Papers

We solve two open problems on ample vector bundles posed by Hartshorne in 1966. We prove that tensor products of ample vector bundles on schemes of finite type over an algebraically closed field are ample in arbitrary characteristic, extending Hartshorne's characteristic-zero result and Barton's projective positive-characteristic theorem. More generally, let f: X -> S be a morphism of schemes. Then we prove that tensor products of f-ample vector bundles are f-ample. Moreover, if E is an f-ample vector bundle of rank r>0 and W is a finite locally free polynomial GL(r,S)-module of positive rank with W_0 = 0, then E(W) is f-ample. In particular, Gamma^n E is f-ample for every n>0. Finally, adapting a construction of Ejiri-Fujino-Iwai, we show that in every characteristic a smooth quasi-projective surface carries a non-ample extension of an ample line bundle by itself.

Let $(M^n,g)$ carry a conformal product structure whose adapted Weyl connection $D$ preserves orthogonal distributions of ranks $p,q\ge2$. We express the Faraday form $d\theta$ directly in terms of the Weyl curvature of $g$. If $S$ is the orthogonal involution determined by the splitting and \[ \mathcal K_W(X)=\sum_i W_{X,e_i}(Se_i) \] for any local orthonormal frame, then \[ (d\theta)^\sharp = \frac{n-2}{4(p-1)(q-1)}[\mathcal K_W,S]. \] No Ricci-curvature assumption is required. The same curvature calculation gives a companion formula for the symmetric part of $\nabla\theta$, \[ S\Theta^s-\Theta^sS+\theta^\sharp\wedge S\theta^\sharp = \frac{p-q}{n-2}(d\theta)^\sharp +\frac1{n-2}[S,\Ric^\sharp], \qquad \Theta=\nabla\theta. \] When the Ricci tensor is block diagonal with respect to the product splitting, these formulas lead to explicit identities for the two components of the Lee form. In dimension four, the rank-$(2,2)$ splitting determines an ambi-Hermitian pair. The classical ambi-Hermitian curvature identities imply local closedness when the Ricci tensor is invariant under both complex structures; we also recover this conclusion from the mixed Weyl trace above. Finally, if $(M^4,g)$ is compact and Einstein, no specialness assumption is needed: $D=\nabla^g$. Hence the universal cover is the product of two simply connected surfaces with the same constant Gaussian curvature.

We give explicit formulas for the spectra of the intertwinors on the spinor bundle over $S^1 \times S^{n-1}$, $n$ even, with the standard Lorentzian metric $g=-g_{{}_{S^1}}+g_{{}_{S^{n-1}}}$. As a special case, we construct conformally covariant differential operators of all odd orders with the leading term a power of the Dirac operator.

In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that the image of an intertwining operator is always semi-simple and give a lower bound on the order of its pole, which is conjectured to be an equality as well as an upper bound in terms of the singular support of certain perverse sheaves. Moreover, we are able to reduce the conjecture of Lapid and Mínguez on the shape of irreducible subrepresentations of induced representations to a computation of characteristic cycles. The theory of mixed Hodge modules plays a crucial role in the proofs.

This paper proposes a preconditioned ordinary hyperbolic smoothing modified L-BFGS framework for finite minimax problems. To mitigate the deterioration of Euclidean conditioning as the smoothing parameter decreases, a problem-derived hyperbolic majorization preconditioner is incorporated into the initial L-BFGS metric; it combines component-gradient Lipschitz curvature with the curvature induced by the hyperbolic smoothing Jacobian, is uniformly positive definite, and provably majorizes the Hessian of the smoothed objective. To exploit strict negative secant curvature, three correction strategies are introduced—direct sign reflection, a Euclidean nearest-point correction, and a $B_k^{-1}$-metric nearest-point correction—for which explicit formulas are derived and Clarke-stationary accumulation points are obtained for the idealized continuation. For a fixed smoothing parameter, Q-linear convergence of the HMLBFGS objective values is obtained under a Polyak–Lojasiewicz condition, local strong convexity further yields R-linear convergence of the iterates, and the corresponding full-memory modified BFGS methods attain local Q-superlinear convergence under the usual unit-step assumptions. Numerical experiments demonstrate the effectiveness of the proposed preconditioned methods and their numerical advantages over the selected comparison methods.

