Papers

Let $\Gamma<\mathrm{SU}(n,1)$, $n\ge2$, be Zariski-dense and contain a regular loxodromic element $A$. The splitting-field problem asks how far traces and spectra determine a field of definition for $\Gamma$. We show first that traces together with the spectrum of $A$ always give a linear realization over $K=\mathbb{Q}(\textrm{tr}\Gamma,E(A))$. We also prove that every complex-irreducible subgroup of $\mathrm{SU}(n,1)$ with real trace field is conjugate into $\mathrm{SO}(n,1)$; in particular, no discreteness hypothesis is required for this real-trace conclusion. In higher dimension, however, linear descent does not by itself give the standard unitary form: a Hermitian similarity obstruction remains, described over imaginary quadratic fields by Landherr's local invariants. In $\mathrm{SU}(3,1)$ we give a Zariski-dense example for which the one-spectrum field is $\mathbb{Q}(i)$ but does not yield the standard unitary form. This leads naturally to the total spectral field. We prove that the obstruction disappears for arithmetic groups and under suitable local openness hypotheses, and we give an even-degree local criterion. In dimension four we analyze the discrete branch through explicit Schottky groups and rank-one local geometry. There the full Bruhat cocycle recovers anisotropic Levi holonomy and forces a non-split rotational word. The remaining quasi-split rank-two case is not reached by this method, so the general $\mathrm{SU}(3,1)$ problem remains open.

The ribosome flow model (RFM) is a nonlinear compartmental model for unidirectional transport along an mRNA chain. We study a generalized RFM in which the classical linear free-space factor is replaced by a monotone nonlinear steric-hindrance response. After recalling the global equilibrium and periodic entrainment properties needed for the analysis, we derive an exact weighted edge-dissipation identity for the gain of entrainment (GOE). This yields a global no-GOE criterion formulated through the curvature of the nonlinear hindrance law. In particular, the quadratic hindrance model admits no positive GOE in any dimension and for arbitrary periodic forcing amplitudes, with equality only when all rates share a common multiplicative modulation. In one dimension, we obtain a sharp convexity-based classification and construct positive-GOE examples when the relevant convexity fails. In higher dimensions, the second variation is governed by a symmetric Jacobi matrix whose diagonal terms encode local curvature and whose off-diagonal entries represent nearest-neighbor transport coupling. This leads to an exact Riccati criterion, a dimension-uniform condition for heterogeneous chains, and a closed-form critical dimension in the homogeneous case. We also exhibit a nonlinear hindrance for which low-dimensional chains have no positive second-order GOE, while a higher-dimensional chain admits a realizable positive GOE. These results reveal a dimension-induced gain mechanism that is absent from the classical RFM.

A generalization of the exact overlaps conjecture predicts a formula for the dimension of self-similar measures in terms of the random walk entropy and the Lyapunov exponent. We prove the exact overlaps conjecture and its generalized version in dimension one. The proof relies on a new entropy inequality quantifying the information lost under addition of independent random variables in terms of a quantity averaging variance across scales.

In this paper, we strengthen the connection between the combinatorics of difference sets and the dynamics of group rotations. Our main result shows that sets which have non-empty intersection with all difference subsets of a commutative semigroup possess local Bohr structure. This generalizes results of Bergelson, Furstenberg, and Weiss and Host and Kra from the integers to arbitrary commutative semigroups. We accomplish this by A) utilizing a recent result showing that the regionally proximal relation is an equivalence relation for minimal actions of commutative semigroups and by B) describing a new, DeMorgan-type algebra on Furstenberg families that allows for efficient manipulation and computation. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.

The regionally proximal and equicontinuous structure relations are fundamental relations in topological dynamics that capture the equicontinuous behavior in a topological dynamical system and its factors. For minimal actions of abelian groups, these relations are known to be equivalence relations and are known to coincide. In this paper, we generalize these facts to semigroup actions: for minimal actions of commutative semigroups, the regionally proximal and equicontinuous structure relations are equivalence relations and the two coincide. We also develop the machinery of natural extensions for commutative semigroup actions that act by surjections, concluding that the maximal equicontinuous factor of a minimal action of a commutative semigroup is the same as the maximal equicontinuous factor of the group action into which it embeds. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.

