Papers

In this paper, we study the sharp Sobolev-type inequality associated with the Born–Infeld energy on the lattice graph \(\mathbb Z^N\), \(N\geq3\): \[ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,\alpha} \left(\sum_{x\in\mathbb Z^N}|u(x)|^\alpha\right)^{\frac{N}{N+\alpha}}, \] for \(u\in D^{1,2}(\mathbb Z^N)\cap\ell^\alpha(\mathbb Z^N) \) satisfying \(|\nabla_{xy}u|\leq1\) for every \(x\sim y\), where \(C_{N,\alpha}\) denotes the optimal constant. We determine the exact positivity threshold and prove that \(C_{N,\alpha}>0\) if and only if \(\alpha\geq2^*:=2N/(N-2)\). At the Sobolev critical exponent \(\alpha=2^*\), we identify the optimal constant as \[ C_{N,2^*}=\frac12\mathcal S_2, \] where \(\mathcal S_2\) is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists \(\varepsilon_0=\varepsilon_0(N)>0\) such that \(C_{N,\alpha}\) is attained for every \(2^*<\alpha<2^*+\varepsilon_0\), with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on \(\mathbb Z^N\) and from the nonhomogeneity of the Born–Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a \(Q_1\) discrete-to-continuum comparison and by establishing a strict separation as the \(\ell^\alpha\)-norm tends to infinity.

We prove the uniqueness conjecture on the twisted Kähler-Einstein currents in big cohomology classes, by solving a problem of Bo Berndtsson posed in his work on the uniqueness theorem of Käher-Einstein metrics.

We study numerical semigroups whose defining ideals admit determinantal presentations by the $2\times2$ minors of a $2\times n$ matrix. A conjecture of Cuong, Kien, Matsuoka, and Truong relates such presentations to arithmetic progressions of pseudo-Frobenius numbers. We prove a numerical criterion for this determinantal presentation, reducing the conjecture to a problem on factorizations of maximal elements of an Apéry set in the numerical semigroup. As an application, we verify the conjecture in embedding dimension four under a unique factorization assumption on an Apéry set.

Let $A$ and $B$ be nonzero unital $C^*$-algebras with units $1_A$ and $1_B$, respectively, and let $T\colon A\to B$ be a bijection satisfying \[ \|T(a+b)\|=\|T(a)+T(b)\|, \qquad \|T(ab)\|=\|T(a)T(b)\|, \qquad T(a^*)=T(a)^* \] for all $a,b\in A$. We prove that there exist a central symmetry $u$ in $B$ and a real $*$-isomorphism $\Phi\colon A\to B$ such that \[ T(a)=u\Phi(a) \qquad(a\in A). \] Moreover, $u$ and $\Phi$ are uniquely determined by $T$. Conversely, if $u\in B$ is a central symmetry and $\Phi\colon A\to B$ is a real $ * $-isomorphism, then $T=u\Phi$ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map $uT$ to preserve the full product, not merely the Jordan product.

We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.

In this paper, we prove Koszul duality for symmetric algebras of connective complexes consisting of locally free sheaves of finite rank, which is not necessarily bounded below. As an application, we construct an equivalence between derived categories of derived loop spaces and the symmetric algebra of the tangent complex for arbitrary separated schemes of finite type over a field $k$ of characteristic zero. This can be regarded as a generalization of the classical duality between derived loop spaces and cotangent bundles by Kapranov in the smooth cases. We also give a geometrical interpretation of our equivalence in terms of derived formal stacks.

Focusing on the pullback asymptotic behavior exhibited by the system, this paper studies the three-dimensional (3D) non-autonomous Boussinesq equations with fractional Laplacian. For $ \alpha > \frac{5}{4} $ and $ \beta > \frac{5}{8} $, by adopting the $ \ell $-trajectory method introduced in \cite{MP2002}, we demonstrate the existence of both a finite-dimensional pullback attractor and a pullback exponential attractor for the process $ \{L(t, \tau)\}_{t \geq \tau} $ that corresponds to \cref{eq0101} in the phase space $ X_{\ell} $. Furthermore, we establish the existence of a pullback exponential attractor for the process $ \{U(t, \tau)\}_{t \geq \tau} $ within the original phase space $ H_\sigma^\alpha(\Omega) \times H^{\beta}(\Omega) $.

