Papers

We prove that the closed unit ball of the real Banach space $\ell_\infty$ is plastic, that is, every non-expansive bijection from the unit ball onto itself is an isometry. The main step is to show that every non-expansive bijection of this ball maps extreme points to extreme points. This is done by using elementary coverings of the unit ball by balls of radius one. The conclusion then follows from a theorem by Fakhoury. The same argument also shows that, for arbitrary $\Gamma$, every non-expansive bijection of $B_{\ell_\infty(\Gamma)}$ preserves extreme points.

We disprove the odd-dimensional extension, conjectured by Moraga and recorded by Kollár and Zhuang, of the rank bound for faithful abelian $p$-group actions on smooth Calabi–Yau varieties. The main result of this paper was obtained by ChatGPT 5.5 pro, and the Danus system based on the Rethlas system.

Recently, omnitrees were introduced as a flexible space partitioning tree that improves upon the benefits of both octrees and k-d trees: Omnitrees' efficient encoding of anisotropic refinements holds particular interest for applications with anisotropic features and high dimensionality. These include, but are not limited to, computer graphics, databases, machine learning, and physics simulations. The present paper defines new operations on the omnitree encoding that extend its capabilities from the existing refinement to also include coarsening and therefore fully adaptive compression. It demonstrates natural integration of omnitrees with wavelets, which conserves moments of the stored function by design. For omnitrees, the wavelet coefficients can be interpreted as local refinement priorities, which can be used to guide the adaptation process. We derive algorithms for coarsening and downsplit that are guided by wavelet coefficients, and show their application to a large dataset of 3D shapes, as well as the continuous-valued density field of a cloud. The comparison to OpenVDB, a widely-used data structure for sparse volumetric data in computer graphics, enables a demonstration of the practical benefits of omnitrees even for moderately anisotropic three-dimensional data. Compared to OpenVDB, objects can be stored using up to 28x less space, and asymptotically show savings that exceed theoretical expectations. Using lossy compression, the cloud dataset can be compressed by $\approx5\times$ compared to OpenVDB, with negligible loss of visual quality. This demonstrates the potential of omnitrees for efficient storage and processing, and motivates further research into their applications in various domains.

Steinerberger curvature encodes the global distance geometry of a graph through an equilibrium measure. In this paper, we derive explicit curvature formulas for undirected Cayley graphs of dihedral groups $D_n$ and generalized quaternion groups $Q_{4m}$. We then extend Steinerberger curvature to strongly connected simple digraphs by introducing in-curvature and out-curvature, reflecting the asymmetry of directed distances. For these directed curvatures, we establish structural properties, including negativity criteria and a permutation relation between in- and out-curvature. Our main results are directed analogues of the Bonnet–Myers, Cheng and Lichnerowicz theorems, together with reverse Bonnet–Myers inequalities for directed diameter and out-radius, and an upper bound for in-radius in terms of total curvature.

Given a nonnegative matrix $X$, a factorization rank $r$ and {a positive integer $p$}, entrywise power matrix factorization (EPMF) looks for a low-rank matrix $X_r$ such that $X = |X_r|^{\circ p}$ (exact case) or $X \approx |X_r|^{\circ p}$ (approximate case), where $(\cdot)^{\circ p}$ denotes the componentwise exponent. EPMF includes the modulus model ($p=1$) and componentwise square factorization ($p=2$) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix $X$ to obtain a rank-$r$ matrix, which we refer to as the low-rank matrix signing (LRMS) problem. We first show that LRMS, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank (Math. Prog., 2015). We then show that LRMS can be solved in polynomial time when $r$ is fixed. Moreover, when the rank $r$ is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter $r$; in fact, the running time is fixed-parameter linear in the number of entries of the input matrix. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when $r=2$, the smallest nontrivial case.