Three classical problems-linear assignment, graph matching, and optimal transport-share a common structure: they seek an optimal matrix-valued coupling on a matrix polytope. We study regularized coupling maps that encompass two families of regularization: entropy/KL regularization, which exponentiates the input score matrix and projects it onto the relevant matrix polytope via the KL-projection; and the Frobenius/Euclidean regularization, which performs the Euclidean projection of the scaled input matrix onto the same polytope. We trace how these two geometries yield complementary algorithmic tools for graph matching (CSGO, ASM, FRAM) and optimal transport (IP-EOT, PSN). The regularized coupling map serves as a direction generator in graph matching and as a solution representation in optimal transport, with the regularizer determining the geometry and the outer task determining the precision standard.

This paper studies nontrivial Yamabe gradient solitons occurring as complete immersed hypersurfaces with constant scalar curvature in space forms $\left(\overline{M}^n(C), \overline{g}\right)$. For solitons with non-vanishing gradients, we establish a structural characterization of the soliton by analyzing a regular level set $\Sigma$ of the soliton function. In particular, imposing a suitable condition on the traceless part of the second fundamental form, $\Phi^{\Sigma}_{\overline{M}}$ of $\Sigma$ as a submanifold of the space form, we prove that the soliton either decomposes as a Riemannian product of $\mathbb{R}$ and a totally umbilical submanifold of the space form, or satisfies a sharp lower bound on $\sup\limits_\Sigma\left|\Phi^{\Sigma}_{\overline{M}}\right|$. Furthermore, we show that the lower bound is attained if, and only if, the soliton is isometric to a Riemannian product of $\mathbb{R}$ and a parallel submanifold of the space form.

In this paper, we study various results on numerical range of weighted composition operators on Hardy space of Dirichlet series. Certain conditions are provided under which the numerical range contains zero and some circular or elliptic disks. With the help of invariant subspaces of the operator, we also determine the numerical range of reductive weighted composition operators on this space. Additionally, we discuss some results on the Davis-Wielandt shell of composition operators with some natural questions that arise from our findings.

Let $E/K$ be an elliptic curve over a number field $K$, and let $[C]\in H^1(K,E(\overline{K}))$ be a homogeneous space under $E$. Suppose that $C$ has period $n$ and index $d$. O'Neil asked whether one can always choose a lift of $[C]$ to $H^1(K,E[n])$ whose period-index obstruction has order exactly $d/n$. We give a negative answer to this question by constructing an explicit family of examples.Taking $K=\mathbb{Q}(\zeta_8)$, we prove that there exist infinitely many pairwise non-isomorphic elliptic curves $E/K$ such that each $E$ admits infinitely many homogeneous spaces $[C]\in H^1(K,E(\overline{K}))$ with period $8$ and index $16$, but the period-index obstruction of every lift of $[C]$ to $H^1(K,E[8])$ has order exactly $8$.

We study the value of information in predicting the evolving state of a finite Markov chain. At each stage, a decision maker chooses a state and observes only whether the current state of the chain matches her choice; the resulting information is used to make a prediction on the state at the final stage of the problem. We show that, when the chain starts from an invariant distribution, the optimal terminal value is non-decreasing with the number of observations and converges at a uniform exponential rate. We introduce the predictive learning index, which measures whether all attainable informational value is extracted after finitely many observations, and show that both finite and infinite indices may occur. In contrast, for nonstationary initial distributions, the value may strictly decrease with the horizon.

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cos\theta)^3$ where $e$ is the relative eccentricity of the primaries and $\theta$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

We develop a method for constructing abelian model structures on the category Q,AMod of Q-shaped modules from cotorsion pairs in AMod, where Q is a small preadditive category satisfying certain conditions and AMod denotes the category of left A-modules for any ring A. More precisely, we construct two cotorsion pairs in Q,AMod from a given cotorsion pair in AMod. This leads to a construction of projective model structures on Q,AMod under the condition that Q has no cycles. We further apply this method to the category Dif(A) of differential left A-modules, viewed as a category of Q-shaped modules for a suitable choice of Q. In this case, the induced cotorsion pairs are shown to be compatible, thereby giving rise to abelian model structures on Dif(A).

Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of $N$ points incurs an $\mathcal{O}(N^{2})$ computational cost, which is prohibitive for large-scale datasets. Kernel approximation techniques can be applied to bring the computational cost down to $\mathcal{O}(N)$. The random Fourier features (RFF) technique, based on sampling from the spectral density of the kernel function, has become popular to speed up kernel estimators for machine learning applications. Unfortunately, it is restricted to positive definite kernels, while the majority of kernel functions popular in KDE, such as the parabolic kernel, do not satisfy this property. To overcome this limitation, this article introduces the signed random Fourier features (SRFF) technique. It is a generalization of RFF compatible with indefinite kernels whose inverse Fourier transform is absolutely integrable. The motivation for introducing this method is to speed up KDE in the case of multivariate compact kernels, which are generally not positive definite. We detail how to implement SRFF for both product kernels and isotropic kernels. For the class of Kuttner-Golubov kernels $K(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=(1-\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert ^{\alpha})^{\beta}\mathbf{1}_{\{\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert \leq1\}}$ where $\boldsymbol{x}_{i}\in\mathbb{R}^{d}$, $\boldsymbol{x}_{j}\in\mathbb{R}^{d}$, $\alpha>0$, $\beta>0$, which includes the triangular, parabolic, biweight, triweight, and other kernel functions of interest for KDE as particular examples, we provide an explicit acceptance-rejection algorithm to sample from its signed spectral density. Our numerical tests on a dataset of one million points confirm the computational efficiency and accuracy of SRFF for large-scale KDE.

We prove two structural results about factorization systems on $\infty$-categories. Firstly, we classify factorization systems on the total spaces of a class of fibrations of $\infty$-categories in terms of factorization systems on the base and the fibers. This will be interpreted as an un/straightening equivalence for $\infty$-categories equipped with a factorization system, using Juran's double $\infty$-categorical framework. Along the way, we develop the general theory of lifting through faithful functors of $n$-uple $\infty$-categories. Secondly, we prove that the forgetful functor from $\infty$-categories equipped with a factorization system to $\infty$-categories is a monadic right adjoint.

Let $\mathcal F=\{2K_2,C_4,C_5\}$, so that $\operatorname{Forb}(\mathcal F)$ is the class of split graphs. Resolving a conjecture of Wang and Tang, we prove that the class of split graphs is generalized spectrally closed and that it admits no walk-realizable $\mathcal F$-supporter of any finite order.

For the 3D semilinear Euler-Poisson-Darboux equation $\square u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\geq1$, $\mu>0$ and $p>1$, it is conjectured that there is a critical exponent $p_{crit}(3,\mu)=\max\{p_s(3+\mu), p_f(3)\}$ with the Strauss exponent $p_s(3+\mu)=\frac{\mu+4+\sqrt{\mu^2+16\mu+32}}{2(\mu+2)}$ and the Fujita exponent $p_f(3)=\frac{5}{3}$ such that when $p>p_{crit}(3,\mu)$, the small data solution $u$ exists globally, otherwise, when $1<p\le p_{crit}(3,\mu)$, the solution $u$ can blow up in finite time. It is pointed out that for $1<p\le p_{crit}(3,\mu)$, the blowup of solution $u$ has been shown. However, it is still open for the global existence of small solution $u$ when $p>p_{crit}(3,\mu)$. Note that $p_{crit}(3,\mu)=p_s(3+\mu)$ for $0<\mu<\frac{14}{5}$ and $p_{crit}(3,\mu)=p_f(3)$ for $\mu\geq\frac{14}{5}$. In the recent paper [16], the authors have obtained the global small solution $u$ for $p>\max\{\frac{5}{3},1+\frac{2}{\mu}\}$ and $\mu\geq\frac{14}{5}$. In this paper, by utilizing the hypergeometric Riemann representation and establishing some delicate pointwise spacetime weighted estimates, we prove the global existence of small data radial solution $u$ in the remaining range of $\frac{5}{3}<p\leq1+\frac{2}{\mu}$ and $\frac{14}{5}\leq\mu<3$. Therefore, for the radially symmetric case and $\mu\geq\frac{14}{5}$, the global existence problem of small data solution $u$ is solved when $p>p_{crit}(3,\mu)=\frac{5}{3}$.

This paper studies global existence for the Cauchy problem of nonlinear Klein-Gordon equations in three space dimensions, strictly within the regularity regime of classical local existence. We prove it via higher-order and lower-order energy estimates. The proof relies on two key ingredients. The first is due to the work of Georgiev and Popivanov, which reduces quadratic nonlinearities to cubic terms plus ghost-energy-controllable terms, securing lower-order estimates. The second is a sharp Klainerman-Sobolev-type inequality without the scaling operator, established herein, which yields enough time decay for derivatives up to second order; integration by parts then controls derivative loss terms in the higher-order estimates.

