Papers

We analyze Reflected UAS routing for heterogeneous multi-server queues at fixed parameters under subcritical load. The deterministic surrogate is a reflected ODE on the nonnegative orthant, not the unconstrained drift equation. This reflected ODE has a unique boundary equilibrium characterized by a scalar consistency equation and a convex-potential representation; all trajectories converge to it. The older argument lifting deterministic Lyapunov descent to CTMC stability fails: the exact generator applied to the deterministic potential produces a boundary term absent from the reflected-ODE descent identity. We give a direct Foster-Lyapunov drift inequality for the CTMC using a weighted-quadratic function, bypassing the failed lift. At the benchmark parameter point, the boundary equilibrium matches the numerical attractor to machine precision, and the default Reflected UAS policy has lower mean queue length than UAS and JSSQ across independent seed blocks.

We prove Schur-positivity for a family of symmetric functions.

This paper is an extended version of two talks I gave during workshop “Loops'13" in Bedlewo in June 2023. In the first talk I gave a historical introduction to Knot Theory. In the second, I traced my journey toward Yang-Baxter homology and this talk has a partially survey and a partially novel character.

For a family $\mathcal F\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal F}(R)=|\{F\in\mathcal F:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal F)=\sum_{R\in\binom{[n]}r}d_{\mathcal F}(R)^p$; write $co_p(\mathcal F)=\ell_{k-1,p}(\mathcal F)$ for the codegree power sum. We introduce a method that uses convexity to extend sharp bounds for degree sums and sums of squared degrees to real powers, while retaining all equality cases. The method bounds $x^p$ by quadratic polynomials or by a continuous function that is linear on each of two intervals. These functions agree with $x^p$ at the degrees of the proposed extremal family, so the argument requires no bounds for sums of higher powers. For families with bounded matching number, we instead use a bound for $\sum_R\max\{d_{\mathcal F}(R)-s,0\}$ together with the degree sum. We give three exact applications. First, a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\geq2$ in the sharp range $n\geq(t+1)(k-t+1)$, with all equality cases determined. This extends the quadratic theorem of Wu and Zhang to real exponents and answers a problem of Zhou and Yuan throughout the sharp Erdős–Ko–Rado range. Second, among intersecting families with $n\geq2k$, a full star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with all equality cases determined. Third, if $\nu(\mathcal F)\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal F)$ is uniquely maximized, up to permutation, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or assumptions that $n$ is sufficiently large with an explicit linear range valid for arbitrary $k$.

One of the outstanding questions in the theory of elephant random walks as observed by Gut and Stadtmüller (2023), is to determine how much memory is needed for a phase transition between the diffusive, critical and superdiffusive regimes to persist. To investigate this memory breakpoint, we introduce the tampered memory elephant random walk, in which the memory is partitioned into two disjoint sets $D_n$ and $D_n^c$, which may be deterministic or random. On $D_n^c$ the dynamics is the same as an elephant random walk, while on $D_n$ the increments are replaced by independent innovations. The resulting walk is thus driven by two competing components: elephant random walk and an independent simple random walk corresponding to the innovations. We first establish a law of large numbers when the increasing collections $\{D_n\}_{n \ge 1}$ and $\{D^c_n\}_{n \ge 1}$ have a renewal structure with exponential moments. We then identify a sharp threshold that governs the persistence of the phase transition for deterministic memory partitions. We show that if $\{D_n\}_{n \ge 1}$ is non-random increasing collection with increasing complement $\{D^c_n\}_{n \ge 1}$ such that $\lim_{n \to \infty} \frac{\lvert D^c_n\rvert}{n} >1/2$, then a phase transition into diffusive, critical and superdiffusive regimes persists, whereas for $\lim_{n \to \infty} \frac{\lvert D^c_n \rvert}{n}<1/2$, there is only the diffusive regime with $\mathcal{O}(\sqrt{n})$. The case of $\lim_{n\to \infty}\frac{\lvert D^c_n \rvert}{n}=1/2$ is also characterised. Thus, one-half emerges as the sharp breakpoint for the persistence of anomalous diffusion in this competitive setting. We conjecture that the same threshold governs the case when $\{D_n\}_{n \ge 1}$ is random. Our proofs rely on stochastic approximation applied to the two dependent competing components of the walk, representing the retained memory and the innovations.

