Papers

We develop a discrete adiabatic framework for Gibbs-state preparation in classical and quantum optimisation algorithms. For finite time-inhomogeneous reversible Markov chains, we derive a tracking theorem that bounds the distance from the evolving distribution to the instantaneous stationary state by combining two effects: contraction of previously accumulated error by the Markov dynamics and the new error generated as the stationary distribution changes from one step to the next. For Gibbs paths, this leads to explicit step-complexity bounds and to gap- and fluctuation-adapted schedules controlled by the Markov-chain spectral gap and the energy-fluctuation scale $V(\beta)=\sqrt{\operatorname{Var}_{\pi_\beta}(E)}$. In the weak-damping regime, the resulting schedule takes the form $\dot{\beta}\propto\delta(\beta)/V(\beta)$. We then apply the discrete quantum adiabatic theorem to temperature-dependent Szegedy quantum walks. The quantum-walk eigenphase gap satisfies $\Delta_w=\Theta(\sqrt{\delta})$, where $\delta$ is the spectral gap of the underlying Markov chain. By analysing the finite-difference terms entering the theorem, we show that this square-root spectral amplification translates into a quadratic improvement in the gap dependence only for suitably gap-adapted schedules. Under the corresponding regularity conditions, the quantum adiabatic cost scales as $O(\delta_{\min}^{-1/2})$, compared with the classical $O(\delta_{\min}^{-1})$ relaxation scale.

In this paper we prove the stability and the continuity of the free boundaries of two-phase and one-phase weak solutions to the Stefan problem in any dimension. Our proof has two main ingredients. The first one is an energy conservation principle for weak solutions that we derive for general domains in both the one-phase and the two-phase settings; this energy conservation principle provides the modulus of continuity for the displacement of the free boundary and is a flexible tool for analyzing long-time behavior of solutions. The second key ingredient for the control of the displacement of the free boundary is a new lower density bound for the melting region at free boundary points that move with maximal velocity. We also discuss some applications, showing a series of qualitative and quantitative results on the asymptotic behavior of weak solutions.

Freeway on-ramp merges are major sources of congestion, causing significant economic and environmental costs. While Deep Reinforcement Learning (DRL) offers a promising solution for ramp metering, existing approaches rely primarily on aggregated macroscopic data. Connected vehicles (CVs) provide vehicle-level observations that can complement aggregate traffic measurements, but their limited penetration produces incomplete microscopic information. This paper proposes a hybrid observation representation combining macroscopic traffic measurements with a two-channel grid encoding observed CV presence and speed. A Dueling Double Deep Q-Network processes these inputs to select ramp-metering green durations. The controller is trained under varying traffic demands and CV penetration rates and evaluated against ALINEA and macroscopic-only DRL variants in SUMO. Across 50 matched evaluation scenarios, the hybrid controller under partial CV visibility reduces the reported total travel time by 11.4 % and mean spillback duration by 84.9 % relative to ALINEA. Evaluating the same trained policy with full CV visibility yields a further travel-time reduction of approximately 1.6 %. Analysis across penetration rates suggests that the performance gap decreases as microscopic observations become more complete. These results support the use of complementary macroscopic and sparse microscopic observations for learning-based ramp metering. The source code implementation of the model is available at: https://github.com/youcefMehamlia/Multimodal-DRL-RMC

We show that a convex surface in Euclidean 3-space is a sphere if the area cut off from it by a slab of some fixed width depends only on the direction of the slab. This result generalizes theorems of Blaschke and Stamm, which assumed the slab-area property for every width. It holds also for any smooth closed embedded surface. The proof is based on an averaging identity, which relates slab areas to the Newtonian potential of the surface, the electrostatic characterization of spheres by Mendez and Reichel, and analytic microlocal properties of the Radon transform, which are used to show that the slab-area property forces real analyticity, via theorems of Kashiwara and Boman.