Feature caching accelerates diffusion sampling by replacing expensive network evaluations with predictions from previously computed activations. However, forecasts based only on past features cannot directly incorporate changes in the current denoising state. We investigate whether inexpensive, freshly computed features can serve as observations for correcting these predictions. We introduce an observation-corrected Koopman framework for accelerating frozen diffusion models. Using calibration trajectories, we identify finite-dimensional, time-dependent Koopman approximations that jointly describe the increments of shallow and deep network features. During accelerated sampling, these operators predict the evolution of expensive deep features, while innovations in the observed shallow features correct the predicted state. Periodic full evaluations refresh the observer, and all generative-model parameters remain unchanged. This formulation enables controlled comparisons of temporal prediction and observation correction. Across three 10,000-image runs per dataset, our method reduces paired Inception-feature MSE by $19.9\%$ on CIFAR-10 and $11.9\%$ on a ten-class ImageNet subset relative to channelwise affine prediction under the same four-partial-step schedule. Matched ablations attribute additional reductions of $4.54\%$ and $4.67\%$ to observation correction. The observer achieves $1.89\times$ and $1.85\times$ measured speedups over DDIM-50, supporting improved reference-sampler fidelity without retraining the denoiser.

Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.

Many mechanical systems, such as the Newtonian $n$-body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery's variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with $q$-fold rotation symmetry, and to counter-rotating double choreographic loops in planar $(N+3)$-body problems.

We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota–Müllner–Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.

We study adaptive stabilization of known control-affine systems by treating the current parameterized critic itself as a Lyapunov candidate. Its total derivative contains a parameter-velocity term that can compensate unfavorable state evolution. A projection onto the intersection of a bounded admissible velocity set and one decay halfspace enforces parameter invariance and Lyapunov decrease simultaneously. We give an exact pointwise feasibility test, a conditional state-stability theorem, and a counterexample showing why a positive critic family containing a stabilizing member does not imply persistent feasibility. A convex endpoint constraint provides a distinct sampled implementation. A generic periodic polynomial-factor critic with a local Riccati anchor is evaluated on a force-limited cart-pole without mechanical-energy features, replay training, waypoints, or controller switching. The frozen controller passes stabilization and numerical decay checks on 600 sampled initial states across two draw sets; 400 matched continuous trajectories also pass. Independent reintegration audits and a conservative local feasibility certificate are distinguished from a global or machine-arithmetic guarantee. Compared with nonlinear model-predictive control, the measured implementation uses less CPU but has higher cost, settling time, and cart travel. The results support a conditional adaptive stabilization mechanism.

We construct area-preserving irrational pseudo-rotations of the closed disk that fail to be $C^0$-rigid. For every Brjuno rotation number we obtain $C^1$ examples, and they may be chosen arbitrarily $C^1$-close to the rigid rotation with the same rotation number. For $0<\beta<1$, a weighted Brjuno condition gives $C^{1,\beta}$ examples that are arbitrarily close in the $C^{1,\beta}$ topology; this weighted range contains every rotation number with denominator-growth exponent strictly smaller than $1/\beta$ and also some numbers at the critical exponent. For a suitable Brjuno–Liouville rotation number, we also obtain an example whose derivative growth is not bounded by any polynomial.

Bacterial co-infection, particularly with Streptococcus pneumoniae, is a major driver of morbidity during influenza epidemics, and this burden falls disproportionately on sub-Saharan Africa, where both pathogens circulate year-round at high levels and pneumococcal carriage remains common and persistent across the region. Yet existing mathematical models rarely track a host's bacterial carriage status jointly with their full influenza history through recovery. We develop a twelve-compartment ODE model that classifies each host by viral status (susceptible, infected, recovered) and bacterial status (susceptible, carrier, actively infected, recovered), allowing carrier-to-active progression to depend on prior influenza exposure. We establish positivity, boundedness, and a positively invariant region, then derive the basic reproduction number $\mathcal{R}_0$ and quasi-endemic equilibria of a single disease, together with invasion reproduction numbers $\mathbf{Inv}^{\mathbf{B}}$ and $\mathbf{Inv}^{\mathbf{V}}$ quantifying each pathogen's ability to invade a population where the other is already endemic. Center-manifold analysis shows that the bacterial subsystem and the full coexistence equilibrium undergo a backward bifurcation precisely when carrier transmission exceeds the carrier's total exit rate, $\beta_c>\kappa_S+\theta+\mu$, so that reducing $\mathcal{R}_0$ below one is not sufficient for elimination, while the viral subsystem bifurcates forward. A global sensitivity analysis (Latin Hypercube Sampling with partial rank correlation coefficients) shows that the viral and bacterial thresholds are governed by largely separate parameters, while the bacterial invasion potential is strongly shaped by viral transmission and recovery, giving a quantitative signature of cross-pathogen synergy.