We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean $\vee$-systems and real polyhedral Kähler (PK) arrangements. We prove that every irreducible Euclidean $\vee$-system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean $\vee$-systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean $\vee$-systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean $\vee$-system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from $\vee$-systems. As a consequence, we prove that the Schreiber–Veselov catalog is complete for irreducible rank-three Euclidean $\vee$-systems with at most $27$ vectors.

For the Erdős–Frankl–Pach problem on uniform set systems of bounded VC-dimension, the Ahlswede–Khachatrian/Mubayi–Zhao construction has long served as the standard lower-bound benchmark. We develop a recursive lifting method that goes beyond this benchmark in every dimension \(d\ge3\), proving that for every \(d\ge3\) and \(n\ge d+3\), \[ M_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+M_{d-3}(n-5). \] The proof is elementary and proceeds through explicit trace obstructions. We also record a further recursive improvement in the concluding remarks.

We construct an affine geometry for commutative $\mathbb F_1$-algebras in the framework of Connes and Consani. Using their localization construction, we equip the Deitmar spectrum of the underlying multiplicative monoid $A(1_+)$ with a sheaf $\mathcal O_A$ of $\mathbb F_1$-algebras. We prove that $\mathcal O_A(D(f))\cong A_f$ and, in particular, that global sections recover $A$. The resulting spectrum construction yields an anti-equivalence between commutative $\mathbb F_1$-algebras and absolute affine schemes. We compare our choice of covering families with the Connes–Consani topology and explain its role in affine reconstruction. Finally, we describe base change to $\mathbb Z$, under which the absolute affine scheme associated to the Eilenberg–MacLane algebra $HR$ recovers the classical affine scheme $\operatorname{Spec}R$.

Splitting methods are a natural choice for the numerical time integration of partial differential equations, and arbitrary high order splitting schemes exist for Schrödinger equations with periodic boundary conditions. However, in the presence of non-periodic boundary conditions, we show that they suffer in general from an order reduction, even for smooth initial conditions. The reason for such order reduction phenomena are so-called compatibility conditions, which are not preserved by classical splitting schemes. In this paper, we introduce a family of modified splitting methods for one-dimensional linear Schrödinger equations with homogeneous Dirichlet boundary conditions, which achieve an arbitrary high order, and do not suffer from any order reduction. This is illustrated with a fourth order splitting scheme considering initial conditions with various regularity properties.

We answer two questions of Qi [J. Reine Angew. Math. 830 (2026)] on the metric space of saturated filtrations, one affirmatively and one negatively. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

Second-order optimization methods, such as Newton's algorithm, achieve fast local convergence and high accuracy, but their practical use is often limited by high computational costs. To mitigate this issue, variants such as inexact and quasi-Newton methods are widely used. A complementary and promising approach to improve the efficiency of the method is to employ mixed precision arithmetic, using different floating-point precisions for different operations, based on their impact on the convergence and accuracy of the method. In this work, we perform an error analysis of Newton's method accounting for different sources of inexactness, including approximations and rounding errors. We present a convergence analysis for the generated sequence, establishing bounds on the convergence rate and attainable accuracy. This theoretical framework covers quasi-Newton and inexact Newton methods, and is leveraged to propose mixed precision algorithms. We present a wide set of numerical experiments to illustrate our theoretical results and the behavior of Newton's method and its approximate variants in mixed precision floating-point arithmetic.

The Functional Autoregressive Model (FAR) generalizes the multivariate AR(1) model in Time Series Analysis to functional data. It serves as a historical foundational point in the study of functional time series and remains a fundamental and widely used model for dependent functional data. The process-the observed data-generated by the FAR model forms a Hilbert-valued Markov chain. This paper investigates the non-asymptotic prediction mean square error and derives a lower bound. This lower bound is established in the specific context of non-i.i.d. data and depends on the mixed smoothness of the functional time series and of the unknown correlation operator driving the FAR model. Instead of the standard functional PCA regularization, a ridge-type estimator is proposed, which avoids the preliminary estimation of the spectrum of the covariance sequence associated with the process. A non-asymptotic upper bound is derived for this estimate, which matches the lower bound up to multiplicative constants. Furthermore, a detailed study of the estimate's bias reveals connections between functional smoothness parameters and regularly/rapidly varying functions, which are common in extreme value theory. Simulation results corroborate the theoretical main theorems.

