Papers

In this paper we study the least positive integer $G(n)$ such that the integers $a\leq G(n)$ with $(a,n)=1$ generate $(\mathbb Z/n\mathbb Z)^\times$. We prove that, for every $\varepsilon>0$, \[ \sum_{n\leq x}G(n)\ll_\varepsilon x(\log x)^{8/3+\varepsilon}, \] improving the previously known bound $\ll x(\log x)^{97}$. We also prove that $G(n)\leq(\log n)^2$ for almost all $n$, unconditionally. Thus, for almost all $n$, we obtain the same logarithmic exponent $2$ as in the classical pointwise bound under GRH.

Policy learning in digital experimentation faces three challenges: weak signal-to-noise ratios, rich covariate spaces, and massive data volumes. We formalize this regime by modeling treatment-effect estimates from increasingly fine covariate partitions as Gaussian observations with bounded signal-to-noise ratios. We establish that, in general, the optimal treatment policy is not learnable in this setting. Even learning the optimal policy value suffers from impractically slow rates. However, when treatment effects vary smoothly, we derive minimax-adaptive policies based on linear smoothers that achieve vanishing welfare regret. We demonstrate the practical value of our framework by applying it to large-scale real-world experiments at Netflix, showing that personalized linear-smoothing policies can dominate unpersonalized policies even in this challenging empirical setting.

We consider the Dirichlet problem for the Lorentz–Minkowski mean curvature operator in a radial setting, $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu g(u) \quad \text{in } \mathcal B_R, \qquad u = 0 \quad \text{on } \partial\mathcal B_R, $$ on a ball of radius $R$ centered at the origin $\mathcal B_R \subset \mathbb{R}^N$, where $\lambda, \mu \geq 0$ are parameters, and the nonlinearity $g$ satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as $\mu \to +\infty$, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function $u_\infty$, with $|u_\infty'|=1$ a.e., having the same nodal properties. We then show that, when the number of zeros is appropriately prescribed, both alternatives are actually realized. More precisely, building on the multiplicity result in [A. Boscaggin, F. Colasuonno, and R. Ziegele, Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics, arXiv:2607.15956 (2026)], we refine the shooting construction therein to obtain a family of small solutions converging to zero and, through a backward shooting argument, a family of large solutions converging to a saturated piecewise affine profile.

A centrally excited random walk on ${\mathbb Z}^d$ moves like simple random walk, except that its first step from each site has a drift of fixed size toward the origin. Kozma (2007) conjectured that after $n$ steps the visited set approximates a ball with radius of order $n^{ 1/(d+1)}$. We prove this in every dimension $d\geq2$ and show that the rescaled numbers of visits converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that a drift opposite to a subgradient of a norm produces the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most $ n^{1/6}$ times a power of $\log n$, and the exponent $1/6$ cannot be lowered.

For every $1<p<\infty$, the Hilbert transform on $L^p(\mathbb{R};\mathcal{S}_p)$ and the ordered Hilbert Schur multiplier on $\mathcal{S}_p$ have norm $C_p=\cot(\pi/(2p^*))$, where $p^*=\max\{p,p/(p-1)\}$. The scalar Hilbert transform and the Schur multiplier have the same canonical completely bounded norm. This proves the Gohberg-Krupnik sharp Volterra conjecture. A simultaneous finite-dimensional realization also resolves the Riesz-Titchmarsh conjecture and Laeng's Conjecture 5.7 for all shifts. The norm identities extend to finite families of real Hilbert-transform pencils with $\ell^p$ direct-sum ranges. The proof uses a spectral-defect identity with two nonnegative remainders, based on Heinävaara's tracial joint spectral measure. It classifies trace inequalities for finite signed real ridge sums and identifies the least homogeneous superharmonic majorant as the pointwise least additive eigenvalue correction. The associated angular cone is characterized by nonnegative potentials and realized by weak-* limits of weighted angular measures of finite nilpotent matrices.

We construct deformations of contraction algebras of type E to semisimple algebras. We consider the class given by the potential x^3+xy^3+p(y), where p(y) is a non-zero polynomial of order larger or equal to 4.

