Papers

We develop a semiclassical theory of tunnelling of Dirac fermions through an n-p-n junction in monolayer graphene subjected to a perpendicular magnetic field. Electrostatic and magnetic fields are assumed to be smooth functions of a single spatial coordinate, supported on a finite interval, vanishing outside it, and thus ensuring asymptotically free states. In contrast to earlier studies restricted to constant magnetic fields and exactly solvable electrostatic potential profiles, we consider a general electrostatic potential forming an n-p-n junction and an arbitrary magnetic field, and formulate the corresponding scattering problem. Within the semiclassical approximation, and under an additional assumption on the incidence angle, the problem reduces to a connection problem for a pair of degeneracy points, treated using results from our earlier work. We obtain explicit expressions for the reflection and transmission coefficients, including their phases, as functions of energy and incidence angle. Furthermore, we derive semiclassical conditions for Fabry-Pérot resonances and "magic" angles, and analyse the resulting interference pattern. Numerical results demonstrate the angular dependence of transmission induced by the magnetic field.

We study the counting problem for non-uniform intersecting families in extremal set theory. Let $J(n,k)$ denote the number of intersecting families $\mathcal{F}\subset 2^{[n]}$ such that every member of $\mathcal{F}$ has size at most $k$. Extending recent counting results for uniform intersecting families, we prove that for $n\ge 2k+2+2\sqrt{k \log k}$ and $k \rightarrow +\infty$, \[ J(n,k) =(n+o(1)) 2^{\sum_{i=1}^{k} \binom{n-1}{i-1}}. \] This result reveals that typical non-uniform intersecting families of bounded size are trivial, i.e., almost all such families share a common fixed element.

Learning kernels in operators from data is a fundamental task that arises in nonlocal continuum mechanics, operator learning, and interacting particle systems. A central question is how the roughness of input data impacts the accuracy of kernel recovery. We quantify the roughness of the input data via its spectral decay exponent and analyze how it determines the degree of ill-posedness of the inverse problem and, consequently, the convergence rates of the Tikhonov-regularized estimator in the small-noise limit. Within this framework, we identify a phase transition between an under-rough regime, in which rougher data improves recovery, and an over-rough regime, in which further roughening leads to slower rates. These theoretical findings are supported by numerical experiments ranging from idealized settings to more realistic configurations, with quantitative agreement in the former and broad consistency of the main trends in the latter.

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan–Shub–Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$\lambda$, through the profile maximum likelihood estimator, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$\kappa$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac–Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta–Fyodorov representation, one from a Ben Arous–Dembo–Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$\kappa$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger–Ben Arous–Černý complexity function in the high-dimensional limit.

Motivated by the recent constructions of Golay Complementary Array (GCA), we extend the "Three-stage construction" proposed by Fiedler and Jedwab to a more general form. All the known 2-phase Golay Complementary Sequences (GCSs) of length $2^a10^b26^c,a,b,c\ge 0$ and 4-phase GCSs of length $2^{a+u}3^b5^c11^d13^e$ where $a,b,c,d,e,u\ge0,b+c+d+e\le a+2u+1,u\le c+e$ can be constructed in a uniform method from five 2-phase seeds of length 2,10,10,20,26 and five 4-phase seeds of length 3,5,8,11,13. Furthermore, additional 2-phase and 4-phase GCSs of many lengths are produced and a closed form for numbers of 2-phase and 4-phase GCSs is given.

We study a continuous-time robust Bayesian portfolio optimization problem under drift uncertainty of risky assets. The investor learns unknown asset drifts through Bayesian filtering while considering uncertainty around posterior estimates via discrepancy-based ambiguity sets, including Wasserstein and $L^p$ distances. To address the resulting time inconsistency, we introduce a feedback-type ambiguity framework that reformulates ambiguity conditionally on observable states. This leads to a modified Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation characterizing the value function and the optimal strategy. For a semi-explicit solution example, we use the exponential utility to derive a reduced semilinear parabolic PDE and establish existence of classical solutions via a verification theorem.