We provide in this text a new framework for a family of combinatorial Hopf algebras. We adopt for this the language of species and twisted bialgebras (also known as Hopf monoids). We define the category of combinatorial twisted bialgebras (briefly, CTBs), a class of twisted bialgebras satisfying strong combinatorial conditions of compatibilities of the product and the coproduct with the underlying combinatorial structure. The considered morphisms also have to respect the combinatorial structure, in some sense. Many classical examples fit into this framework, including Hopf algebras of graphs, rooted trees, posets, hypergraphs, symmetric functions, boolean maps, etc. Most of known combinatorial Hopf algebras can be obtained from a CTB by application of a Fock functor, or by duality. A central result is the introduction of a universal CTB denoted by $\mathbf{Exch}$, built from collections of compositions satisfying a certain exchange condition. It has a special character $\epsilon$, such that the pair $(\mathbf{Exch},\epsilon)$ plays a role analogous to that of quasi-symmetric functions for graded and connected Hopf algebras: it is the terminal object in the category of CTBs equipped with combinatorial characters. This leads to universal morphisms that encode important information about the combinatorial structure of coproducts. The natural order on the set of compositions on a fixed set is used to define three important sub-objects $\mathbf{Exch}\_-$, $\mathbf{Exch}\_+$ and $\mathbf{Exch}\_{+\hspace{-2mm}+}$ of $\mathbf{Exch}$. We also studies polynomial invariants associated with CTBs. It establishes a correspondence between characters and polynomial invariants. Well-known invariants such as the chromatic polynomial of graphs and the Ehrhart polynomials of posets arise naturally within this framework. We show that, for a given degree, only finitely many polynomial invariants can occur. We show that these universal polynomial invariants can be used to give cancellation-free formulas for the antipode or the Eulerian idempotent, under conditions on the CTB, and use this for an example based on rooted trees, coming from a Hopf algebra used in linguistics. Results are also obtained for families of polynomial invariants with the alternating sign condition, or satisfying a reciprocity principle. We also extend the theory to combinatorial twisted double bialgebras (briefly, CTDBs), which involve two interacting coproduct structures. These generalize previously known examples arising from graphs, posets, hypergraphs, and rooted trees. The twisted approach preserves the combinatorial richness of these structures while providing a unified algebraic framework. We show that not any CTB can be made a CTDB, leading to the introduction of the notion of quasi-cocommutativity. In particular, we show that $\mathbf{Exch}\_+$ is not a CTDB, and construct a sub-object which is, in a unique way, and maximal for the inclusion. In the cocommutative case, simplifications naturally occur, and we show that in this case, a CTB of hypergraphs plays a central role, especially two sub-objects based on connectedness systems and antichain covers. We also show that, in the cocommutative case, the universal polynomial invariants are always chromatic polynomials of hypergraphs, no matter which CTB is considered. Finally, forgetting the combinatorial underlying structure, we study the algebraic structure of $\mathrm{Exch}$, with the help of a twisted version of Loday and Ronco's rigidity theorem for infinitesimal bialgebras.

A unit-distance graph is a graph which admits a realisation in Euclidean space in which every edge has unit length. Imposing further geometric conditions on the non-edges gives a family of natural subclasses. Requiring that no two vertices lie at distance less than one gives penny and marble graphs: the contact graphs of collections of equal radii $d$-dimensional spheres with non-overlapping interiors for $d=2,3$. Requiring instead that the straight-line drawing in the plane be non-crossing gives matchstick graphs. Since any motion of a realisation must preserve these extra conditions, the rigidity and flexibility properties of the resulting frameworks differ from those of classical bar-joint rigidity theory. In this note we analyse various rigidity problems for penny and marble graphs, matchstick graphs and unit-distance graphs. In particular we answer a recent open problem on penny and marble graph rigidity and establish a link between penny graphs and the concept of NAC-colourings.

The Bradley-Terry model is a parametric model for ranking from pairwise comparisons. Existing asymptotic theory for the maximum likelihood estimator (MLE) in regimes where the number of objects grows often requires homogeneity assumptions or compatibility conditions on comparison graphs, which limits its applicability to many practical settings. In this work, we establish uniform consistency of the MLE under general deterministic comparison designs. Our pairwise error bound consists of a pair-specific term governed by the effective resistance between the corresponding objects and a global term governed by the spectral gap of the unnormalized graph Laplacian. The bound is tight up to logarithmic factors for certain graph sequences. For any prespecified sequence of pairs satisfying an additional balancing condition, we further establish asymptotic normality of the corresponding estimated utility differences. As an application, we obtain uniform consistency for independent-edge random graph designs whenever the minimum edge probability exceeds the Erdős-Rényi connectivity threshold by a logarithmic factor, as well as asymptotic normality under additional balancing conditions.

