Papers

In this paper, we study compact vacuum static spaces of dimension $\ge 4$ with nonnegative sectional curvature. First, we prove that an $n$-dimensional compact vacuum static space is isometric to a standard sphere provided it has positive sectional curvature and the complete divergence of its Weyl curvature tensor is nonnegative. Under the slightly weaker condition of nonnegative sectional curvature, we show that such a vacuum static space must have parallel Ricci curvature tensor. Second, we examine the critical point equation arising from the critical metrics of the total scalar curvature functional on the space of Riemannian metrics restricted to unit volume and constant scalar curvature. We demonstrate that a compact Riemannian manifold with nonnegative sectional curvature and nonnegative complete divergence of the Weyl curvature tensor, which admits a nontrivial solution to the critical point equation, must be Einstein and is isometric to a standard sphere. Our results can be viewed as extensions of the results in [4].

Integrated sensing and communications (ISAC) infrastructure can provide an external perception layer for aerial robots, but dynamic blockage makes the set of informative sensing links at the next update uncertain. A single predicted availability pattern can miss low-information outcomes, whereas protecting all patterns wastes power and may become infeasible when all links are blocked. We propose an outage-aware conformal robust power-allocation framework for multi-BS UAV tracking. A history-aware joint predictor maps radar and tracking histories to next-slot availability probabilities, and adaptive prediction sets retain plausible correlated patterns. Tracking constraints are enforced only for retained non-outage patterns, while the all-blocked pattern is treated separately as a physical sensing outage. With per-slot beam directions fixed, the resulting scalar-power allocation is a semidefinite program. A reliability bound relates serviceable-slot tracking failure to deployed-policy miscoverage and optimization infeasibility. Across paired closed-loop simulations over five random seeds, the method attains a 0.01% serviceable-slot constraint-violation rate and uses 47.0% less power than protection over all non-outage patterns.

Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem has a unique minimizer in the case where S is a slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof relies on a refinement of Smirnov's decomposition of divergence-free measures [62] which is of independent interest.

Computing local Stackelberg equilibria in nonlinear sequential games is challenging because gradient-based best-response methods may diverge, cycle, or converge to stationary points that do not satisfy the required equilibrium conditions. These failures can result from curvature asymmetry, unstable response mappings, and lack of invariance with respect to the follower's best-response manifold. This paper introduces the Stackelberg Sliding-Mode Algorithm (SSMA), a control-theoretic framework that recasts Stackelberg equilibrium computation as a manifold-stabilization problem. The follower and leader first-order optimality conditions define Stackelberg sliding manifolds, and sliding-mode dynamics drive the iterates to these manifolds in finite time. The follower curvature and leader reduced-curvature conditions determine manifold attractivity, making valid local Stackelberg equilibria attractive while rejecting stationary points that fail the second-order equilibrium conditions. The resulting approach integrates finite-time manifold stabilization, Stackelberg-consistent dynamics, and curvature-based equilibrium selection within a unified framework. Across 500 nonconvex Monte Carlo trials, SSMA achieved 100% success, reached the prescribed tolerance in a median of 26 iterations, compared with 456 for the nearest fully successful baselines, and attained terminal residuals near \(10^{-16}\), while converging to a valid local Stackelberg equilibrium.

We study uniform approximation for operator learning on compact subsets of Banach spaces. We bound the recovery dimensions and the number of network parameters in terms of compact-set complexity, the target gauge, and recovery stability. For bounded linear operators with a separable domain, we characterize gauge approximability by coordinate decompositions with a common vanishing gauge and arbitrarily small additive errors. Their partial sums give a fixed family of nonlinear encoder–decoder pairs. We prove that nonlinear recovery with a common vanishing gauge and a rank bound depending only on the model dimension is equivalent to the uniform approximation property (UAP). We also obtain linear recovery estimates with the same rank bound and an operator version for the linear factors of nonlinear targets. For Barron operators taking values in a Banach space $Y$ of type $p\in(1,2]$ and admitting a Bochner-integrable density, sampling gives the rate $N^{-(1-1/p)}$ for the expected $L^p(\lambda;Y)$ error, where $\lambda$ is a probability measure on the input compact set. For fixed finite-dimensional factors and a class of Hölder maps, we obtain matching upper and lower parameter bounds at fixed sufficiently large depth. Classical sampling and truncation examples provide explicit recovery maps, and a nonlinear integral operator illustrates the parameter bounds.