In 2016 J. Koenigsmann proved that $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\not\in\mathbb Z\iff \exists x_1,\ldots,x_{n}\in\mathbb Q\,[P(t,x_1,\ldots,x_{n})=0].$$ In this paper we show that we may take $n=7$ which improves the previous record $n=10$ obtained by Daans in 2024. (Actually we even extend this to any global field.) This, together with a previous result of Z.-W. Sun, implies that there is no algorithm to decide for any $F(x_1,\ldots,x_{16})\in\mathbb Z[x_1,\ldots,x_{16}]$ whether $$\forall x_1,\ldots,x_9\in\mathbb Q\exists y_1,\ldots,y_{7}\in\mathbb Q\,[F(x_1,\ldots,x_9,y_1,\ldots,y_{7})=0].$$

We consider fourth-order and second-order partial differential operators localized on domains of the sphere in arbitrary dimension. These operators arise as weighted compositions of the linearized Einstein constraint operators and their adjoints, and played a key role in our resolution of the optimal localization problem in general relativity, also referred to as the gravitational shielding problem. To control the asymptotic behavior of solutions to Einstein's constraints in our companion paper (preprint arXiv:2312.17706), we introduced the notions of harmonic, radial, and shell stability. Harmonic stability controls the borderline harmonic modes, radial stability governs the radial evolution of spherical averages, and shell stability controls the coupled radial-angular evolution of solutions. In the present paper, we establish that these stability properties follow from weighted Poincaré, Korn, and Hardy inequalities. Furthermore, in arbitrary dimension, we investigate the behavior of the associated geometric constants, and conclude that the stability conditions hold for a broad class of localization functions; the theory applies to arbitrarily small localization domains, corresponding to gluing cones with arbitrarily small aperture. At the opposite extreme, our conditions also hold on the entire sphere, corresponding to the absence of localization. This completes, for gluing cones of arbitrarily small aperture in every dimension, the program initiated by A. Carlotto and R. Schoen on gravitational shielding and the construction of solutions enjoying super-harmonic decay estimates. In our proofs, we introduce Hamiltonian and momentum functionals, which we call shell functionals, and show that they enjoy monotonicity and semi-coercivity properties; their structure also suggests possible analogies with functionals arising in other curvature-related geometric problems.

In this article, we present an explicit family of invertible $n\times n$ matrices over $\mathbb Z_2$ whose CNOT and row complexity is at least $4n-\text{o}(n)$; equivalently, reducing these matrices to the identity requires at least $4n-\text{o}(n)$ elementary row operations. Moreover, the same complexity lower bound holds in the stronger computational model where the CNOT gates are replaced by arbitrary local linear logic gates, namely arbitrary invertible linear transformations acting on pairs of coordinates. Let $G_n$ denote the permutation group generated by local logic gates acting on the set of binary strings of length $n$. We prove that $G_n$ is naturally isomorphic to the group of all invertible affine transformations of the vector space $\mathbb Z_2^n$, thus reducing the problem of estimating the quantum complexity of permutations in $G_n$ to the row reduction complexity of invertible matrices over $\mathbb Z_2$. As an application, we show that the permutations associated with our explicit matrices have quantum complexity at least $4n-\text{o}(n)$.

We introduce nonnuclear Bunce-Deddens algebras, defined as reduced crossed product C*-algebras associated with free odometer actions of nonamenable countable discrete residually finite groups on a Cantor space. These are nonnuclear counterparts to the generalised Bunce-Deddens algebras introduced by Orfanos. For large classes of groups, we show that they share many regularity properties, including real rank zero, stable rank one, strict comparison of positive elements, and admitting a unique tracial state. In many cases, we deduce that they are even selfless and hence pure. Finally, we compute the K-theory of Bunce-Deddens algebras over nonabelian free groups.