Universal maps (maps whose renormalized iterations approximate every map in a given class) are locally generic in several spaces of diffeomorphisms [Bonatti–Díaz 2003, Turaev 2015]. In fluid dynamics, steady Euler flows whose Poincaré maps are universal are also known to be locally dense [Berger–Florio–PeraltaSalas, 2023]. Motivated by Arnold's vision [Arnold, 1966] that the complexity of orbits in celestial mechanics and that in fluids should be comparable, in the present paper we address the question of the existence of universal maps in celestial mechanics. We give a partial answer by proving a weak, finite-dimensional form of universal dynamics. More concretely, we show that every symplectic embedding of the disk into $\mathbb{R}^2$ can be approximated, with arbitrary precision, by a renormalization of a Poincaré map of a restricted planar circular $(n+1)$-body problem, for suitable $n$ and appropriate choice of the masses of the primaries. The proof relies on Gonchenko–Shilnikov–Turaev theory [Turaev 2003, Gonchenko–Turaev–Shilnikov 2007] and requires control of the dynamics near homoclinic tangencies of arbitrarily high order; the techniques developed in [Garrido–Martín–Paradela, 2025] are essential for such control.

We study entropy realization for continuous actions of infinite countable amenable groups. We introduce an approximate product property for amenable group actions that permits a small proportion of tracing mistakes on prescribed pairwise disjoint, sufficiently invariant finite sets. We prove that this property implies entropy-denseness and almost entropy-approximability of every invariant measure. The construction combines zero-entropy exact tilings with a finite-block estimate for the complexity of tracing mistakes. Under asymptotic entropy expansiveness, almost entropy-approximability upgrades to entropy-approximability. For every $0\leq\alpha<h(X,G)$, ergodic measures of entropy $\alpha$ then form a residual subset of the invariant measures whose entropy is at least $\alpha$. In particular, the set of ergodic measure entropies equals $[0,h(X,G)]$.37A35, 37B40, 37B05

We answer two questions about the heavy-ball method on smooth strongly convex functions. The first concerns Polyak's tuning. Its dimension-uniform worst-case exponential rate equals the largest growth factor of an interpolable mixture of rotating geometric sequences, and we determine this factor for every condition number. The optimal mixtures form four algebraic families; the classification combines exact polynomial certificates, a rational parametrization, an exact solution in a quintic number field at the critical three-cycle, and verified interval continuation. Two frequencies suffice, one function in dimension five attains the rate at every horizon, the upper and lower bounds differ by constant factors, and the method converges on the whole class if and only if $\kappa<9+4\sqrt5$, the threshold located by Badithela and Seiler. The second question concerns arbitrary step sizes and momenta, in particular the regions of the parameter plane between the computed Lyapunov and cycle regions. We describe the phase diagram through the angle $\Theta(\omega)$ between the characteristic polynomials of the two extreme quadratics. The method converges when $\Theta(\omega)>\omega/2$, has a $k$-periodic orbit when $\Theta(2\pi j/k)\le\pi/k$, and the remaining regions are rotation-number gaps organized by Farey fractions. A single strict certificate at one growth exponent bounds the rate, so convergence can be certified at the threshold exponent. In the first level of the gaps, fans of Farey fractions accumulating at a resonant endpoint carry periodic orbits with several harmonics, explicit fan-lag inequalities certify convergence, and both behaviors occur in the regions left open before.

For every quartic Salem number $\beta\in(1,2)$, we prove that the Bernoulli convolution $\nu_{\beta^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \[ 0.99999<D_2(\nu_{\beta_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(\nu_{\beta_2^{-1}})<1-10^{-75}, \] where $\beta_1<\beta_2$ are the two quartic Salem numbers in $(1,2)$.

We study iterated function systems with large overlaps via the Kajiwara-Watatani C*-algebra and its KMS states. Our main contributions are threefold. First, we introduce Markov partitions for iterated function systems and give an algorithm that constructs a finite Markov partition whenever one exists. Second, for contractive systems admitting a finite Markov partition, we construct an intrinsic analogue of the Hutchinson measure that is independent of any labelling or weighting of the maps. It is the unique Borel probability measure satisfying a self-similar integral equation dictated by the set-valued dynamics of the system, and it recovers the Hutchinson measure with uniform weights when the system satisfies the open set condition. Finally, we give a complete classification of KMS states for the gauge action on the Kajiwara-Watatani algebra of every such system. The critical inverse temperature is the exponential growth rate of the intrinsic orbits, which can be strictly smaller than the natural logarithm of the number of maps, and the unique KMS state at this temperature is given by our intrinsic Hutchinson measure. These systems include many with large overlaps not covered by the results of Izumi-Kajiwara-Watatani.

