Papers

Motivated by majority vertex-colorings of graphs and digraphs and majority edge-colorings of graphs, we introduce two concepts of strong majority colorings. A strong majority vertex-coloring of a graph $G=(V,E)$ is a mapping $c:V\rightarrow C$ such that for every vertex $v\in V$ and every color $\alpha\in C$, at most half of the neighbors of $v$ have color $\alpha$. The strong majority number of $G$, denoted Maj$(G)$, is the least number of colors in such a coloring. We show that Maj$(G)$ can be arbitrarily large and prove a tight upper bound Maj$(G)\le 2\Delta(G)+1$ for every graph $G$ without pendant vertices. A strong majority edge-coloring of a graph $G$ is a mapping $c:E\rightarrow C$ such that for every edge $e\in E$ and every color $\alpha\in C$, at most half of the edges adjacent to $e$ have color $\alpha$. The strong majority index of $G$, denoted Maj'$(G)$, is the least number of colors in such a coloring. It is shown that there is an upper constant bound for Maj'$(G)$ of all admissible graphs $G$. We conjecture that this constant is as small as 4 and confirm this conjecture for numerous graph classes.

Given a closed normal subgroup $H$ of a topological group $G$, we address the question of whether the left coarse structure on the quotient group $G/H$ equals the quotient of the left coarse structure on $G$. We provide a counterexample among Polish groups, namely, the mapping class group of the Loch Ness monster surface seen as a quotient of the mapping class group of the punctured Loch Ness monster surface, and establish both equivalent and sufficient conditions for when this holds in special settings. The latter are formulated in terms of liftings of bounded sets, existence of transversals and metrisability of the left coarse structure of $G$ restricted to $H$.

In Poisson Boolean models with deterministic ball grains, the directional visible range from an uncovered point is known to be exponentially distributed in Euclidean and real hyperbolic space. We show that the same phenomenon holds on every simply connected non-compact homogeneous harmonic manifold. The geometric mechanism behind this fact is the affine-linear growth of tube volumes around geodesic segments. As a consequence, we identify the finiteness regime for the expected volume of the visible region, including a geometric interpretation of the critical threshold in the positive-entropy case. We also construct explicit complete non-homogeneous Riemannian manifolds showing that exact exponentiality is tied to exact tube linearity: superlinear tube growth leads to Weibull-type tails, while asymptotic tube linearity still yields an exponential decay rate.

Modern optimal control theory involves adjoining the already known equations of motion of a dynamic system to the objective function using dynamic costates; this is done in order to constrain the optimal control solutions to satisfy the equations of motion. The use of costates increases the number of variables and hence increases the complexity of the problem. On the other hand, variational methods of analytical mechanics finds the equations of motion by minimizing an action functional of the dynamic system, realizing control forces as external input to the system. In this paper a new disruptive approach for computing the optimal control is presented. This approach adopts the variational methods of analytical mechanics to derive equations for the control, in addition to the equations of motion. This is achieved by recognizing the control actuator as part of the dynamic system. In addition to the kinetic energy and potential energy, the action functional in this new approach includes additional energy terms that represent the control energy of the system. Two different methods are presented to write the modified action functional. The proposed approach is a significant departure from the modern optimal control theory, and it eliminates the need for costates when solving for the control. In this paper, a case study is presented to demonstrate the new approach.

In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-\epsilon})u^{2_{s}^{\sharp}-2-\epsilon} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}\Omega,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}\Omega,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\partial\Omega,\end{aligned} \right.\end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $0<s<1$, $n\in(2s,\min\{6s,n+2s\})$, $\epsilon>0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the spectral fractional Laplacian. For a general domain $\Omega$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $\epsilon\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.

This paper presents a framework for assigning intrinsic geometric structures to topological groups using only the data provided by their topological and algebraic structure. The geometrisation spits into small-scale and large-scale components, formalised respectively through local Lipschitz and quasimetric categories that, in turn, are definable from the canonical left uniform and left coarse structures of the group. For Polish groups, the paper characterises metrisability of the left coarse structure in terms of local boundedness, countable coverings by bounded sets, and the existence of compatible coarsely proper left-invariant metrics. It then introduces minimal metrics, which determine local Lipschitz structure, and maximal metrics, which determine quasimetric structure, and provides intrinsic characterisations of both. When both structures exist, they combine into a single canonical Lipschitz structure. Our framework is subsequently applied to specific examples such as homeomorphism groups, non-Archimedean Polish groups and automorphism groups of Fraïssé limits.