Let $S$ be a Cohen-Macaulay normal domain containing a field $k$ with an action of a finite group $G$ with $|G|$ invertible in $k$. We prove that $G$-equivariant noncommutative crepant resolutions of $S$ induce noncommutative crepant resolutions of $S^G$ if the $G$-action is small. We also discuss similar results for equivariant maximal modification algebras. We introduce the notion of $G$-maximal modifying ($G$-MM) module, and we show that $G$-MM module over $S$ induces MM $S^G$-module in some special cases. Furthermore, we prove that Gorenstein MMAs of three-dimensional $\mathbb{Q}$-Gorenstein Cohen-Macaulay rings are derived equivalent. As an application of these results, we prove the existence of an MM module and derived equivalences of MMAs for three-dimensional terminal singularities of index two.

We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip $S_a=\{(x,y)\in\mathbb{R}^2:0<y<a\}$ endowed with the metric $g_{S_{a}}=\frac{\pi^2}{a^2\sin^2(\pi y/a)}(dx^2+dy^2)$, and on its unit tangent bundle $T^1S_a$ endowed with the Sasaki metric $g^S$. On the strip, we show that the Gauss curvature is constant equal to $-1$, that the Lie algebra of Killing fields is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, that every Ricci soliton is trivial (Killing potential, $\lambda=-1$), and that a $2$-Killing field which is conformal, or which is the potential of a hyperbolic Ricci soliton with $\lambda\neq0$, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of $(T^1S_a,g^S)$ is four-dimensional and isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$, that $g^S$ is not Einstein but is $\eta$-Einstein with constant coefficients, $\operatorname{Ric}^S=-\tfrac32\,g^S+2\,\eta\otimes\eta$, and that $(T^1S_a,g^S)$ admits no Ricci soliton. For hyperbolic Ricci solitons we prove non-existence for Killing, left-invariant, and $\theta$-invariant potentials with conformal projection, and we reduce the general $\theta$-invariant case to a system of equations on the strip; the classification for an arbitrary potential is left open. Finally, we show that the $\eta$-Ricci solitons of $g^S$ have a Killing potential and $\lambda=\tfrac32$, $\mu=-2$, and that the $\eta$-hyperbolic Ricci solitons obtained in the same classes of potentials are the trivial ones, with a Killing potential, $\mu=-\tfrac32$ and $\nu=2$.

We introduce a smooth radial approximation of the covariogram of a convex body. The construction replaces the intersection \(K\cap(K+x)\) by the radial \((-p)\)-mean of \(K\) and \(K+x\), producing a family \(g_{K,p}\) which converges pointwise to the classical covariogram as \(p\to\infty\). For each fixed \(p>0\), the function \(g_{K,p}\) is smooth near the origin, and its Hessian is computed explicitly in terms of the operator defining the polar \(L_2\)-projection body. We study the simultaneous limit in which \(p\to\infty\) and the level deficit is \(\delta=p^\alpha\), \(\alpha<0\). The normalized level sets exhibit three regimes: the polar projection body appears for \(\alpha>-1\), the polar \(L_2\)-projection body appears for \(\alpha<-1\), and a critical transition body appears at \(\alpha=-1\). This transition body is described by an explicit \(\log\cosh\)-type boundary integral, placing the critical regime naturally within the framework of Orlicz projection bodies.

Distributed control requires choosing how far information travels and how long it is retained. For networked linear quadratic regulation (LQR), we use System Level Synthesis to bound these resources for a specified performance loss per node relative to centralized control. For locally coupled systems under uniform regularity assumptions, including stabilizability and detectability, we prove that the centralized optimal state and input responses to disturbance impulses satisfy exponential decay bounds in both time and spatial distance. On networks with polynomial neighborhood growth, sufficient communication ranges and memory horizons grow logarithmically with the inverse tolerance, independently of network size. Truncating these responses gives local finite-memory filters with stability and performance guarantees under a computable small-gain condition. Under an additional uniform local feedback condition, we also obtain controllers that confine nominal disturbance effects in space and eliminate them in finite time. Numerical experiments illustrate how performance and disturbance containment requirements guide architecture selection.

