi–Eckmann manifolds of complex dimension \(3\). Although these manifolds are mutually non-biholomorphic, we show that their integral Bott–Chern cohomology rings are nevertheless isomorphic. A cohomological distinction is then obtained via a characteristic map in cohomology, connecting singular, Dolbeault, and integral Bott–Chern cohomologies.

We classify the maximum scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^5)$, proving that every such set is of pseudoregulus type or of Lunardon-Polverino type. Building on the reduction obtained by Lia, Longobardi and Zanella in [S. Lia, G. Longobardi and C. Zanella, Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$, Algebraic Combinatorics 9 (2026), 327-355], we show that the two remaining candidate families contain no scattered linear sets. Our approach reduces these candidates to two normal forms and establishes their non-scatteredness through the existence of rational points on associated algebraic varieties.

We prove that each of the inequalities $P^{+}(p+1)>P^{+}(p-1)$ and $P^{+}(p-1)>P^{+}(p+1)$ holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.

LVMB manifolds are a large class of compact complex non-Kähler and non-algebraic manifolds, constructed from an algebro-combinatorial datum. We provide an upper bound on their algebraic dimension and prove it to be sharp. Comparing our result with the lower bound found by Meersseman, we obtain the exact algebraic dimension of an LVMB manifold in many relevant cases. Pursuing our analysis further, we also obtain a description of the algebraic reduction of an LVMB manifold. From this, it emerges that Meersseman's lower bound coincides with the dimension of the rational component. This bimeromorphic invariant of an LVMB manifold is then computed in terms of the nonrationality degree of the associated toric quasifold. This is an invariant recently introduced by Battaglia and Prato, for which we provide an explicit formula.

We construct complete canonical logical bases for qubit CSS codes by assigning quaternary coefficients to binary circulant permutation matrix pair-partition (CPM-PP) checks, constructing and normalizing logical representatives, and expanding the result into binary matrices. When the two check systems have invertible submatrices on disjoint column sets, cofactor representatives are canonically paired by the inverse of a single pairing polynomial. The resulting pairs span the entire logical space. When the pairing polynomial is a cyclic-shift monomial with coefficient one, normalization preserves the binary weights of the representatives. We state the construction for general block dimensions and CPM size, work through a corresponding example, and report the parameters, check ranks, and basis weights of seven binary codes. Representative examples have parameters $[[320,80,14]]$, $[[448,112,18]]$, and $[[2048,512,24]]$. All three have maximum check weight 10 on both the X and Z sides. Their canonical logical representatives have binary weights 25, 27, and 27, respectively, on both the X and Z sides.

Let \(X\) be a rationally smooth complex affine variety with a torus action and an attractive fixed point \(x\). Suppose that \(X\setminus\{x\}\) is \(p\)-smooth and that its integral equivariant cohomology has no \(p\)-torsion. We prove that the order of its total \(p\)-primary cohomology torsion is the \(p\)-part of the reduced numerator of the equivariant multiplicity at \(x\). In particular, this resolves the torsion-order conjecture of Juteau–Williamson and its local version for normal slices in Schubert varieties, using the equivariant torsion-freeness result of Fiebig–Williamson.

We show that the degree $d$ Jouanolou foliation $\mathcal J_d$ on $\mathbb P^2$ is structurally stable for every $d \geq 2$. We prove the existence of a neighbourhood $\mathcal U_d$ of $\mathcal J_d$, such that every foliation $\mathcal F \in \mathcal U_d$ satisfies: the Fatou set is a connected disc bundle over a compact Riemann surface of genus $d(d+1)/2$, no leaf is dense in $\mathbb P^2$, and all but countably many regular leaves are biholomorphic to the unit disc. Finally, we prove transverse perfectness of the Julia set on the regular locus for $d \geq 2$.

We consider the problem of minimizing a sparse quadratic function over the unit hypercube. In binary quadratic programming, treewidth of the interaction graph is a central parameter for tractability: bounded treewidth yields polynomial-time solvability. Motivated by this fact, we investigate whether treewidth plays a similar role when the binary domain is replaced by the unit hypercube. We show that the situation is strikingly different. If the interaction graph is a forest, we give a strongly polynomial-time algorithm based on dynamic programming and a structural analysis of the resulting univariate value functions, which are shown to be concave and piecewise quadratic with linearly many pieces. However, the problem becomes strongly NP-hard already when the interaction graph has treewidth two. We then identify substantially more general polynomial-time solvable classes whose interaction graphs may have unbounded treewidth. Our approach exploits the fact that there exists an optimal solution in which every variable with a nonpositive coefficient for its square term is binary-valued, thereby separating the problem into a combinatorial part and a genuinely continuous part. We show that tractability can be recovered by controlling the complexity of these two parts and their interaction, rather than the treewidth of the entire interaction graph.