Communication is a central bottleneck in distributed optimization, but its effect is often summarized by the spectral gap of a chosen mixing matrix. Since this gap depends on link weights as well as topology, it can obscure the intrinsic effect of the network. We clarify this distinction by relating graph diameter to the inverse spectral gap. We establish a universal one-sided bound that is tight up to constant factors, while constructions across several graph families show that no topology-only converse or universal two-sided scaling law exists. This motivates a class-based view of network optimality: diameter-dependent guarantees can be assessed uniformly over graph classes, whereas spectral-gap optimality requires both the graph–matrix pair class and the admissible algorithm class to be specified. Based on the resulting diameter-based minimax benchmark, we introduce Tree-Routed Gradient Tracking (Tree-RGT), a distributed method for stochastic nonconvex optimization. The method pipelines model dissemination and gradient aggregation over a rooted shortest-path spanning tree. Each iteration uses one round of one-hop communication and one stochastic-gradient evaluation per agent, without inner consensus or multi-gossip. Under smoothness and unbiased bounded-variance oracles, Tree-RGT achieves uniformly optimal network dependence over connected graphs of prescribed size and diameter. It recovers centralized stochastic scaling after $\mathcal{O}(nD_{\mathcal{G}}^2)$ transient iterations, where $n$ is the number of agents and $D_{\mathcal{G}}$ is the graph diameter. This network-dependent transient-iteration bound improves upon or matches those reported for representative distributed methods. Thus, topology-aware routing attains diameter-optimal guarantees without relying on a prescribed mixing matrix.

In this paper, we develop a systematic construction framework of hyper-derivative algebraic geometry codes via local expansions, extending hyper-derivative Reed-Solomon codes from the rational function field to general algebraic function fields. Using the residue theorem, we determine their Euclidean duals and illustrate that the duals naturally reverse. We further give criteria for reverse self orthogonality and reverse self duality in terms of two classes of bilinear forms. Finally, we provide an asymptotic bound on the rate and relative distance via function field towers.

Vehicular federated learning (VFL) enables privacy-preserving collaborative model training for intelligent transportation systems, where communication resource allocation and gradient sparsification techniques have been explored to reduce communication overhead. However, vehicle mobility leads to rapidly varying channel conditions and transmission capacity, rendering predetermined resource allocation and sparsification decisions ineffective. In this paper, we propose FedPGT, a progressive gradient transmission scheme for VFL over time-varying channels, where vehicles progressively transmit high-magnitude gradient entries in response to instantaneous channel conditions. We establish a convergence bound that characterizes the impact of transmitted gradient entries and reveals diminishing-return behavior governed by a power-law decay. Motivated by this result, we formulate a stochastic optimization problem for online decision-making, where the main challenge lies in a cumulatively coupled, non-separable objective. To handle this challenge, we introduce per-slot surrogate transmission variables to decouple the long-term dependence across time slots and convert the original objective into an additive per-slot optimization problem, enabling a Lyapunov drift-plus-penalty approach for online scheduling. We further develop a low-complexity resource allocation algorithm for efficient online implementation. Experimental results demonstrate that the proposed scheme achieves a 3.65% accuracy improvement on the CIFAR-10 image classification task and a 12.66% reduction in average displacement error on the Argoverse trajectory prediction task compared with state-of-the-art baselines, demonstrating its applicability to diverse learning tasks under highly dynamic vehicular environments.

Let $k$ be an arbitrary field and let $S=k[x,y,z]$ be the polynomial ring with three variables $x,y,z$. We study integrally closed $(x,y,z)$-primary ideals of $S$ that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup $H$ of embedding dimension three, we obtain such ideals as inverse images $I_h=\varphi_H^{-1}(t^hk[t]\cap k[H])$. If $\ell$ is the least degree of a defining relation of $k[H]$, then $I_\ell$ is monomial, whereas $I_{\ell+1}$ has a binomial generator in the cases considered here. We determine $I_{\ell+1}$ for numerical semigroups of embedding dimension three and multiplicity three or four. The six-generated cases arising in multiplicity three and in the symmetric multiplicity-four case form two explicit families. For every ideal $I$ in these families, we prove that its Rees algebra is a Cohen–Macaulay normal domain. In the non-symmetric multiplicity-four case, $I_{\ell+1}$ is seven-generated; for $H=\langle4,9,15\rangle$, we prove that its Rees algebra is again a Cohen–Macaulay normal domain.