We study second-order methods for convex stochastic optimization, where gradients and Hessians are available only through stochastic estimates with variances $\sigma_1^2$ and $\sigma_2^2$, respectively. First, we propose the Stochastic Contracting Cubic Newton method. At each iteration, it minimizes a cubic model with additional quadratic regularization and then contracts the step toward the current point. After $T$ iterations, the method achieves the expected convergence rate $\mathcal{O}(\sigma_1/\sqrt{T}+\sigma_2/T+1/T^2)$. Building on this construction, we develop an accelerated variant achieving $\mathcal{O}(\sigma_1/\sqrt{T}+\sigma_2/T^2+1/T^{7/2})$, matching the known lower bounds of Agafonov et al. (2024) in all three terms.

We develop a localized hybrid finite element–random feature method (FEM–RFM) for elliptic interface problems on unfitted background meshes. The central idea is to use the random feature method (RFM) only in the narrow band formed by interface-cut cells of a uniform Cartesian background mesh,while retaining standard $Q_p$ finite elements on the remaining uncut cells. Within the band, local RFM representations are used on the two physical sides of the interface. The prescribed jumps in the solution and normal flux are incorporated through residual equations involving the two RFM representations on the physical interface $\Gamma$, while the finite element method (FEM) and RFM components are coupled across a separate grid-aligned artificial interface $\Gamma_c$ through trace and variational normal-flux residuals. A Schur reduction eliminates the interior finite element unknowns through sparse solves while retaining their Galerkin equations as equality constraints. The remaining band and interface residuals are minimized in a reduced least-squares problem involving only the coupling nodal values and random-feature coefficients. The method is assessed numerically on a range of two- and three-dimensional interface problems involving high coefficient contrasts, nonsmooth and multiple interfaces, and irregular outer boundaries, together with an application-oriented thermal-homogenization problem for a periodic bicontinuous composite.

The study of the distribution of real sequences in the unit torus has a long and rich history. There are many notions which describe the “pseudo-random” behavior of a given deterministic sequence (often of arithmetic origin), such as equidistribution (which describes the distribution on a “global scale”) or the pair correlation (which is a “local-scale” statistics). Another popular and important statistics, which can be applied to all ranges from global via intermediate to local, is the number variance $V_N(s)$, which describes the mean-square fluctuation of the number of elements of an initial segment of the sequence in an interval of length $s$, when moving this interval around the torus. In the terminology of quantum chaology, where such statistics are used to describe the distribution of the energy levels of a system, pseudo-random behavior is called Poissonian behavior. Since the number variance is an important indicator to qualify pseudo-random behavior, it is quite surprising that the literature does not seem to contain a precise description of the distribution and of the almost sure asymptotic behavior of the number variance in the truly random case (i.i.d.\ random points), even in the comparatively simple setup of uniformly distributed points on the unit torus. In the present paper we calculate the distribution of the number variance with high precision throughout a wide range of the parameter $s$, and determine the precise threshold (in terms of $s$) where the “Poissonian” asymptotics $V_N(s) \sim N s$ breaks for a typical realization of a random sequence. Key technical input comes from the Komlós–Major–Tusnády theorem and the Karhunen-Loève decomposition.

We study near-surface tension instabilities in bilayered cylinders composed of an incompressible core coated with a thin compressible film. In theory, circumferential wrinkles with axial wavefronts may appear on the outer surface when the cylinder is stretched. We analyze these instabilities through modelling, numerics, finite element simulations, and controlled experiments. We show that the wrinkle amplitude and wavelength depend on the Poisson ratio of the coating and the ratio of the film and substrate shear moduli. We find that the lower the film's Poisson ratio, the earlier the wrinkles emerge under tension. This observation motivates us to use auxetic films (with negative Poisson ratios) with bespoke microstructures in the finite element simulations and the experiments. For the latter, we fabricated 3D-printed soft cylinders with stiff auxetic outer layers and subjected them to axial loading. Although simulations predicted clear wrinkle formation, experimental observation proved more elusive due to the small amplitude of the wrinkles.

