Papers

We prove that the notion of mass recently introduced by Mazurowski and Yao for continuous metrics coincides with Huisken's isoperimetric mass on smooth Riemannian $3$-manifolds with nonnegative scalar curvature whose metric has sharp $C^0$-asymptotically flat behavior. As a consequence, we deduce a Riemannian Penrose inequality for the mass parameter of asymptotically Schwarzschildian $3$-manifolds.

The usual settings for discussing the notion of stochastic objects are very different from Kolmogorov's original structure-free setting, in which there is no predefined external structure on the objects (such as the lengths of binary strings). A natural modification of stochasticity, obtained by replacing finite sets as models with probability measures, is quasi-stochasticity. The structure-free setting is common in studies of quasi-stochasticity, and known results show that the universal machine generates non-quasi-stochastic objects with exponentially small probability. This paper shows that, on the other hand, the universal machine generates nonstochastic objects surprisingly often. It also systematically explores numerous connections between stochasticity and quasi-stochasticity in the structure-free setting. In particular, the definition of quasi-stochasticity can be modified in a natural way modelled on Kolmogorov's definition so that non-quasi-stochastic objects are produced as often as nonstochastic ones; the results for this modification imply all the results of this paper showing that nonstochastic objects are frequent.

We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector $v$ of the 3-dimensional Minkowski space. We prove that for each vector $v$, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.

This paper considers the local convergence for solving general nonlinear equations. We establish randomized quasi-Gauss–Newton methods based on the approximation of the Gram matrix, addressing both the underdetermined and the overdetermined settings. For the underdetermined case, we show that our methods achieve local superlinear convergence to the optimal solution. For the overdetermined case, we show that our methods achieve local condition-number-free convergence to the stationary point of the nonlinear least-square formulation of the problem. In contrast, existing quasi-Newton methods mostly focus on square systems and additionally require the initial Jacobian estimate to be sufficiently close to the exact one.

We show that the strength as a spherical design of the spherical embedding of a $P$- and $Q$-polynomial association scheme with at least three classes with respect to a $Q$-polynomial idempotent is at most five, provided that the multiplicity is at least three. We also identify the examples that attain this upper bound on the strength. Our result improves on Suda's earlier upper bound of eight [J. Combin. Des. 19 (2011)], and is considered dual to the results of Lewis [Discrete Math. 223 (2000)] and Miklavič [Electron. J. Combin. 32 (2025)] concerning the girth of a $Q$-polynomial distance-regular graph with diameter and valency both at least three. To establish our upper bound, we introduce and discuss a polynomial method that works by constructing an appropriate polynomial that vanishes at every point of the spherical embedding.

This paper establishes that the system of ordinary differential equations arising from the sequential and distributive dual futile cycle has the structural capacity for Hopf bifurcations, whenever it is endowed with general parameter-rich kinetics, but it loses such capacity if it is endowed with mass action kinetics. The proof of the latter fact relies on a Routh-Hurwitz approach, improved by few preliminary structural considerations, but a decisive contribution came from ChatGPT Sol 5.6, which provided a nontrivial positivity certificate. A second purpose of the paper is indeed to document such AI-contribution, exploiting this well-studied toy-model, and to offer an example of the utilizability of such tools in the context of computer algebra and positivity certificates for large polynomials.

We investigate some of the properties of the Levi map and related objects for nondegenerate CR submanifolds in $\mathbb{C}^N$ of codimension d.

This paper develops a comprehensive stochastic variational framework for surface gravity waves. Starting from Luke's variational principle for irrotational, incompressible free-surface flow, we introduce a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic noise term representing unresolved scales. A path-wise variational principle yields the stochastic counterparts of the Laplace, Bernoulli, and kinematic boundary conditions. To close the system, a second variational principle in expectation is formulated, providing evolution equations for the noise correlation functions. The resulting coupled system preserves the Hamiltonian structure of the Zakharov–Craig–Sulem formulation. We further analyze explicit solutions via WKB approximation and show that the noise correlation functions satisfy a Hamilton-Jacobi equation with ray-tracing dynamics. This framework provides a rigorous foundation for reduced stochastic models of ocean waves and for the kinetic theory developed in the companion paper.

