Papers

In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding $n-2$ to the $n$th Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism—the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings—to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN

In this paper, we study the presence of perfect powers in orbits of polynomials, or of semigroups generated by a finite family of polynomials, defined over a number field. In particular, we correct and generalize a result of Ahmadi et al. (2012), showing that multiplicative shifts of perfect powers in orbits can be detected by a finite algorithm. We also give a finiteness result on elements in orbits satisfying certain multiplicative relations.

We prove the transcendence of numbers that have $\beta$-expansions in algebraic bases $\beta$ belonging to dendric shifts with purely discrete spectrum, under certain combinatorial conditions on their directive sequences of substitutions. This includes Arnoux-Rauzy sequences on arbitrary alphabets and Cassaigne-Selmer sequences. The proof uses Rauzy fractals and the subspace theorem.

Adapting the circle method developed by Frantzikinakis–Klurman–Moreira, we establish a sufficient criterion for a set of natural numbers to be multiplicatively recurrent. We apply this criterion to Piatetski–Shapiro sets with small exponents and to regular irrational nil-Bohr sets. We also study the value distribution of unimodular completely multiplicative functions along these sets.

We study non-autonomous basins of $F_j$ automorphisms of $\mathbb C^2$ satisfying $$a_j\|p\|\leq\|F_j(p)\|\leq b_j\|p\|\qquad (\forall p\in \mathbb B)$$ where $0<a_j<b_j<b<1$. We show that if $b_j^m<a_j$ for some integer $m$ and all $j>0$ then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have $\prod_{i=1}^{j-1}b_i\leq b_j^m$ then the basin is biholomorphic to $\mathbb C^2$. For the quadratic Hénon family $$F_n(z,w)=\bigl(z^2+c^{q_n}w,c^{q_n}z\bigr), \qquad 0<c<1,$$ we study how the asymptotic behavior of the sequence $q_n\to\infty$ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to $\mathbb C^2$ or is a Short $\mathbb C^2$, thereby isolating possible sequences for which other complex structures may occur.

We examine how local motif structure and global network topology jointly shape spiking dynamics in stochastic neuronal networks. Using networks of Izhikevich neurons with Erdős-Rényi (ER) and scale-free (SF) background connectivity, we compare motif-embedded networks with synapse-count-matched, non-motif controls under noise- and stimulus-driven protocols. Motif embedding increases noise-induced coherence in both topologies and provides a smaller improvement in signal transmission, while SF-based networks attain greater absolute coherence and transmit signals faster and more reliably than ER-based networks. Motif-rewiring experiments show that this benefit depends on the specific arrangement of intra-motif connections rather than on strong local connectivity alone: randomizing the motif skeleton reduces coherence overall, although rewiring the non-recurrent feed-forward loop and bi-parallel motif can increase it. Targeted ablation shows that the SF advantage depends in part on hub integrity; removing high-degree excitatory neurons impairs coherence and signal transmission more than matched random node or edge removal. Together, these results indicate that local motifs and global hubs make distinct, complementary contributions to coherent spiking dynamics.

Neuronal networks exhibit complex dynamics shaped by connectivity and stochastic input. Empirical studies show that neuronal networks contain recurring subgraphs, or motifs, but the collective influence of different motif types after embedding in large stochastic networks remains less well understood. We construct a spiking network composed of six representative structural classes and examine how intrinsic noise, coupling strength, inter-motif connectivity, network size, and neuronal heterogeneity shape synchronization. Both motif- and network-level coherence peak at intermediate noise intensities, consistent with coherence resonance. Bidirectionally coupled pairs (M2) and the type-2 recurrent feed-forward loop (M3c) consistently exhibit high motif-level coherence. Rewiring these motifs while preserving local synapse number and strength produces some of the largest reductions in network coherence, showing that connection arrangement contributes beyond strong local coupling alone. M2 and M3c also exhibit frequent spike doublets, linking short inter-spike intervals with elevated coherence. Relative to a degree- and weight-matched random control, the motif-structured network reaches greater coherence at weaker noise. Increasing network size enhances coherence until the response begins to saturate, whereas applied-current heterogeneity lowers peak coherence and shifts the optimum toward stronger noise without erasing the relative differences among motif classes. These results show that local connection arrangement remains dynamically consequential after embedding and can shape noise-driven coherence at the population level.

