Papers

In this work, we revisit a continuous-time two-compartment pharmacokinetic model of human ethanol metabolism originally proposed by Levitt and Levitt. We first establish the positivity and boundedness of the solutions, investigate the existence and uniqueness of a positive equilibrium, and analyze its local and global asymptotic stability. As a result, the global dynamics of the ethanol metabolism model is completely characterized, thereby complementing and extending the analytical results reported in the original benchmark study. Second, we extend the original continuous-time model by replacing the Michaelis–Menten metabolism rate with a general class of metabolism-rate functions that includes many well-known monotone and nonmonotone forms. This extension enhances the flexibility of the model and enables it to capture a wider range of realistic metabolic scenarios. We then investigate the global dynamics of the generalized continuous-time model. Finally, numerical experiments are conducted to support the theoretical findings. The numerical results provide further evidence for the theoretical results.

Bernoulli-type waveforms for modified Emden nonlinear oscillators of arbitrary natural power $q$ are obtained through a generalized commutative factorization approach. These oscillators display a well-defined odd-even dynamical dichotomy, which is discussed in detail: the odd-$q$ cases entail isochronous oscillators whose period $T = 2\pi/\omega$ is independent of amplitude and initial conditions, while the even-$q$ cases display underdamped behavior. The Lagrangian formulation is presented in the Lurie's dissipative description. The isochronous regime and the period of the solutions in the odd case are also confirmed through a generalized polar-coordinate analysis in the spirit of Sabatini's work. The absence of periodic orbits for even $q$ is shown to be a consequence of the Bendixson-Dulac criterion applied to the radial velocity function. Explicit waveforms and their phase portraits are presented for $q = 1, 2, 3, 4$, along with the non exponential envelope formulas for the damped cases and singular-region bounds for the isochronous ones. A few possible applications are also mentioned.

Let $\mathcal{H}_{\mathcal{Q}}$ be the field of unary germs at $0^+$ definable in the quasianalytic o-minimal structure $\mathbb{R}_{\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of $\mathcal{H}_{\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that $\mathbb{R}_{\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic $\mathbb{R}_{\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary $\mathbb{R}_{\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of $\mathcal{H}_{\mathcal{Q}}$.

Boumal, Criscitiello and Rebjock (BCR) proved that if $M$ is a contractible, connected and complete Riemannian manifold, then every smooth function $f\colon M\to R$ satisfying the global Polyak–Łojasiewicz inequality (PŁI) is necessarily of the form $f = f^* + \|\phi\|^2$ with $\phi$ a submersion. Informally, minimizing such a function amounts to solving a nonlinear least-squares problem in new coordinates. The global PŁI hypothesis fails, however, in many problems of interest, among them continuous-time LQR policy optimization in optimal control and a standard formulation of logistic regression. A hierarchy of weakened PŁ inequalities has been introduced in order to cover such problems, and more generally to study the effect of noise and adversarial perturbations on gradient flows. This note shows that, with minor modifications, the same reduction to a nonlinear least-squares problem holds under a substantially weaker hypothesis, “semiglobal” PŁI, which is satisfied in both of the examples just mentioned. That condition asks that $f$ satisfy an estimate $\|\nabla f(x)\| \ge \alpha\bigl(f(x)-f^*\bigr)$ for all $x$, with $\alpha$ merely positive definite and bounded below by a positive multiple of $\sqrt{s}$ for small $s>0$.

In this Brief Communication, we establish the notion of critical damping for generic two-degree-of-freedom system. For given set of masses and stiffnesses, the critical damping corresponds to the real eigenvalue with maximal multiplicity, equal to minus geometrical mean of the eigenfrequencies. This case corresponds to the fastest possible asymptotic decay rate for generic initial conditions. The damping matrix for the critical case is unique up to reflection of one modal coordinate and, generically, non-diagonal. Quite surprisingly, for large difference of the eigenfrequencies, it is also not positive definite. Therefore, physical realization of the critical case will require active elements that provide negative effective damping.