We investigate ground states of the stationary magnetopolaron in $\mathbb R^3$ with a constant magnetic field. When the strength $|b|$ of the magnetic field is sufficiently small, we prove the uniqueness and nondegeneracy of ground states, up to magnetic translations and phase shifts. Applying the uniqueness result, we further derive rigorously the symmetry and monotonicity of ground states for sufficiently small $|b|>0$. The second-order asymptotic expansion of the ground state energy is also derived as $b\to 0$.

For a graph $H$, the chromatic threshold $\delta_\chi(H)$ is the infimum of $c>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $cn$ is bounded by a constant depending only on $H$ and $c$. Allen, Böttcher, Griffiths, Kohayakawa, and Morris proved that if $\chi(H)=r\geq 3$, then $\delta_{\chi}(H)\in\{\frac{r-3}{r-2}, \frac{2r-5}{2r-3}, \frac{r-2}{r-1}\}$. Liu, Shangguan, Skokan, and Xu introduced the bounded-VC chromatic threshold $\text{VC}(H)$ by restricting the host graphs to have bounded VC-dimension. We determine this parameter for graph $H$ with $\chi(H)\ge 3$. More precisely, let $\mathcal{M}(H)$ be the decomposition family of an $r$-chromatic graph $H$, then \[ \text{VC}(H)= \begin{cases} \dfrac{r-3}{r-2},&\text{if $\mathcal{M}(H)$ contains a forest},\\[4pt] \dfrac{r-2}{r-1},&\text{otherwise}. \end{cases} \]

Joint analyses across multiple institutions are increasingly important in biomedical and epidemiological research, particularly for rare diseases where datasets are typical small. However, privacy regulations and institutional policies often prevent the sharing of individual-level patient data. In this paper we present an accurate and single-communication federated inference algorithm. Single-communication federated inference enables statistical analyses through a single exchange of summary statistics between participating centers and a coordinating server, preserving privacy while reducing communication and computational costs compared with iterative federated learning. We extend a recently proposed single-communication federated inference strategy that is based on second-order Taylor expansions by using third-order expansions to better approximate local log-likelihood functions. The proposed method is evaluated through simulation studies based on real data and compared with existing federated inference strategies. The simulation studies assess the performance of the proposed method, with a particular focus on scenarios involving small local sample sizes, where quadratic approximations may fail to capture skewness and other higher-order characteristics of the log-likelihood function. They demonstrate that incorporating higher-order information of the log-likelihood function improves the accuracy while preserving the privacy, communication efficiency, and scalability required for collaborative biomedical and epidemiological research.

We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined by the rank when the rank is $3$ or $4$, but we provide a characterization in each case. The resulting classification of facial exposedness in fact agrees with the corresponding classification of spectrahedrality for completely positive cones.

We study wave propagation through a time-modulated thin heterogeneous layer. The layer is assumed to have a thickness of order $\es\ll 1$ and its material properties exhibit rapid oscillations on multiple spatial and temporal scales. We aim to rigorously derive the effective model and the corresponding interface conditions for wave propagation through the limiting interface. To perform the homogenization together with dimension reduction, we generalize the notion of two-scale convergence for thin layer introduced by Neuss-Radu and Jäger (2007) to a multiple space-time scale framework. One of the main analytical difficulties arising in the homogenization analysis is that, in general, a uniform energy estimate cannot be obtained for a wave equation with time-varying coefficients. Therefore, we identify two physically relevant classes of coefficients for which we can derive a uniform energy bound. These include coefficients with traveling wave-type modulations of their properties. Using the energy estimates and the multiscale convergence for thin layer concept, we derive the effective model, which consists of linear wave equations in the bulk domains coupled through a nonstandard jump condition at the interface. This jump condition is governed by a wave-type dynamical equation with effective macroscopic coefficients determined by suitable cell problems of elliptic and hyperbolic type. Finally, we discuss the uniqueness of the effective model.

In this paper, we consider the spectrum of the one dimensional non-self-adjoint Dirac operator with matrix-valued coefficients whose entries are PT-symmetric periodic functions. We prove that the spectrum of the considered operator contains a large part of the real axis.

