Papers

For various definitions of fractal dimension that are finitely stable, including upper box dimension, upper intermediate dimensions and Assouad spectra, we show that, given a compact subset $E$ of a metric space, typically $R^n$, there is a convergent sequence of points contained in $E$ of the same dimension as $E$ itself. Moreover, under certain conditions it is possible for a sequence to witness the dimension of $E$ for many definitions of dimension simultaneously, for example in a self-affine set $E$ there is a single convergent sequence that has the same Assouad spectrum or intermediate dimension values as $E$ itself.

We determine the exact weak limited augmented Zarankiewicz numbers $z_{wL}(m,3)$ for all $m\ge 3$: \[ z_{wL}(m,3)= \begin{cases} m+3+\left\lceil \dfrac{m}{2}\right\rceil+1, & 9\le m\le 15,\\[2mm] m+3+\left\lfloor \dfrac{2m-4}{3}\right\rfloor, & m\ge 16, \end{cases} \] with $z_{wL}(3,3)=6$, $z_{wL}(4,3)=8$, and $z_{wL}(m,3)=2m$ for $5\le m\le 9$. In particular, \[ \lim_{m\to\infty} \frac{z_{wL}(m,3)}{m} = \frac{5}{3}. \] The proof is fully analytic, relying on a uniform base classification, two constructive lower-bound families (staircase and $5m/3$), and a sharp upper-bound argument based on a peeling lemma and the analysis of two W2-sensitive boundary cases. Numerical MILP computations were used only as proof-mining tools to identify the structural lemmas; the final theorem is unconditional. We also extend the known range of the original limited numbers $z_L(m,3)$ through $m=13$, where the gap to $z_{wL}(m,3)$ is only 2 or 3.

We develop an ODE/IM-based formulation for the thermal partition function and the spectral zeta function of the homogeneous Hermitian and PT-symmetric oscillators. For both classes of systems, the quantization condition can be expressed using the counting function $a(E)$, which can be solved via the Destri-de Vega equation of the integrable model. We then express the partition function and spectral zeta function as contour integrals involving the counting function, thereby providing a direct bridge between quantum spectral functions and integrable models.

Given a graph $G=(V,E)$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex $u$ adjacent to $v$ with $f(u) = 2$, or two distinct vertices $x,\; y$ both adjacent to $v$ with $f(x)=f(y)=1$ (Chellali et al. 2016). Every graph $G$ satisfies $\gamma_{\{R2\}}(G) \leq 2\gamma(G)$, where $\gamma_{\{R2\}}(G)$ denotes the minimum weight of a $\{2\}$-Roman dominating function of $G$ and $\gamma(G)$ is the domination number of $G$. \emph{$\{2\}$-Roman graphs} are those for which the equality is reached (Klostermeyer et al. 2019). A characterization of $\{2\}$-Roman trees was given by Henning et al. in 2017. In 2025, Ferrari et al. characterized the $\{2\}$-Roman property by the existence of a minimum $\{2\}$-Roman dominating function of $G$ that assumes only $0, 2$-values. Afterwards in 2025, Bešter Štorgel et al. introduced the problem of recognizing $\{2\}$-Roman graphs, proved polinomiality for middle graphs, and characterized \hbox{$\{2\}$-Roman} split graphs that can be decomposed with respect to the split join operation into two smaller split graphs. Recognition complexity is still open for general graphs. In this paper we study the \hbox{$\{2\}$-Roman} property on graphs that can be decomposed into two smaller graphs with respect to the join and union operations, allowing to completely characterize the $\{2\}$-Roman property. The 4-path is the non trivial connected non $\{2\}$-Roman graph with the fewest number of vertices and edges. We classify the $\{2\}$-Roman property within specific families of non decomposable graphs with a limited number of 4-paths which are present in the decomposition of partner limited graphs; these are well-labelled spiders, the graphs in ZOO and some special split graphs.

The multiple-unicast conjecture asserts that network coding offers no throughput advantage over routing in undirected networks. Its validity is known to imply fundamental lower bounds in computational complexity. We propose a Session Interaction Framework that reduces the conjecture to a central equivalence: the conjecture holds universally if and only if every irreducible core is independent. This result transforms the global feasibility problem into a two-stage process. First, to make the reduction phase tractable, we provide simplified sufficient conditions for session dominance, offering geometric criteria to iteratively simplify complex session sets. Second, for the remaining "irreducible core," we propose a Session Decoupling Theorem, reducing the conjecture's validity for a session set to its independent subsets. Topologically, we prove that sessions separated by high-cost cuts or cut-vertices are guaranteed to be independent. By integrating these reduction and decomposition mechanisms, our framework offers a systematic methodology to verify the conjecture across general network topologies.