We construct real-analytic diffeomorphisms of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) that lift We study real-analytic word maps of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) obtained by evaluating Nielsen automorphisms of the free group \(F_2\). Fried studied these word maps and computed their topological entropy. For each maximal torus \(T\subset\SU(2)\), the product \(T\times T\) is invariant and carries the corresponding toral dynamics. Their union is the commuting locus. We describe its regular part as a twisted conjugacy-sphere bundle over the punctured pillowcase and determine the local topology at the four central singularities. Periodic toral points give periodic conjugacy two-spheres, which obstruct Anosov hyperbolicity on \(S^3\times S^3\). Normalized images of the coordinate three-cycles converge to stable and unstable Ruelle–Sullivan eigencurrents supported on the commuting locus. The construction extends to \(\Sp(1)^n\). The higher-rank commuting locus has dimension \(n+2\), periodic conjugacy two-spheres persist, and the action on \(H_{3k}((S^3)^n)\) is \(\Lambda^k A\). For \(n\geq3\), IA-automorphisms can act nontrivially on the \(\SU(2)\)-character variety although they are trivial on abelianization. The degree-three eigencurrents again have a Ruelle–Sullivan description; higher exterior-power classes lead to a compactness problem for normal currents. We also relate the nonuniqueness of Nielsen lifts to Fried's computation of the mapping class group of \(S^3\times S^3\), where the group \(\Theta_7\) of homotopy seven-spheres appears in the kernel of the action on homology.

Recent work produced a linearly dependent system of twelve time–frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $\alpha=\frac13+10^{-12}\sqrt2$ and $\beta=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $\lambda\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(\alpha,\beta/2)f=\lambda f$, where $W$ denotes the Weyl time–frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb R)$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.

Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.

We prove that for almost every $\alpha\in\mathbb{R}$, the \textit{fully inhomogeneous $p$-adic Littlewood Conjecture} holds; namely, $$\forall\, \delta\in\mathbb{R},\;\forall\, \kappa\in\mathbb{Z}_p, \quad \liminf_{|q|\rightarrow+\infty}\, q\,\langle q\alpha+\delta\rangle\,|q+\kappa|_p=0.$$ Here, $\langle\cdot\rangle$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic norm. Moreover, we prove this conjecture for every quadratic irrational $\alpha$. This gives an affirmative answer to the $p$-adic version of a question posed by Cassels in [Cas59].

We develop an algebraic framework for central configurations of the $n$-body problem with homogeneous potentials, grounded in the exterior algebra of the configuration space and the normalized shifted Brehm–Wintner–Conley (BWC) matrix $S$. Relating the kernel of $S$ to the Plücker coordinates of the configuration, we generalize the determinantal equations obtained by Williams (1938) for the planar five-body problem to central configurations of any dimension and any number of bodies, and derive the Dziobek–Williams equations $\det(S_I^J)=\kappa\, z_I z_J$, which exhibit the compound matrix $S^{(t)}$ as a rank-one matrix. Introducing the Brehm–Wintner–Conley dimension $\operatorname{bwc}(x)=\operatorname{rank}(S)$ (an integer invariant that stratifies central configurations and measures vertical degeneracy), together with the Plücker–BWC coordinates attached to it, we prove that each stratum of central configurations with fixed dimension and Brehm–Wintner–Conley dimension admits a base-point-free map into a Veronese variety, factoring through a Grassmannian invariant; on the Dziobek stratum this recovers the Dziobek–Veronese geometry previously introduced by the author. We further describe universal determinantal relations satisfied by the minors of $S$ and expand explicitly the resulting systems for the planar five- and six-body problems. We also interpret the mass-weighted entries of $S$ as an equilibrium stress: under the MacMillan–Bartky sign condition a strictly convex central configuration underlies a cable–strut tensegrity, and a theorem of Connelly then forces $S$ to be negative semidefinite with nullity three, so that vertical degeneracy cannot occur in this regime.