The logarithmic Brunn–Minkowski and logarithmic Minkowski inequalities are proved for convex bodies whose centroids are at the origin, with equality precisely for independent positive dilations of common direct summands. This characterizes when centered convex bodies have the same cone-volume measure. The \(L_p\) Brunn–Minkowski and Minkowski inequalities are also obtained for every \(p>0\), with equality only for positive dilates, and the corresponding uniqueness of centered \(L_p\) surface area measures is established.

To quantize the reduction of a coisotropic submanifold of a Poisson manifold, Cattaneo and Felder constructed a homotopy Poisson algebra together with a deformation quantization. Its degree-zero cohomology is the reduced Poisson algebra, while the full homotopy structure retains information about the formal embedding even when the reduced space is singular or trivial. On the integration side, Cattaneo showed that a coisotropic submanifold integrates to a Lagrangian subgroupoid of any symplectic groupoid integrating the ambient Poisson manifold. It is now recognized that both constructions fit into shifted symplectic geometry: the Lagrangian subgroupoid presents a $1$-shifted Lagrangian morphism of differentiable stacks, and the homotopy Poisson algebra describes the associated $0$-shifted Poisson structure. However, such morphisms form only a special class of $1$-shifted Lagrangians. We extend this picture to $1$-shifted Lagrangians and, more generally, $1$-shifted coisotropics in $1$-shifted symplectic differentiable stacks. We establish an integration theorem for their infinitesimal description in terms of twisted Dirac structures and construct a natural Poisson bracket on invariant functions. Under regularity hypotheses, we equip the associated Lie algebroid complex with a flat $P_\infty$-structure and a curved $A_\infty$-quantization, independent of the auxiliary choices up to isomorphism. The induced bracket in degree zero recovers the reduced Poisson bracket, and suitable cohomological vanishing conditions yield a deformation quantization of invariant functions. This extends the Cattaneo–Felder picture to more general reduction procedures, with the same infinitesimal data governing integration and quantization.

We construct effective finite group actions on smooth projective complex curves for which Conjecture A of Polishchuk and Van den Bergh fails. For every $h\geq1$, an explicit $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ has quotient of genus $h$ and exactly one ramification orbit, with inertia of order three. The fixed quotient for the three-cycle class consists of two points, but its derived category admits no $\C$-linear exact fully faithful embedding into the equivariant derived category. More generally, when the coarse quotient has positive genus, we prove that the maximum number of pairwise completely orthogonal exceptional objects is $\sum_i\lfloor e_i/2\rfloor$, where $e_i$ are the inertia orders. We also characterize conjugacy-class decompositions with embeddings linear over the coarse quotient: such a decomposition exists if and only if $|[g]\cap H_i|\leq\lfloor e_i/2\rfloor$ for every nonidentity conjugacy class and every branch inertia group $H_i$. The sufficiency proof uses coloured paths of type $A$ and derived reflection functors, and produces kernels on the natural fibre products. The known ordered curve decomposition still yields a decomposition indexed by the connected components of the fixed quotients.

In a recent paper, a mathematical analogy was found between two physically distinct classes of two-dimensional free boundary problems in inviscid fluid mechanics. One class involves free surface flows with constant vorticity in the bulk but without boundary surface tension, the other comprises free surface flows without vorticity in the bulk but with non-zero boundary surface tension (capillarity). Several examples were given of pairs of physically-distinct free boundary problems that amount to solving the same mathematical problem, albeit with the possibility of a single constraint arising for the capillary flows. The present work shows that this analogy extends to another pair of physically-distinct problems: a hollow vortex equilibrium in a shear flow with a third order irrotational strain and a steady bubble with surface tension in a simple straining flow. For both problems, we construct an isolated exact solution when a parameter $\gamma=0$ and provide numerical solutions for non-zero $\gamma$. An interesting feature is that non-trivial conformal mapping identities emerge from the class of problems with capillarity that apply equally for the hollow vortex problems when the parameters are appropriately constrained.

We show that the Fomin–Kirillov algebras $FK_n$ are infinite-dimensional for $n\geq6$. This answers a question posed by Fomin and Kirillov in 1997. To prove this, we demonstrate that the Nichols algebras supported on the conjugacy class of transpositions of the symmetric group are infinite-dimensional for both relevant 2-cocycles. For this, we develop a new method of partially semisimplifying a Nichols algebra along an automorphism of order $p$ in characteristic $p$.