The Liouville problem for the three-dimensional stationary Navier–Stokes equations remains open, even for axisymmetric \(D\)-solutions. In this paper, we obtain two results based on decay in the cylindrical radial variable \(r=|x'|\). (i). Using a new pointwise Calderón–Zygmund estimate adapted to cylindrical geometry, we improve the decay estimates of Carrillo–Pan–Zhang (2020, JFA) and prove \[ |\nabla u_r|+|\nabla u_z| \lesssim r^{-5/4}[\log(\mathrm e+r)]^{5/4}, \quad |\omega_r|+|\omega_z| \lesssim r^{-9/8}[\log(\mathrm e+r)]^{9/8}, \quad r\gg1. \] (ii). We develop a new approach to Liouville theorems that improves the axisymmetric criteria of Wang (2019, JDE) and Zhao (2019, Nonlinear Anal.). Without any symmetry assumption, we show that a \(D\)-solution is trivial if one of the following holds: \[ (\mathrm a).\,\sup_{{|x'|=r,\, z\in\mathbb R}} |u(x',z)| \leq Cr^{-2/3}[\log(\mathrm e+r)]^{-\gamma}; \quad (\mathrm b).\, \sup_{{|x'|=r,\, z\in\mathbb R}} |\omega(x',z)| \leq Cr^{-5/3}[\log(\mathrm e+r)]^{-\gamma}, \] for $r\geq1$, where $\gamma>1/3$.

We consider a Poisson process $\eta$ on a general Borel space with intensity measure $\lambda$ and a stopping set $Z$ depending on $\eta$. Generalizing earlier results we prove a spatial Markov property for $Z$. In fact we establish this property for the larger class of finitely insertion tolerant random sets, which include pointwise limits of stopping sets. Using the spatial Markov property of $\eta$, we derive several distributional identities for the restrictions of $\eta$ and $\lambda$ to $Z$ and the complement of $Z$. An important special case in Euclidean space is the convex hull of a finite Poisson process. In this case our results generalize classical (and also more recent) identities connecting the number of vertices and the volume of the convex hull.

This note studies the distributed fault diagnosis problem in partially-observed discrete event systems, where the system is monitored by a group of agents to cooperatively diagnose faults within a finite number of steps. The novelty of this work is the creation of a methodology to verify when the faults can be diagnosed even in the presence of transmission delay impairments. To address this scenario, a new distributed diagnosability condition is proposed, which extends decentralized diagnosability conditions proposed in the literature. Such distributed diagnosability condition is then verified via a novel structure named delay recorder and a new diagnosis function. Theoretical analysis shows that the verification method can successfully determine whether the faults can be diagnosed.

We establish high-energy asymptotic expansions of Titchmarsh–Weyl functions for one-dimensional Schrödinger and Dirac operators in regions whose boundaries approach the spectrum at a prescribed polynomial rate. For bounded potentials with bounded derivatives, the expansions remain uniform up to these boundaries, with an explicit loss in the remainder determined by the rate of approach. We treat both self-adjoint Dirac operators and non-self-adjoint Dirac operators with skew-adjoint potential matrices, keeping track of the distinct half-planes in which the two scalar Weyl coordinates are naturally defined. As an application of the Schrödinger expansion, we verify the high-energy hypothesis in Kotani's construction of KdV flows: for every odd integer $p\geq3$, each real-valued $q\in W^{2p-1,\infty}(\mathbb{R})$ generates a global classical solution of the member of the KdV hierarchy indexed by $(p+1)/2$.

In this note, we observe that the class of unbounded subnormal operators on quaternionic Hilbert spaces is a solution of quaternionic q-oscillator problem whose complex case is addressed by Szefraniec in [27]. We also address the connection of unbounded subnormal operators on quaternionic Hilbert spaces, quaternionic quantum harmonic q-oscillator and quaternionic quantum Economics using model theory of unbounded subnormal operators. Moreover, we study model theory for sub-2-normal operators on quaternionic Hilbert space.

