Papers

In this paper, we show that the quantitative data underlying $K$-stability, originally developed in the study of canonical metrics, also detect the existence of positive scalar curvature Kähler metrics in a fixed class.

Let $S$ be a smooth quartic del Pezzo surface over a field $k$, of characteristic zero. We prove that a universal torsor $\mathcal T$ over $S$ is $k$-rational, provided $\mathcal T$ has $k$-points. As an application, we obtain examples of stably rational smooth cubic hypersurfaces over $\mathbb Q$ in every dimension greater than 2.

Let $B_{m}$ and $D_{n}$ be the denominators of the $m$th and $n$th convergent of the real numbers $\alpha$ and $\beta$, respectively. We introduce an abnormal method in the theory of simultaneous Diophantine approximation to the pair $\alpha,\beta$. Namely, the question of finding simultaneous approximation is turned into study of small solutions of a linear Diophantine equation $xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}$. The Thue-Siegel's lemma guarantees the existence of a non-zero integer vector $(x,y,z,v)$ in such a way that $|x|,|y|,|z|,|v|$ are bounded above by $\big( B_{m} + B_{m+1} + D_{n} + D_{n+1} \big)^{1/3}$. Thereby we can construct an integer $q:=xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}\ge 1$, a common denominator, which by the theory of continued fractions gives simultaneous approximations to $\alpha$ and $\beta$. We give also a variety of explicit constructions without the Thue-Siegel's lemma. Let $\|\alpha\|:=\underset{k\in\mathbb{Z}}\min\{|\alpha-k|\}$. For a class of numbers, including particular equivalent numbers $\alpha$ and $\beta$, we show there exist a real number $\kappa=\kappa(\alpha,\beta)>1/2$ and infinitely many explicitly constructible positive integers $q$ such that $\|q\alpha\| \le \frac{1}{q^{\kappa}}$ and $\|q\beta\| \le \frac{1}{q^{\kappa}}$. As the result improves Dirichlet's theorem on simultaneous approximation it also confirms the classical Littlewood conjecture for such a pair $\alpha,\beta$. In addition, we present general criteria for the classical Littlewood conjecture as well as for its $p$-adic counterpart.

We introduce the prescribed scalar curvature measure equation on a compact Kähler manifold. For a Kähler class of positive total scalar curvature, we prove that the following are equivalent: the existence of a positive scalar curvature Kähler metric, solvability of this equation for every admissible measure, $d_1$-coercivity of the associated functionals, and uniform geodesic stability along finite-energy $d_1$-geodesic rays. As a consequence, in each fixed Kähler class, the space of positive scalar curvature Kähler metrics is either empty or contractible. We further prove that every Kähler class on a positive-dimensional compact smooth toric Kähler manifold contains a torus-invariant metric of positive scalar curvature. Therefore, for every Kähler class, the prescribed scalar curvature measure equation admits a smooth solution for every admissible measure, unique modulo constants.

Weather forecasting and climate projection frequently use multi-model ensembles (MMEs) to improve short-term forecasts by averaging across models. However, this practice is often not well justified or validated. Using reservoir computing (RC) as a computationally efficient alternative to large-scale physical models, we assess the validity of the MME approach for chaotic time series. By training multiple randomly constructed RCs on the same dataset, we create a multi-model ensemble in which each model has its own unique error. These model errors lead to very different forecasting performances, with forecast error distributions that exhibit heavy tails. The arithmetic mean across forecasts from multiple models for the same target is usually closer to the ground truth than most individual forecasts, and further improvement is achieved by weighted arithmetic means where the weights are constructed based on each model's test-set performance. We show that iterated forecasts over many time steps deviate from the ground truth along the unstable manifold of the target point, in both directions, so that, if forecast errors were independent and had zero mean, the arithmetic mean forecast should approach the true target like $1/\sqrt{\nens}$ where $\nens$ is the size of the multi-model ensemble. We observe deviations from this behavior, which we attribute to the tails of the error distribution of random RCs.

There is a fascinating array of interrelated questions studying which structures can be guaranteed in digraphs of large minimum out-degree. These often have intriguingly simple statements, yet seem surprisingly difficult to approach. A well-known example is Lichiardopol's conjecture (2014), stating that there exists a function $g:\mathbb{N}\rightarrow \mathbb{N}$ such that every digraph with minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of distinct lengths. In this paper, building on earlier work of the second author, we confirm this conjecture in full generality. We also generalise this result to a weighted setting. Our proof uses and combines many ingredients from structural digraph theory such as butterfly minors, directed tangles, a directed analogue of the Tangle-Wall Theorem due to Robertson and Seymour as well as a local variant of the Directed Flat Wall Theorem due to Giannopoulou, Kawarabayashi, Kreutzer and Kwon. These techniques, which are somewhat atypical in the study of minimum degree conditions, may be of independent interest and may find further applications.