We study variational inequalities governed by a point-to-set maximal monotone operator in a real Hilbert space and constrained by a convex inequality \(C=\{x\in\Hi:c(x)\le0\}\), where the defining function \(c\) is continuous and not necessarily differentiable. The proposed method uses only projections onto intersections of half-spaces and avoids the metric projection onto \(C\). Feasibility is handled by subgradient cuts and, when a trial operator point is infeasible, by a Slater correction based on a fixed strictly feasible point. The variational inequality is represented by Minty-type separating half-spaces generated at feasible graph points of the operator, and a Haugazeau half-space is added to obtain best-approximation convergence. Under a Slater-corrected feasible-separation condition, together with explicit exact, approximate and finite-candidate oracle realisations, the whole sequence converges strongly to \(P_{S^*}(x^0)\), the projection of the initial point onto the solution set. We also derive best-iterate \(O(N^{-1/2})\) residual estimates for the step residual, feasibility violation and Minty gap. The analysis is stated directly for point-to-set maximal monotone operators, while the concrete oracle realisations include finite-dimensional single-valued models. We record the consequences of strong monotonicity in the point-to-set setting and provide numerical comparisons on nonsmooth and large-scale constraints, including maxima of convex quadratics, a discretised optimal-control problem, mixed-norm sparse recovery, a Cournot–Nash capacity equilibrium, and a genuine point-to-set \(\ell_1\)-subdifferential example.

In kinetic theory, interactions between particles are typically assumed to be "all-to-all", meaning that any pair of randomly selected particles may, in principle, interact. This assumption originates from the theory of colliding gas molecules; however, it may be less appropriate for describing other forms of interaction, such as social interactions. These are more naturally characterised as "some-to-some", reflecting the existence of preferential connections between agents. In this paper, we consider homogeneous Boltzmann-type equations on finite graphs that model such networks of preferential interactions, and we rigorously derive their dense graph limit as the number of agents tends to infinity. We also investigate the long-time behaviour of the limiting equation in the case of linear pairwise interactions, characterising the emergent equilibrium distributions and relating them to their counterparts in the classical "all-to-all" setting.

We show that the topological structures, such as loops, voids, and higher-dimensional holes of unknown objects (of flow of an object in space-time) can be recovered from noisy and indirect measurements. More precisely, we describe how the part of the persistent homology of a space can be determined from a noise-prone and discretized model space when there is a quasi-isometry between the original space and the space modeling indirect measurements. The result not only guarantees the existence of the structures but also provides size bounds for them. The structure is studied using persistent homology, and the results assume the existence of a quasi-isometry between a model space and the noisy measurements. We explore imaging problems, particularly X-ray imaging and EIT, that are well-suited to this framework.

We study the vanishing viscosity limit for the incompressible Navier-Stokes equations with anisotropic viscosity in bounded domains, analyzing certain classes of symmetric flows: plane parallel, pipe parallel and circularly symmetric. By anisotropic viscosity, it is meant here that the viscosity coefficient in the direction normal to the wall is different than that in the direction tangential to the wall. Using boundary layer theory and semigroup techniques, we establish the validity of the vanishing viscosity limit in the energy norm, that is, in $L^2$ in space uniformly in time, for all three classes of flows, with explicit convergence rates. We further obtain higher-order estimates under suitable assumptions on the anisotropic viscosity coefficients. In particular, we consider both the case in which the tangential viscosity coefficient goes to zero faster than the normal one and, conversely, the case when the normal coefficient vanishes faster then the tangential one. Our results extend previous works on isotropic viscosity and provide new examples where the vanishing viscosity limit can be rigorously justified in the anisotropic setting.