Given a maximal order $\mathfrak{O}$ in a rational definite quaternion algebra, and a prime $\ell$ coprime to the discriminant of $\mathfrak{O}$, this paper considers subsets of $\mathfrak{O}$ consisting of elements with $\ell$-power norms that together generate $\mathfrak{O}$ as a $\mathbb{Z}$-algebra. We prove two theorems about the existence of such sets: the first states that $\mathfrak{O}$ is generated by elements of norm $\ell^k$ for any $k$ larger than an explicit bound, and the second states that there is a generating set for $\mathfrak{O}$ consisting of at most three elements, each with norm a power of $\ell$. We discuss implications for the study of supersingular isogeny graphs. As steps towards these theorems, for quaternion orders $\mathcal{O}$ that are not necessarily maximal, we also prove structural results about the algebra of Brandt matrices for $\mathcal{O}$ and explicit bounds on the coefficients of the theta function of $\mathcal{O}$. Computational experiments are also discussed.

Hierarchical Archimedean copulas (HACs) admit redundant tree representations: when adjacent internal nodes have equal parameters, contracting the connecting edge leaves the copula unchanged. This creates a structural overparametrization. For homogeneous one-parameter families, we remove this redundancy and identify the resulting reduced model class with the ultrametric fan. We treat the Clayton, Frank, Ali-Mikhail-Haq, Gumbel and Joe families. Using the negative logit of normalized Kendall's tau as a common node-height coordinate, reduced hierarchies form polyhedral strata, binary hierarchies index maximal cones, and multifurcating hierarchies lie on common contraction faces. The induced cophenetic metric generates the same topology as uniform convergence of copulas. We also show that informative pairwise dependence summaries act componentwise on cophenetic coordinates and hence provide global coordinates for the reduced model. This representation yields a global estimator obtained by inverting estimated pairwise summaries and projecting onto the fan, without preselecting a hierarchy. We prove consistency and convergence-rate transfer. At binary truths, the unthresholded estimator recovers the hierarchy with probability tending to one; at multifurcating truths, it asymptotically selects a binary refinement, while vanishing thresholding consistently recovers the reduced hierarchy. Under a joint central limit theorem for the pairwise summaries, the projected estimator has a root-$n$ limit given by Euclidean projection of a Gaussian vector onto the tangent fan. The limit is Gaussian at binary hierarchies and generally non-Gaussian at multifurcating ones. For empirical Kendall's tau, we verify the required regularity conditions and obtain the corresponding singular limit.

We study a Monge normal form of a surface in $\mathbb{R}^4$ based on the principal axes of its curvature ellipse. The curvature-ellipse-based Monge normal form is obtained using only Euclidean motions of the ambient space and local reparameterizations of the surface, and preserves the Euclidean invariants of the curvature ellipse, including its semi-axis lengths up to their ordering. We also give applications of this normal form to the $2$-jet geometry of surfaces in $\mathbb{R}^4$ and to the geometry of their orthogonal projections.

We prove a sharp Szegö-Weinberger-type inequality for balanced doubly connected domains in $\mathbb S^2$, $\mathbb R^2$ and $\mathbb H^2$. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the area of the smallest component of the complement. Equality holds only for the rotationally symmetric annulus. We also show that the balancing assumption is essential: if we drop it, the inequality fails in general.

We show that no meaningful version of the Cone Conjecture holds for non-generic Sakai surfaces.

We disprove the classical version of the Borsuk-Ulam-flavored conjecture, due to Baum-Dąbrowski-Hajac, that free actions $X\times \mathbb{G}\to X$ on non-empty compact Hausdorff spaces by non-trivial compact groups admit no equivariant maps $X*\mathbb{G}\to X$. Specifically, among non-trivial compact abelian $\mathbb{G}$, the conjecture fails precisely when $\mathbb{G}$ is torsion-free profinite.