Integrated sensing and communication (ISAC) is a cornerstone of sixth-generation (6G) networks, yet conventional fixed-antenna systems lack the spatial adaptability to cope with dynamic users and targets. The emerging pinching antenna (PA) offers a flexible, low-cost solution by dynamically reconfiguring radiation points along waveguides, introducing large-scale spatial degrees of freedom. This article develops a unified architectural perspective for PA-enabled ISAC. We first discuss the unique advantages of PAs over existing flexible solutions, and then propose a PA-enabled ISAC framework that accommodates both uplink and downlink communication while being compatible with passive and active sensing targets. Within this framework, we identify representative application scenarios, discuss major design challenges, and highlight critical enabling techniques. A numerical case study demonstrates how PA reconfigurability affects the communication-sensing rate trade-off. We also outline open issues to guide further PA-ISAC research for future 6G networks.

The topology of a bounded three-dimensional domain can strongly affect electromagnetic fields. In a topologically complex domain, a curl-free field may not have a globally single-valued scalar potential and a divergence-free field may also fail to have a global vector potential. These topological effects lead naturally to harmonic vector fields in the Helmholtz decomposition. This paper gives a geometric and constructive study of such fields using vector analysis, circulation integrals, cutting surfaces, scalar Laplace problems, Stokes' theorem, and Green's identity. The first Betti number is interpreted through independent handle-type circulations, whereas the second Betti number counts enclosed voids. Direct proofs are given for the dimensions of the Neumann and Dirichlet harmonic-field spaces. The analysis is also extended to anisotropic lossless media. The results provide a simple topological interpretation of harmonic fields and physical DC modes in bounded electromagnetic resonators.

We study the vacuum structure of maximally supersymmetric Yang–Mills theory in two dimensions. At low energies, the theory decomposes into superselection sectors consisting of orbifold conformal field theories and massive vacua. The sectors are indexed by the cuspidal data appearing in Lusztig's generalized Springer correspondence. This suggests an enhancement of the correspondence to holomorphic factorization algebras.

Let $S=k[x_1,\ldots,x_n]$ and let $I=I_\Delta\subsetneq S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I/I^q$, but more generally the shifted quotients $I^r/I^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}/I^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge3$, the module $I^r/I^q$ is Cohen–Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}/I^{(q)}$ is Cohen–Macaulay if and only if $\Delta$ is a matroid, where loops are allowed. Thus, the Cohen–Macaulayness of a single nonexceptional window forces the Cohen–Macaulayness of every window in the corresponding filtration. The unique exceptional pair is $(r,q)=(1,2)$: at this conormal level, we show that the Cohen–Macaulayness of $I/I^2$ forces $I^2=I^{(2)}$, and hence $I/I^{(2)}$ is Cohen–Macaulay.

We study when the logarithm $\psi$ of a positive finite exponential sum on $\mathbb{R}^d$ satisfies $\det\nabla^2\psi=C\exp(\langle b,\theta\rangle-\lambda\psi)$. This is the Kähler–Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis–Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler–Einstein metric is a product of projective spaces with matched multiples of the Fubini–Study metrics, immersed by a complete Veronese–Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.

A $\lambda$-translating soliton is a hypersurface in $\mathbb{R}^{n+1}$ satisfying $H=\langle \mathbf{T},\nu\rangle+\lambda$; equivalently, it has constant weighted mean curvature with respect to the log-linear density $e^{\langle T,X\rangle}$, and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed $\lambda$-translating soliton with $\lambda>0$ and $\inf_{\Sigma}H>\lambda$ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic $\lambda$-translating solitons ($\lambda\geqslant 0$) with bounded gradient, and two rigidity theorems.

We introduce a one-parameter family of notions between double ergodicity and metric ergodicity for measure-class preserving (i.e., nonsingular) actions of countable groups on standard probability spaces, providing infinitely many new invariants distinguishing weakly mixing actions. We call these properties (essentially) $k$-chaining, for $k \in \mathbb{N}$. We apply this framework to study boundary actions of free groups of finite rank $r \ge 1$, where the boundary is equipped with a stationary Markov measure. We prove that in this context, weak mixing is equivalent to $(2r-1)$-chaining, as well as to strict irreducibility of the transition matrix of the Markov measure.

We prove that the Enots Wolley sequence contains every positive integer with at least two distinct prime divisors. Suppose, toward a contradiction, that some eligible integer is omitted, and consider its finite set of prime divisors. The local rules then severely restrict how terms involving these primes can occur: after a finite initial segment, terms divisible by some but not all of them can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite conclusion at large scales. From almost every term divisible by all of the chosen primes, it produces enough smaller earlier terms divisible by only some of them; a weighted double count makes this excess quantitative and yields a contradiction. It follows that any omission would force every sufficiently late term to have a prime divisor in one fixed finite set. Prime recurrence and a disjoint-cover argument rule out such a finite obstruction, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.