The full shadow of a complex polynomial $P$ of degree $d\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\ge1$, $0\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\to\infty$ and the support of their limiting normalized root-counting measure. For $m/n\to\alpha\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.

The iteration of $z\mapsto z^2+c$ over the octonions has been drawn for more than thirty years, and it is known that the octonionic Julia sets are bodies of revolution about the imaginary part of the constant. Real eight-dimensional composition algebras, however, are much richer than the octonions: beside the Hurwitz algebras there are the symmetric composition algebras, the para-octonions and the Okubo algebras, which have no unit and whose automorphism groups are $G_2$ and $PSU(3)$ respectively. In this work we study, to our knowledge for the first time, the quadratic dynamics over these algebras, and we do it in an experimental spirit: an atlas of 134 four-dimensional renderings, measured quarter by quarter, suggests the statements, which we then prove and test against new, predicted pictures. We show that the symmetries of the Julia sets of the Okubo algebra are governed by the root decomposition of $\mathfrak{su}(3)$ with respect to the plane of $1$ and $i$: for a constant in that plane the continuous symmetry is a maximal torus, which acts by rotations on three root planes and grows to $U(2)$ on the Weyl walls; the automorphism group of the symmetrized product is $PSU(3)\rtimes\mathbb{Z}_2$. We then prove that the Mandelbrot set with seed $0$ of every symmetric composition algebra is the pull-back of a single planar set through the invariants $n(c)$ and $n(c,c\diamond c)$: the tricorn in the compact case, the tricorn and a hyperbolic tricorn in the split case. The result is quite surprising since the para-octonionic and the Okubonic Mandelbrot sets turn out to be the same planar figure saturated by two different groups. Finally, we show that every bilateral map over a symmetric composition algebra is an octonionic quadratic map in disguise, and that at the Petersson angle $\theta=90^\circ$ the dynamics acquires symmetries that do not come from automorphisms.

We study a strongly damped stochastic wave equation on the three-dimensional torus driven by finite-rank pure-jump Lévy noise obtained from subordinated Brownian motion. Lyapunov estimates and an asymptotic compactness decomposition yield invariant measures, and a small-noise event gives weak irreducibility. To prove uniqueness, we combine conditional Malliavin calculus with terminal observability and estimates for the full variational flow starting from high modes. A control constructed on alternating stopping blocks yields the e-property under a fixed-rank balance condition involving high-mode decay, spectral leakage, and the random clock. Consequently, the invariant measure is unique and ergodic, and the Cesàro averages of the transition laws from every initial state converge weakly to it. We verify this condition for sufficiently small positive cubic coefficients at each fixed forcing operator of sufficiently large rank. Under a separate explicit smallness condition, synchronous coupling proves the e-property and weak convergence of the transition laws, without a lower bound on the forcing rank.

A classical theorem by Mañé states that compact metric spaces admitting expansive homeomorphisms must be finite-dimensional. It is a natural question, raised by Morales and Sirvent, whether this dimensional constraint extends to the measure-theoretic notion of expansiveness. In this paper, we answer this question in the negative. We prove that the shift map on the Hilbert cube is expansive with respect to the product Lebesgue measure. Furthermore, we establish a general sufficient condition: any homeomorphism admitting a full-support expansive measure is densely measure-expansive. As a consequence, we show that the set of expansive and topologically stable measures for the shift map contains a dense $G_\delta$ subset, revealing a rich measure-theoretic structure despite the infinite dimensionality of the phase space.

Quasi-nonsingular relative orbital elements (qnsROE) are among the most widely used state representations for formation flying and rendezvous in near-circular inclined orbits. They admit closed-form perturbed state transition matrices (STMs) and a direct geometric mapping, which underpins eccentricity/inclination (E/I)-vector separation for passive safety. Existing eccentric extensions of the qnsROE resolve this geometry in the radial–tangential–normal frame, where a residual semimajor-axis difference makes the E/I separation margin phase dependent. This paper introduces eccentric ROE that reduce to the qnsROE as \(e\to0\) and whose first-order geometry is resolved in the velocity–normal–binormal (VNB) frame, where the secular drift is confined to the velocity direction. Their first-order position and velocity maps yield a closed-form, phase-independent binormal–normal separation bound with an explicit margin for semimajor-axis residuals. Closed-form Keplerian and secular-\(J_2\) STMs and a decoupled impulsive control input matrix are derived for the same elements, and the elements are recovered as the physically normalized integration constants of the Cartesian equations of relative motion in the VNB frame. Together, these results provide a unified ROE framework for formation flying in general elliptic orbits.

