For integers $n \ge 3$ and $r \ge 1$, let $Γ_{n,r}$ be the alternating-oriented digraph obtained by gluing $r$ directed $n$-cycles along a single edge in a staircase pattern, and let $A_{n,r}$ be its adjacency matrix. A canonical $n$-layer partition puts $A_{n,r}$ into an $n$-cyclic block form and isolates a cyclic product core $K_{n,r}$, so the nonzero spectrum of $A_{n,r}$ is obtained from that of $K_{n,r}$ by taking $n$th roots. We show that $K_{n,r}$ is totally nonnegative and irreducible, and hence its nonzero eigenvalues are real, positive, and simple. It follows that all nonzero eigenvalues of $A_{n,r}$ are simple and occur in $\exp(2πi/n)$-orbits, forming unions of regular $n$-gons in the complex plane. A one-step Schur complement yields a three-term recursion in $r$ for the characteristic polynomials $Φ_{n,r} \in \mathbb{Z}[x]$. This determines both the multiplicity of the eigenvalue $0$ and the number of nonzero eigenvalues, and leads to a generating function with cubic denominator. Applying a Tran-type confinement theorem gives the uniform bound $ρ(A_{n,r}) \le (27/4)^{1/n}$ and the sharp limit $\displaystyle\lim_{r \to \infty} ρ(A_{n,r}) = (27/4)^{1/n}$ for each fixed $n$. Finally, specializing at $x=1$ relates $Φ_{n,r}(1)$ to Padovan spiral numbers and yields a complete classification of rational nonzero eigenvalues.

Objective: Real-time adaptive proton range verification systems based on produced neutrons require accurate information on their non-isotropic momentum distributions within short times, for which Monte Carlo (MC) methods are too computationally expensive. We present a surrogate model based on Fourier Neural Operators (FNO) for fast prediction of angle- and energy-resolved proton transport and neutron production within proton therapy. Approach: We treat the irradiated phantom and the proton beam's state as depth-evolving series, respectively of different materials, and of spatial, angular and energy phase space density distributions. The task is solved auto-regressively by learning changes in the distributions of protons and those of produced neutrons. For training and evaluation, two datasets of 47 MC simulations featuring different primary intensities were produced. Simulated geometries were extracted from a thoracic CT scan as series of laterally homogeneous materials. Main Results: An average relative $L^2$ discrepancy of 0.067 and 0.137 was achieved by the predicted proton and neutron distributions, respectively. This corresponded to an average gamma passing rate in the spatial distributions of 99.95$\%$ and 99.40$\%$. Training with higher primary intensities led to improvements between 12$\%$ and 30$\%$ in density metrics. Inference over depths of 40 cm at a resolution of 0.5 mm required on average 23.17 s per beam. Significance: The proposed proton beam surrogate generates accurate spatial and momentum distributions of neutrons at MC-level accuracy within seconds, while demonstrating robust generalization with respect to irradiated geometry and beam characteristics. This approach can be used for prototyping and operation of range verification systems, other tasks such as neutron dose estimation, and can be extended to include other kinds of secondary emissions.

We investigate the Maximal Entropy Simple Symmetric Exclusion Process (MESSEP) on a discrete ring with L sites and N indistinguishable particles. Its eigenfunctions are Schur polynomials evaluated at the L-th roots of unity, yielding an explicit spectral decomposition. The analysis relies on this eigenstructure and on the link between Schur polynomials and irreducible characters of the symmetric group, which forms the core algebraic tool for the scaling limits. In the low-density regime, where N is fixed and L tends to infinity, the rescaled dynamics converge to the Unitary Dyson Brownian Motion (UDBM). The electrostatic repulsion then emerges as an entropic force, providing a canonical microscopic derivation of the UDBM. In the hydrodynamic regime, where N is equivalent to $α$L with $α$ P p0, 1q, the empirical measure converges to a density solving a nonlinear, nonlocal transport equation. Its moment generating function satisfies a complex Burgers-type equation. As $α$ tends to 0, this equation coincides with that governing the spectral distribution of the Free Unitary Brownian Motion (FUBM), thereby bridging discrete entropic exclusion dynamics and free unitary hydrodynamics. Overall, the MESSEP provides a unified canonical discrete framework connecting unitary Dyson motion and free unitary Brownian motion through nonlinear hydrodynamic limits, with Schur and character theory as the central algebraic structure.