This paper addresses the control of multi-input strict-feedback nonlinear systems subject to a joint capacity constraint, in which the admissible input set is a coupled subset of the individual actuator limits. Unlike existing constraint-handling methods that enforce actuator bounds channel by channel and may unnecessarily suppress admissible control directions, we develop an Anisotropic Joint-Admissibility-Preserving Input Realization (AJ-APIR) framework that explicitly exploits the geometry of the joint constraint. The proposed realization constructs a state-dependent gain matrix whose spectral decomposition separates the commanded input into normal and tangential directions relative to the constraint boundary. The normal component is attenuated as the boundary is approached, while the tangential component is preserved, which allows the admissible control effort to be redistributed without loss of tracking authority. Integrated with a backstepping controller, the AJ-APIR framework guarantees forward invariance of the joint admissible set for all time. We establish exponential convergence of the tracking error to zero together with uniform boundedness of all closed-loop signals, and characterize the resulting command-demand behavior under the joint constraint. Simulation results for a representative second-order, two-input nonlinear system subject to a power-budget constraint demonstrate the efficacy of the proposed method to enforce the joint input constraint.

Whether a forced-oscillation source can be uniquely localized depends jointly on the available measurements and the candidate intervention dictionary. This paper considers a single unknown constant-amplitude sinusoid acting through one of physical intervention channels. Because its amplitude and phase are unknown, each candidate harmonic response is observable only up to a nonzero complex scalar and therefore defines a projective ray in measurement space. A necessary-and-sufficient condition is derived for exact localization, together with a weighted nearest-ray locator and a deterministic recovery bound based on projective separation. The paper proposes a framework distinguishing four mechanisms of source localization failure: mechanism mismatch, feature-projection loss, structural nonidentifiability, and poor conditioning or model error. Numerical studies on the Kundur two-area system illustrate well-separated recovery, mechanism-dependent terminal decisions, measurement-induced ambiguity, and the sensitivity of nearly collinear signatures to model perturbations. Then, IEEE–NASPI Contest models are used to further examine broad candidate dictionaries, terminal-feature source coverage, and ambiguity removal through measurement augmentation. The results show that correctly locating the source bus does not by itself establish unique identification of the internal forcing channel or robustness to measurement and model uncertainty.

Let $\Gamma<\mathrm{Isom}(\mathbb H^n)$, $n\geq2$, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $\Gamma$ has hitting measure singular with respect to Lebesgue measure on $\partial\mathbb H^n$. The proof extends the method of Kosenko–Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.

Linear-fractional rotational interval exchange transformations appear naturally when investigating circle diffeomorphisms with breaks using the renormalization group approach. Earlier, we achieved an important rigidity result on circle diffeomorphisms with a single break by means of establishing a time-reversing duality for the corresponding commuting pairs of linear-fractional maps. In this paper, we extend the established duality to the case of multiple breaks, where a commuting pair from the previous case is replaced by a `commuting collection' of an even number of linear-fractional maps, which act according to a rotational interval exchange scheme of special type, while satisfying certain commutation relations.

In physics-based simulations, the evaluation of nonlinear constraint forces represents a primary source of computational complexity. Within the framework of projective dynamics, the solver alternates between estimating these forces and updating vertex positions for the subsequent frame. This work investigates the potential of snapshot-based subspace methods to accelerate the computation of constraint terms while maintaining the fidelity of nonlinear dynamics. We focus specifically on the Discrete Empirical Interpolation Method (DEIM), examining whether it can identify a reduced set of constrained mesh elements that enable accurate interpolation and approximation of forces across the full system. Our experiments demonstrate that DEIM is neither intuitive nor consistently effective for constraint projection reduction. In particular, its performance degrades when constraints involve multiple vertices. To validate these observations, we present comparisons between full-order and reduced-order simulations on both surface meshes embedded in three dimensions and volumetric meshes. Furthermore, we report reconstruction errors for various classes of constraints, providing quantitative evidence of the method limitations in this context.

We give a conformal mating description for the class $\Sigma_d$ of hyperbolic Schwarz reflections with connected and full filled Julia set; this parallels the recent development in several non-hyperbolic settings. We show that the escaping dynamics of $\Sigma_d$ can be characterized independently by the class of annular Schwarz reflections we introduce. Our main result is that each $\sigma\in\Sigma_d$ decomposes into a unique hyperbolic polynomial factor and a unique annular Schwarz reflection factor and, conversely, any two such factors may be mated to determine a unique such Schwarz reflection. We also show that annular Schwarz reflections converge to Nielsen maps in suitable settings, connecting the hyperbolic and non-hyperbolic mating descriptions.

We construct a smooth complex projective rational surface whose automorphism group is discrete and not finitely generated. It is obtained by blowing up a point on a branch curve of a rational quotient of a product Kummer surface. Every point with transcendental coordinate gives such a surface. The proof uses leading jets along the branch curve, the complete family of blowups and the cubic intersection of the family, which are not being considered in relevant earlier works.