The Kemeny rule aggregates rankings by minimizing their total Kendall-tau distance from an aggregate order. We prove that Kemeny Score is NP-complete for exactly three unweighted rankings, even when every candidate pair is split $2$-to-$1$. On the same profiles, the winner, unique-winner, and possible- and necessary-precedence problems are $\Theta_2^p$-complete, while recognizing a Kemeny-optimal or uniquely Kemeny-optimal aggregate is coNP-complete. The hard instances induce tournaments of majority dimension exactly $3$. The reduction also determines the exact maximum-cut value from the optimal Kemeny score and recovers a maximum cut from any Kemeny-optimal aggregate. For every fixed $q\geq3$ and $\lceil q/2\rceil\leq s\leq q$, minimum pairwise support $s$ yields a sharp dichotomy: the score problem is NP-complete, the winner and precedence problems are $\Theta_2^p$-complete, and the recognition problems are coNP-complete when $3s\leq2q$; for $3s>2q$, the majority tournament is transitive and its unique topological order is the unique Kemeny-optimal aggregate. Exact support $s$ suffices in the hard case when $s>q/2$, and supports in ${s,s+1}$ suffice when $s=q/2$. These results give complete fixed-profile-size classifications and transfer to Slater orders, permutation medians, and maximum-likelihood central rankings in the Mallows model. Finally, a six-copy construction proves NP-completeness of both Kemeny Score and Kendall–Tau Center for three pairwise-equidistant rankings that still split every pair $2$-to-$1$. For $N$ output candidates, their common distance is $\frac23\binom N2$, the largest possible for an equidistant triple. The construction gives affine formulas for both optimal values, characterizes all Kemeny-optimal output orders, and shows that the output has a unique Kemeny-optimal order and a unique center exactly when the input has a unique Kemeny-optimal order.

We establish the minimal model program for klt adjoint foliated surfaces and use it to study the boundedness of polarized adjoint foliated surfaces. We prove that $\epsilon$-lc adjoint foliated surfaces with nef adjoint canonical divisor and a nef and big integral polarization form a bounded family, provided that the volume of their sum is bounded from above. As applications, we establish boundedness and effective birationality for adjoint foliated surfaces of general type, as well as a uniform positive lower bound for their volumes. Finally, for $\epsilon$-lc Calabi–Yau adjoint foliated surfaces, we prove that the volumes of the canonical divisors of rank one foliations on the underlying surfaces belong to a fixed discrete set.

This paper continues the development of (elementary) 2-topos theory, a foundational theory based on an axiomatization of the 2-category of categories. We prove that any 2-topos contains a model of intuitionistic ZF set theory, and we use this to show that, if appropriate size restraints are not imposed, then a Burali-Forti type paradox can be deduced from the 2-topos axioms. The set-theoretic universe is produced as a special case of a general construction giving an internal category of models in a 2-topos of an arbitrary finite higher-order theory, which in turn is carried out using a notion of "topos sketch". As a byproduct of our construction, we also make contact with the subject of "algebraic set theory", introducing a novel approach to the construction of set-theoretic universes from a "category of classes".

In various disparate settings one studies how random processes give way to parabolic partial differential equations and in turn to functional inequalities involving gradients and variational calculus. Particles subject to gradient forces, the fields which move them, the empirical measures which they form, the smooth laws which those measures approximate, and differentiable functionals of these laws constitute a hierarchy of descriptions of diffusion. Intervening on this hierarchy is a choice of how a conintuity operator converts particle velocity to the evolution of measures. Here a central object is constructed mediating two different such descriptions. The space of smooth positive probability densities on a closed Riemannian manifold is treated as a Fréchet manifold whose full continuous cotangent space consists of nonconstant distributions and whose regular cotangent space consists of sufficiently regular nonconstant functions. Beginning from the operator on this space arising as the infinitesimal lift of diffeomorphisms of the base manifold, two different Hilbert completions of this space of densities account for a wide class of objects relevant, leading to a partial universalisation of the microscopic–mesoscopic–macroscopic hierarchy in Markov analysis.

We study linear disturbances localized near planar surfaces moving at constant velocity $v$ in relativistic media. Depending on the physical setting, the surface may represent a moving obstacle, a thermal boundary, or an external source, providing a unified description of boundary layers, wakes, and the asymptotic tails of shock waves. The central result is a propagator representation of the interface solution that yields a geometric characterization of these phenomena. Using a Laplace-transform formulation, we show that the solution is a superposition of modes with purely imaginary frequency and wavenumber. For a given interface velocity, the admissible modes are selected by the line $i\omega=vik$ in the $\{i\omega,ik\}$ plane. As $v$ varies, this line sweeps across the spectrum, providing a unified geometric description of interface-localized solutions for arbitrary interface velocities. We illustrate the formalism with applications to relativistic hydrodynamics and kinetic theory.