Many dynamical-state reconstruction problems seek to infer a high-dimensional spatial state from measurements at only a few locations. Because the governing dynamics couple evolution across space and time, temporal measurements can contain information about state components beyond the sensor locations. This work develops a theoretical framework comparing the information content of a temporal measurement history with that of a specified instantaneous spatial sensor array. The temporal history and spatial reference array are represented by prior-whitened information operators that account for prior variability, sensor geometry, and measurement noise while preserving the directional distribution of uncertainty reduction. A general space-time information-exchange theorem gives necessary and sufficient conditions for retaining a prescribed fraction of the spatial-reference information in every state direction. It answers three practical questions. First, how many temporal measurements and how much total information are required? Second, is the prescribed fraction attainable, or does a limiting information ceiling rule out every finite history? Third, when attainable, what history length is sufficient? Explicit history-length bounds are derived for contractive, spectrally separated unitary, and simultaneously diagonalizable dissipative linear dynamics, and for stochastic linear estimation under Gaussian process and measurement noise. For nonlinear systems, global delay-map injectivity is combined with a uniform local Fisher-information condition. Numerical experiments test directional exchange lengths, information ceilings, and finite-memory bounds, and assess nonlinear delay-map distinguishability and local Fisher-information predictions. The framework provides an explicit basis for determining when temporal measurements can reduce spatial sensor coverage for full-state reconstruction.

A mathematical model is analyzed for a population of fish exhibiting cannibalism during its larval stage. The population is divided into three classes (vulnerable, cannibals, and non-vulnerable). It is assumed that initially, the entire population is vulnerable, and as they develop, individuals transition into one of the other classes. In addition, the interaction between the non-vulnerable and the cannibals is a predator-prey relationship with a Beddington-DeAngelis functional response. The global stability of the equilibrium point is demonstrated and numerical simulations show how food quality influences the cannibalistic behaviour of the population and its survival. Furthermore, it is shown that the solutions fit the data obtained experimentally for Atractosteus tropicus in its larval state. This model can be adapted for other fish species that exhibit cannibalism.

This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\omega)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,\omega)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,\omega)$.

We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schrödinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schrödinger equation fails when one allows small perturbations of the paraboloid, which can be viewed as a higher-dimensional extension of the phenomenon proved by Fu, Ren, and Wang. Our approach uses new lower bounds for incidence estimates originally proven by Cairo and Zhang, for which we provide an alternative proof based on homogeneous dynamics. Moreover the estimates are essentially sharp at the decoupling endpoint for the paraboloid $p = \frac{2(d+2)}{d}$.

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that, for lattices in connected semisimple real Lie groups with finite center and no compact factors, the scalar-expectation case of the corresponding classification is equivalent to ordinary ITAP.

Solar Radiation Pressure (SRP), the force exerted by photons emitted by the Sun, is one of the main non-gravitational perturbations affecting spacecraft trajectories, making its accurate modeling essential for high-fidelity orbit propagation. While physically-based ray-tracing models improve SRP accuracy, their computational cost becomes a limitation during numerical integration, where the SRP force must be evaluated repeatedly throughout the trajectory propagation. This paper investigates the integration of high-fidelity SRP models into orbit propagation through two complementary contributions: a Vulkan-based GPU implementation that accelerates direct SRP evaluation, and an extension of the SRP model to account for the dynamic orientation of solar panels. These contributions are evaluated independently through orbit propagation experiments, while a precomputed SRP interpolation strategy (SPAD) is included as an alternative approach for reducing computational cost through offline sampling and interpolation. Numerical validation shows that the Vulkan implementation preserves the accuracy of the original OpenGL-based method, showing relative differences below $5\times10^{-4}$ while achieving speed-ups of up to 9.4 for individual SRP computations and up to 15.2 for complete orbit propagation, particularly for geometrically complex spacecraft. The movable solar panel model shows that neglecting panel motion can produce significant long-term propagation errors, especially for spacecraft with large articulated solar panels, while introducing only a moderate computational overhead. Based on the experimental evaluation, this work concludes with practical guidelines for integrating high-fidelity SRP models into orbit propagation frameworks, identifying the scenarios under which interpolation-based methods are sufficient and when online high-fidelity SRP computation is justified.