The undirected multiple-unicast conjecture [LL04] asserts that network coding offers no throughput advantage over multicommodity flow. We refute this conjecture by constructing a deterministic linear network code over $\mathbb{F}_9$ on a 182-vertex bipartite subgraph of the point-line incidence graph of $\mathrm{PG}(2,9)$. The construction supports $157$ independent unicast sessions at common coding rate at least $1$, while every fractional multicommodity flow has common rate at most $147/157$. By the amplification theorem of~[BGS17], this yields a family of undirected multiple-unicast instances with coding gap $\Omega((\log n)^\varepsilon)$ for some $\varepsilon>0$. We also introduce a nondeterministic model of network coding based on locally verifiable certificates, which guides our construction and may be of independent interest. Building on the high-girth graph and error-correcting code framework of [BH25], we use GPT-6 to find a nondeterministic counterexample based on a new choice of Reed–Solomon local codes, and then convert this example into a causal code using an edge orientation and local search.

We introduce a weighted adjacency-degree matrix \(A_{fg}(G)\) with edge weights \(f(d_i,d_j)\) and diagonal entries \(g(d_i)\), unifying adjacency, Laplacian, signless Laplacian, \(A_\alpha\), ABC, Randić, Sombor and related matrices. For dendrimer trees \(D_{n,k}\) and Bethe trees \(B_{n,k}\), the characteristic polynomial is factorized through one recursive sequence \(P_{fg,n}\). When \(f^2(1,k)=f^2(k,k)\), this sequence admits a Chebyshev reduction to \( U_{j+1}(x)+\delta U_j(x)=0, \) covering \(L/L^+\) and \(A_\alpha\); explicit cosine spectra occur only in special cases, such as the adjacency matrix and the \(L/L^+\) end factor of \(B_{n,k}\). We derive positive-semidefinite energy formulas, Gershgorin criteria, spectral-gap estimates, interlacing and non-interlacing results, partial eigenvalue-coincidence information, and McClelland- and Koolen–Moulton-type bounds. Several known results are recovered as special cases of this unified framework.

This paper proposes a progressive Bayesian multiple-input multiple-output (MIMO) sensing framework for multiple targets over multiple stages. We consider a practical yet challenging scenario where the targets' angles are unknown and random parameters to be estimated, while the targets' reflection coefficients are unknown nuisance parameters. With an initial prior probability density function (PDF) for the targets' angles, we progressively update the prior PDF for each sensing stage as the posterior PDF obtained from the previous stage, based on which Bayesian transmit beamforming optimization is performed to minimize the sum posterior Cramér-Rao bound (PCRB) in estimating the targets' angles and Bayesian sensing is performed with the help of new observations in this stage. To analytically characterize the intractable and high-dimensional posterior PDF with low complexity, we propose a variational Bayesian inference based approach which derives a surrogate posterior PDF in closed form with only polynomial complexity over the number of targets, in sharp contrast to existing numerical calculation approaches with exponential complexity. Numerical results validate the efficacy of our proposed framework in progressively refining sensing performance.

The Fourier Neural Operator (FNO) learns solution operators of partial differential equations (PDEs) through Fourier-space kernel parameterization, but frequency truncation can limit the learning of high-frequency variations. AM-FNO and SirenFNO generate kernels for all grid modes from spectral coordinates using shared networks, making coordinate encoding and generator design important. Recent work on implicit neural representations (INRs) has proposed constructing frequency interactions through explicit feature composition rather than relying on subsequent MLPs to form them implicitly. Building on this approach, we propose CAFE+FNO, which incorporates Content-Aware Frequency Encoding+ (CAFE+) into Fourier kernel generation. CAFE+ combines Fourier–Chebyshev features through parallel affine branches and a Hadamard product, forming interactions within and across the two feature families. A kernel MLP maps the resulting representation of each normalized spectral coordinate to a complex channel-mixing matrix. Each layer shares its generator across all stored modes, making the number of trainable parameters independent of the number of modes for a fixed architecture. We compare CAFE+FNO with existing FNO variants on five PDE benchmarks and conduct ablation studies on basis configuration, multiplicative composition, and bandwidth learnability. Code and experimental configurations are available at https://github.com/fabsk101/CAFEPlusFNO.git.

Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.

Many mechanical systems, such as the Newtonian $n$-body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery's variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with $q$-fold rotation symmetry, and to counter-rotating double choreographic loops in planar $(N+3)$-body problems.