We use finite-field calculations and mixed-characteristic deformation to construct nef divisors on very general complex blow-ups of the plane. A lifting and smoothing criterion converts a reducible nodal divisor over $\mathbb{F}_{127}$ into a smooth genus-$10$ curve of class $117H-37(E_1+\cdots+E_{10})$, retaining control of its first-order obstruction map. Its normal bundle in a suitable deformation is semistable, so Petrakiev's multiplicity estimate yields the nef class $27379H-8658(E_1+\cdots+E_{10})$ at ten very general points. Their multipoint Seshadri constant is therefore at least $8658/27379$, improving Petrakiev's bound $228/721$. The geometric input is an existence criterion in characteristic zero verified by computations in positive characteristic.

An oscillon in scalar field theory in $(1+1)$-dimensions is an oscillating, localised, almost periodic classical solution that slowly radiates and decays to the vacuum solution. We calculate its momentum flux integrated over a period, which would be zero for an exact breather. For an oscillon of small amplitude $\eps$, truncated at order $\eps^N$, we show that the integrated momentum flux is of leading order $\eps^{N+4}$ for even $N$ and $\eps^{N+5}$ for odd $N$, except for $N=1$, where it is of order $\eps^4$. It is therefore very small.

Let $n$ and $N$ be integers with $2\leq n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map that admits a $C^{N-n+1}$-smooth extension to the boundary. We prove that if the CR Gauss map of its boundary restriction is generically degenerate, then $F$ is holomorphically equivalent, under composition with automorphisms of the source and target balls, to $V_m\oplus 0$ for some positive integer $m$, where $ V_m(z)=\big(\sqrt{\frac{m!}{\alpha!}} z^\alpha\big)_{|\alpha|=m} $ is the degree-$m$ Veronese map. This resolves a problem posed by Xiaojun Huang.

We study the fundamental solution operators associated to Kolmogorov equations with symmetric non-local diffusion with rough kernels. We prove that these operators are induced by measurable kernels and prove sharp upper bounds in terms of a kinetic profile governed by the interplay of transport and non-local diffusion. Our proof combines the small-jump with loss decomposition of P.-A. Meyer with the ideas of Nash. The proof is purely analytic.

In this work, we propose and analyze low-order nonconforming face-based finite element methods for elliptic problems on complex hybrid three-dimensional meshes, composed of arbitrary polytopal cells. The focus is on scalar problems, but the approach can be extended in a tensor manner to design generalizations of the classical Crouzeix–Raviart and Rannacher–Turek methods for mixed formulations. We adopt a nonparametric construction of the local approximation spaces. We demonstrate that it avoids the degradation of approximation properties commonly induced by parametric mappings on distorted meshes. Using these approximation spaces, we define finite element schemes for very general cells. We further demonstrate that, on strongly degraded meshes, an approach inspired by discontinuous Galerkin methods enables removing consistency errors, even in the presence of potentially curved faces. Explicit examples of the construction of the polynomial space are given for prismatic (of triangular basis) and pyramidal (of quadrilateral basis) cells. The resulting schemes are shown theoretically and numerically to be stable and convergent, with an optimal convergence rate.

We study the $\Gamma$-convergence of a general class of anisotropic phase-field functionals modelling cohesive fracture in the multidimensional, vector-valued, and geometrically nonlinear setting. The limiting energy comprises a quasiconvexified bulk term, a Cantor part governed by its recession function, and a cohesive surface energy density defined via an asymptotic cell problem. After establishing the structural properties and equivalent formulations of the limiting surface density, we show that, whenever the recession function of the elastic energy is given by the square of an operator norm compatible with the phase-field anisotropy, the vectorial cell problem explicitly reduces to a one-dimensional variational problem. As a consequence, we extend the cohesive law reconstruction procedure of \cite{alessi2025phasefieldpart2} to the vectorial framework, explicitly designing phase-field potentials that recover prescribed classes of anisotropic, jump-dependent traction-separation laws of the form $g(\zeta,\nu)=\gamma(\nu)g_{\scal}(|\zeta|)$.