Let $K=K(n,m,\Gamma)$ be a Bedford-McMullen carpet with $n>m\geq2$ and $1<\#\Gamma<nm$, and let $\mu$ be a self-affine measure on $K$ with strictly positive weights. We prove that if a similitude $f$ satisfies $\mu(K\cap f(K))>0$, then its orthogonal part is diagonal or anti-diagonal. This holds without any arithmetic restrictions on the defining bases. For isometries, we further prove that if $\mu(K\cap f(K))>0$ then $f(K)$ and $K$ differ only by a translation, thereby establishing a strengthened form of the 2010 conjecture of Elekes, Keleti and Máthé in this planar self-affine context.

Hamiltonian systems exhibit rich dynamical behaviors, among which escape is a typical phenomenon. Below the escape threshold, trajectories initiated inside the central potential well remain bounded. Once the energy exceeds this threshold, and the initial conditions lie within the escape basin, the system eventually evolves along unbounded orbits toward infinity. Predicting the escape dynamics is important for understanding the evolution of these systems. In this study, we employ the next-generation reservoir computing framework, a powerful machine learning architecture widely used in dynamical prediction, to address this challenging issue. To capture the intrinsic features of the system, we train the model using data from multiple energy values. Numerical experiments on two Hamiltonian systems show that, with only three time series as training data, each corresponding to an energy value below the escape threshold, the model is capable of not only predicting the escape threshold itself, but also reliably predicting the structure of the escape channels. As a primary contribution of this work, these findings demonstrate that escape dynamics can be inferred from subcritical data alone in the two systems studied, offering a promising paradigm for data-driven studies of Hamiltonian systems.

We construct a two-parameter family of polynomial vector fields in $\mathbb{R}^3$ and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on $\mathbb{R}^3$. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.

Let $P_h$ be a semiclassical pseudodifferential operator on a closed Riemannian manifold $(M,g)$, with principal symbol $p$. We consider perturbations $P_h+h^\alpha Q$, with $0<\alpha\le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^\beta$, with $2\beta <\alpha$. Assuming that the classical dynamics of the unperturbed system on a regular energy surface $p^{-1}(E)$ is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order $|\log h|$. This formula relates the distribution of energy levels of the quantum system near the energy $E$ to the periodic orbits of its classical counterpart. The perturbation $h^\alpha Q$ affects only the phases appearing in the trace formula.

We prove that every countable Borel equivalence relation containing a finite index Borel treeable subequivalence relation is Borel treeable. This answers a long-standing question of Jackson–Kechris–Louveau. Our proof is based on a median subalgebra filtration construction on products of trees, together with an extension of the median graph treeing algorithm of Chen–Poulin–Tao–Tserunyan to certain well-behaved increasing unions.

This work investigates the number of limit cycles bifurcating from the periodic orbits of a linear center when it is perturbed by piecewise holomorphic systems with three zones separated by two external circles. We analyze how different geometric configurations of these zones influence the number of limit cycles. Moreover, we show the existence of at least sixteen limit cycles in fourth-degree piecewise polynomial holomorphic systems under certain configurations.

This paper presents a model-free method for turbine repositioning control in floating offshore wind farms. The method uses a distributed online optimization framework to mitigate the wake effect. Conventional approaches rely on centralized, model-based control architectures, which suffer from poor practical feasibility. To overcome this limitation and address nonconvexity and inaccessible gradient information, we incorporate physics-informed knowledge of wake dynamics into the control system design. We demonstrate the effectiveness of the proposed method by comparing it with a centralized, model-based baseline in mid-fidelity simulations. Results indicate that the proposed method achieves slightly better performance while significantly reducing computational cost.

We establish a counterpart, for individual finite open covers, of the global entropy-mean-dimension comparison of Lindenstrauss and Weiss. More precisely, the mean dimension of a finite open cover is quantitatively bounded in terms of its topological entropy and cardinality. In particular, a finite open cover with zero entropy has zero mean dimension. The same comparison holds for actions of countable amenable groups and in the sofic setting. It follows that every mean dimension pair is an entropy pair and, in particular, that uniformly positive mean dimension (UPMD) implies uniformly positive entropy (UPE). The cover and pair implications answer Questions 2.27 and 4.9 of García-Ramos and Gutman, respectively. The proof associates a polytope to a minimal subcover and combines the Lindenstrauss-Weiss compression argument with Maurey's empirical method to estimate the volumes of its coordinate projections.