We give a local geometric criterion ensuring that the edge-connectivity of a hypergraph equals its minimum incidence degree: every locally finite connected $r$-uniform linear hypergraph with $r\ge3$ and nonnegative Lin–Lu–Yau curvature has this property. Among the various extensions of Ollivier–Ricci curvature to hypergraphs, we work with the equal-edges random walk on hypergraphs \cite{CoupetteEtAl2023}. Moreover, we show that both uniformity and linearity are essential: if either assumption is removed, there exist positively curved hypergraphs for which the gap between minimum incidence degree and edge-connectivity is arbitrarily large. For arbitrary locally finite simple hypergraphs, we also determine the dependence on idleness completely: every idleness function is piecewise affine with at most three affine pieces and is affine on the universal interval $[1/2,1]$. These results extend the corresponding theory for graphs \cite{BourneEtAl2018}. They also provide two useful tools below: the $2$-section reduction underlying the edge-connectivity argument and a limit-free formula used in the sharpness constructions.

We analyze the stability of the equilibria and bifurcations of a planar system of ordinary differential equations describing the product-resource interaction in a sustainable supply chain. While periodic behavior in supply chain models has often been documented either in systems of higher dimensions or in delayed systems, here a Hopf bifurcation arises in a planar and delay-free system. We show that the interior equilibrium loses stability through a supercritical Hopf bifurcation as the environmental capacity exceeds a critical threshold that depends on the maximum production rate, the resource level at which production reaches half its maximum rate, and the demand and remanufacturing rates. We derive this threshold explicitly and provide estimates for the amplitude and period of the periodic solution close to the Hopf bifurcation, by means of the resulting Hopf normal form. We then further substantiate our analytical results through numerical simulations.

We introduce several conjectures which mainly deal with restrictions on the Betti tables of licci ideals. We focus on a series of questions that compare the number of generators of homogeneous licci ideals in polynomial rings to the maximal last shift in their graded free resolution. We prove these conjectures in a large number of cases.

We formulate several open questions regarding the tropicalization of linear ODEs, aiming primarily to develop methods for calculating the radius of convergence of their classical solutions. To this aim it is of foremost importance to characterize the classes of equations that admit a finite differential tropical basis, as introduced in (Fink and Toghani, 2022). Our initial exploration examines the second- and third-order cases.

We study static, spherically symmetric axion–dilaton solutions of the tree-level four-dimensional string effective action. Using the sigma-model structure of the scalar sector, we show that the full family of static, spherically symmetric and asymptotically flat solutions is obtained as the $SL(2,\mathbb R)$ orbit of the pure-dilaton FJNW solution, and we characterize the associated axion–dilaton charge space. We then analyze the perturbative regime of these solutions by comparing the onset of $\alpha'$ curvature corrections with that of string-loop corrections. Finally, we apply the string–black-hole correspondence criterion to the FJNW family by evaluating the local $\alpha'$ curvature diagnostic at the physical size $R_{\rm typ}$ of a highly excited string state. With the normalization fixed on the Schwarzschild branch, Schwarzschild saturates the nominal $\alpha'$ threshold at the correspondence surface, whereas every scalar-haired FJNW branch lies above it. More generally, independently of the common overall normalization within the matching prescription, every nonzero-hair branch has a larger curvature diagnostic at the correspondence surface than Schwarzschild. Away from the correspondence point, scalar-haired exteriors may remain under $\alpha'$ control at sufficiently weak local self-gravity and may provide perturbatively controlled long-distance fields of highly excited string states whenever the local string coupling also remains weak.

In this paper, we study the strong averaging principle for multiscale time-inhomogeneous stochastic systems driven by multiplicative $\alpha$-stable processes with $\alpha\in(1,2)$. Based on Khasminskii's discretization approach, we first establish that the fast component processes with a frozen slow variable admits a periodic measure. We then prove the strong convergence of the slow subsystem to an averaged system that depends on the time scale $\varepsilon$. For any fixed $\varepsilon$, if the reciprocals of the two periods $\tau_1$ and $\varepsilon \tau_2$ are rationally linearly independent, an important consequence is that the averaged system has random quasi-periodicity. Furthermore, by applying the ergodic theorem, we prove the strong convergence of the slow subsystem to another averaged system, a time-inhomogeneous SDEs independent of the time scale $\varepsilon$. Our result is also novel even in the time-homogeneous case for a fully coupled multiscale system with multiplicative $\alpha$-stable noises. Finally, we apply the result to a climate-weather system.