Pangenomics uses graph-based models to represent and study the genetic variation between individuals of the same species or between different species. In such variation graphs, a path through the graph represents one individual genome. Subgraphs that encode locally distinct paths are therefore genomic regions with distinct genetic variation and detecting such subgraphs is integral for studying genetic variation. Biedged graphs is a type of variation graph that use two types of edges, black and grey, to represent genomic sequences and adjacencies between sequences, respectively. Ultrabubbles in biedged graphs are minimal subgraphs that represent a finite set of sequence variants that all start and end with two distinct sequences; that is, ultrabubbles are acyclic and all paths in an ultrabubble enter and exit through two distinct black edges. Ultrabubbles are therefore a special case of snarls, which are minimal subgraphs that are connected with two black edges to the rest of the graph. Here, we show that any bidirected graph can be transformed to a bipartite biedged graph in which lowest common ancestor queries can determine whether a snarl is an ultrabubble. This leads to an O(Kn) algorithm for finding all ultrabubbles in a set of K snarls, improving on the prior naive approach of O(K(n + m)) in a biedged graph with n nodes and m edges. Accordingly, our benchmark experiments on real and synthetic variation graphs show improved run times on graphs with few cycles and dead end paths, and dense graphs with many edges.

We establish the asymptotic sharpness of a Nikolskii type inequality proved by A. Baranov and R. Zarouf for rational functions $f$ in the Wiener algebra of absolutely convergent Fourier series, with at most $n$ poles, all lying outside the dilated disc $\frac{1}λ\mathbb{D}$, where $\mathbb{D}$ denotes the open unit disc and $λ\in[0,1)$ is fixed. More precisely, this inequality tells that the Wiener norm of such functions is bounded by their $H^{2}$-norm -- i.e., their norm in the Hardy space of the disc -- times a factor of order $\sqrt{\frac{n}{1-λ}}$. In this paper, we construct explicit test functions showing that this bound cannot be improved in general: the inequality is asymptotically sharp as $n\to\infty$, up to a universal constant, for every fixed $λ\in[0,1)$.

We present experimental study of nonradiative double electron capture processes in collisions of 95 and 146 MeV/u bare xenon ions with krypton and xenon gaseous atoms at the HIRFL-CSR storage ring. Angular distributions of the characteristic Kα radiation of the down-charged projectile ions Xe52+* are measured, which are closely related to the magnetic sublevel population of the excited 1s2l_j states of Xe52+*. It was found that the Kα1 radiation shows pronounced anisotropic and is sensitive to the collision energies and the target atoms, whereas the Kα2 radiation gives rise to isotropic. Moreover, obviously difference in the anisotropy parameters of Lyman-α1 of Xe53+* ions and Kα transitions of Xe52+* ions separately following nonradiative single and double electron capture into the L-shell levels of projectiles is obtained and discussed.

The uncontrolled proliferation of cancer cells and their interaction with healthy tissue poses a major challenge in oncology. This manuscript develops and analyzes mathematical models that describe tumor response to radiotherapy by incorporating the Linear Quadratic model for cell survival. To improve therapeutic efficiency, the theory of optimal control is introduced on a system of coupled differential equations, allowing for the comparison of constant versus optimized radiation strategies. The analytical study of these models provides insights into the expected dynamics under different treatment scenarios, while numerical simulations validate the theoretical results and highlight the benefits of optimal control in reducing tumor burden with minimized collateral damage.

Remote plasma-assisted vapour deposition under nitrogen (RPAVD-N2) is introduced as a single-step, solvent-free, room-temperature strategy to integrate iron(II) phthalocyanine (FePc) into carbon nanofiber (CNF) scaffolds for high-performance pseudocapacitive electrodes. In this process, CNFs are activated by low-energy N2 remote plasma and subsequently exposed to sublimated FePc, which undergoes controlled plasma polymerisation to form conformal, nitrogen-rich FePc-derived coatings while preserving Fe-N coordination. By tuning the plasma power, the degree of crosslinking, defect generation and molecular fragmentation is precisely controlled. Structural and spectroscopic analyses reveal progressive incorporation of amine, nitrile and oxygenated functionalities while maintaining the Fe-N coordination environment, with 30 W power providing the optimal balance between structural integrity and defect density. Plasma processing enhances the capacitance by nearly one order of magnitude compared to sublimated FePc films, underscoring the critical role of plasma-induced molecular integration. The FePc30W@CNFs electrode delivers 80.9 F/g at 0.25 A/g (areal capacitance 0.92 mF/cm2 at 2.9 mA/cm2), achieves 7.42 Wh/kg at 225 W/kg, and retains 86.5% of its initial capacitance after 6000 cycles. These results demonstrate that remote plasma polymerisation enables robust, high-rate and durable phthalocyanine-based electrodes, establishing RPAVD as a scalable platform for next-generation energy-storage materials.