Let $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$, Knuth and Buckholtz proved $d(p)=\operatorname{lcm}(p-1,4)$ together with \[ d(p^r)\mid p^{r-1}d(p), \qquad s(p^r)\le r, \] and Ramassamy conjectured that both bounds are attained for every $r\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring $S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x^2+1)$. More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove \[ d(p^r)=p^{r-1}d(p) \qquad \text{for every odd prime $p$ and every $r\ge1$}, \] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving \[ s(5^5)=4<5. \] Finally, we prove that $5^5$ is the smallest odd prime power for which $s(p^r)\ne r$, and based on our findings we further conjecture \[ s(p^r)\ge r-2 \] for every odd prime $p$ and every $r\ge2$.

In sparse functional data analysis, where $n$ curves are each observed at $m$ random points, the covariance surface undergoes a sharp phase transition: if the covariance has smoothness $\beta$, the risk drops from the two-dimensional nonparametric rate to the parametric rate $n^{-1}$ once $m$ exceeds $m^*_n \asymp n^{1/(2\beta)}$. We determine what a symmetry of the domain does to that transition. A cyclic group of order $q$ preserving process and design displaces the threshold to $n^{1/(2\beta)}q^{-1/2}$; the exponent is a square root because symmetry acts on the number of usable pairs, which enters the variance quadratically, while the parametric floor is untouched by group averaging. The displacement saturates: once the orbit is finer than the bandwidth the reduction factor is $\min(q,c_K/h)$, by Poisson summation and positive definiteness of the kernel autocorrelation, and beyond that point the rate collapses to the one-dimensional nonparametric rate. Hence no rotation symmetry, even the full circle group, lowers the threshold below $n^{1/(4\beta)}$; this floor is attained from above by our estimator and from below, up to a polynomial factor, by our lower bounds. Symmetry thus takes one halfway on a logarithmic scale from the classical threshold to constant sampling. The uniform lower bound rests on a positivity-preserving packing of the stationary sub-class; closing the gap is left open. If the symmetry is only approximate, the risk acquires an approximation term and the design plane splits into three regimes, one unreachable by additional sampling; the symmetry spectrum governing it is explicit in Fourier coordinates, and a hold-out procedure selects the symmetry level with leading constant one below saturation and within an absolute constant beyond. All laws above are confirmed numerically. Reparametrisation of the domain, by contrast, leaves the threshold unchanged.

We study stochastic hypoviscous Navier–Stokes equations on the torus with dissipation $(-\Delta)^\gamma$ for $\gamma \in (\frac{1}{2},1]$ and multiplicative noise. Relying on stochastic maximal regularity results for the linear equation, we establish local well-posedness for this problem in arbitrary dimensions with initial data in a range of scaling-critical Besov spaces. With the aid of $L^q$-energy estimates for the vorticity equation, we also prove global well-posedness of the stochastic hypoviscous Navier–Stokes equation in 2D with linear multiplicative noise.

We prove global existence and modified scattering for solutions to relativistic Vlasov–wave systems without null structure, arising from small distribution functions. When the homogeneity of the force field in the momentum variable is the same as in the Vlasov-Maxwell system, the scalar fields are allowed to be large and our method requires stretched exponential velocity decay. When it matches that of the Einstein-Vlasov system, the scalar fields are required to be small and the distribution function to have slightly super-exponential velocity decay, which in particular includes Maxwellian decay. Moreover, we show that the smallness assumption on the scalar fields cannot in general be removed, even for compactly supported initial data.

We study exact low-dimensional reformulations for a regularized spectral approximation problem and its weighted variant. In the unweighted setting, weak-majorization monotonicity of the fidelity term enables an exact reformulation over singular values. In the weighted setting, we establish such a reformulation under compatibility and simultaneous diagonalization conditions. We further show, via a counterexample, that this exact reformulation can fail in the absence of simultaneous diagonalization.