Optimal Control Problems (OCPs) have been applied to a variety of applications. Such problems are often transcribed into a Nonlinear Program (NLP) for which solution methods typically solve a Linear Quadratic (LQ) subproblem. To efficiently solve this LQ subproblem, the Riccati recursion exploits the block-diagonal structure that arises from discretizing an OCP using a Multiple Shooting method. However, reformulating general dynamics constraints to fit the required problem structure to use the Riccati recursion introduces additional micro-structure within these block matrices that remains unexploited. This work demonstrates how structure-exploiting solvers such as Fatrop can benefit from exploiting not only the block-diagonal structure but also the micro-structure within those blocks. We propose a structure-exploiting LU decomposition algorithm and integrate it within the Riccati recursion. Numerical results on randomized linear systems and two OCP examples demonstrate a reduction in computation time of the search direction of up to 20%, with the largest speedup observed for relative zero block sizes around 0.25 and large stage-wise equality constraint jacobians.

We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa–Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp Hölder regularity $C^{7/9}$. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved $H^1$-type quantity excludes the $C^{1/3}$ cusp associated with Burgers-type gradient blow-up, the precise Hölder exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent $7/9$, rather than the $3/5$ exponent known for the Camassa–Holm and Hunter–Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter–Saxton-type leading equation. Its self-similar profiles determine the $C^{7/9}$ cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.

A de Bruijn polyomino has colored cells and includes exactly one instance of each possible coloring of another polyomino with those colors. This generalizes the notion of a de Bruijn sequence. We are interested in the de Bruijn polyominoes of minimum size. As a step toward finding these, we introduce the problem of finding the smallest polyominoes containing at least $N$ translated copies of the polyomino $p$, allowing overlaps. We say these are $(p,N)$-dense and have size $a_{p,N}$. For certain pairs of polyominoes $p$ and $q$ of equal size, we show there are polyomino transformations relating $a_{p,N}$ and $a_{q,N}$. Using these transformations, we identify classes of polyominoes which share the sequence $(a_{p,N})_{N=1}^\infty$. We give closed forms for $(a_{p,N})_{N=1}^\infty$ for most polyominoes with up to five cells. Leveraging these results we introduce novel de Bruijn polyominoes, as well as other colored polyforms with de Bruijn-like properties.

We develop an entropy-stable and physical-constraint-preserving discontinuous Galerkin spectral element method for symmetry-reduced general-relativistic hydrodynamics on prescribed stationary spacetimes. Using a local orthonormal transformation, the fluid variables are expressed in a form for which the relativistic hydrodynamic algebra and the admissible set are independent of the spatial metric, while the spacetime geometry enters through stationary coefficients. This separation allows entropy-conservative special-relativistic fluxes to be combined with a compatible discretization of the geometric source terms. On affine tensor-product meshes, the resulting DGSEM is conservative and satisfies a semidiscrete entropy inequality, while the transformed variables provide a convex framework for physical-constraint preservation. For practical stabilization, we use a geometry-only causal speed that is sufficient for both classical local Lax–Friedrichs entropy dissipation and the physical-constraint-preserving Lax–Friedrichs splitting. The fully discrete method combines this stabilization with SSP Runge–Kutta time stepping, oscillation elimination, and conservative local-orthonormal-state scaling. Numerical experiments cover smooth and strongly shocked special-relativistic flows, an axisymmetric jet, stationary Michel accretion, Schwarzschild Bondi–Hoyle flow, and four Kerr accretion cases. The results demonstrate the designed high-order accuracy in smooth regimes and robust performance for demanding relativistic flows on curved stationary backgrounds.

Brenti conjectured that the Eulerian triangle is totally positive. In this paper, we prove that the Eulerian triangle is totally positive of order 3, thereby providing a partial affirmative answer to Brenti's conjecture. In addition, we establish total positivity of order 3 for several families of generalized Eulerian triangles.

Risk-averse design optimization with conditional value-at-risk (CVaR) constraints requires accurate tail estimates, but repeated high-fidelity (HF) evaluations are costly. We propose a multifidelity (MF) tail-region correction method that uses dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) as a global surrogate and directs a limited HF budget to responses that most strongly influence CVaR. Covariance of the estimated DD-GPCE coefficients quantifies finite-sample prediction uncertainty and defines a confidence-interval-based tail region. Within this region, a two-stage strategy explores locations with high prediction uncertainty and then exploits locations with large correction-induced shifts. A Tikhonov-regularized residual expansion using the same polynomial basis yields a unified tail-corrected surrogate. For a ten-bar truss, the method produced a design close to the crude-MCS reference with approximately 72 to 274 times fewer HF evaluations than the considered MF importance sampling configurations. For a suction valve, it reduced mass by 1.05% and bending and bulging CVaR by 17.68% and 21.53%, respectively, while satisfying the valve-opening requirement. The results demonstrate accurate, HF-sample-efficient CVaR-constrained design through localized MF correction.