In this paper, we study finite-mode Gaussian quantum Markov semigroups with a faithful invariant Gaussian state. We present an algebraic characterization of the complete modified logarithmic Sobolev inequality relative to the fixed-point algebra, and we obtain the corresponding optimal constant under the assumption that the drift evolution $e^{t\mathbf{Z}}$ converges as $t\to\infty$. We establish hypercontractivity for such quantum Markov semigroups with Hurwitz drift, in the sense that every fixed pair $1<p<q<\infty$ is attained at sufficiently large times. We find that the usual reverse hypercontractive curves with a positive rate from time zero can fail in general, but every fixed pair $1/2 < p < q < 1$ is attained at sufficiently large times under the Hurwitz assumption. We also systematically investigate the $p$-logarithmic Sobolev inequality and show that the $p$-logarithmic Sobolev constant can be negative for $0 < p < 1/2$.

In this article, we show that the characterization of reversible rings and various other rings related to them such as i-reversible rings, semi-commutative rings and i-domains in terms of properties related to idempotent elements known in the literature can be generalized to exactly the same properties for quasi-idempotent elements.

We introduce a topological abelian group $\mathrm{K}_1^\Delta(A)$ for every $\mathrm{C}^\ast$-algebra $A$, which is functorial in $A$ and invariant under stabilisations. $\mathrm{K}_1^\Delta$ can distinguish group actions indistinguishable by equivariant (K)K-theory and traces — we illustrate this on the irrational rotation algebra by exhibiting strongly outer actions of $\mathbb Z$ which, via $\mathrm{K}_1^\Delta$, are not (stably) cocycle conjugate. These actions are moreover trivial in $\mathrm{KK}^{\mathbb Z}$ and on the Hausdorffised unitary algebraic $\mathrm{K}_1$ group $\overline{\mathrm{K}}_1^{\mathrm{alg}}(A)$. The Hausdorffisation $\overline{\mathrm{K}}_1^{\Delta}(A)$ of $\mathrm{K}_1^\Delta(A)$ forms the final part of the total stable invariant $\mathrm{\underline{K}T}_{\mathrm{s}}(A)$, which we also introduce. We prove that $\mathrm{\underline{K}T}_{\mathrm{s}}$ classifies $*$-homomorphisms up to approximate unitary equivalence between certain stable $\mathrm{C}^\ast$-algebras which contain a non-zero projection. $\mathrm{K}_1^\Delta(A)$ is new even in the unital setting, but under further assumptions on $A$ recovers the unitary algebraic $\mathrm{K}_1$ group $\mathrm{K}_1^{\mathrm{alg}}(A)$, whereas the Hausdorffisations $\overline{\mathrm{K}}_1^{\Delta}(A)$ and $\overline{\mathrm{K}}_1^{\mathrm{alg}}(A)$ agree for every unital $A$.

We obtain improved Fourier restriction and Kakeya maximal estimates in $\mathbb R^4$. More precisely, by proving the two-ends incidence estimates $\mathbf{TE}\left(\frac{133}{44},\frac{177}{88},\frac34\right)$ and $\mathbf{TE}\left(\frac{386453}{125988},\frac{386453}{125988},\frac{358951}{377964}\right)$, we extend the restriction range to $p>2+ \frac{176}{221}$ and the Kakeya maximal estimate up to dimension $3.06737$. The implications to the known lower-bounds for the Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$ and the Bochner–Riesz conjecture are also studied.