We study dynamical systems arising from reaction networks under mass-action kinetics. For certain choices of the rate constants (parameters), such systems are complex-balanced (vertex-balanced), which guarantees the existence of a unique positive equilibrium. Moreover, this equilibrium is asymptotically stable (admitting a global Lyapunov function) and linearly stable. In a series of recent papers, Craciun and collaborators introduced and studied disguised complex-balanced systems, that is, mass-action systems that are dynamically equal to auxiliary complex-balanced systems and therefore inherit their strong stability properties. Determining the parameter values for which a given system is disguised complex-balanced is a nontrivial algebraic problem. In this work, we show that the defining conditions for disguised complex-balanced equilibria naturally give rise to parametrized systems of polynomial inequalities. Using the framework for positive algebraic geometry developed by Müller and Regensburger, we reformulate these systems as binomial equations (on the disguised complex-balanced flux cone). Computing the disguised complex-balanced parameter locus can be viewed as a quantifier-elimination problem, and our approach eliminates the concentrations (state variables) from the problem. We illustrate our results using the running example of a recent paper by Boros et al.

This paper is mainly concerned with the large-time dynamics of bounded nonnegative solutions of reaction-diffusion equations on the real line with nonlinearities mainly of the bistable-type. We first consider initial data of the type $\alpha \mathbf{1}_I$, namely scalar multiples of indicator functions of intervals $I$. For each amplitude $\alpha$, the existence of a threshold length $L^*(\alpha)$ separating the extinction and the persistence of the solutions at large time is known. We here address the question as to whether the limit of the threshold sizes $L^*(\alpha)$ as $\alpha\to+\infty$ is positive or zero. We provide sufficient conditions under which this limit is positive, and others under which it is zero. Secondly, when the limit is positive, we show that some fragmented initial data, which are equally distributed as $\alpha \mathbf{1}_I$, give rise to solutions persisting at large time, whereas the solutions emanating from $\alpha \mathbf{1}_I$ go to extinction at large time. This result shows an unexpected favourable effect of the fragmentation of the initial datum on the large-time dynamics. Last, we consider the mass concentration principle for parabolic equations, which states that nonnegative solutions can be controlled from above in an integral sense by the solutions emanating from the Schwarz symmetrically decreasing rearrangements of the initial data. Since the pioneering results of [Bandle, 1976] and [Alvino, Trombetti, Lions, 1990], this principle is known to hold in bounded domains with Dirichlet boundary conditions under different types of assumptions of the coefficients of the equation. We show that this mass concentration principle is not valid in general, even for simple equations of the type $\partial_t u = \partial_{xx} u + f(u)$.

We investigate two-dimensional steady subsonic Euler flows with a contact discontinuity in infinitely long nozzles. An Euler–Lagrange transformation reduces the problem to a scalar quasilinear elliptic equation in divergence form on a fixed strip, with discontinuous coefficients and the contact discontinuity conditions incorporated into the weak formulation. By solving this elliptic problem, we establish existence, uniqueness, and regularity of the flow on the maximal interval of wall amplitudes containing the straight-nozzle configuration. These results apply to general piecewise smooth subsonic upstream data and dispense entirely with the monotonicity, convexity, and small-perturbation assumptions imposed in earlier studies. In particular, the contact discontinuity curve is of class $C^{2,\alpha}$. We further investigate the downstream asymptotic behavior and show, under the corresponding decay assumptions on the nozzle walls, that the convergence rate of the flow matches that of the walls.