We construct an explicit pre-$9$-angulated category which is not $9$-angulated, thereby giving a genuinely higher counterexample to the implication from pre-$(n+2)$-angulated to $(n+2)$-angulated. The underlying additive category is the category of finitely generated projective right modules over the preprojective algebra $\Pi(A_5)$ over $\mathbb F_2$. The construction is obtained by taking an odd power of the twisted complete comparison used by Chen-Liu-Lu-Zhang in their pre-triangulated counterexample and by showing that the resulting pre-$9$-angulation fails the higher mapping-cone axiom.

We give solutions to several problems in combinatorial group theory recorded as open in the literature. Our main source is the Kourovka Notebook, although we also consider questions from elsewhere. All solutions were found with the assistance of large language models and checked by the authors.

In 1927 J.E. Littlewood proved that, for bounded harmonic functions on the unit disc, the Fatou-type result, on the existence of almost everywhere boundary values through approach regions that are nontangential, will fail for any system of tangential approach regions that has the following two additional properties: It consists of curves ending at the various boundary points, and it is rotationally invariant. However, in 1984 A. Nagel and E.M. Stein, elaborating results of Rudin (1979) and Nagel, Rudin and Shapiro (1982), proved the existence of rotationally invariant systems of tangential sequences along which a Fatou-type result holds, i.e., along which every bounded harmonic function converges a.e. to its nontangential boundary values. Moreover, they extended their result to higher-dimensional Euclidean half-spaces. The Nagel-Stein result has prompted the question of formulating and proving a Littlewood type theorem that can also be applied to tangential approach regions which are sequential. In this paper we prove a Littlewood-type theorem for bounded harmonic functions in higher-dimensional Euclidean half-spaces, for systems of tangential approach regions which includes the sequential ones. This is the first result of this kind, apart from a recent result of ours whose setting is the unit disc. Indeed, in all the other results of Littlewood type, the tangential approach regions were required to be curvilinear or at least to possess a certain topological property that excluded the possibility that they could be sequential.

We characterize the ambiguity in reconstructing bounded measurable sets from their Fourier transforms sampled on a lattice. For a full-rank lattice \(\mathcal L\subset\R^d\), we prove that two bounded measurable sets have identical indicator Fourier transforms on \(\mathcal L^*\) if and only if they are equidecomposable, modulo null sets, into finitely many measurable pieces by translations in \(\mathcal L\). If the sets are connected finite unions of convex bodies and no two distinct points of either set are congruent modulo \(\mathcal L\), we show that this equidecomposition reduces to a single translation by a vector of \(\mathcal L\). In particular, two such sets that are symmetric about the origin must coincide. We also construct counterexamples in dimensions two and three to a conjecture of the second author, which had a similar hypothesis, but lacked the anti-aliasing hypothesis. Together with the first author's counterexamples in dimensions at least four, these show that the conjecture fails in every dimension $d\geq 2$. The positive results here are an outgrowth of refinements of this conjecture.

In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient $\mathcal E_{\rm orb}(X,T)$ of orbit-Lipschitz almost additive potentials, where we extend the theory of maximizable sets and countable maximizable families developed by W. Huang, O. Jenkinson, L. Xu and Y. Zhang [Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. Math. 245 (2026), 1–63], and establish a global structural theorem. For a countable maximizable family, global TPO holds if every non boundary member has $X$-extendable TPO and the boundary region has empty interior. As an application, we construct a compact system with global TPO for which $\mathcal E_{\rm orb}(X,T)$ is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential $\Phi$, we also develop relative TPO theory on its Lipschitz leaf. When $\Phi=0$, this framework reduces to classical Lipschitz TPO. We prove the corresponding leafwise structural theorem and give a non additive rank-one matrix example on a full shift.

We study integer-interval deformations of Coxeter arrangements of types $A$, $B$, and $D$, assigning an integer interval to each root direction through an integer-valued root function. We introduce a local compatibility condition on this root function. For compatible root functions, subject to additional local bounds in types $B$ and $D$, we derive characteristic-polynomial formulas as binomial sums over cyclic orders in type $A$ and signed permutations in types $B$ and $D$. The proofs use the finite-field method and reduce hyperplane avoidance to consecutive-gap inequalities. These formulas recover, in particular, the classical Catalan and Shi characteristic polynomials.

In this paper we investigate the existence of a maximal-cardinality \(t\)-resolvable quantum Latin square (\(t\text{-}\text{RQLS}(v)\)). We determine the existence of a maximal-cardinality \(1\text{-}\text{RQLS}(v)\) with \(17\) possible exceptions based on the established research result and construct a maximal-cardinality \(2\text{-}\text{RQLS}(n^2)\) for every integer \(n\geq6\).