In this note, using a result of Bernardi, Holden and Sun, we give an explicit geometric relation between the perimeter of the peeling process along the percolation interface of a triangulation, and the corresponding Kreweras walk. The relation naturally extends to the branching peeling exploration. This sheds light on the relation between the growth-fragmentation process and correlated Brownian excursions discovered by Da Silva, Powell and Watson.

Let $q=p^f$, and let $A\leq\mathbb{F}_q^\times$ be a multiplicative subgroup with $\mathbb{F}_p(A)=\mathbb{F}_q$. We prove that a proper subgroup $A$ is a generalized arithmetic progression (GAP) if and only if $|A| \in \{1, 2, 4\}$, and we determine when the full group $\mathbb{F}_q^\times$ is a GAP. For certain families of subgroups, we obtain the stronger conclusion that $A$ is additively irreducible. In particular, if $|A|>4$ and $p^e\equiv-1\pmod{|A|}$ for some $e\ge1$, then $A$ admits no nontrivial sumset decomposition. We also prove that every $c \neq 0$ has fewer than $|A|/2$ representations as a sum (or difference) of two elements of $A$ whenever $[\mathbb{F}_q^\times:A] \ge3$ and $|A| \ge 5$, which may be of independent interest.

We give negative solutions to the two basic completeness problems for locally convex Lie groups: whether a Lie group modelled on a Mackey-complete locally convex space must be regular, and whether it must at least possess a smooth exponential function. Both counterexamples have complete model spaces. We first construct a contractible complex analytic BCH–Lie group $H$, modelled on the complete Silva space $\mathbb C^{(\mathbb N)}$, for which $\exp_H$ is a global homeomorphism although $H$ is not $C^0$-semiregular. The failure is witnessed by smooth controls converging to zero in a fixed finite-dimensional subspace; their evolutions exist uniquely on $[0,1)$ but have no endpoint at time $1$. The obstruction is a multiplicative graded escape in the principal unit group of a complete continuous inverse algebra. We then suspend one such control: the translation action on $C^\infty(S^1,H)$ converts the time-dependent obstruction into a single element of a semidirect-product Lie algebra. The resulting Lie group is modelled on a complete Hausdorff locally convex space and contains an element which generates no one-parameter subgroup. Hence it admits no exponential function. Thus completeness forces neither non-autonomous evolution nor autonomous exponentiation.

We give a self-contained development of the first-order block calculus in unsigned sequent-style notation: each node of the refutation tree carries a finite block $\Pi = \Gamma \cup \neg[\Delta]$, negation is governed by explicit rules, and a branch closes on a complementary pair of literals. The calculus is Smullyan's, and so in substance are the theorems; what is offered here is a different arrangement of them. The structural properties are established in the order of dependence familiar from G3-style sequent calculi: closure on arbitrary formulae is admissible, weakening and the substitution of parameters are admissible with preservation of the height, every rule is height-preserving invertible, and cut is admissible, the last being derived from the first three rather than conversely. Soundness, completeness under a fair strategy, countable compactness and the countable model property follow, together with a syntactic criterion under which every fair construction terminates. The correspondence is then proved, in both directions and with cut included, with Gentzen's LK in its usual presentation with explicit weakening, which requires lemmas on parameters that set-based Gentzen systems do not need.

We introduce a construction of algebraic coherators for Grothendieck $\infty$-groupoids using the algebraic small object argument, replacing previous approaches we have used based on distributive series of monads with a more direct method for freely adjoining coherence data. Given a controlled theory, we define unreduced and reduced Grothendieck realizations, producing $\infty$-Lawvere theories and extending this construction functorially to connected diagrams of controlled theories. We apply this framework to construct globular models for monoidal $\infty$-groupoids, symmetric monoidal $\infty$-groupoids, coherent $\infty$-groups, and Picard $\infty$-groupoids. We define canonical semi-model structures on categories of models over $\infty$-Lawvere theories and formulate a generalized pushout conjecture that implies the existence of these semi-model structures and the Homotopy Hypothesis for Grothendieck $\infty$-groupoids.

We prove that every ironed two-qubit gadget whose KAK-derived parameter satisfies $a=5/9$ has, on the complete graph $K_n$ with $n\geqslant 5$, the same second-moment spectral gap as the iSWAP gadget. The central step is a representation-theoretic localisation theorem: the largest strictly negative eigenvalue of the associated $\mathfrak S_n$-invariant operator always occurs in the highest-spin $\mathrm{SU}(2)$ summand. A local positive-semidefinite decomposition separates every spin sector except the two highest. This settles a conjecture of Kong, Li, and Liu.