Let Gamma be a convex-cocompact group of isometries of a CAT(-1) space X and let Y to X/Gamma be a Galois cover with a word-hyperbolic group of deck transformations. We show that, for almost every geodesic xi with respect to the Bowen-Margulis-Sullivan measure on the lift of X/Gamma, there exist m,sigma > 0 and a standard Brownian motion B(s) such that, for any lambda > 1/4, d(p(g(s,xi)), o) = ms + sigma B(s) + o(s^lambda), where g(s,xi) denotes the geodesic flow acting on the geodesics of Y and p the canonical projection to Y. The result is a consequence of an almost sure invariance principle for random walks on hyperbolic groups with dependent increments. Its proof makes use of a new Ruelle operator theorem for skew products and Martin boundary techniques for random walks with dependent increments.

Let $\mathcal M$ be a diffuse semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$. We prove that, for every nonzero Banach space $Y$, the projective tensor product $L_1(\mathcal M,\tau)\widehat{\otimes}_{\pi}Y$ has the operator Daugavet property. Moreover, the witnessing operators may always be chosen contractive. This extends the operator Daugavet phenomenon from atomless vector-valued $L_1$-spaces to the semifinite noncommutative setting and yields further Daugavet-type consequences for projective symmetric tensor products, all without approximation assumptions.

We study $\widetilde{H}$-cobordisms of distinguished homology handles, introduced by Kawauchi in 1976 using infinite cyclic covers. Despite the extensive development of gauge-theoretic and Floer-theoretic invariants since Kawauchi's work, none were previously known to distinguish smooth and topological $\widetilde{H}$-cobordism. In this paper, we construct asymptotic invariants of distinguished homology handles by applying real Seiberg–Witten theory to finite cyclic covers. Using these invariants, we show that the kernel of the natural map from the smooth $\widetilde{H}$-cobordism group to its topological counterpart contains a subgroup isomorphic to $\mathbb{Z}$. We also define spin versions of these groups and show that the kernel of the corresponding natural map contains a subgroup isomorphic to $\mathbb{Z}^\infty$.

We establish explicit sufficient conditions for multiplicity of positive solutions of the Hénon equation in the unit ball and for uniqueness in one dimension. In the Sobolev-subcritical range, a spherical harmonic function argument shows that the radial solution has Morse index greater than one and therefore differs from a ground state. In even dimensions, a degree-four perturbation in a symmetry and monotonicity cone yields a sufficient condition for nonradiality in the supercritical range. The one-dimensional uniqueness criterion follows from a comparison of weighted Dirichlet eigenvalues and a concavity estimate.

We prove that the Scott topology commutes with arbitrary products of countably presented frames and that every such product has a sober Scott space. In particular, this holds for arbitrary products of countable complete Heyting algebras. For a family of consonant spaces, Scott-product compatibility of their open-set lattices is equivalent to consonance of their topological sum. Countable generation does not suffice: a countably generated spatial frame can have a non-sober Scott space and fail the product identity for its square. We also show that the cardinal spectra of Scott non-sober frames and spatial frames are upward closed and, under the Continuum Hypothesis, consist of all uncountable cardinals. Finally, every Artinian $T_0$ web space is a $B$-space; if it is also a $d$-space, it carries the Scott topology of an algebraic dcpo. Consequently, every Artinian meet-continuous dcpo is algebraic. This yields a dichotomy theorem: a dcpo with a non-sober Scott space must fail meet continuity or Artinianity.

We study an optimal control problem for a two-equations Keller-Segel system on a bounded domain with Neumann boundary conditions, where the chemotactic sensitivity is modulated by a control $f(x,t)$ acting on the taxis flux $\nabla\!\cdot(f\,u\,\nabla v)$. We prove the well-posedness of the state equation under low regularity assumptions. The analysis is based on the study of an $\varepsilon$-regularized problem, the application of Schauder's fixed point theorem, $\varepsilon$-independent, positivity and a priori estimates, the passage to the limit as $\varepsilon \to 0$, and a uniqueness argument. For a cost functional combining trajectory tracking, taxis penalization, control regularization, and terminal objectives, we establish the existence of optimal controls and derive the first-order optimality conditions through a Lagrange multiplier framework, leading to an adjoint system and a variational inequality for the optimal control. A numerical scheme based on spline collocation in space and Runge-Kutta time-stepping is set and implemented. Numerical experiments are performed to simulate the free and controlled dynamics in conservative and non-conservative cases.