We provide generalized linearization theorems for skew-product flows on abstract Banach bundles over locally compact metric spaces, allowing for a unified treatment of uniform and nonuniform, exponential and strong exponential dichotomies. This is accomplished by constructing an operator whose fixed points are exactly the conjugacies that are bounded perturbations of the identity, and by giving criteria under which this operator is a contraction. Under stronger assumptions we refine the argument to obtain Hölder regularity of the conjugacy. Finally, we apply our results to recover known results for non-autonomous differential equations and to obtain new linearization results for control-affine systems on manifolds.

Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitude. Recent Transformer-based neural operators attempt to capture global dependencies, but often at the cost of quadratic attention complexity. Transolver resolves this problem by projecting physical states into a reduced slice space for attention computation. Although fast, this projection sacrifices fine spatial information. Moreover, by operating in this reduced space with shared weights across attention heads, it may constrain the model's flexibility, thereby limiting its capacity to capture complex phenomena. To address these issues, we propose the Cluster Attention Neural Operator (CANO), which reformulates attention via a novel cross-attention mechanism that dynamically clusters queries while preserving full-resolution keys and values. This avoids slice compression loss and removes weight-sharing limits. At the same time, the model remains fast without losing global interactions. Empirically, CANO achieves state-of-the-art performance across canonical PDE benchmarks, covering fluid and solid dynamics (e.g., Navier-Stokes, Airfoil, Plasticity), irregular unstructured geometries (e.g., Pipe Turbulence, Composites), and long-term temporal rollouts. Across solid deformation and turbulent flow benchmarks, CANO achieves lower errors than baselines and exhibits strong geometric adaptability and temporal consistency.

Distinguishing periodic from chaotic dynamics in a time series is a fundamental challenge in both physics and engineering. Yet, end-to-end learned architectures must discover both a representation and a decision boundary from data, at substantial cost. We introduce TopTimeNet, which decouples these tasks: a fixed, non-learned stage extracts a $42$-dimensional geometric and topological descriptor from Takens delay embeddings and persistent homology, and a lightweight learnable stage performs classification. On a benchmark of $49$ nonlinear dynamical systems, a $1{,}638$-parameter configuration matches the mean accuracy of one with $33\times$ more trainable parameters. Additionally, this approach delivers mean accuracy comparable to convolutional neural networks and surpasses the average performance of converged Transformer models, while requiring three to four orders of magnitude fewer trainable parameters. Robustness also depends sharply on where noise is introduced: TopTimeNet degrades gracefully under perturbations to its precomputed features, but degrades sharply when noise is introduced into the raw signal and the full feature-extraction pipeline is recomputed, showing that robustness to perturbations of the precomputed features does not imply robustness of the complete raw-signal-to-prediction pipeline. These results show that decoupling fixed geometric and topological feature construction from a lightweight discriminative stage can achieve comparable classification accuracy with substantially fewer trainable parameters.

In this paper, we investigate weakly persistent centers for several classes of planar cubic differential systems, including systems with a nondegenerate linear center and systems with a nilpotent center. Necessary parameter relations are obtained by computing singular point quantities or quasi-Lyapunov constants, while their sufficiency is established by suitable analytic constructions. For the general complex cubic system, we obtain necessary and sufficient conditions for the origin to be a weakly persistent center, and the corresponding result for the associated general real cubic system follows by imposing the conjugacy relations between the complex coefficients. We further characterize the weakly persistent isochronous centers of the general complex cubic system by combining the weakly persistent center conditions with the vanishing of the linearization quantities and constructing time-preserving analytic linearizations.

In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function $W$, whose graph $\Gamma_W$ has the box dimension $2+\log_N\lambda$, we obtain \[ \dim_A^\theta(\Gamma_W) \leq \frac{2+\log_N\lambda-\theta}{1-\theta}, \qquad \theta\in \left(0,\log_N\frac{1}{\lambda}\right). \] and if $\lambda^2 N <1$, we get \[ \dim_A(\Gamma_W)\geq 1 +\log_N\left(\frac{1}{\lambda}\right). \] For the classical Takagi function $T$ with graph $\Gamma_T$, we show that \[ \dim_A^\theta(\Gamma_T)=1, \theta\in(0,1), \] and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.

We study the Hilbert geodesic flow on a closed convex real projective $n$-dimensional manifold. We use the thermodynamic formalism for this dissipative Anosov flow to obtain a geometric description of the SRB measure. We also deduce new counting properties for the induced representations of the fundamental groups in $SL(n+1, \R)$.