This paper aims to provide a direct combinatorial proof of the Gessel-Wachs formula for $q$-derangement numbers in the setting of decorated permutations, without using the $q$-binomial inversion formula. Decorated permutations, introduced by Postnikov in his study of the totally nonnegative Grassmannian, provide a natural framework for Chen's signed fixed-point model. Our proof is based on a major-index generating function for decorated permutations with a fixed number of signed fixed points, together with a sign-reversing and descent-set-preserving involution, thereby answering a question raised by Chen. This involution was discovered through human–AI collaboration. Moreover, our framework yields an immediate proof of a result of Désarménien and Wachs concerning the equidistribution between descent classes of derangements and descent classes of desarrangements. We also supply combinatorial proofs of two recurrence relations for the $q$-derangement numbers in the setting of desarrangements with the aid of the insertion lemma for ordinary permutations.

Let \(\mathcal{B}_n\) be the Boolean lattice of all subsets of \([n]\) and let \(\mathcal{P}_{n;\ell,u}\) be the subposet of \(\mathcal{B}_n\) induced by the consecutive levels \(\ell,\ell+1,\ldots,u\). We determine $\nu_{n;\ell,u}$, the maximum size of a family of pairwise disjoint maximal intervals in $\mathcal P_{n;\ell,u}$, whenever \(u\le ({n+\ell^2})/({\ell+1})\). This completely settles Engel's problem~[Combin. Probab. Comput., 1996]. The proof is constructive. We also record consequences for weakly cross-intersecting set-pair systems and discuss the three-level case.

It is well known that if $(K\ltimes N, K)$ is a Gelfand pair with $N$ a nilpotent Lie group and $K$ a compact subgroup of the automorphism group of $N$, then $N$ is at most $2$-step. The notion of Gelfand pairs is extended to the notion of generalized Gelfand pairs, where $K$ is not necessarily compact. Gallo and Saal constructed an example of a generalized Gelfand pair $(K\ltimes N, K)$ with $N$ $3$-step and $K$ non-compact. In this work, we construct a family of generalized Gelfand pairs $(K_d \ltimes N_d, K_d)_{d\ge 1}$, where $N_d$ is $3$-step and $K_d$ is isomorphic to $\mathbb{R}^{d+1}$; the case of $d=1$ recovers the example of Gallo and Saal.

We investigate a subspace of Bridgeland stability conditions on $\PP^n$ satisfying the so-called Li condition. These are the stability conditions whose restriction to a smooth projective subvariety $X \subset \PP^n$ is again a stability condition. We then show that, when $X$ is a threefold, the restricted stability conditions coincide with those obtained via the double-tilt construction introduced by Bayer-Macrì-Toda. As an application, we prove the weak BMT conjecture.

We devise and analyze a pressure-robust residual-based $hp$-a posteriori error estimator for $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin (dG) methods for the Stokes problem on two- and three-dimensional polytopal Lipschitz domains. The estimator provides an upper bound and a local lower bound for the velocity error in the energy norm, both robust with respect to the viscosity and independent of the pressure. Our analysis relies on a decomposition of the error into conforming and nonconforming parts. The nonconforming error is bounded using a partition-of-unity framework combined with local Helmholtz decompositions on vertex patches. The conforming error is analyzed by means of the generalized Bogovskiĭ operator of [14] in both two and three dimensions, yielding two pressure-independent residual-based estimators associated with different interpolation operators. In the first approach, the upper bound for the conforming error consists of five error indicators and a data oscillation term. Four of these indicators exhibit $p$-optimal scaling, while the remaining one is suboptimal by a factor of $p^{1/2}$. In the second approach, the upper bound involves only two residual indicators together with the data oscillation term, at the expense of losing one order in $p$. Moreover, a pressure-robust local lower bound is established using $H^2$-bubble functions inspired by techniques developed for fourth-order PDEs. Numerical results in two and three dimensions confirm the reliability, efficiency, and pressure-robustness of the proposed estimators.