A normalized analytic function $f$ defined on the unit disk $\mathbb{D}$ is called $\alpha$-spirallike if $f(\mathbb{D})$ is an $\alpha$-spirallike domain, that is, for every $z\in f(\mathbb{D})$, the $\alpha$-spiral $ze^{-e^{i\alpha}t}$, $t\ge0$, is contained in $f(\mathbb{D})$, or equivalently, if $\operatorname{Re}\!\left(e^{i\alpha} {zf'(z)}/{f(z)}\right)>0$ for all $z\in \mathbb{D}$. The 0-spirallike functions are the usual starlike functions. The function $f$ is parabolic starlike, if $\operatorname{Re}\!\left(\frac{zf'(z)}{f(z)}\right)>\left|\frac{zf'(z)}{f(z)}-1\right|$, and lemniscate starlike if $|(zf'(z)/f(z))^2-1|<1$ respectively. In this paper, we determine the radii of parabolic starlikeness and lemniscate starlikeness of $\alpha$-spirallike functions. Our approach combines geometric characterizations of the underlying regions with the resultant method for eliminating auxiliary variables, leading to explicit equations for the sharp radii. Some known radius results are recovered as special cases.

Spectral Barron spaces are Fourier-Lebesgue-type spaces that arise in the approximation of functions by neural networks. Conventional spectral Barron spaces are mostly considered on the whole space or on a bounded domain with certain smoothness assumptions on the boundary. In this work, we investigate spectral Barron spaces for periodic functions, thereby extending the study of spectral Barron spaces to high-dimensional tori and, more generally, to locally compact Abelian groups. These extensions are carried out by considering Fourier series expansions that are isometrically isomorphic to weighted sequence spaces built upon the $\ell^1$ sequence space. We discuss the well-posedness of PDEs in these newly introduced spaces. Two specific inverse problems for evolution equations are investigated, yielding conditional stability within the framework of spectral Barron spaces on the torus.

We prove that a smooth irreducible proper variety over a number field admits polylogarithmic-size coverings of rational points of bounded height if it admits a tower of finite étale coverings satisfying several conditions. One of the conditions is a height comparison on twists of the finite étale covers. We prove that these conditions hold for abelian varieties.

We analyse solutions of the flow equations of the renormalisation group for the n-point Schwinger functions of scalar field theory in four-dimensional Euclidean space with nonpolynomial boundary conditions. We show that there exist UV asymptotically free solutions of this type which are Bose symmetric, invariant under the Euclidean group, analytic in the flow parameter and the momenta, and satisfy the Osterwalder-Schrader growth condition.

The design of high-quality nanoparticles requires the production of materials with narrow particle size distributions, as these strongly influence the functionalities of the final product. To achieve this goal, it is useful to consider a coupled process chain in which nanoparticles are first synthesized and subsequently fractionated by chromatographic separation. We introduce a novel mathematical optimization model for the integrated optimization of coupled nanoparticle synthesis and separation processes. The proposed model simultaneously accounts for process performance and economic objectives through a cost function that incorporates both production costs and achievable profit defined by the particle quality. For efficient evaluation of the synthesis model, we employ the exact method of moments. The resulting non-convex process-design problem is solved numerically using local optimization. Optimization results of our approach are demonstrated for the prototypical design of gold nanoparticles. The computational results show that optimized solutions of the coupled synthesis–separation process are capable of accurately representing both process stages while achieving optimized particle quality. The proposed model highlights the potential of mathematical optimization for the design of coupled nanoparticle production process-chains.

A relaxed maximum-based normal $S$-iteration method (RMNSI) is proposed for solving generalized absolute value equations (GAVEs). The method combines a maximum-based fixed-point formulation with a constant relaxation parameter and avoids the selection of an auxiliary matrix. Assuming that $A+B$ is non-singular, both stages require linear systems with the same coefficient matrix $A+B$, allowing a single factorization to be reused throughout the iteration. A single spectral condition is established that guarantees unique solvability of the GAVE and global convergence from an arbitrary initial vector. An admissible range of the relaxation parameter is characterized, and $R$-linear convergence and error estimates are derived. Since the global condition can be conservative, a local convergence analysis based on the solution sign pattern is also presented. Numerical experiments on complementarity-derived, dense mixed-sign, and asymmetric ridge-regression problems are presented, and comparisons with several recent methods demonstrate the competitive efficiency and accuracy of RMNSI. The results further indicate that the preferred relaxation parameter depends mainly on the matrix structure and diagonal shift, while its dependence on the problem dimension is generally weak.