We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations. Unlike previous approaches based on positive-definite signatures, QCA employs a split-signature construction that enables a natural realization of quantum states and operators while simplifying computational implementation. We further present an implementation of QCA using the GAALOP software and show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally. As an application, we demonstrate the use of QCA in quantum game theory, where the real-algebraic formulation provides computational advantages for modeling entangled strategies and quantum interactions. The proposed framework establishes a practical bridge between the abstract formalism of quantum computation and efficient geometric algebra implementations.

We investigate the impact of large synaptic delays on the emergence of periodic dynamics in inhibitory neuronal networks, within the framework of the NNLIF model. Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficitently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Cesàro mean toward a limit function determined solely by these pseudo-equilibria.

In response to the growing demand for sustainable and efficient urban deliveries, this study introduces a two-echelon vehicle sharing and repositioning problem for containerized delivery operations within a hyperconnected urban logistics system. We leverage a Physical Internet (PI)-enabled three-tier logistic hub network, comprising gateway, local, and access hubs, to facilitate efficient flows. By adopting containerized delivery, vehicles can rapidly swap standardized modular containers at hubs to reduce handling time. Moreover, inspired by the PI concept of open resource sharing, we determine optimal service routes within a two-echelon structure that jointly utilizes heterogeneous vehicle fleets and enables dynamic vehicle relocation across hubs. We formulate this problem as a multi-period integer program that integrates path-based service vehicle planning with arc-based container routing. To address real-world large-scale instances, we propose a decomposition-based heuristic with capacity-aware flow assignment, which partitions the problem into subproblems structured by echelons and regions. A case study on the Atlanta metropolitan area demonstrates the effectiveness of the proposed model and solution approach. Experimental results show that the two-echelon hyperconnected delivery system reduces CO2-eq emissions by 45.0% and total costs by 16.8% at full market share compared to a traditional single-echelon alternative, while enabling vehicle repositioning further lowers costs by up to 17.7%.

The classical Frenet frame is defined by a concrete construction from the tangent, principal normal, and binormal vectors of a regular space curve. However, this construction breaks down at singular points and at points where the curvature vanishes. Motivated by this observation, we reconsider the Frenet frame from an axiomatic viewpoint and identify the fundamental properties that characterize it independently of its classical construction. Based on the theory of frontals on the unit sphere and Legendre duality, we introduce a generalized Frenet frame for singular space curves. Furthermore, we introduce the notion of a frame sequence, which gives rise to a family of $k$th-order Frenet frames and their corresponding curvatures and torsions, indexed by $k \in \mathbb{Z}$. This viewpoint provides a unified framework encompassing the Frenet and Bishop frames of space curves as well as the evolute-involute correspondence for spherical frontals. Moreover, explicit recursive formulas are derived, revealing that the curvatures and torsions at each level encode, respectively, the magnitude and rotational behavior of the invariants at the preceding level.

Menger's theorem is one of the most fundamental results in graph theory. It states that if a graph $G$ does not contain $k$ disjoint paths between two given sets $X$ and $Y$ of vertices in $G$, then there is a set of at most $k-1$ vertices that intersects every path between $X$ and $Y$. Nguyen, Scott, and Seymour gave a counterexample to the conjectured natural coarse variant in which the paths are required to be pairwise at distance at least $d$, and, conversely, there is a set of at most $k-1$ bounded-radius balls intersecting every path between $X$ and $Y$. In other words, the coarse Menger property does not hold in general. We prove that graphs whose cycles space is generated by cycles of bounded length do have the coarse Menger property. As a corollary, we show that many natural graphs and geodesic metric spaces have the coarse Menger property. These include hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded face size, and complete Riemannian planes.

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erdős-Rényi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

We study the one-loop torus vacuum of Type IIB Superstring theory through sector-resolved modular integrals. Building on the i\varepsilon-prescription and the E_s-regularized modular-integral framework of Manschot and Wang [1], we construct regularized sector functionals for the closed oriented torus before the final GSO projection. The construction keeps the unprojected spin-sector data explicit and fixes the compact-domain and cusp contributions within a single modular prescription. We also independently cross-check the result with the Lorentzian-inversion reconstruction of modular integrals by Baccianti et al. [2] This provides a first direct regularized construction of the unprojected sectors of the Type IIB Superstring torus vacuum.