We develop a high-order fully discrete ALE-SAV finite element method for sharp-interface two-phase Navier–Stokes flow that satisfies a discrete energy dissipation law. We introduce an SAV reformulation to ensure the stable treatment of the surface and gravitational potential energies, together with an energy-stable ALE transfer operator that controls the density-weighted kinetic energy across moving meshes. The resulting scheme is linear and satisfies a modified energy-dissipation inequality. Numerical experiments, including rising-bubble benchmarks, demonstrate the robustness of the method and confirm second-order convergence in time and arbitrarily high-order convergence in space.

In this note, we prove that for n > 2 a harmonic map $\mathbb{S}^n \to \mathbb{CP}^m$, of index n+1 is pseudo horizontally weakly conformal, that $n$ must be odd. In particular, for even n every non-constant harmonic map $\mathbb{S}^n \to \mathbb{CP}^m$ has index at least n+2. We also present a harmonic map from $\mathbb{S}^4$ to $\mathrm{Gr}_{\mathbb{C}}(2,4)$ of index 5 that is not pseudo horizontally weakly conformal, so the statement does not extend to Grassmannians.

We prove a formula for the total variation of SBV functions in sub-Riemannian Heisenberg groups arising as the limit of certain BMO-type functionals.

For probability measures on $\mathbb{R}^d$ supported in fixed compact sets, we prove that quadratic optimal transport plans are quantitatively stable in Wasserstein distance under perturbation of both marginal measures, assuming that one of the initial measures satisfies an upper Ahlfors regularity condition with exponent strictly greater than $d-1$. Under the same assumption, we also prove novel quantitative stability results for optimal transport maps. Furthermore, we prove a coercivity theorem which states that any probability measure on the product space must be quantitatively close to the set of optimal plans, if it has similar marginals and a similar transport cost to the optimal value. Finally, we prove a quantitative uniqueness theorem which acts as a quantitative counterpart to Brenier's theorem. The Wasserstein diameter of the set of optimal plans is controlled by the distance of one marginal measure to a regular measure for which uniqueness holds. In this way, "almost uniqueness" of optimal plans is quantified by the source measure being "almost regular". Examples are provided which prove that the exponents of all estimates are sharp.

Let $\Gamma$ be a countable discrete group and let $A$ be a unital $\Gamma$-simple $\Gamma$-$C^*$-algebra. We prove that $A\rtimes_r\Gamma$ has the generalized Powers averaging property if and only if the canonical map $A\rtimes_r\Gamma\to I_\Gamma(A)$ is the unique $\Gamma$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_\Gamma(A)\rtimes_r\Gamma$, with the averaging coefficients still belonging to $A$.

We associate to each compact subset $K$ of a closed symplectic manifold $M$ an $A_\infty$-category $\Fuk_K$, called the Fukaya category with support on $K$. The objects are tautologically unobstructed global Lagrangians endowed with appropriate decorations. The dependence on $K$ is detected by the morphism complexes which can be computed by completed telescopes associated with acceleration data for $K$. In particular, these vanish for Lagrangians disjoint of $K$. For $K=M$ this recovers the usual Fukaya category of tautologically unobstructed objects. These categories admit restriction functors under inclusions of compact subsets. We prove a Mayer–Vietoris descent theorem for weakly involutive covers. The construction is carried out over the Novikov ring and is established whenever classical transversality methods apply.

We prove that the postcritical set of every quadratic polynomial with a neutral fixed point has zero Lebesgue measure. In particular, no arithmetic condition on the rotation number is required. Our proof uses sector renormalization and the pseudo-Siegel bounds of Dudko-Lyubich to derive uniform distortion estimates for the renormalization change of variables. For sector renormalization towers, we also prove the same zero area statement for the associated postcritical set and the optimal Brjuno criterion for the interior of the associated Mother Hedgehog.