We prove two new cases of the Stuck–Zimmer conjecture. First, we settle the Stuck–Zimmer conjecture for ergodic probability preserving actions of irreducible lattices (both uniform and non-uniform) in connected semisimple real Lie groups with finite center, no compact factor and real rank at least two. We furthermore extend the Stuck–Zimmer theorem to all irreducible probability preserving actions of such groups whenever one simple factor is locally isomorphic to $\mathrm{SL}_2(\mathbb R)$. In particular, this settles the Stuck–Zimmer conjecture for irreducible actions of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$. The proof relies on a new stabilizer rigidity criterion using local spectral gap of projected stabilizers in a single simple factor. Here local spectral gap is understood in the sense of Boutonnet, Ioana and Salehi Golsefidy.

Despite the availability of excellent vaccines and antiviral medicines, hepatitis B virus (HBV) remains a major public health problem. In this work, we develop a mathematical vaccination-treatment model for HBV transmission to investigate threshold conditions for illness persistence as well as the level of intervention. The analysis gives the characterisation of both disease-free and endemic equilibria and determines the basic reproduction number, $\mathcal{R}_0$, by the next-generation matrix approach. The model indicates that $\mathcal{R}_0=1$ is a critical threshold dividing disease-free from endemic states and characterises a forward transcritical bifurcation regarding the transmission parameter. Numerical equilibrium continuation and stability assessments confirm the analytical results. A two-parameter study of transmission and vaccination rates identifies threshold geometries and assesses impacts of important parameters using sensitivity analysis. The model is calibrated with India-specific data, such as HBsAg prevalence, HBV incidence and mortality rates and fits epidemiological targets closely through limited nonlinear calibration. Additionally, multi-start optimisation and practical identifiability profiling are used to investigate parameter uncertainty. Robustness analyses reveal that the calibrated $\mathcal{R}_0^*=1.1744$ is associated with a super-threshold regime; however, parameter flexibility permits both sub-threshold and super-threshold regimes. This combined analytical and data-driven framework connects epidemic thresholds, bifurcation structures, intervention efficacy, and parameter uncertainty. It provides a quantitative framework for analysing HBV persistence and evaluating control options, especially in the absence of epidemiological evidence.

The extreme therapeutic resistance of malignant gliomas motivates the development of multimodal treatment strategies. We present a mechanistic mathematical framework to investigate the coupled dynamics of radiotherapy, temozolomide, and chimeric antigen receptor (CAR) T-cell therapy. The model is formulated as a nine-dimensional impulsive dynamical system that captures proliferative-quiescent tumor states, treatment-specific resistance, tissue damage, TMZ pharmacokinetics, and the lymphotoxic interferences exerted by conventional regimens on engineered T cells. Mathematical analysis establishes the existence of biologically meaningful solutions, maps invariant structures, and identifies threshold conditions under which sustained therapy locally stabilizes the tumor-free equilibrium. After study-matched benchmarking against clinical trials, including the standard Stupp protocol, we deploy heterogeneous virtual patient cohorts to evaluate five combined CAR-T–Stupp sequences. At the population level, the immunotherapeutic adjunct yields a consistent but modest survival extension, increasing median overall survival by $0.8$–$0.9$ months with minimal sensitivity to temporal ordering or scheduling perturbations. Crucially, these population-level medians mask profound individual heterogeneity: under combined protocols, $19\%$–$22\%$ of virtual patients extend survival by more than $60$ days, and $\sim 3\%$ gain over a year. A low intrinsic tumor proliferation rate $r_1$ emerges as the primary determinant of response, bolstered by robust T-cell expansion and attenuated microenvironmental suppression. These findings suggest that the clinical promise of multimodal integration depends on identifying responsive tumor phenotypes for personalized treatments rather than searching for a single universally optimal timeline.

A deep classifier is defined not only by the decision it produces, but also by the sequence of transformations through which that decision is formed. Treating this evolution as a dynamical system across layers provides a natural framework for asking how decision geometry emerges through depth and how far back we can trace a boundary's dynamical signature. We model a feed-forward classifier as a finite, nonautonomous discrete dynamical system, with layers playing the role of discrete time steps. We study the Finite-Time Maximum Lyapunov Exponent (FTMLE) of the data samples' dynamical trajectory through depths of the classifier. The FTMLE measures the rate of convergence/divergence of nearby trajectories. We move the observation endpoint backward from probabilities to logits and then to hidden representations. For Gaussian classes, we prove that probability-level FTMLE carries a clear geometric signature of the decision boundary, with its dominant direction aligned with the boundary normal. Moving one step backward to the logits, we prove this relationship is no longer universal but depends critically on how the classifier is trained, particularly on the choice of loss function. Moving further backward to the hidden representation, the connection becomes more conditional: boundary-related FTMLE can persist, but only under identifiable structural conditions. We propose geometry-aware fine-tuning for restructuring the classifier's hidden FTMLE, and propose conditions for guaranteed concentration of high hidden FTMLE near the decision boundary. Through our numerical results, we show the generality and validity of our theoretical results. Understanding the evolution of data samples as traveling through the layers of classifier provides a principled foundation for identifying where boundary-relevant sensitivity emerges and for developing layer-aware regularization strategies.