Mediation analysis is essential for decomposing the causal effect of a treatment into direct and indirect pathways. However, many practical settings rely on the stringent assumption that recanting witnesses, defined as treatment-induced mediator-outcome confounders, are either absent or fully known a priori. Such a requirement is often untenable, especially when these variables remain unobservable due to measurement difficulties or privacy constraints. In this paper, we leverage proximal causal inference to develop three novel identification strategies to address the challenge of identifying path-specific effects in the presence of unknown recanting witnesses. Building on this, we develop a semiparametric inference framework that derives the efficient influence function and proposes a proximal multiply robust estimator, which remains consistent if at least one set of nuisance models is correctly specified. When all nuisance models are correctly specified and converge at appropriate rates, the estimator is asymptotically normal and achieves the semiparametric efficiency bound. We provide a minimax optimization-based debiased machine learning procedure for point estimation and constructing valid confidence intervals. The performance of the proposed methods is demonstrated by simulation studies and a real data application.

For a manifold $\overline{M}$ with boundary $\partial M$ and interior $M$, we introduce and study a weakening of the concept of projective compactness for torsion-free linear connections on $M$, which we call projective pre-compactness. Via the Levi-Civita connection, this concept applies to pseudo-Riemannian metrics on $M$. This is motivated by scattering theory and general relativity (GR), via asymptotic forms of metrics used in these areas. In the general setting of a projectively pre-compact connection $\nabla$ we show that, assuming weak asymptotic conditions on the Ricci curvature, there is an induced projective structure on the boundary. Under a slightly stronger condition on Ricci, we show that the standard tractor bundle and its normal tractor connection arise naturally on this boundary structure. The key ingredient to this is that $\nabla$ admits a smooth extension to the boundary as a linear connection on the tensor product of Melrose's b-tangent bundle with a density bundle, which then restricts to the boundary tractor bundle. A projectively pre-compact pseudo-Riemannian metric (satisfying the conditions on the Ricci curvature) is then shown to induce a holonomy reduction of the boundary projective structure to an indefinite orthogonal group. This endows the boundary with a decomposition into so-called curved orbits, which are either open or embedded hypersurfaces, representing space-like, time-like and light-like infinities in a GR context. We introduce and study a new asymptotic form for such metrics which is available near any boundary point and relate it to an asymptotic form used in general relativity, which is only available near boundary points in the open curved orbits. We show that, in that region, projective pre-compactness essentially is equivalent to the asymptotic form from GR, and projective compactness is equivalent to vanishing of the mass aspect.

This paper considers the problem of a low-thrust spacecraft pursuit-evasion differential game with an arbitrary terminal relative speed constraint. It addresses the terminal phase of the engagement for two relatively close spacecraft near a circular orbit. The problem is formulated as a linear-quadratic zero-sum differential game, with soft constraints on the terminal relative position and velocity, and running costs on the players' control efforts. An analytical, closed-loop, minimum-fuel-consumption optimal guidance law is derived for each player, forming a saddle-point solution. It is proven that any terminal speed can be achieved by properly choosing the weighting parameters of the cost function. To verify the optimality of the solution, a conjugate point analysis is performed when the cost function velocity weighting matrix is either positive or negative definite. The negative-definite case arises at high terminal speeds and is seldom seen in the literature. The performance of the derived guidance law is evaluated in simulations for different target maneuvers and compared to a state-of-the-art optimal-control-based guidance law. The simulations show that the derived guidance law satisfies the constraints and offers a substantial advantage over the optimal-control-based guidance law when the target is optimally evading.