Post-hoc conditioning of pretrained diffusion models can be addressed using Sequential Monte Carlo (SMC) methods. By evolving an interacting particle system, SMC-guided diffusion samplers combine unconditional reverse-diffusion dynamics with sequential reweighting to approximate conditional distributions. Nevertheless, even in the infinite-particle limit, the implemented sampler may differ from the ideal conditional target because of errors in the diffusion model, its numerical implementation, and the guidance mechanism. We characterize how these local errors propagate through forward-smoothing kernels, which jointly account for the reverse dynamics and the remaining conditioning information. This yields non-asymptotic error bounds that capture both finite-particle fluctuations and approximation errors arising from initialization, numerical integration, score approximation, and potential design. In doing so, we extend stability guarantees for diffusion models to the conditional setting. Finally, we apply our framework to several state-of-the-art SMC-guided diffusion algorithms, providing a unified theoretical perspective on their approximation mechanisms and sources of error.

Rainfall-induced landslides are governed by the interaction between subsurface water flow and soil mechanics, requiring robust numerical methods for the simulation of variably saturated porous media. In this work, we consider a semi-coupled hydro-mechanical model based on Richards' equation and linear elasticity and propose a numerical framework based on a stabilization-free Virtual Element Method for its spatial discretization. The proposed approach naturally accommodates general polygonal meshes while avoiding problem-dependent stabilization terms, whose design may become challenging when heterogeneous and strongly non-linear coefficients are involved. The approach is combined with a mass-lumping technique to improve stability in the treatment of the storage term and with Nitsche's method to weakly impose seepage-face and infiltration boundary conditions, allowing for the automatic switching between Neumann and Dirichlet conditions. Time integration is performed using the backward Euler scheme, while non-linearities are handled through a Picard iteration. Numerical experiments demonstrate the stability and robustness of the proposed methodology and show its effectiveness in simulating rainfall infiltration and evaluating slope stability through the Local Factor of Safety.

We study the local Lipschitz regularity of local minimizers for a class of degenerate orthotropic functionals with $\varphi$-growth, where $\varphi$ is a general N-function. Unlike standard isotropic functionals, the ellipticity of the associated Euler-Lagrange equation degenerates separately in each coordinate direction, presenting significant anisotropic difficulties. Furthermore, the general N-function setting lacks the algebraic scale invariance available in the classical orthotropic $p$-Laplacian case. Despite these structural difficulties, we prove that local minimizers are locally Lipschitz continuous. Our approach relies on a regularized approximation scheme, mixed-direction Caccioppoli inequalities, and a carefully designed Moser-type iteration that incorporates an interpolation argument to bridge the gaps between consecutive integrability exponents.

We study the Cauchy problem for biwave type equations with coefficients depending only on time. Assuming $\mathcal{C}^\kappa$ regularity of the principal symbol and suitable Levi conditions on the lower order terms, we establish well-posedness in Gevrey spaces.

In this paper, we study the asymptotic behaviour for distribution of supercritical branching process in a Markovian environment with linear fractional offspring distributions. Phase transitions in the behaviour of the process is discovered, through investigating the strongly, intermediately and weakly supercritical cases.

We propose a new formulation of the multi-component short pulse (MCSP) equation that includes the coupled complex short pulse (CCSP) equation as a reduction. Using Hirota's bilinear method, we construct its $N$-soliton solutions in Pfaffian form. We then derive integrable semi-discrete and fully discrete analogues of the MCSP equation admitting Pfaffian $N$-soliton solutions. The resulting fully discrete system provides a practical self-adaptive moving mesh scheme for numerical simulations. For the parameter sets considered, numerical simulations demonstrate excellent agreement between the numerical and exact solutions, confirming the robustness and high accuracy of the proposed scheme.