We prove the redundancy of a family (the so-called melonic family) of large-$N$ moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator $C$ at large-$N$ has an expectation value that depends on $C$ exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.

We investigate the dynamics of an oscillating water column, which is a wave energy converter where water waves interact with a fixed partially immersed structure and an air chamber. Considering the shallow water regime, the fluid motion is governed by either the one-dimensional nonlinear shallow water equations or the Boussinesq-Abbott equations, while the air pressure variation in the chamber acts as a spring force on the fluid. The presence of the structure and the air chamber introduces constraints into the fluid equations. Assuming conservation of the total fluid-elastic energy in the absence of structural damping, we reformulate the constrained equations as transmission problems between the open water and the chamber and show their local well-posedness. We also consider a different modeling approach in which the upper part of the fluid in the chamber is treated as a rigid layer (water column) free to move vertically above the lower fluid. For this configuration, we reformulate the dynamics as wave-spring-mass systems, where the motion of the water column is driven by the shallow water waves outside the chamber. We establish their local well-posedness and show that the effective buoyancy period of the water column motion is determined by the competition between the added mass and the stiffness of the spring force.

Human movement drives the spatial spread and persistence of many infectious diseases, yet existing theory and real-time operational tools for inferring the instantaneous reproduction number R(t) often assume static and/or homogeneously mixing populations and cannot describe how individuals generate and acquire infections heterogeneously based on their movement patterns within a day. Renewal equations underpin many such popular estimators of R(t), and here, we develop a network-based modelling framework from which we derive new mechanism-led renewal equations and control indicators for outbreaks of infectious diseases. These equations directly integrate within-day human movement to rigorously define a family of instantaneous reproduction numbers; inward, outward, and type R(t) for individual locations, R(t) between locations, R(t) at meeting locations, and R(t) for the entire mobility network. These quantities correct for the unsuitability of existing location-specific R(t) estimators that operate in closed, static populations. Applying our framework to epidemics on diverse types of networks alongside mobile phone data, we demonstrate how our new framework's outputs provide new, multi-scale control indicators at the network, location, and transmission corridor scales, and can be used to design targeted disease control interventions including the strength, type, and length of intervention required across space and time. We capture the biasing effects of different existing ways to measure location-specific and network-level transmission potential without capturing within-day human movements. This generalisable framework redefines reproduction numbers in real-world outbreaks that are shaped by individuals moving across connected locations, enabling more spatially and temporally precise interventions.

We study cutting-plane generation for binary polynomial optimization (BPO), whose feasible region is the multilinear set of a hypergraph. Strong inequalities for this set—such as two-links, flowers, and odd $\beta$-cycles—are classically hand-derived for fixed support patterns. Instead, we propose a decision-diagram (DD) approach: for any chosen support, it separates a facet-defining cut in the local multilinear polytope and lifts it back to the original problem. We utilize a novel compact DD encoding based on a recursive formulation that represents only the vertex variables and implicitly encodes the hyperedge variables inside the state representation. From this DD encoding, we also obtain: (i) an extended formulation of the multilinear polytope, (ii) a width characterization via valid antichains that uncovers a new polynomially solvable class of hypergraphs, and (iii) a certificate for the facetness of the generated cuts. We explore three different support strategies for cut generation that reuse, expand, or partition local structures into section hypergraphs. Our empirical results show that our DD-based methodologies achieve a larger gap closure at the root node with fewer cuts than existing procedures and, in turn, markedly accelerate branch-and-bound procedures.

In this paper, we establish upper bounds for the second eigenvalue of the Schrödinger operator $L=\Delta+|\sigma|^2+k$ on $k$-dimensional closed submanifolds of the compact projective spaces $\mathbb F P^m$, where $\mathbb F\in\{\mathbb R,\mathbb C,\mathbb H\}$, as well as the Cayley projective plane. The estimates are obtained by combining the standard spherical embeddings of these spaces with a conformal test-function argument. In the complex, quaternionic, and Cayley cases, the resulting bounds involve correction terms that record the position of the tangent spaces of the submanifold relative to the corresponding geometric structures. We also derive universal estimates depending only on the dimension of the submanifold and the underlying division algebra. We show that equality in the sharp estimates forces the submanifold to be totally umbilical. Finally, using known classifications of totally umbilical submanifolds, we compute the second eigenvalue for the standard real, complex, quaternionic, and Cayley models and identify the totally geodesic examples that attain the sharp upper bounds.