We study synchronization in the Kuramoto model on finite graphs. We prove that there is an absolute constant $\eta>0$ such that every finite simple graph $G$ on $n$ vertices with minimum degree at least $(3/4-\eta)n$ has no local minima of the Kuramoto energy other than the fully synchronized states. This strictly improves the previous $3/4$ upper bound and refutes a conjecture of Bandeira, Kireeva, Maillard, and Rödder.

When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Simó, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.

With additional spatial degrees of freedom provided by the active intelligent reflecting surface (IRS), interference alignment (IA) can be achieved at low cost. In this letter, we propose a real IA scheme for an active IRS-aided system. The proposed scheme only requires the IRS to know the instantaneous channel coefficients under the assumption of blocked direct links. To maximize the achievable sum rate subject to individual minimum rate requirements and transmission power constraints, we propose a rate-profile learning-based algorithm. The algorithm uses offline-trained achievable rate profiles to decouple the original problem into multiple feasibility subproblems, which are then solved by generalized eigenvalue decomposition. Simulation results demonstrate that our proposed algorithm outperforms the conventional weighted minimum mean square error algorithm, while requiring significantly less program execution time.

A set of positive integers $R$ is called a set of nice recurrence if for any measure preserving system $(X,\mu,T)$, for all measurable $A\subseteq X$, and each $\varepsilon>0$, there exists $n\in R$ such that $\mu(A\cap T^{-n}A)\geqslant \mu(A)^2 - \varepsilon$. Answering a long-standing question of Bergelson, we show that sets of nice recurrence have the following Ramsey property: any finite colouring of a set of nice recurrence admits a monochromatic set of nice recurrence.

The elastic modulus of porous materials is commonly described using power-law scaling relations with relative density, where the scaling exponent is often interpreted in terms of the underlying deformation mechanism. However, in highly disordered porous networks, changes in density are generally accompanied by changes in network topology, which can substantially modify the apparent scaling behavior. In this paper, we propose a topology-informed framework that separates the intrinsic mechanical contribution from the effects of network structure. Three representative topological descriptors are considered: the mean coordination number, the fraction of the load-bearing backbone, and the tortuosity of the load paths. For each case, the corresponding density-dependent contribution to the apparent modulus-scaling exponent is derived and analyzed. The results show that variations in connectivity, mechanical participation of the solid phase, and load-path efficiency can all lead to apparent scaling exponents exceeding the intrinsic exponent associated with the local deformation mechanism. These effects are particularly pronounced at low relative densities, where network topology evolves most strongly. The framework provides a physically interpretable basis for understanding anomalous modulus-density scaling in disordered porous materials and highlights the need to consider topology explicitly alongside relative density.

A partial permutation of $[n] := \{1,\dots,n\}$ is a bijection $g: I \to J$ between two subsets $I,J \subseteq [n]$. Given a partial permutation $g$ of $[n]$, let $a_g \in \mathbb{C}[\mathfrak{S}_n]$ be the group algebra sum of those permutations $w \in \mathfrak{S}_n$ which extend $g$. Informally, a partial permutation $g$ is obtained by forgetting some data in a genuine permutation. The forgotten symmetric functions are the least-studied of the six `standard' bases for the ring of symmetric functions. We show that forgotten symmetric functions arise naturally in class function evaluations on partial permutations.

Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.

We introduce a novel visco-elastic Mullins-Sekerka system with a prescribed constant contact angle at the boundary. The system is derived as an $H^{-1}$-$H^1$-type gradient flow of an energy consisting of the perimeter together with capillary, elastic, and second-gradient contributions. Building on the framework of Hensel and Stinson (Arch. Ration. Mech. Anal. 248, 2024), we introduce a measure-valued solution concept featuring a sharp De Giorgi-type energy-dissipation inequality. Moreover, we establish existence of solutions via an implicit time discretization scheme, and prove existence of $BV$ solutions under an energy-conservation hypothesis.

Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $ f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C $ such that $ f^{(-1)} $ is monotone, continuous and it acts as a left-inverse of $ f $. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.

We use Hamiltonian Floer theory to prove a cuplength result about the existence of periodic solutions of particle-field systems with a Gaussian random field. As a concrete model we study stochastic electrodynamics, which approximates quantum electrodynamics up to first order in $\hbar$, and is even exact in the case of low-order Hamiltonians or when the particles are treated classically.