In this work, for a network of differential equations with a prescribed graph structure, our goal is to show how to select the Laplacian of the network in order to obtain the most favorable outcome insofar as synchronizability. That is, we will want to: (i) guarantee asymptotic stability of a synchronous solution (as measured by a negative value of the master stability function), (ii) minimize the normalized spread of the Laplacian eigenvalues, and (iii) have a transient as short as possible. Within the class of tridiagonal Laplacians, we give both necessary and sufficient conditions for satisfying our three criteria above, and give extension to banded Laplacians as well. Finally, we give extensive numerical results to elucidate our theoretical results and to compare to existing works.

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$ \frac1N\sum_{n=1}^N T_1^n f_1\,T_2^n f_2\,S^{\Omega(n)}g $$ associated with three commuting transformations by isotropy factors and nilpotent structures in $\mathbb{Z}^2$-actions.

Let $(X,T)$ be a topological dynamical system and let $h(T,K)$ denote the topological entropy of a compact set $K\subset X$. We settle a question and a conjecture raised by Huang, Ye, and Zhang (2014) concerning hereditary lowerability. First, we show that every system with finite topological entropy is hereditarily lowerable: for every nonempty compact set $K\subset X$ and every $0\leq h\leq h(T,K)$, there is a compact set $K_h\subset K$ such that $h(T,K_h)=h$. This gives a negative answer to their Question 2'. Second, we prove that if $(X,T)$ admits an ergodic invariant measure with infinite entropy, then $(X,T)$ is not hereditarily lowerable. More precisely, we construct a compact set $K$ with infinite entropy such that every compact subset of $K$ has entropy either zero or infinity. This proves the conjecture stated immediately after Question 2'.

For Anosov diffeomorphisms the geometric approach for constructing Sinai-Ruelle-Bowen (SRB) measures works by pushing forward under the dynamics the leaf volume on a local unstable manifold or any measure that is equivalent to the leaf volume. Such a measure is called a reference measure. In applications one often needs to use a reference measure that is absolutely continuous with respect to leaf volume. In this paper we show that a reference measure can be chosen from a substantially more general class of measures which are the restriction of the leaf volume to any measurable subset of a local unstable manifold of positive leaf volume. Moreover, we do this in a more general settings of diffeomorphisms which are effectively hyperbolic on a set of positive volume. Such diffeomorphisms were introduced by Climenhaga, Dolgopyat and Pesin in \cite{CDP} where they used the geometric approach to construct SRB measures. Our results applies also to diffeomorphisms that are uniformly partially hyperbolic or more generally possess a dominated splitting.

Building on the vector-Zak and cohomological framework developed by Faulhuber, Petersen, van Velthoven, and Voigtlaender in their twelve-point counterexample, we give a computer-assisted four-point counterexample with a nonzero complex-valued Schwartz window. Every symplectic triangle determinant of the explicit configuration has absolute value below one, placing it in the intrinsically subcritical regime.

This paper considers planar mathematical billiards with smooth billiard tables. We prove that through any point outside a convex smooth caustic, there passes a unique billiard table (Lemma 2). Furthermore, we describe a billiard table having a nephroid as its caustic.

In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proof of this result. We also point out other uses of the cluster graphing construction in the literature, for instance in showing nonuniqueness at $p_u$ in arXiv:1509.00247 [math.GR]. The purpose of this short note is to make folklore proofs known to experts more accessible to the wider community of probabilists and measured group theorists.

Recent advances in combinatorial dynamical systems that generalize the classic discrete Morse theory have prompted algorithmic studies of combinatorial vector fields. In this regard, authors in [7] recently proposed the concept of Conley-Morse persistence barcode that summarizes the continuation of invariant sets in an evolving vector field through homological persistence. They proposed an algorithm to compute this barcode using a filtration of the so called index pairs on a poset called transition diagram. The algorithm becomes costly due to multiple runs of zigzag persistence it executes on filtrations of `unwieldy' structures of index pairs. We overcome this difficulty by replacing the index pairs with blocks, which are structurally much simpler. These replacements need reversal of certain relations in the transition diagram resulting in a much simpler algorithm. The algorithm works by updating matrix decompositions akin to computing `vineyard' in standard persistence.