Many fractals $G$ admit a harmonic immersion into $\R^n$, i.e. an immersion which minimises a natural energy under fixed boundary conditions; we look for harmonic immersions of the Sierpinski gasket into the hyperbolic plane. We show that, given any three points $\tilde A$, $\tilde B$, $\tilde C$ in the hyperbolic plane there is a harmonic map bringing the three points $A$, $B$, $C$ of the boundary of the gasket to $\tilde A$, $\tilde B$, $\tilde C$ respectively. Moreover, if the points $\tilde A$, $\tilde B$, $\tilde C$ are sufficiently close in the hyperbolic distance, then the harmonic map is unique and depends differentiably on $\tilde A$, $\tilde B$, $\tilde C$. Lastly, we show that, if the harmonic map $\phi$ is injective, then it brings geodesics of the gasket $G$ into geodesics of $\phi(G)$.

Most quantum gravity theories propose that the fundamental concept of space-time is mostly compatible with quantum theory in noncommutative (NC) space. In the present paper, we revisit the notion of entanglement induced by NC deformations of phase space. The positive partial transpose (PPT) criterion for separability of bipartite Gaussian states is extended to a general class of Bopp's shift. In particular, we have considered both the position-position and momentum-momentum noncommutativity, with deformation parameters $\theta$ and $\eta$, respectively. It turns out that $\theta$ and $\eta$ induce the entanglement. We have directly applied the formalism for an anisotropic two-dimensional harmonic oscillator. Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space. It turns out that the bipartite Gaussian state is almost always entangled in deformed space. To implement the theoretical idea, we provide an outline for a gedankenexperiment to identify the signature of phase-space noncommutativity, i.e., quantum gravity. In particular, the gedankenexperiment is devised to test the separability of supposedly separable Gaussian states in the usual commutative space, through the covariance matrix, which is constructed via measured output photocurrents after interaction of input Gaussian states and reference states. If the experiment shows that the supposedly separable states are actually entangled, then the entanglement is created through the intermediate background noncommutative space, which is a signature of the quantum nature of gravity.

Prey aggregation is widely regarded as a defense against predation, yet we show that in disease-structured populations subject to predator-induced fear and demographic Allee thresholds, aggregation can paradoxically accelerate ecosystem collapse. We develop and analyze a susceptible-infectious-predator model incorporating dual fear responses – together with a sublinear aggregation-based predation term and an Allee effect. Critically, we derive an explicit upper bound on the extinction time that decreases as predator pressure increases or aggregation strengthens, quantifying for the first time how behavioral and demographic parameters jointly determine the speed of ecological collapse. This finite-time extinction subsequently triggers a cascade collapse of the infected prey and predator populations, driving the entire ecological community to extinction. Bifurcation analysis reveals transcritical, saddle-node, and Hopf bifurcations as fear intensity, aggregation strength, and Allee threshold vary. Two-parameter continuation further identifies the precise regions of the fear–Allee parameter plane in which stable coexistence, oscillatory coexistence, predator exclusion, and finite-time extinction occur, demonstrating that stronger aggregation monotonically enlarges the finite-time extinction region while weaker aggregation supports a richer landscape of coexistence dynamics. These results demonstrate that behavioral defenses operating at the population level can generate abrupt ecological tipping points when they interact with disease dynamics and demographic vulnerability.

In this paper, we develop a framework based on differential forms that enables us to deal with the Hall problem on domains in any dimension $n\ge2$. In the case of smooth bounded domains of $R^n$, we prove local existence of mild solutions in subcritical spaces.

In this study, the set of orthomorphisms of a finite group G of order 3M with a normal subgroup of order M has been partitioned into classes. Additionally, a number of relationships between the orders of these classes have been established. Ultimately, this approach has been used to theoretically determine all 3840 orthomorphisms of alternating groups on four symbols by just counting 30 orthomorphisms, a problem that had been unsolved since 1992.