We define an enlargement of the class of exponential periods by applying the rapid decay cohomology theory of Hien to a larger class of integrable connections on varieties than those twisted by a regular function. The main and motivating examples are given by those associated to $E$-operators. This delivers a method to realise a class of absolutely convergent integrals involving $E$-functions as matrix elements of a period pairing, which we call $E$-periods. Furthermore, we generalise rapid decay cohomology to singular varieties and prove a version of Nori's basic lemma in this setting.

We introduce a second-order mean-field description of the four-dimensional Euclidean $\phi^4$ model within the Wilson–Polchinski renormalization-group framework. The construction is based on the connected amputated Schwinger functions evaluated at the symmetric momentum configurations $ (p,-p,\ldots,p,-p)$, which are decomposed into a momentum-independent component, a component quadratic in $p$, and a higher-order remainder. The first two components are chosen to satisfy a closed nonlinear hierarchy. We prove the existence of solutions to this hierarchy for arbitrary positive bare coupling and establish their convergence to the Gaussian fixed point as the ultraviolet cutoff is removed. In particular, both the momentum-independent and the quadratic momentum sectors are asymptotically trivial.

We construct a divergence-free velocity field on the three-dimensional torus that is time-periodic and consists of three alternating piecewise-affine shears. For sufficiently large shear amplitude, every non-zero divergence-free initial field in $L^p$ grows exponentially under the ideal induction equation. The proof establishes a uniform cone condition for the time-one map, and combines it with a bunching inequality to rule out nontrivial divergence-free fields lying almost everywhere in the stable bundle. Additionally, we show that for the time-reversed velocity, whose time-one map is the inverse of the original one, there exist nontrivial, bounded, divergence-free initial configurations taking values almost everywhere in its two-dimensional stable bundle. The corresponding solution decays exponentially in every $L^p$.

We study the simultaneous vanishing-viscosity and vanishing-resistivity limit of steady incompressible MHD flows in a disk. The boundary velocity is a small nonaxisymmetric perturbation of a rigid rotation with mean angular speed \(\alpha\), while the prescribed tangential magnetic trace has mean \(\beta\). Assuming \(\alpha\neq0\) and the non-Alfvénic condition \(|\alpha|\neq|\beta|\), we construct solutions converging on compact interior subdisks to a rigidly rotating ideal MHD core with constant vorticity and out-of-plane current density. A new MHD–Wood law determines the two core rotations from the boundary data: the velocity core is selected by a coupled kinetic–magnetic balance, whereas the magnetic core is fixed by the imposed mean circulation. Consequently, zero circulation gives complete interior magnetic expulsion, while nonzero circulation leaves a uniform magnetic rotation after the nonaxisymmetric modes are confined to a thin boundary layer. This provides a fully coupled realization of Prandtl–Batchelor selection and flux expulsion; unlike classical kinematic models, the magnetic field actively changes the flow and need not be weak. The proof combines a non-Alfvénic coercive theory for a periodic MHD boundary layer, global matching of the two fields, and a coupled stability estimate adapted to the magnetic boundary condition. A separate conditional rigidity argument, using exact viscous identities and local convergence but no interior asymptotic expansion, explains the same core structure for a broader single-eddy family.

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that \[ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha} <1. \] By his variation formula, this inequality is sufficient for every positive zero of $\pi_n$ to move to the right as $c$ increases. For $0<c<1$, a first-crossing argument proves the conjecture throughout $0<\alpha<\beta$. Together with the earlier regions recorded by Gautschi and established by Milovanović, this settles $\beta\geq0$. In the negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, an ensemble Ward bound yields the region $c^2\leq3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\geq1$, $-1<\alpha<\beta$, and $0<c\leq1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.

We study, under the Generalized Riemann Hypothesis (GRH), the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character $\chi_{8p}$, where $p$ ranges over odd primes. We establish the correct order of magnitude for the unsmoothed $m$-th moment for all real $m\geq 4$, and a sharp upper bound of order $XY^{m/2}\,\, (\log X)^{m(m-3)/2}\,\,$ for the smoothed $m$-th moment for all integers $m\geq 4$. A matching lower bound for all even integers $m\geq 4$ shows that this bound is optimal.