We study a family of liftings $\mathfrak U(\lambda)$ of the super Jordan plane over the infinite cyclic group in characteristic zero. For a nonzero lifting parameter $\lambda$, we identify a natural subalgebra $\mathfrak B$ with the bosonization of $U(\mathfrak{osp}(1|2))$, also known as $sl_{-1}(2)$, and show that the lifting $\mathfrak U(\lambda)$ is obtained from $\mathfrak B$ by an Ore localization at a single distinguished element. This structural description provides a unified approach to the representation theory and ideal structure of $\mathfrak U(\lambda)$. We classify the finite-dimensional simple modules of $\mathfrak B$ and prove that every finite-dimensional $\mathfrak B$-module is completely reducible; the localization then yields the corresponding results for $\mathfrak U(\lambda)$. We determine the prime, primitive, and completely prime ideals of both algebras. Finally, we compute their classical quotient rings, showing in particular that the classical quotient ring of $\mathfrak U(\lambda)$ is a $2\times2$ matrix algebra over the skew field of fractions of the tensor product of its centre and the first Weyl algebra.

Start with the number $2$. At each move, combine two numbers already made, but the operations must be used in the fixed repeating order $+,\times,-,\div$. We ask for the fewest moves needed to make every integer from $1$ to $N$. Since $2$ is already one of the numbers we want and each move makes at most one new number, at least $N-1$ moves are needed. We show that $N-1$ moves are also enough for every $N\ge 9$. Conventional induction cannot work, because a division that comes right after a completed interval $\{1,\dots,P\}$ produces numbers already made. Instead we extend a completed interval $\{1,\dots,P\}$ to $\{1,\dots,3P\}$ all at once, which counting shows is the smallest multiplicative extension $P\to kP$ that can work, and then adjust the last few moves to reach every other $N$. For $9\le N\le 33$ we give explicit sequences, found by computer search.

We prove that the peripheral spectrum of every positive operator on a complex Banach lattice is cyclic, thereby resolving the cyclicity conjecture without additional growth assumptions. Our proof is based on a spectral domination theorem derived from Vesentini's theorem on the subharmonicity of the spectral radius: when two positive operators have the same spectral radius and one dominates the other, the peripheral spectrum of the smaller operator is contained in that of the larger one. This extends an earlier result of Räbiger and Wolff. After passing to a suitable lattice extension, an averaging procedure based on torsion operators produces a decreasing sequence of positive minorants which retain the spectral radius and become asymptotically rotationally self-similar. An ultrapower argument turns this asymptotic relation into exact rotational self-similarity, allowing us to deduce cyclicity from spectral domination. The same approach also yields an alternative proof of Lotz's cyclicity theorem for Abel solvable operators.

It is proved that there exists a language $L$ such that its $\omega$-power has infinite rank in the difference hierarchy of open sets. The same holds for the decreasing difference hierarchy of $\Pi^0_2$ sets.

\noindent Let \(\Om\subset\R^2\) be a smooth bounded simply connected domain. We prove that all positive solutions of \(-\Delta u=\theta ue^{u^2}\), with \(\theta>0\) and zero Dirichlet boundary data, have Dirichlet energy bounded by a constant depending only on \(\Om\). Consequently, the Trudinger–Moser functional has no nonnegative constrained critical points at sufficiently large prescribed energies. The proof combines a weighted Pohozaev identity obtained after conformal change of variables, concentration analysis of normalized measures, and local radial comparison. The comparison hypotheses are obtained from local integral estimates, without an a priori energy bound. An estimate for the exterior Green potential then excludes sequences of unbounded energy.

We establish sharp two-point function asymptotics for the two dimensional Villain model at sufficiently low temperatures, confirming a central prediction in the Berezinskii-Kosterlitz-Thouless theory. We characterise the power law exponent through the renormalisation group flow of a marginal line defect observable in the dual high-temperature Discrete Gaussian model.

Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the kernel parameters are routinely estimated by the maximum likelihood estimators (MLEs), as implemented by standard GP software. Despite this widespread use, the asymptotic behavior of the MLEs remains poorly understood under fixed-domain asymptotics, even for the RBF kernel. The main difficulty arises from the increasingly strong dependence among densely sampled observations and the nonlinear dependence of the covariance matrix on the kernel parameters. In this paper, we address this gap by providing, to the best of our knowledge, the first complete asymptotic characterization of the joint MLE of the spatial variance, lengthscale, and nugget variance under fixed-domain asymptotics. We establish consistency, derive convergence rates for all three parameters, prove joint asymptotic normality, and show that these rates are minimax optimal.