We study the Brownian dynamics and linear response of a particle with inertia moving in a 2-dimensional helical landscape imprinted on a cylindrical surface. In the harmonic well approximation, the deterministic motion separates into free propagation along the screw direction and harmonic motion in the transverse screw-normal direction. We show that for isotropic damping this simplification survives in the Langevin description, whereas anisotropic damping along the axial and angular directions couples the stochastic dynamics and destroys separability. The resulting anisotropic model is formulated as a linear Ornstein-Uhlenbeck process in phase space with a zero mode associated with diffusion along the screw coordinate, so that in an infinite system the full phase-space dynamics does not relax to a stationary distribution. To treat transport in this setting, we construct the stationary dynamics in the stable subspace obtained after projecting out the zero mode. This leads to a linear response theory for this system and yields closed analytical expressions for stationary time-correlation functions and the dynamical mobility tensor in both the time and frequency domains. The off-diagonal elements of the mobility tensor describe cross-response between axial forcing and angular motion, and between applied torque and axial transport. Consistent with time reversal symmetry, these cross mobilities are equal and provide a direct dynamical signature of the helical geometry. In addition, a simultaneous application of driving in both the axial and angular direction reveals asymmetry in energy dissipation rate due the helical landscape.

We study Rota–Baxter operators on vertex algebras using the integrated $\lambda$-bracket formalism. A Rota–Baxter operator produces a deformed vertex algebra structure, and the difference between the deformed and original brackets yields a two-cocycle in vertex algebra cohomology. This generalizes the classical relation between Rota–Baxter operators and Hochschild two-cocycles. We also characterize when this two-cocycle is trivial, showing that non-scalar operators give rise to non-trivial cohomology classes.

Let $\mathbb{P}$ be an algebraic number field. We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense groups $H\leq \mathrm{SL}(n,\mathbb{P})$, $n$ prime. That is, we present algorithms to find the set of congruence quotients of $H$ modulo all maximal ideals of a finitely generated subring $R$ of $\mathbb{P}$ such that $H\leq \mathrm{SL}(n,R)$. The algorithms have been implemented in GAP. Potential applications are illustrated by a range of experiments in degree $2$, with a special focus on Bianchi groups.

We investigate phaseless inverse scattering problem for the Schrödinger equation and develop reconstruction methods based on the inverse Born series (IBS). We consider three types of phaseless data: the far-field total field, the total field and the far-field scattered field. For phaseless total-field data, we extend the IBS framework and analyze its radius of convergence. In the far-field setting, we propose a Fourier-based reconstruction method that exploits the scattering reciprocity between incident and observation directions to recover Fourier coefficients of the scattering potential. For phaseless scattered-field data, we employ a polarization-based approach to recover phase information and enable IBS reconstruction. Numerical experiments are conducted to validate the proposed methods.

We determine which connected sets of $N$ vertices in the square lattice support the most spanning trees, using all lattice edges between the chosen vertices. Every maximizer minimizes the number of edges joining the set to the rest of the lattice. Its shape is a rectangle with at most two straight strips of vertices removed at opposite corners. For $N\ge18$, the maximizer is unique up to translation, rotation, and reflection: an explicitly determined rectangle with one possibly empty corner strip removed along its longer side. The proof combines comparisons of Laplacian determinants with geometric comparisons of corner deletions. For $N=n^2$, we prove a sharp stability inequality that bounds the symmetric difference from a translated $n\times n$ square in terms of the loss in logarithmic tree count. Thus, nearly optimal tree counts force the vertex set to be close to a square. We also derive an exact product formula and a recurrence for the number of spanning trees in rectangles with a single corner strip removed.