For the purposes of proving Smale's mean value conjecture, one may restrict consideration to an explicitly described family of so-called Schlicht normalized polynomials. For Schlicht normalized polynomials whose leading coefficient decays sufficiently slowly, we show that the conjectured upper bound $1$ becomes asymptotically valid as the degree $d$ tends to infinity. The key and novel input is a refined Koebe $\frac{1}{4}$-type theorem of Cunningham, tailored to Schlicht functions whose images have bounded logarithmic capacity. For the dual mean value conjecture, we obtain an improvement of Eremenko's Markov-type inequality for regions bounded by polynomial lemniscates on which the polynomial is univalent. As a consequence, we improve the best known unconditional lower bound $\frac{1}{d^2}$ of Dubinin to $\frac{(d-\frac{1}{2})^{1/d}}{d^2}$ for all $d \ge 2$. The strengthened Markov-type inequality constitutes one of the main technical contributions of this paper. Its proof combines the solution to a related extremal problem for the logarithmic capacity of polynomial lemniscates, with the quasi-conformal deformation method introduced by Eremenko and Hayman to establish the connectedness of extremal polynomial lemniscates arising in the Erdos, Herzog and Piranian's problem on maximal lemniscate length.

We study the problem of stable spectral differentiation of functions in Sobolev spaces from noisy data. We introduce a class of admissible Fourier multipliers under simple and directly verifiable conditions and show that the corresponding regularized differentiation operators achieve minimax optimal stability rates. The results extend the previous $L^2$ based results to Sobolev spaces $H^{s,p}(\mathbb{R}^n)$, $1<p<\infty$. The analysis relies on multiplier estimates and applies to a wide class of multipliers, including Gaussian, spectral cutoff, and Tikhonov-type regularizations. Numerical examples demonstrate the behavior of several admissible spectral multipliers.

Quantum computing offers potential advantages for solving partial differential equations (PDEs). However, most existing quantum PDE solvers primarily focus on preparing quantum states for solutions, while the efficient recovery of classical information from these states remains less explored. Motivated by the readout limitation, we propose a quantum-classical hybrid framework for Bayesian PDE inversion problems: The quantum processor evolves the PDE and evaluate the loss function with sampling noises, while the classical computer tunes the hyper-parameters in the Gaussian Process Regression to explore the next trial candidate. To match the quantum solvers for linear and semi-linear autonomous evolution PDEs, we suggest to use a normalized quantum-state loss as the data-misfit function and evaluate the new misfit by combining quantum PDE solvers with the Hadamard test, thereby allowing us to extract useful classical information using only a limited number of quantum state copies without reconstructing the full solution vector. The analysis of error propagation and overall complexity of loss evaluation under a prescribed accuracy shows that the new data-misfit function outperforms the conventional L2-loss under quantum measurements. Quantum circuit simulations of 1D and 2D linear convection diffusion equations under approximate and finite sampling loss evaluations, together with classical numerical experiments on a nonlinear forced viscous Burgers equation, demonstrate the feasibility of the proposed approach for parameter inversion even when the loss evaluations are affected by sampling noise. This framework may provide a viable quantum-assisted scheme for PDE-based inverse problems and elucidate the potential of quantum PDE algorithms in addressing a complete quantum-to-end optimization stack.

In this paper, we prove that for a smooth complete intersection $X\subset \mathbb{P}^n$ of type $(d_1,\dots,d_c)$ over a number field $k$, there exists an effective bound $N_{\mathrm{AP}}(d_1,\dots,d_c)$ such that if $n\ge N_{\mathrm{AP}}(d_1,\dots,d_c)$, then $X$ satisfies arithmetic purity of strong approximation. We establish a connecting lemma that promotes weak approximation to strong approximation under a suitable connectivity hypothesis via rational curves over $k$. We construct the required connecting conics by combining the deformation theory of rational curves with results from the circle method.

We study the maximization of the minimum eigenvalue under combinatorial and integrality constraints. We propose a new approach based on branch-and-bound combines entropic smoothing of the minimum eigenvalue function and Frank-Wolfe methods over concave relaxations of the constraints, thereby exploiting combinatorial structure through linear optimization oracles. We establish approximation and convergence guarantees, including for truncated gradients computed from partial eigendecompositions, and introduce rank- and eigenvalue-based pruning and duality-based variable fixing. We evaluate the method on E-optimal experimental design and maximum algebraic connectivity problems and compare it with SCIP-SDP. The results show that our approach is particularly effective for large-dimensional instances and problems with additional combinatorial structure, whereas SCIP-SDP performs better on moderately sized instances with simpler constraints.