We study intermediate coverings of spectral families, Cantor sets, and projections of measures. Two perturbation families of a fixed ReLU tangent kernel have identical source thresholds. Their Brownian images have equal box thresholds and distinct intermediate thresholds, almost surely along prescribed geometric scales. For Cantor sets with delayed contractions, we obtain entropy formulas for the intermediate dimensions of the sets and their fractional Brownian images. These formulas yield endpoint asymptotics and a sharp separation theorem for the first phase transition. An oscillating variant has distinct lower and upper intermediate dimensions while preserving the Assouad spectrum. Finally, almost every orthogonal projection preserves both intermediate dimensions of a nonzero finite compactly supported measure whose quasi-Assouad dimension does not exceed the target dimension.

We introduce positively transverse reaction networks and show that, with mass-action kinetics, no bounded trajectory with positive initial condition has a boundary omega-limit point. These networks are characterized by a geometric property of a collection of polytopes defined by a subset of complexes of the network.

We investigate weighted perimeters of convex bodies in $\mathbb{R}^n$. Our main result states that the weighted perimeter is continuous with respect to the Hausdorff metric for every radial weight function. We also establish monotonicity under inclusion under suitable assumptions on the radial weight, and show that these assumptions are sharp. These results are applied to show the existence of minimizers for a shape optimization problem.

In this paper, both the pointwise space-time structures of the solution for a three-dimensional model describing gas dynamics in thermal non-equilibrium and the corresponding background stabilities within the $L^p$ framework are investigated under the small initial perturbation assumption and the irrotationality condition. Our main tools are the modified Green's function according to the irrotationality condition, a priori energy method and a refined decay estimates of products composed of the entropy function and high-order derivatives.

In this paper, we study the rational realization of the gap-two product $K(\Q,n)\times K(\Q,n+2)$ as the classifying space $\B(X)$ for a simply-connected $\pi$-finite space $X$ with rational homology of finite type. We prove that $K(\Q,n)\times K(\Q,n+2)$ can be realized as $\B(X)$ up to rational homotopy equivalence if and only if $n=3,4$ or $4m+2$ for $m\geq 1$. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such $X$ is unique for $n=3$ or $4m+2$ for $m\geq 1$; there exist infinitely many rational homotopy types of such $X$ for $n=4$.

We present a spacetime that serves as a horizonless mimicker of extremal black holes. Unlike previously studied mimickers of black holes with non-degenerate horizons, whose geometries are of the wormhole type and consist of two asymptotically flat regions connected by a very long throat, the mimicker of an extremal black hole possesses a single asymptotically flat region at one end and an infinitely long throat at the other. In the throat region, the spacetime approaches the direct-product geometry $M_{1,1} \times S^2$, where $M_{1,1}$ is two-dimensional Minkowski spacetime and $S^2$ is the two-sphere. We investigate scalar-field dynamics and the fate of the Aretakis instability. Although Aretakis-like modes arise in a local analysis of the throat, they do not lead to a physical instability of the mimicker. A scalar perturbation initially localized in the throat gives rise, at late times, to a sequence of signals with a characteristic separation $t_{\rm delay}\sim 1/\lambda$, where $\lambda\ll 1$ is a deformation parameter. This timescale is parametrically larger than the echo time $t_{\rm echo}\sim \ln(1/\lambda)$ characteristic of non-extremal black-hole mimickers. We refer to this sequence as a long-delay signal train. Geometrically, the periodicity of the long-delay signals is associated with periodic geodesics on $S^2$. We then extend the construction to extremal Kerr. In this case, the throat region approaches a generalized product geometry involving $M_{1,1}$ and a two-dimensional surface $\Sigma_2$. The corresponding signals are expected to form a quasi-periodic signal train, reflecting the quasi-periodic geodesic motion on $\Sigma_2$. We also comment on the generalization to wormhole mimickers of non-extremal black holes and on the implications of these results for unitarity in the context of the AdS/CFT correspondence.