MLegS is an open-source Modern Fortran code package for solving linear and nonlinear time-dependent partial differential equations on radially unbounded domains. It builds on an earlier in-house prototype, recasting it as a modular solver with a unified interface for fields, transforms, and spectral operators. Within this framework, the numerical formulation maps $0\leq r<\infty$ onto a finite interval, where associated Legendre functions provide the radial basis, while Fourier series represent the periodic azimuthal and axial directions. The mapped radial basis represents radial infinity without imposing an artificial boundary condition at finite radius while incorporating the regularity required at $r=0$ according to azimuthal order. The implementation distributes the corresponding field layouts using the Message-Passing Interface (MPI), with strong- and weak-scaling measurements extending to 512 cores across four compute nodes. Beyond parallel performance, three examples assess the solver across diffusion, reaction-diffusion, and vortex dynamics: closed-form diffusion solutions establish spectral spatial convergence and the design orders of both semi-implicit integrators; a Fisher-Kolmogorov-Petrovsky-Piskunov front matches the linearized solution and propagates at the classical speed within its logarithmic correction; and vortex-pair simulations based on toroidal-poloidal decomposition yield a measured Crow growth rate $2.7\%$ above the theoretical prediction and demonstrate the reconnection of the vortex pair.

Chatterjee and Sloman proved that a bounded measurable similarity function with sufficiently small average Gromov hyperbolicity admits a tree representation with small mean approximation error. Their argument uses a weighted version of Szemerédi's regularity lemma and does not yield explicit quantitative bounds. Here, we establish a tighter relation between average hyperbolicity and mean tree approximation error. For a similarity function $s:S\times S\to[0,b]$, we prove that $$\operatorname{Tree}(s) \leq (63/e)^{1/3} \sqrt[3]{b^2 \operatorname{Hyp}(s)} \leq 2.8512 \sqrt[3]{b^2 \operatorname{Hyp}(s)}.$$ The proof uses a simple pivoting construction inspired by \KwikCluster. We also discuss the optimal dependence on average hyperbolicity, including a square-root lower bound, and connections with ultrametric fitting.

For a real symmetric matrix $H$ and distinct vertices $a,b$, we classify exponents $k$ for which $H^k$ has perfect state transfer (PST) from $a$ to $b$. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to $H$ and every positive even exponent reduces to $H^2$. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of $P_7$ has PST from vertex $2$ to vertex $6$ only at exponent $2$.

We completely characterize the second-order coding rate region of the two-user Gaussian multiple-access channel at points in the relative interior of the sum-rate face, under maximal per-codeword power constraints. For any fixed average error probability $0 < \varepsilon < 1/2$, a nontrivial mixture of power splits reduces the sum-rate dispersion and strictly improves upon standard inner bounds based on a constant power split. Thus, a constant power split, although sufficient to attain the first-order capacity region, is insufficient for second-order optimality in this regime. Our achievability proof builds on the constant composition and coded time-sharing framework of Scarlett, Martinez, and Guillén i Fàbregas, with an optimization over admissible power splits. A matching second-order converse establishes the optimality of the resulting coded time-sharing strategy.

We study the existence of periodic solutions and invariant tori bifurcating from a zero-Hopf equilibrium in a Hopf-Langford type system. Using second-order averaging theory, we establish explicit parameter conditions under which the system admits a periodic solution near the origin, together with its stability. We then apply recent results relating invariant tori to Neimark-Sacker bifurcations of the associated Poincaré map to obtain a smooth bifurcation curve along which a unique invariant torus surrounds the periodic solution. The relevant first Lyapunov coefficient is computed explicitly as a power series in the perturbation parameter, and we show that, under the conditions considered, this coefficient is always positive; consequently, whenever it exists, the bifurcating torus is unstable, regardless of the specific parameter values chosen within our hypotheses. This provides a structural obstruction to the existence of asymptotically stable tori in this family of systems. We illustrate our results with two numerical examples, one exhibiting an isolated asymptotically stable periodic solution with no bifurcating torus, and another exhibiting a periodic solution surrounded by an unstable invariant torus, the latter also illustrated through its Poincaré map.