It is not a priori clear which of the Gauduchon connections is better suited to LCK and Vaisman manifolds. We thus investigate the geometry of these connections through Einstein problems and more general analytic and cohomological conditions on their Ricci tensors. Our results single out the Bismut connection as the privileged one for the non-Kähler lcK and Vaisman geometry. We therefore study the {second-Bismut–Einstein} equation and provide a characterization of these structures on Hopf manifolds.

We provide necessary and sufficient conditions for the specializations of a vertically parametrized polynomial system to attain the maximal number of complex nonzero solutions. To each vertical system, we attach a pair consisting of a projective simplicial toric variety and a toric vector bundle. The toric vector bundle allows for a homogenization of the polynomials in the Cox ring of the toric variety, providing a homogeneous ideal with the same zeros over the torus as the original system. We show that the maximal number of isolated solutions is attained if and only if this ideal has no solutions in the faces of the toric variety and prove that this happens for generic values of the parameters. In addition, we provide a novel formula for this generic number of zeros as an alternating sum of mixed volumes over a family of polytopes attached to the toric vector bundle.

We study 2D compressible Euler equations linearized around monotone shear flows $(U(y),0)$ on $\mathbb{T} \times \mathbb{R}$. The shear rate $U'$ is strictly positive, not necessarily close to any constant, and varies sufficiently slowly. For every fixed Mach number $M > 0$, we prove that the density and the irrotational velocity obey algebraic growth bounds, whereas the solenoidal velocity undergoes componentwise inviscid damping. Although a non-uniform shear couples the transverse Fourier frequencies and precludes the full Fourier reduction available for Couette flow, we are still able to recover the Couette rates without loss. The proof hinges on two new ingredients: a time-dependent pseudodifferential energy that restores a coercive structure for the variable-coefficient shear dynamics, and terminal-time-dependent higher- and lower-order weighted energies that capture the long-time effects of shear mixing.

Let $K$ be a field of characteristic zero, and let $\phi(x)\in K[x]$ be a polynomial of degree at least 2. Denote the $n$-th iterate of $\phi$ by $\phi^n$. The $n$-th dynatomic polynomial of $\phi$ is defined by $$\Phi_{\phi,n}(x) := \prod_{k\mid n}(\phi^k(x)-x)^{\mu(n/k)}.$$ In this paper, we specialize to the case $\phi(x) = x^d$. We first establish several properties of $\Phi_{\phi,n}$ that are analogous to those of cyclotomic polynomials. We then combine these properties with known results on resultants of cyclotomic polynomials to determine the resultants of dynatomic polynomials. In particular, for $1\leq n< m$, we show that $\operatorname{Res}(\Phi_{\phi,n},\Phi_{\phi,m}) = 1$ if and only if $n\nmid m$. When $n\mid m$, we obtain the explicit formula $$\operatorname{Res}(\Phi_{\phi,n},\Phi_{\phi,m}) = \Phi_{m/n}(d^n)^{\nu_d(n)},$$ where $\Phi_j$ is the $j$-th cyclotomic polynomial, $\nu_d(1) = d-1$, and $\nu_d(n) = \operatorname{deg}\Phi_{\phi,n}$ for $n>1$.

In this paper we prove that the Generalised Cone Conjecture holds for generic Sakai surfaces, which completes the proof of the Generalised Cone Conjecture in dimension two. Furthermore, we show that there exists a non-generic Sakai surface for which no meaningful variant of the conjecture holds.

We investigate a combinatorial notion of orientability for abstract GKM graphs and its connections to graph cohomology in the sense of Guillemin–Zara. In particular, we prove that orientability of the GKM graph is equivalent to Poincaré duality of the rational (non-equivariant) graph cohomology algebra. As an application, we prove that orientable GKM graphs remain connected after removing any single vertex.