Inspired from some works on constructions of algebraic structures from other ones, we purpose to give some construction theorems on BiHom-Poisson conformal superalgebras which unify the notion of the Z2 -graded Poisson algebras, the BiHom-Poisson algebras and their conformal setting. So, we introduce BiHom-Poisson conformal superalgebras and show that they are closed under twisting. We then prouve that any regular BiHom-associative conformal algebra leads to BiHom-Poisson conformal algebra via the commutator bracket. Then, we point out that any BiHom-Poisson conformal superalgebra together with Rota-Baxter operator give rises to another BiHom-Poisson conformal superalgebra. Next, we show that tensor prod- uct of any BiHom-associative superalgebra and any BiHom-Poisson conformal superalgebra is also a BiHom-Poisson conformal superalgebra. Contruction from Hom-post-Poisson conformal superalgebras is also given.

We derive the triangular prism equations (TPE) for near-group fusion rules of type $G+|G|$, where $G$ is a finite abelian group. We prove that spherical categorification with trivial Frobenius–Schur indicator over any field in which $|G|$ is invertible is equivalent to the solvability of the TPE system. For every cyclic group, we construct an explicit solution to the TPE system using hyperbolic gamma functions, yielding both a spherical category with negative simple-object dimensions and, after Galois conjugation, a pseudo-unitary category with positive dimensions. For scalar solutions with positive dimensions, we establish an exact singular-value formula and a uniform gap, which provide explicit sufficient criteria for unitarity. Finally, we show that the unitarity of the cyclic Galois-conjugated coefficients for arbitrary orders corresponds to a specialization of a real-multiplication conjecture implied by the order-one abelian Stark conjecture, revealing a deep connection between categorical unitarity and Hilbert's twelfth problem for real quadratic fields.

In this paper, we treat the problem of determining when a sequence $\{c_n\}_{n=0}^{\infty}$ with the exponential generating function $F(t)$ satisfies the congruence $c_{n+(p^h-1)p^k}\equiv c_n \pmod{p^{k+r}}$ for large $n$ and when its canonical interpolating functions are locally analytic. This type of congruence is often observed for several number-theoretic or combinatorial sequences, such as Bernoulli numbers, Euler numbers, and Fibonacci numbers. Traditionally, such congruences are proved by using $p$-adic integrations. In this paper, we present another approach to this problem. Instead of using $p$-adic integrals, we investigate an algebraic structure of the set of generating functions and establish some conditions on $F(t)$ that are equivalent to or sufficient for the congruence and smoothness of the interpolating functions. We also provide concrete applications of our framework to Genocchi numbers, generalized Euler numbers, and so on, which are outside the scope of the typical $p$-adic integration method.

Erdős and Sós conjectured that every $n$-vertex $3$-uniform hypergraph whose link graphs are all bipartite has at most $(1/4+o(1))\binom n3$ edges. We disprove this conjecture by constructing counterexamples with edge density at least $0.250000356>1/4$ for all sufficiently large $n$.

In this paper, we study $r$-neutralized Pfister–Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On $r$-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower $r$-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper $r$-neutralized topological entropy (iii) We obtain upper bounds for $r$-neutralized PS entropy under different smoothness assumptions and lower bounds for $C^{1+\alpha}$ diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For $C^\infty$ diffeomorphisms and $C^{1}$ Anosov diffeomorphisms, the $r$-neutralized PS entropies equal $h_\mu(f)+r\dim M$ for every ergodic measure $\mu$. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for $C^2$ surface diffeomorphisms.