Reachability analysis plays a central role in low-thrust spacecraft trajectory optimization by identifying which target states can be achieved under constraints on time, thrust, and propellant. Classical approaches construct reachable sets by solving many optimal control problems over grids of terminal states, requiring extensive forward simulations with fixed initial conditions. While effective, this approach is computationally expensive and becomes impractical for high-dimensional systems or strongly nonlinear dynamics, such as those encountered in cislunar environments or solar sail missions. This work introduces a dual formulation of the reachability problem. Instead of computing reachable sets directly, we determine, for fixed transfer time and boundary conditions, the maximum allowable initial mass (or, for solar sails, a scalar sail-strength parameter) that permits a successful transfer. A target is reachable if the spacecraft's initial mass does not exceed this threshold. This reformulation reduces reachability assessment to a scalar optimization problem for each target, producing a smooth scalar field that encodes equivalent feasibility information to classical reachable sets. We develop indirect maximum-initial-mass (MIM) formulations for both electric low-thrust and solar-sail dynamics and show how they can serve as efficient reachability oracles. Building on this formulation, we construct data-driven surrogate models to approximate the MIM-based reachability indicator. We investigate fully connected neural networks and demonstrate that residual networks provide the best trade-off between accuracy, training stability, and model complexity. The resulting surrogates enable rapid reachability evaluation while preserving the numerical advantages of the dual formulation, offering a practical tool for preliminary mission design and feasibility assessment.

We study the solution of symmetric positive-definite linear systems by way of families of full- and limited-memory methods. Our contributions are threefold. We first derive new relationships between the conjugate-gradient method (CG) and quasi-Newton methods of the Broyden class that refine existing results, and clarify when those methods generate the same iterates and enjoy quadratic termination. We extend this perspective to the limited-memory BFGS (LBFGS) method. Next, we examine how DIOM, a limited-memory variant of the full orthogonalization Krylov method (FOM), is akin to LBFGS in that it provides a memory lever that is critical in practical performance. Finally, we generalize LBFGS and DIOM to the computation of trust-region steps for unconstrained, potentially nonconvex, optimization. We report numerical experience on positive-definite linear systems and unconstrained optimization problems. The results show that memory is a key algorithmic lever: LBFGS and DIOM are consistently more robust than CG and often achieve comparable accuracy with fewer Hessian-vector products. They emerge as viable alternatives to CG when high accuracy is desirable or when operations with the Hessian are at a premium. The limited-memory SR1 (LSR1) method can be competitive in full-memory form, but its limited-memory variant suffers from discarded curvature information.

This paper introduces a framework for Chance-Constrained Optimization with Complex Variables, addressing complex linear programming for both individual and joint probabilistic constraints in the complex domain. We first analyze the 3CP model in the density-based setting under the assumption that the random parameters follow a Complex Elliptically Symmetric distribution. The framework is then extended to distributionally robust settings, which include a moment-based model where the moments are known or bounded; a support-based model, where the ambiguity set contains distributions supported on norm-bounded uncertainty sets; and a data-driven model where moments are estimated empirically. The individual constraints are transformed into a convex deterministic second-order cone problem. We employ copula theory to the joint probability constraints and derive both upper and lower approximations. Finally, we demonstrate the proposed framework on the minimum variance distortionless response beamforming problem in signal processing. We further evaluate empirical out-of-sample rates and show that the observed behavior closely matches the prescribed probabilistic guarantees.

We study finite-dimensional spaces of rational one-forms on a projective manifold by means of their integrable locus.

The generalized Zernike family $H_{(N)} = p_1^2 + p_2^2 + \sum_{n=1}^N \gamma_n\,(q_1 p_1 + q_2 p_2)^n$ is a parametric family of two-dimensional superintegrable Hamiltonians, admitting $N$ integrals of motion of degree $N$ in the momenta. A theorem of Nozaleda, Tempesta, and Tondo guarantees that canonical separation coordinates (Darboux–Haantjes coordinates) exist for any such system; the challenge is to construct them explicitly. This paper solves the problem for $N = 2$ – the classical Zernike system, which is canonically equivalent to the isotropic harmonic oscillator on flat space or on a space of constant curvature – covering all four known separation types: polar, two Cartesian-type, and elliptic. The key structural fact is that the Haantjes operators associated with all integrals of $H_{(2)}$ have no momentum-dependent off-diagonal block (lift form). We prove that this implies the separation coordinates are reachable by an extended point transformation: the new positions depend only on the old positions, with no momentum entering the coordinate change. In the polar and Cartesian-type cases the new position coordinates involve at most a square root of a single-variable rational function; in the elliptic case they are given by the two roots of a quadratic polynomial in the original coordinates, and the resulting branch structure introduces a fourth regular singular point in the quantum separated ODE, placing it in the Heun class, in agreement with results of Atakishiyev, Pogosyan, Vicent, Wolf, and Yakhno. For $N \geq 3$ we prove an obstruction: no lift-form Haantjes operator can generate an integral independent of the angular momentum. The separation coordinates for higher Zernike Hamiltonians therefore require momentum-dependent canonical transformations, whose explicit construction is the subject of future work.