The goal of the present paper is to investigate the exact decay rate of the tail probability $\mathbb{P}(|X_1-x_0|>R)$ for large $R$, where $X_t$ is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition $x_0$. In the first place, under the assumption of uniform ellipticity, we establish a general $(2H+1)$-Weibull lower tail estimate in Young's regime of $H\in(1/2,1)$ and a general Gaussian lower tail estimate in the rough regime of $H\in(1/4,1/2)$. These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with $C_b^\infty$-coefficients could fail to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any $2\times 2$ periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general $(2H+1)$-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.

Alternating projections (AP) near a clean intersection of smooth manifolds can converge slowly when the Friedrichs angle is small. We develop the Fixed-Point Manifold Acceleration (FPMA) framework for {local convergence and polynomial acceleration of} smooth maps that fix a manifold pointwise. {Under suitable regularity and uniform normal spectral stability, FPMA establishes local R-linear convergence and a smooth limiting map inducing a local retraction. At fixed points, consistent polynomial acceleration with $p(1)=1$ preserves the selected limit's first-order response to normal initial perturbations whenever the transformed normal blocks are uniformly spectrally stable. Under sufficient smoothness, we give a curvature criterion for second-order induced retractions and show that stable consistent polynomial acceleration preserves both the quadratic retraction expansion and this criterion. {As a special case, we establish new local R-linear convergence guarantees for fixed-degree Chebyshev-accelerated exact AP and second-order induced retractions. For uniform Friedrichs angle $\theta_F\in(0,\pi/2)$, acceleration strictly improves the local convergence rate per AP evaluation. The large-degree limiting spectral factor at fixed $\theta_F$ predicts $O(\theta_F^{-1})$ evaluations for fixed relative error reduction as $\theta_F\downarrow0$, versus $O(\theta_F^{-2})$ for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}

We prove that every subspace $V\subset\ell_\infty^N$ of codimension one or two with absolute projection constant $\lambda(V)<4/3$ satisfies $$ d(V,\ell_\infty^{\dim V}) \leq\frac{\lambda(V)}{4-3\lambda(V)}. $$ We conjecture that the codimension assumption can be removed and that the same estimate holds for every subspace $V\subset\ell_\infty^N$ with $\lambda(V)<4/3$.

In this paper, we introduce and study a special function, referred to as a variant of the fractional gamma-regularized zeta kernel, in which the lower incomplete gamma function appears in the numerator, contrasting with the fractional hypergeometric zeta function, where it appears in the denominator. We investigate its analytic properties, convergence behavior, and its connection to the Riemann zeta function. A representation is derived in terms of a multiplicative modification of the Riemann zeta function, and comparisons are made with known zero-free regions and deformation phenomena. We demonstrate that this function defines a deformation of the Riemann zeta function, which interpolates to the alternating zeta function in a specific limit. Additionally, a Dirichlet-type series representation is provided. Potential extensions, including a functional equation and asymptotic analysis, are proposed for future work.

This paper proves propagation of energy moments for the three-dimensional plasma-charge model, namely, the repulsive Vlasov-Poisson system with a moving point charge. For every \(K>2\), an initially finite \(K\)th energy moment remains finite on every bounded time interval, without any higher-order moment assumption, smallness condition on the plasma mass, or initial separation from the point charge. The constructed global weak solution conserves total energy for almost every time. The proof relies on a critical space-time density estimate and a direct control of the electric field along the moving charge.

The dynamics of chemical reaction systems with two chemical species can be described (under mass-action kinetics) by planar autonomous systems of ordinary differential equations (ODEs) with right-hand sides containing polynomials. While their behaviour close to hyperbolic critical points is well understood, their dynamics has not been fully characterized for strongly non-hyperbolic critical points, where the linearization close to the equilibrium is given by the zero matrix. In this paper, the behaviour of such chemical reaction systems is investigated. Considering second-order chemical reaction networks with a positive equilibrium point, we show that only nine geometric equivalence classes are possible out of the sixteen classes for general ODE systems. Examples of chemical reaction networks in each class are presented. All sixteen geometric equivalence classes can be realized by chemical systems if the non-hyperbolic equilibrium point is located at the origin.

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_\gamma $, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $\gamma\in [0,n)$, between the classical Lorentz sequence spaces $\lambda_p(v)$ and $\lambda_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_\gamma x$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_\gamma$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.