We complete the Braverman-Kazhdan-Ngô program over non-Archimedean local fields assuming local Langlands conjecture for tempered representations and a natural assumption on the $\gamma$-factors of non-supercuspidal discrete series. In particular, the program is complete unconditionally for general linear groups over non-Archimedean local fields.

Robert Smithson's 1970 land artwork Spiral Jetty, located in the north arm of Utah's Great Salt Lake, provides a fixed remote-sensing target whose visual complexity reflects hydroclimatic conditions. A companion study analyzed 1,744 co-registered Landsat 4-9 and Sentinel-2 image chips spanning every year and month from 1984 to 2025. It found robust relationships between coarse-scale permutation entropy, mean intensity, and the third principal component of ResNet50 avg-pool embeddings, and lake elevation, regional temperature, and cumulative CO2. It also showed that image complexity leads lake stage by about three years and that the long-term trend is non-monotonic, with decline after 2015. Building on those findings, this paper develops a two-stage forecasting pipeline. Stage 1 applies IPCC AR6 SSP1-2.6, SSP2-4.5, and SSP5-8.5 temperature changes to forecast regional temperature, north- and south-arm lake elevation, salinity, and cumulative CO2 for 2030 and 2050. Stage 2a projects 14 image-complexity features using linear and Random Forest regressions trained on the 42-year record. All six scenario-year combinations move the inputs outside the 1984-2025 training distribution. The Random Forest model therefore saturates near the post-2015 dry-playa regime, while a threshold-based hydrological interpretation classifies Spiral Jetty as fully exposed in every scenario. Stage 2b specifies a climate-conditioned latent-diffusion framework using Stable Diffusion XL fine-tuned with LoRA on the 1,744 chips, ControlNet conditioning on the climate vector, and a hydrological physics mask for ensemble image synthesis. Code is provided, and validation results will be released after diffusion training. We conclude by discussing ethical implications of generative speculation on cultural heritage and outlining a roadmap for forecasting at Sun Tunnels, Double Negative, Lightning Field, and Roden Crater.

Dynamical systems subject to non-autonomous perturbations decaying in time arise naturally in many physical contexts, including laser-molecule interactions, epidemiological models, nonlinear oscillatory systems, and celestial mechanics. In this paper, we consider time-dependent perturbations decaying polynomially fast in time of Hamiltonian systems having a lower-dimensional isotropic normally parabolic invariant torus supporting quasiperiodic solutions. We prove the existence of invariant manifolds in the extended phase space, and determine the asymptotic behavior of the associated transverse dynamics. The above results are obtained for Hölder, smooth, and analytic Hamiltonian systems.

The Atlantic Meridional Overturning Circulation is monitored and emulated through reduced indices, but such projections discard thermohaline structure and may require either explicit physical state or memory of the observed index. We compare these strategies in 30 branch-consistent CMIP6 trajectories from eight model families using leave-one-family-out validation. Salinity, temperature and density information improves direct 20-year forecasts, whereas compact scalar memory is top-ranked at every recursive horizon and yields the lowest case-averaged Brier score. A matched ablation confirms that feedback from memory improves long-horizon prediction. Physical state and recent trends also predict future ocean-state changes beyond the emissions pathway, most robustly at five years. NorESM under SSP5–8.5 identifies a forcing-dependent limit of scalar compression, while MIROC shows negative long-horizon transfer. A resolvent analysis explains why stable memory components do not guarantee stability of the complete learned model. Physical augmentation and memory compression therefore serve different AMOC prediction tasks.

Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $\gamma>0$, or if $n\ge3$ and $0< \gamma \leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\hookrightarrow R^\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik–Rosenberg and Li–Mantoulidis. Then, we prove that if $n\ge3$ and $\gamma>4(n-1)/(n-2)$, the space $R^\gamma(M)$ is contractible, and hence nonempty. This solves a homotopy-theoretic strengthening of a conjecture of Gromov (Conjecture 3, Section 6.1.2, "Four Lectures on Scalar Curvature") in the maximal possible coefficient range. Concerning Gromov's conjecture we also treat the equivariant case and the case of manifolds with boundary.