We prove that $(G\circ G^{-1})^{-1}$ is entrywise nonnegative for every real symmetric positive definite matrix $G$ of order at most six. Its entries are strictly positive exactly within the connected components of the nonzero-entry graph of $G$. An explicit rational example of order seven establishes sharpness. We give two complementary structural proofs. The analytic proof combines a trace obstruction with continuation of positive column forms and a bordering identity. The algebraic proof establishes a stronger four-dimensional polarized inequality through an explicit ten-square polynomial certificate. A common framework relates biquadratic positivity, column forms, positive semidefinite precision loads, and inverse relative gain array nonnegativity in consecutive dimensions. Further consequences include exact load-deficit identities, polynomial sum-of-squares certificates with the covariance fixed, and a reverse Schur–Horn majorization inequality with the same sharp dimension bound. In every dimension, sufficiently large powers of the inverse relative gain array are doubly stochastic.

We prove that the $g$-polynomial of every split matroid has nonnegative coefficients, establishing Speyer's conjecture for a class closed under taking minors and containing all paving and copaving matroids. Our proof uses a deletion–contraction identity obtained by constructing an auxiliary split matroid. We first show that every simple, cosimple, connected split matroid $M$ has an element $e$ for which both the deletion $M\setminus e$ and the contraction $M/e$ are connected. More generally, let $M$ be any connected split matroid of rank $k$ on a ground set $E$ with $|E|\ge4$. Suppose that $e\in E$ is such that both $M\setminus e$ and $M/e$ are connected. For every $v\in E\setminus\{e\}$, we construct a connected elementary split matroid $N_{e,v}$ of rank $k-1$ on $E\setminus\{e,v\}$ satisfying \[ g_M(t)=g_{M\setminus e}(t)+g_{M/e}(t)+t\,g_{N_{e,v}}(t). \] Using a fixed total order on $E$, we prescribe the proper cyclic flats of $N_{e,v}$ and their ranks. The identity follows from the covaluative formula of Ferroni and Schröter together with recurrences for its correction polynomials, derived from Ferroni's enumeration of admissible Delannoy paths for Schubert matroids. Since all three matroids on the right have fewer elements, the identity supplies the induction step in the proof of nonnegativity.

Let $q_n$ denote the $n$th product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every $\varepsilon>0$ and all sufficiently large integers $\rho$, \[ \liminf_{n\to\infty}(q_{n+\rho}-q_n) \leq \exp\left((\pi+\varepsilon)\sqrt\rho\right). \]

Let $G$ be a reductive group over a number field $F$. We define $C^*_{\rm aut}(G(\mathbb A))$, the automorphic $C^*$-algebra of $G$, to be the $C^*$-algebraic image of the full automorphic representation of $G(\mathbb A)$ on $L^2(G(F)\backslash G(\mathbb A))$. Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective $*$-homomorphism \[ C^*_{\rm aut}(G(\mathbb A)) \longrightarrow \bigoplus_{[M,\sigma]} K_{C_0(\widehat{A_M})} \left( \operatorname{Ind}_P^G C_0(\widehat{A_M},H_{M,\sigma}) \right)^{W(G,M,\sigma)}, \] where $[M,\sigma]$ ranges over the associate classes of discrete Levi data. When $G$ is $\operatorname{GL}(n)$ or an inner form of it, we prove that this map is an isomorphism of $C^*$-algebras.

We give a complete classification of finite left-simple skew braces. We prove that a non-trivial finite skew brace has no non-trivial proper left ideals if and only if it is either the trivial skew brace of prime order or the almost trivial skew brace associated with a finite non-abelian simple group.

In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter $\alpha > 0$. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size $h$ tends to zero and the penalization parameter $\alpha$ tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.

Surrogate models deployed on real physical systems rarely see data at fixed resolutions: sensor configurations vary across deployments and may evolve over time as sensing infrastructure changes. We introduce the Latent Twin Operator (LTO), a latent-space surrogate for time-evolving PDEs with a Convolutional Conditional Neural Process (ConvCNP)-style encoder and decoder that accepts a context set of $N$ sensor observations with arbitrary placement and can be queried at any resolution. A learned latent evolution map advances the encoded state directly between arbitrary time points, decoupling temporal evolution from the observation and query discretizations. We derive an explicit $\mathcal{O}(N^{-2/(3D)})$ rate for the context discretization error in spatial dimension $D$ under quasi-uniform refinement. We verify the predicted decay empirically on a 2D heat-equation benchmark. Across time-dependent PDE benchmarks, LTO achieves strong accuracy under one-step comparisons and transfers from native to coarser spatial resolutions with fixed parameters. On Navier–Stokes, its direct latent evolution further reduces error over longer prediction horizons relative to recursive evaluation.