It has been found that utilizing the geometric properties of the graph dynamics can bring out crucial information of the data that a statistical analysis will not. The properties of Ricci flow bring out hidden dynamics of the data in the same way as decomposition of smooth manifolds. In this paper we explore an alternative to Ricci flow, where we find that we can bring out similar properties like it using a simpler and well defined Edge Laplacian to practice community detection in graphs. Moreover, the results obtained by the present approach show that a flow driven by the Edge Laplacian yields results similar to those of the Ricci flow. In contrast, computing the Forman-Ricci curvature requires looping over the adjacent edges, while computing Olivier-Ricci curvature involves the computation of Wasserstein distance and probability distribution associated with the graph nodes, making the Edge Laplacian approach computationally more efficient.

We introduce action integrals for partially integrable Hamiltonian systems that interpolate between the non-integrable and completely integrable theory. We show that under the assumption of ergodicity of the Hamiltonian system on the common level sets of its integrals, these actions become constants of motion for the averaged dynamics and, consequently, adiabatic invariants when the Hamiltonian function is allowed to slowly change with time.

In a recent paper published in the Journal of Differential Equations (384, 2024), Cui and Zhao investigated an SIRS epidemic model with the nonlinear incidence rate $\frac{kI^p}{1+\alpha I^q}$, where $p>0$ and $q\geq0$ are arbitrary real numbers. They proved that the codimension of a Bogdanov-Takens bifurcation in this model is at most two, while the codimension of the Hopf bifurcation was left as an open problem. In this work, we investigate this problem while keeping $p$ and $q$ free throughout the analysis. A strategic nondimensionalization and parametrization remove the exponential dependence of the positive equilibrium and convert the generalized-Hopf conditions into algebraic focus equations coupled to explicit semialgebraic admissibility regions. Using algebraic focus reduction, strict semialgebraic admissibility, and validated interval computation, we rigorously prove the existence of admissible nondegenerate generalized Hopf points generating three, four, and five small-amplitude limit cycles. In particular, we rigorously certify a strict-interior codimension-five weak focus and a full-rank local unfolding, thereby proving that five small-amplitude limit cycles can bifurcate from a single positive equilibrium. We further derive exact algebraic obstructions to codimension six and find no admissible candidate in extensive continuation and projected-component searches. These results provide analytical and numerical evidence for the conjecture that five is maximal, but no codimension-six nonexistence theorem is claimed. The same parametrization also yields a simple proof that the Bogdanov-Takens codimension is at most two.

Recently Burniol Clotet proved that two transitive Anosov flows on 3-manifolds with equivalent strong unstable foliations must be conjugate up to time rescaling. We further explore the mechanism of Burniol Clotet's argument and treat the case of higher-dimensional Anosov flows and their isometric extensions by compact Lie groups.

A single-tone large-signal operating point of a nonlinear two-port is a periodic orbit of a periodically forced circuit. When the device has memory, the Floquet exponents of that orbit separate into fast (electrical) and slow (thermal and trapping) modes, and long-term memory is motion on the invariant manifold attached to the slow modes. Existence and uniqueness of that manifold follow from the parameterization method of Cabré, Fontich and de la Llave, applied to the stroboscopic map at the orbit; the manifold is the spectral submanifold of Haller and Ponsioen, without a small-forcing parameter. The manifold is a bundle over the circle of drive phase, its fiber dimension the number of slow exponents, and the dynamic X-parameter kernel of Verspecht et al. identifies its reduced dynamics from the transient after a step in drive amplitude. An exact reduced model has one memory state per slow exponent, the memoryless X-parameter surface is the fixed-point family of the reduced dynamics, and the envelope-domain model is the reduced dynamics driven by the envelope. The hypotheses are verified and the manifold constructed for a GaN HEMT compact model with thermal and trap memory: the trap's time constant at the operating point, $14 \mu$s, is set by the linearization, not by its $6$ ms emission time, and the trap's nonlinearity confines a polynomial representation of the manifold to a few millivolts of drive, so trap memory must be represented over the operating range, by tables or a fitted network, not by an expansion about the operating point.