We develop a resonant-state expansion formulation for acoustic scattering by individual resonators. The scattered pressure and particle-velocity fields are expanded over the resonant states of the system, with excitation amplitudes determined by overlap integrals between the incident field and the resonant states over the resonator volume. Using the acoustic energy flux, we derive expressions for the extinction, scattering, and absorption cross-sections and show that the extinction spectrum can be resolved into contributions from individual resonant states. The formulation is first validated for a homogeneous two-dimensional cylinder, where it reproduces the analytical Mie-theory solution. We then consider a sectorally perturbed cylinder with coupled azimuthal modes and demonstrate agreement with finite-element simulations. Finally, we combine the eigenvalue and scattering formulations for a material-programmed hard-wall annular metaatom and reproduce its scattering spectra and near fields. The developed framework provides a physically transparent modal approach to acoustic scattering by open resonators with reduced symmetry and spatially structured material parameters.

We prove optimal $(n-1)$-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the $n=2$ case.

We discuss three questions from a recent paper of Bhandari, Cunningham, Morrell, Oh and Smith. We obtain an exact local formula for the facet gap between the order and the chain polytope of a finite poset. This gives a classification of the case $\gap(P)=2$ in terms of star elements. For marked chain–order polytopes, the same local weights determine the facet differences between all admissible decompositions.

We prove that weakly stable solutions of Serrin's problem are compact and therefore round balls.

In this paper the tree structure of numerical semigroups is studied. An internal numerical semigroup is a semigroup located in an internal node of the tree. Analogous a leaf numerical semigroup is placed in a leaf node. Internal semigroups with fixed multiplicity, Frobenius number or genus are studied by providing algorithms to construct all of them. Several conjectures are established, for example, in each case (fixed multiplicity, fixed Frobenius number and fixed genus respectively), the results suggest that there are always more internal than leaf numerical semigroups. Finally, numerical semigroups with fixed multiplicity and Frobenius number simultaneously are investigated. In this case with two invariants fixed, moreover closed formulas to count the number of internal and leaf semigroups are provided for some values of multiplicity and Frobenius number.

Resonant-state expansion (RSE) is a powerful modal framework for the perturbative analysis of open resonant systems, providing direct access to complex eigenfrequencies and eigenmodes. While RSE is well developed in electromagnetism, a comparably systematic formulation for acoustics remains less established. Here, we develop a general Green-function-based formalism for acoustic RSE and illustrate it for a class of two-dimensional acoustic resonators. Using the resonant states of an analytically solvable cylindrical reference system as a basis, we derive explicit perturbation matrix elements for uniform, radial, and sectoral variations of density and compressibility, representing homogeneous tuning, graded profiles, and symmetry-induced modal coupling. The resulting complex eigenfrequencies and eigenmodes are validated against exact analytical solutions and finite-element simulations, showing excellent quantitative agreement. The framework provides a systematic and physically transparent approach for analyzing perturbed open acoustic resonators and establishes a basis for resonant-state methods in acoustic metamaterials and non-Hermitian acoustics.

We provide a positive answer to a conjecture proposed by Tohǎneanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical Bézout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.

A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $\sigma$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.

Let \(\psi(n)\), \(\phi(n)\), and \(\sigma(n)\) denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions \(\phi(\psi(n))\), \(\phi(\sigma(n))\), \(\psi(\psi(n))\), and \(\psi(\sigma(n))\). For every fixed \(c>0\), we obtain quantitative upper bounds for the numbers of integers \(n\le x\) satisfying \(\phi(\psi(n))\ge cn\) and \(\phi(\sigma(n))\ge cn\), thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers \(n\le x\) for which \(\psi(\psi(n))\le cn\) or \(\psi(\sigma(n))\le cn\). More generally, we allow the fixed thresholds in the first two problems to be replaced by thresholds involving non-decreasing functions subject to mild growth conditions.

We introduce explicit construction methods for two distinct birational models of elliptic surfaces equipped with a bisection — or, more generally, a relative polarisation of degree two — and double fibres. For Halphen surfaces of index two and Enriques surfaces, we illustrate how these give simple descriptions inside a toric variety.

This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincaré-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.

Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.

For the planar Newtonian 1+N-body problem when the N masses tend to zero, the corresponding relative equilibria become coorbital around the dominant mass. In this work, we focus on convex central configurations in the planar 1+N coorbital problem. For the 1+5 coorbital problem with the homogeneous potential, we prove that any convex coorbital central configuration with symmetric masses must have an axis of symmetry. Furthermore, under explicit restrictions on the angular variables in a homogeneous potential, we prove the linear stability of both convex symmetric 1+5 and convex 1+N coorbital central configurations.

We study the behavior of good involutions under two fundamental constructions of quandles: interaction-free unions and direct products. In particular, we show that the set of good involutions on a quandle decomposes naturally into the set of involutions on its maximal trivial component and the set of good involutions on the complement. Moreover, we construct an example of a connected noninvolutory symmetric quandle with multiple good involutions, which serves as a counterexample to a conjecture by Ta.