For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be a graph of order $n$ with minimum degree $\delta\geq6$, where $n\geq3\delta$. In this paper, we give a sharp spectral radius condition for $G$ to contain $k$ edge-disjoint 2-connected factors and $\left\lfloor\frac{\delta-4k}{2}\right\rfloor$ edge-disjoint spanning trees, where $1\leq k\leq\left\lfloor\frac{\delta}{4}\right\rfloor$ is an integer.

We study the dynamical properties of composition operators acting on Banach spaces of measurable functions. In particular, we study in some detail the composition operators induced by odometers, which allows us to give a variety of new examples and counter-examples. We also get general statements about supercyclicity and frequent hypercyclicity of composition operators on $L\_p$-spaces.

The paper introduces unbounded antilinear operators on Hilbert spaces and develops their fundamental theory. In particular, we establish a closed range theorem, a polar decomposition theorem, and the convexity of the numerical range for antilinear operators. Furthermore, we present several new results on antilinear normal operators and provide necessary and sufficient conditions for the existence of a minimal antilinear normal extension of an antilinear subnormal operator. We further develop a comprehensive characterization of antilinear block operator matrices with purely antilinear entries, establishing necessary and sufficient criteria for their closability through the framework of Schur and quadratic complements.

The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.

Caffarelli's contraction theorem and the analogous Laplacian result in [arXiv:2411.12109, arXiv:2501.11382] are two examples of how log-Hessian bounds on probability densities yield estimates on the derivative of the corresponding Brenier map with optimal dimensional dependence. The main goal of this paper is to extend such phenomenon to a broader class of convex estimates such as norms.

Given a pseudo-free self-similar action of a countable group $G$ on a countable directed graph $E$ with amenable stabilizers of the vertices, we identify the exact conditions under which these stabilizers do not contribute to the ideal structure of the corresponding Exel-Pardo algebra $\mathcal{O}_{G,E}$. Under these conditions, we give a complete description of the primitive ideal space of $\mathcal{O}_{G,E}$ in graph-theoretic terms. Our results apply in particular to certain crossed products $\mathcal{O}_E\rtimes G$, where $G$ acts on $E$ by graph automorphisms. When $G$ is trivial, this recovers Hong-Szymanski's description of the ideal structure of the Cuntz-Krieger algebras $\mathcal{O}_E$. Similar results are then obtained for self-similar actions of groups on row-finite higher rank graphs without sources. In order to obtain these results we formalize the notion of a graded groupoid with essentially central isotropy, which generalizes essentially principal groupoids and groupoids injectively graded by abelian groups. Under the amenability and second countability assumptions, we describe the primitive ideal spaces of the corresponding C$^*$-algebras as topological spaces.

Mixed-precision methods combine low and high precision arithmetics to exploit low precision computational speed and high precision accuracy. Large ODE systems that contain many heterogeneous interactions lead to a high computational cost that could be tackled with mixed-precision solvers. We tested mixedprecision versions of the Bogacki-Shampine 3(2) Runge-Kutta pair over three benchmark systems: coupled linear oscillators, the Kuramoto model and a circadian clock model. Our study is performed in a way that can be adapted to any finite-precision format, software architecture and numerical scheme. We found that mixed-precision solvers can preserve most of the high-precision solver accuracy under a wide range of solver tolerances. Moreover, mixed-precision solver accuracy improves with system size, reaching levels equivalent to high-precision solvers in small system size. We also observed that mixed-precision arithmetic does not impact the number of evaluation in a way that balances the benefit of fast operations in low precision. Taken together, these results show that mixed-precision methods can offer significant computational speed-up at little or no loss of accuracy in large coupled ODE systems.