We study spectral intertwining operators between spectral Eisenstein series $\operatorname{Eis}_{P^\vee}$, $\operatorname{Eis}_{Q^\vee}$ for two parabolic subgroups $P, Q$ of a $p$-adic reductive group $G$ with the same Levi subgroup $M$, inspired by the analogy with the classical intertwining operators between parabolic induced representations of $p$-adic reductive groups. In particular, we construct the normalized (canonical) intertwining operator that satisfies the transitivity and an unnormalized intertwining operator that is adjoint to a rational section of an analog of the Bruhat-Mackey's filtration. Moreover, the normalized and unnormalized intertwining operators differ by a ratio of L-functions, analogously to the Langlands conjecture about classical ones up to units. Finally we prove that the spectral intertwining operators correspond to classical ones up to units under any conjectural categorical local Langlands correspondence.

In this paper, we provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.

In this paper, we introduce the power mean transform $P_{\lambda}(T)$ of an operator $T$ on a Hilbert space, which is a convex combination of some classical operator transforms such as the mean transform $M(T)$, the Aluthge transform $\Delta(T)$, and the Duggal transform $T^D$. In particular, when $T$ is invertible, this transform coincides with the induced Aluthge transform $\Delta_{\mathsf{m}_{f}}(T)$ recently defined by Yamazaki \cite{yamazaki-laa-2021} with $f(x)=(\lambda+(1-\lambda)\sqrt{x})^2$ for $x\in(0,\infty)$ and $\lambda\in(0,1)$. We study basic properties of $P_{\lambda}(T)$ including its spectrum, norm and numerical radius. Moreover, we use the power mean transform to give new characterizations of normal, quasinormal and binormal operators. The questions of Golla et al. \cite{yamazaki-laa-2023} and some new results on the Duggal transform are also mentioned. We obtain a result close to the recent one of Osaka and Yamazaki \cite[Theorem 3.3]{yamazaki-tams-2025} on the iteration of the induced Aluthge transform for centered operators. Finally, we describe the form of bijective maps commuting with the power mean transform of the product of matrices.

Linear-multi-parametric optimization problems are a widely studied class of optimization problems. The objective function in such a problem is affine linear dependent on a parameter vector, and the goal is to compute a set of solutions that contains an optimal solution for every fixed parameter vector. However, this is known to be computationally challenging: The underlying non-parametric problem might be NP-hard, and, in addition, optimal solution sets might have exponential cardinality. Parametric approximation aims at providing polynomial-time algorithms that overcome these challenges. Instead of computing an optimal solution set, the goal is to compute an approximation set that contains only an approximate solution for every fixed parameter vector. Several new parametric approximation algorithms have been developed in recent literature. However, all of these share a common set of assumptions, which limits the class of parametric optimization problems that can be approximated. Namely, they do not allow negative parameter dependencies and have their parameter sets fixed to the positive orthant. We present a new adaptive approximation (and, also, exact) algorithm that can be applied to a wider class of linear-multi-parametric optimization problems. Our algorithm builds upon existing algorithms from both the fields of parametric and multi-objective optimization and generalizes these algorithms. In addition, we provide structural results for the transformation of parameter sets, and demonstrate that, for linear-multi-parametric maximization problems, the assumption of non-negative optimal objective values over the whole parameter set is not sufficient to ensure approximability.

An upper homogeneous (upho) poset is a poset whose every principal filter is isomorphic to the whole poset. Fu–Peng–Zhang conjectured that every finitary upho poset admits a compatible left-cancellative, invertible-free monoid structure whose left-divisibility order coincides with the given order. We disprove this conjecture. For every vertex-transitive graph $G$, we construct a finitary upho poset $P(G,v_0)$ from walks starting at a fixed vertex $v_0$. Applying this construction to the Petersen graph, we show that multiplicability of $P(G,v_0)$ would force the automorphism group of $G$ to contain a regular subgroup. This would imply that $G$ is a Cayley graph, contradicting the fact that the Petersen graph is not Cayley. Hence $P(G,v_0)$ is a non-multiplicable finitary upho poset. We also show that the analogous poset associated with the line graph of the Petersen graph is multiplicable, demonstrating that non-Cayleyness of the underlying graph alone does not determine multiplicability.