Policy iteration (PI) is an important reinforcement learning tool for solving optimal control problems which includes an initialization stage, i.e., the search for an initial stabilizing controller. However, the initialization stage typically relies on complete model information, thereby imposing substantial constraints on the initialization of model-free PI. For stochastic systems with multiplicative noise dependent on state and control, the stability is not ensured by Hurwitz conditions as in the deterministic case, but rather by a Lyapunov-type inequality that incorporates both drift and diffusion terms. Therefore, the corresponding model-free PI initialization problem is more challenging. To this end, a novel spectrum assignment method is proposed to obtain an initial stabilizer for PI in continuous-time indefinite stochastic linear quadratic control. With the help of the Lyapunov-type operator's spectrum, the original system is gradually approximated from the stable auxiliary system by adjusting a cumulative factor, thereby obtaining a stabilizing control gain. Furthermore, by leveraging system data and adjusting the cumulative factor, we design a model-free algorithm that does not rely on an initial stabilizing policy and can achieve optimal control. Finally, simulation results are provided to validate the effectiveness of the proposed methods.

We investigate Hamiltonian curl-force systems with indefinite kinetic energy. We first reconsider Berry's polynomial Hamiltonian curl-force model, whose numerically observed closed trajectories motivated an integrability conjecture. A Painlevé analysis yields a non-principal resonance spectrum, so that the corresponding Laurent series cannot accommodate the required number of arbitrary constants of the general solution. The model therefore fails the standard Painlevé test. We then introduce a four-parameter curl-force family and identify the parameter locus on which this system is integrable. We construct a second Hamiltonian, compatible Poisson tensors, separated complex characteristic variables, and a Lax representation. More generally, the separated form yields polynomial integrable curl-force Hamiltonians of arbitrary degree. We also show that the same construction admits a higher time-derivative potentialisation whose free limit is the degenerate Pais-Uhlenbeck oscillator. Finally, we analyse zero-curl invariant reductions and elliptic periodic solutions, and exhibit an isolated periodic orbit outside the integrable regime. This illustrates that closed trajectories alone do not imply Liouville or Painlevé integrability.

Let $M_n=M(K_{n+1})$ be the graphic matroid of the complete graph, and let $\mathcal{F}_k(M_n)$ be its rank-$k$ flats. We study families $\mathcal{A}\subseteq\mathcal{F}_k(M_n)$ satisfying $\mathrm{rk}(A\wedge B)\ge t$ for all $A,B\in\mathcal{A}$. For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erdős and L.~A. Székely~\cite{ErdosSzekelyHigher}. We prove the corresponding Erdős–Ko–Rado theorem in the explicit linear range $n+1\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\ge2k$. For every fixed $t$, we further prove an Erdős–Ko–Rado theorem under an explicit condition of order $O_t(k^2)$ on the block number $n+1-k$, with equality only for a full $t$-star. We also determine the largest nontrivial intersecting families under an explicit $O(k^6)$ threshold and characterize the unique extremal family up to isomorphism.

The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erdős, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.

This paper investigates when a complex family of K3-surfaces with prescribed simple singularities contains real algebraic surfaces. We develop an explicit and uniform algorithm for detecting real representatives in any equisingular stratum, valid for all polarizations. As a primary application, we resolve the problem for spatial quartics, showing that all real equisingular strata contain real surfaces except for three exceptional types, thereby completing the study of real representability in this case. Our method also revisits the analogous phenomenon in the realm of plane sextics, recovering the unique known real stratum of simple sextics without real representatives.

This is the first part of a two-part work concerning the boundary layer convergence for chemotaxis-Navier-Stokes system in a two-dimensional half-space. In this paper, we investigate the chemotacxis-Navier-Stokes system with the logarithmic singularity under Navier-slip boundary conditions. More precisely, we perform an exact asymptotic expansion for the chemotaxis Navier-Stokes system with viscous coefficient $\varepsilon>0$, and obtain partial boundary layer profiles, establishing the well-posedness of the corresponding boundary layer profiles. Specially, we also establish the local well-posedness of solutions to the supercritical chemotaxis Euler equation (with $\varepsilon=0$) by overcoming the difficulty from the disappearance of diffusion terms.

In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

On imaginary quadratic fields, we establish the $\mathrm{GL}_2$ Bessel $\delta$-method and prove the hybrid Weyl-type subconvexity bound for $\mathrm{GL}_2 \times \mathrm{GL}_1$ twisted $L$-functions in the Archimedean aspect of the Hecke character on $\mathrm{GL}_1$.