We consider the interaction of a general viscous compressible and heat-conducting fluid with an elastic shell located at the boundary of the fluid's domain. The fluid is described by the compressible Navier-Stokes-Fourier equations and the shell evolves in accordance with a viscoelastic beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. Our first result is the local-in-time well-posedness of the underlying coupled system of nonlinear PDEs in smooth function spaces. Eventually, we prove a conditional regularity criterion which is in the spirit of the Beale-Kato-Majda condition for the incompressible Euler equations combined with the boundedness of density and temperature. It rests upon a weak-strong uniqueness result for the underlying system.

This study proposes a precipitation control framework integrating a realistic Numerical Weather Prediction (NWP) model with model predictive control (MPC). At each control instant in MPC, a finite-difference sensitivity matrix is constructed from the NWP model and used as a local linear model of how perturbations to the atmospheric state affect future precipitation. A sparse convex optimization problem is then solved to compute the control input, which is implemented as a perturbation to the atmospheric state. To reduce computational cost in sensitivity analysis, multiple grid points in the NWP model are treated collectively as a single block, and a uniform perturbation is applied to all points within each block. Moreover, a tailored convex optimization problem is introduced to effectively control the accumulated precipitation at the end of a weather event, using a prediction horizon much shorter than the entire event duration while promoting spatially sparse atmospheric perturbations. To evaluate the proposed MPC method, four control methods are compared: (i) initial-only open-loop optimal control (IO-OL), (ii) full-horizon open-loop optimal control (FH-OL), (iii) shrinking-horizon optimal control (SHOC) with a fixed terminal time, and (iv) single-move MPC with a fixed prediction-horizon length. Numerical experiments on a warm bubble benchmark demonstrate that MPC achieves precipitation reduction comparable to SHOC while reducing the total computational time relative to FH-OL and SHOC. Moreover, despite using a linear prediction model, MPC successfully achieves a challenging level of precipitation reduction, even when open-loop optimal control methods, namely, IO-OL and FH-OL, fail because of nonlinear atmospheric evolution. These findings suggest that MPC is a promising control framework for NWP-based precipitation reduction in complex weather events.

Higher-order influence functions, introduced in a series of articles (Robins et al., 2008, 2009a; van der Vaart, 2014; Robins et al., 2016, 2023; Liu et al., 2017), are a unified framework for constructing rate-optimal point estimates of a class of statistical functionals under various complexity-reducing assumptions on the posited statistical model that generates the observed data. Although higher-order (influence functions) estimators are theoretically appealing, they have very limited practical uptake compared to their first-order counterparts. The original higher-order estimators proposed in Robins et al. (2008) and Robins et al. (2017) involve nonparametric density estimation of multi-dimensional covariates, a highly nontrivial statistical and computational problem on its own. The density estimator is, in turn, used in the evaluation of the inverse population Gram matrix $\Omega$ of a set of $k$-dimensional basis transformations of covariates. There, $k$ is allowed to be as large as $o (n^2)$. To partially address this potential shortcoming, Liu et al. (2017) restrict $k$ to $o (n)$ and instead estimate $\Omega$ directly using the inverse sample Gram matrix estimator, but computed from an independent sample often obtained by sample-splitting. Liu et al. (2017) refer to this alternative estimator as the empirical higher-order estimator. Although the empirical higher-order estimator bypasses density estimation, it suffers from numerical instability due to inverting a large-dimensional sample Gram matrix. In this article, for a class of bilinear forms/functionals that often appear in substantive fields, we propose a new stabilized higher-order estimator without sample splitting, which exhibits more stable finite-sample performance compared to the empirical higher-order estimator. We also prove that this new class of higher-order estimators enjoys similar statistical guarantees.

We define homotopy index polynomials and a homotopy type of knotoids in an oriented surface, and consider some basic properties of these polynomials. We show that for planar knotoids the homotopy polynomials are not weaker than previously defined index polynomials, and that the homotopy index polynomials detect non-rotatability of spherical knotoids.