Zero forcing is a process described by a color change rule on the vertices of a graph. In this paper, we maximize the discrepancy between various zero forcing parameters relative to graph order. First, we find an upper bound on the difference in cardinality between minimal zero forcing sets (sets containing no proper zero forcing subset) of maximum and minimum size, and we show that this bound is sharp for an infinite family of graphs. Furthermore, we derive an upper bound for the discrepancy between the maximum and minimum propagation times of the minimum zero forcing sets of any graph, showing this bound is sharp for an infinite family of graphs.

We prove direct and inverse estimates for band-limited biorthogonal wavelet systems in weighted Besov spaces with Muckenhoupt weights. On functions recovered from their wavelet expansion along the dyadic tree of the unit cube, the error of approximation by a prescribed number of terms decays at a rate set by the smoothness-to-dimension ratio, and this exponent cannot be increased. Under a multiresolution condition, Bernstein inequalities hold at a single scale and on every multiresolution subspace, and the linear approximation spaces are identified with weighted Lebesgue–Besov intersections. All constants are explicit; the weight enters only through displayed powers of the Muckenhoupt characteristic.

This work constructs a new function class over the algebra of dual quaternions: dual slice functions, which include one-sided dual quaternionic polynomials. The peculiar properties of dual slice functions therefore provide new tools to understand factorization of dual quaternionic polynomials, which was extensively studied in recent years for its applications to mechanism science. Dual slice functions are strictly related to the well-established theory of slice and slice regular functions over quaternions and dual quaternions.

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{\sigma\to 0^+} \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T \lvert f(\sigma+it) \rvert^2\,dt.\] We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau–Schnee convergence theorem for $\mathscr{H}^2$. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.

Many learning tasks map an input distribution to an output distribution. A natural way to model such an operator is to transform each input sample using a continuous function that may depend on the entire input distribution, and then take the distribution of the transformed samples. This defines a measure-dependent pushforward model and includes measure-theoretic formulations of transformers. We ask when such models can approximate arbitrary continuous operators between spaces of probability measures. We first show that universal approximation fails when atomic inputs are allowed: some continuous measure-to-measure operators that split or redistribute atomic mass cannot be approximated arbitrarily well by deterministic pushforward models. We then introduce the uniform level set condition, which requires a continuous measure-dependent scalarization whose shrinking level set neighborhoods carry uniformly vanishing mass over the input family. This condition is satisfied, in particular, by compact families of absolutely continuous measures. On every compact family satisfying this condition, we prove that any continuous measure-to-measure operator with outputs of finite $p$-th moment can be uniformly approximated, in the $p$-Wasserstein distance, by continuous measure-dependent pushforwards. Combining our theorem with existing approximation results for measure-dependent in-context maps yields universal approximation by measure-theoretic transformers. We also extend the framework to continuously-varying source measures, yielding a corresponding universality result for a class of pushforward models that are closely aligned with cross-attention architectures.

Let $D\subseteq \mathbb{C}^n$ be a bounded $\mathbb{C}$-convex domain with $C^1$ boundary whose outward unit normal admits a modulus of continuity $\omega$ satisfying $\lim_{t\to 0^+}\dfrac{\omega(t)}{\sqrt t}$. We prove that $D$ admits an increasing exhaustion by bounded $C^{\infty}$ strongly linearly convex domains. This, in particular, answers a question posed by Azinberg \cite{azin} in affirmative for a class of domains $D$.

Let $X$ be the good reduction of a projective hyperkähler variety of dimension $2n\geq4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $\operatorname{H}^2_{\operatorname{crys}}(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. The proof combines a Verbitsky slope comparison with a Witt–Euler identity and requires no crystalline torsion-freeness. The same dichotomy holds for primitive symplectic varieties in characteristic $p$, and Hodge-goodness is open in smooth proper families. Hodge-deformations of Hilbert schemes $S^{[n]}$ of $K3$ surfaces ($p>n$) and generalised Kummer varieties $K_n(A)$ ($p>n+1$) remain primitive symplectic, with torsion-free crystalline cohomology and unobstructed mixed-characteristic formal deformations.