We propose a symmetric Nitsche method for the biharmonic Dirichlet problem on a smooth domain whose boundary may cut arbitrarily through the computational mesh. Bulk least-squares stabilization in a boundary strip and tangential boundary penalties control the consistency terms through an explicit lifting of the two Dirichlet traces. This gives coercivity on $H^4(\Omega)$ without inverse inequalities and permits a fixed penalty independent of the cut configuration for sufficiently thin strips. Under suitable regularity assumptions, splines of degree $p\geq4$ give energy and $L^2$ error bounds of orders $h^{p-1}$ and $h^{p+1}$. We also analyze a face-stabilized extension to $C^1$ splines of degree $p=2$ and $C^2$ splines of degree $p=3$. Exact-ellipse experiments examine convergence, cut stability, and parameter sensitivity.

We develop a model-free policy gradient method for discrete-time mean-field control (MFC). In MFC, the policy affects the objective both through the controlled dynamics and through the population distribution. Standard REINFORCE estimators capture the first effect but not the second. We introduce Transport REINFORCE, a transport map-based approach that perturbs a suitable transformation of the population distribution to estimate this missing mean-field contribution. The method applies to both finite and continuous state spaces. In finite state spaces, we perturb the population distribution directly on the probability simplex through a convex combination of the current population weights and random weights. In continuous state spaces, we project the population distribution onto the manifold of Gaussian mixtures, and then randomize it via a transport map that ensures the perturbed law remains within this manifold. We prove consistency of the perturbed objective and gradient as the perturbation vanishes, and derive bias and mean-square error bounds for the resulting sample-based gradient estimator. Numerical experiments on several MFC benchmarks show that Transport REINFORCE improves over standard REINFORCE.

We give a direct information-theoretic proof of the Plünnecke–Ruzsa inequalities for finite subsets of abelian groups. The argument rests on an elementary coupling inequality proved by maximizing joint entropy, without tensor powers or asymptotic constructions. It also yields Ruzsa's sumset triangle inequality and an extension to non-abelian groups.

We give a short proof of the classical weak-disorder result that the Anderson model on the Bethe lattice has purely absolutely continuous spectrum on compact intervals in the interior of the free spectrum. This note continues [6], where we gave a minimal proof of the existence of a nontrivial absolutely continuous component using the cyclicity criterion of [8]. There, Hellinger overlap yields the required non-cyclicity. Here, a stability argument for the forward Green function, followed by the zero–one law for the tree recursion, yields positivity of its boundary imaginary part. This positivity already gives absolutely continuous spectrum throughout the interval; the structural result of [7] excludes singular spectrum and yields purity. We also observe that the same structural argument shortens the final passage from boundary positivity to purity in the recent OpenAI preprint on the weak-disorder Anderson model on $\zz^d$, $d\geq3$ [10].

Error-correcting codes are a foundational technology for representing information robustly against noise in communication and data storage. The binary code packing problem, which seeks to maximize the number of codewords at a fixed code length and minimum distance, is a fundamental problem in coding theory that remains open for many parameter choices. We give explicit binary codes with parameters $(n,M,d)=(22,84,9)$, $(24,196,9)$, and $(25,65,11)$, where $n$, $M$, and $d$ denote the code length, number of codewords, and minimum distance, respectively. These establish the lower bounds $A_2(22,9)\ge84$, $A_2(24,9)\ge196$, and $A_2(25,11)\ge65$, where $A_2(n,d)$ denotes the maximum possible number of codewords, improving the respective lower bounds 80, 192, and 64 in the public table of general binary codes. Most notably, the third result improves the lower bound supplied by a lexicographic construction for the first time in 66 years, while the first two improve bounds that have stood for 21 years. The new codes represent more messages without changing the code length or error-correction guarantees, with gains compounding multiplicatively across blocks. We release all codewords and an exhaustive verifier to enable independent verification.

In this note we study the regularity properties of the \emph{Effort Function $E(\beta)$}. This can be defined as the minimal $L^1$-norm of the control $\tilde \alpha(x) \geq 0$ so that the PDE $$U_t = \Delta U + f(U) - \tilde \alpha(t,x) U, \quad U \in [0,1], %\ \tilde \alpha \geq 0,$$ admits a traveling profile $U(x - \beta t)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control $\tilde \alpha(t,x)$ describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of $f(U)$ and (consequently) on the effort function $E(\beta)$, the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that $E(\beta)$ is in general only $C^1$.