Two nonradial scattering criteria are established for the three-dimensional focusing energy-critical generalized Hartree equation below the ground-state threshold, under a bounded-scale concentration–compactness reduction. The first criterion is formulated in terms of fixed-center occupation windows and quantitative bounds on the motion of the concentration center. The second is based on uniform low-frequency \(L^2\)-decay, which yields finite mass, precompactness in the inhomogeneous energy space, a zero-momentum normalization, and sublinear center drift. The proofs combine a localized Hartree virial identity with a critical Hardy–Littlewood–Sobolev estimate controlling the nonlocal tail.

A framework is developed for the inference of dynamics described by a generalized system of ordinary differential equations. A stochastic gradient method is coined that infers dynamics from observed marginal probability density functions using the joint probability density function of the observable and latent variables. Diffusion and other irreversible processes observed in a low-dimensional state can be recast as deterministic, reversible flows in a sufficiently augmented state space, where the joint density satisfies the hyperbolic Liouville equation. The marginal distribution observed is the projection of these hyperbolic dynamics onto the observed coordinates, with the latent components carrying the randomness and memory. This reframing allows inference for irreversible or stochastic dynamics into the recovery of a deterministic Ordinary Differential Equation (ODE) from marginal observations. Instead of solving the high-dimensional Liouville equation for the joint density, the algorithm exploits its characteristic representation. Particles sampled from the initial distribution are transported along characteristic lines. The Eulerian sensitivity with respect to parameters is obtained by sensitivity propagation along the characteristic lines, with a crossed U-statistic producing an unbiased gradient estimator, which enables stochastic gradient descent. Four experiments validate the method: recovery of a three-mode linear system observed through the marginal of a single mode; a nonlinear Gompertz growth model with a hidden mode; a bistable system whose hidden mode turns a unimodal marginal bimodal; and Stokes–Oseen drag law recovery for particles in a cellular flow. Convergence behavior is analyzed across these settings.

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon–Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert–Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

A compact metrizable system is null if its topological sequence entropy vanishes along every sequence of times. We prove that nullness is equivalent to polynomial maximal pattern complexity for every finite open cover, while equicontinuity is equivalent to sublinear maximal pattern complexity. The first characterization is obtained from finite fat-shattering at every positive scale and polynomial empirical covering of orbit-distance classes. We also construct transitive nonminimal null systems with properties excluded in the minimal setting: one is uniformly rigid and has two fixed points, and another is two-scattering. These results settle several long-standing open problems from the literature on polynomial maximal pattern growth and on the structure of transitive nonminimal null systems.

We study the existence and multiplicity of principal eigenvalues and eigenfunctions for a periodically heterogeneous nonlocal dispersal model with advection. The operator we consider is resolvent-positive but not resolvent-compact; therefore, the classical Krein-Rutman theory cannot be applied directly. When the advection coefficient has a constant sign, we prove the existence and uniqueness of the principal eigenvalue and the corresponding normalized eigenfunction. In sharp contrast, when the advection does not have a constant sign, the problem is more involved and leads to surprising results. Depending on the coefficients of the equation, the principal eigenproblem can either have a unique normalized solution or a continuum of solutions, at the boundary of which there exists a principal eigenvector with a singular measure component. In the latter situation, all the constructed eigenvalues are embedded in the continuous spectrum of our operator. We completely characterize the eigenvalues associated with positive eigenvectors, even when the eigenvector is a Radon measure. Finally, we discuss an application to a nonlinear KPP-type equation with nonlocal dispersal, which possesses a continuum of nontrivial stationary solutions, a different behavior from the classical KPP equation with local diffusion.

We analyze the stability of the equilibria and bifurcations of a planar system of ordinary differential equations describing the product-resource interaction in a sustainable supply chain. While periodic behavior in supply chain models has often been documented either in systems of higher dimensions or in delayed systems, here a Hopf bifurcation arises in a planar and delay-free system. We show that the interior equilibrium loses stability through a supercritical Hopf bifurcation as the environmental capacity exceeds a critical threshold that depends on the maximum production rate, the resource level at which production reaches half its maximum rate, and the demand and remanufacturing rates. We derive this threshold explicitly and provide estimates for the amplitude and period of the periodic solution close to the Hopf bifurcation, by means of the resulting Hopf normal form. We then further substantiate our analytical results through numerical simulations.