The logistic two-gene negative-feedback oscillator is locally asymptotically stable for all biological parameter values, since the trace of the Jacobian is uniformly negative. Real biological oscillators (circadian rhythms, the segmentation clock, Hes1, p53) nevertheless rely on delays. We extend the logistic two-gene model to a delay-differential system with transcriptional delays $\tau_1$ and $\tau_2$, and prove that the equilibrium loses stability through a Hopf bifurcation as the total delay $\tau=\tau_1+\tau_2$ crosses an explicit critical value $\tau_c$. The Hopf frequency $\omega_c$ and $\tau_c$ are computed in closed form from the logistic derivatives; the loop-gain condition $AB>\gamma_1\gamma_2$ is necessary and sufficient; the transversality $\mathrm{Re}(d\mu/d\tau)|_{\tau_c}>0$ admits a parameter-uniform positive lower bound; and the bifurcation persists globally. A sum-of-delays symmetry reduces the analysis to the scalar parameter $\tau$. Numerical simulations confirm three regimes (damped, small limit cycle, relaxation), the supercritical amplitude scaling $A\sim c\sqrt{\tau-\tau_c}$, and the deep-relaxation period asymptote $T\sim 2\tau+C_\infty$ with closed-form offset $C_\infty$. For the symmetric-threshold loop, supercriticality is proved by a Lindstedt–Poincaré reduction yielding closed-form amplitude and frequency laws; for the general asymmetric loop it delivers a closed-form first Lyapunov coefficient and an explicit criticality criterion. Calibrated to p53–Mdm2 data, the closed-form Hopf period matches the observed oscillation within $3\%$, and the standard Hill-function model within a few percent. The analysis extends to cyclic $N$-gene loops, with a closed-form transversality rate valid for every $N$ and – in the symmetric case – an explicit delay-induced-Hopf window $\gamma^N<\Lambda<\gamma^N\sec^N(\pi/N)$.

In this work, we develop a constructive method for deriving four structure relations and a fourth-order linear differential equation satisfied by Laguerre-Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre-Hahn polynomials of class zero, analogous to the Hermite case. In addition, we present results for a semiclassical example of class 1.

We study geodesics in the Brochette first-passage percolation model, where edges on the same axis-parallel line share a common random passage time, inducing long-range dependence. We focus on the maximal transversal deviation H n of geodesics from the origin to ne 1 . We prove existence of geodesics under mild assumptions and establish the order of magnitude of H n depending on the behavior of the passage-time distribution near its infimum. These results yield explicit wandering exponents in this dependent setting.

We present in this work a very short proof for the existence, uniqueness and smoothness in dimensions $d\leq 3$ of the system of reaction diffusion $ \partial\_t a\_i - d\_i \Delta a\_i = (-1)^i (a\_1 a\_3 - a\_2 a\_4)$, where $a\_i \geq 0$ model the concentrations of chemical species undergoing a chemical reaction and diffusing (each with its diffusion rate $d\_i > 0$) in a bounded container.

We classify connected finite acyclic graded quivers $Q$ for which the graded path algebra $kQ$, regarded as a formal dg algebra, is silting-discrete. We prove that $kQ$ is silting-discrete if and only if it is derived-discrete, and that both conditions are equivalent to the underlying graph of $Q$ being of type ADE, or of type $\widetilde{A}$ with unequal clockwise and counter-clockwise total degrees. The key ingredient is an explicit construction of an infinite pre-simple-minded collection in $\\text{pvd } kQ$ in the non-discrete case.

We prove norm inflation phenomena for KdV and KP equations in negative order Sobolev spaces, in the periodic case, as well as on the whole space, on an arbitrarily large scale of negative order Sobolev spaces as target spaces. The proof relies on WKB analysis for a semiclassical version of the equation, in a weakly nonlinear régime, and the creation of the zero Fourier mode by resonant interaction. Unlike in previous similar results, this average mode has a smaller order of magnitude than the initial data, which requires a more detailed WKB analysis.

The dynamics of two-phase flows out of mechanical and thermal equilibrium are described by a partially dissipative first-order quasilinear system with stiff interaction terms associated with fast relaxation scales. In this paper, we analyze from a mathematical point of view the resulting pressure and temperature relaxation singular limit problem for a one velocity Baer-Nunziato model. This leads to a singular limit problem involving two small parameters. We propose a uniform symmetrization of this system which allows us to justify the strong relaxation limit and to establish a convergence rate for classical solutions.