For an offline-first collaborative application to operate in true peer-to-peer fashion, its collaborative features must function even in environments where internet connectivity is limited or unavailable. Each peer may only be interested in a subset of the application data relevant to its workload, and this subset can overlap in different ways with those of other peers. Limitations imposed by access control and mesh network technologies often result in peers being sparsely connected. Reasoning about consistency in these systems is hard, especially when considering transactional updates that may alter different sets of data in the same transaction. We present \textsc{IntersectionAtomicity} and \textsc{IntersectionCC} as models to reason about offline-first collaborative applications that are sparsely-connected and rely on partially replicating different subsets of a broader set of data. We then use these models to propose a set of guidelines to help developers design their application with atomicity and consistency guarantees.

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3Σ$ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincaré section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.

We study the indivisibility of a quantum-corrected AdS black hole involving two kinds of global monopoles, regular or phantom ones to declare the effects from the quantum fluctuation, monopole factors and the negative cosmological constant on the evolution of black holes. We focus our attention on the possibility that this kind of black holes break into their own two parts because of the second law of thermodynamics. We derive and calculate the entropies of the initial black hole and the broken parts respectively and the entropy difference relates to the black holes structure including the quantum-gravity oscillation, global monopoles and the influence from AdS environment. It is found that the two kinds of global monopoles both keep the black holes intact instead of splitting because the entropy difference is negative. The considerable quantum fluctuation compels the defference to be positive and the fragmentation of the black holes may happen in the case that the mass of one fragmented black hole is tiny and the other one's mass is huge. The AdS radius can increase the whole value of change in entropy and the difference near the midpoint of mass ratio will become positive, so the large enough radius may lead the isolated black hole to become two parts nearly equal in the mass.

We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the cosmological polytope. Moreover, we construct two triangulations of the dual cosmological polytope in terms of maximal and almost maximal tubings of the underlying graph. Though the existence of the first triangulation was already suggested by Arkani-Hamed, Benincasa and Postnikov, the second is completely new and, in particular, gives rise to a new expression of the canonical form of the cosmological polytope.

The integrity of plasma-facing components (PFCs) in tokamaks is critically challenged by transient events such as runaway electron (RE) impacts. We report the first systematic analysis of the thermal damage to tungsten-based PFC tiles comprising the SPARC outboard off-midplane limiters that is induced by RE beams formed during vertical displacement events. Parametric scans in RE impacting characteristics as well as energy-pitch distribution functions from the Dream code are employed for calculations of the volumetric heat loads. A realistic panel design is adopted to enhance the fidelity of the thermal analysis. The PFC thermal responses are compared in terms of in-depth temperature profiles and damage characteristics, such as melt depth and vaporization losses.

We consider the non-homogeneous first-order differential equation with hysteresis described by the Krasnoselskii-Pokrovskii rate-independent hysteresis operator. Existence and uniqueness of solutions as well as the boundedness of solution in response to a bounded input are proved. The global stability of the equation is also investigated. Periodic solutions and their stability are studied in addition. The differential equation under analysis constitutes the so-called inversion-free feedforward control, which was proposed for mitigating arbitrary rate-independent hysteresis effects in the actuated systems. The experimentally identified non-smooth and non-strictly monotonic hysteresis of a magnetic shape memory alloy (MSMA) actuator serves as the case study. The performed analysis is settled in a series of theorems which are illustrated by numerical examples.

Cloud-edge collaboration enhances machine perception by combining the strengths of edge and cloud computing. Edge devices capture raw data (e.g., 3D point clouds) and extract salient features, which are sent to the cloud for deeper analysis and data fusion. However, efficiently and reliably transmitting features between cloud and edge devices remains a challenging problem. We focus on point cloud-based object detection and propose a task-driven point cloud compression and reliable transmission framework based on source and channel coding. To meet the low-latency and low-power requirements of edge devices, we design a lightweight yet effective feature compaction module that compresses the deepest feature among multi-scale representations by removing task-irrelevant regions and applying channel-wise dimensionality reduction to task-relevant areas. Then, a signal-to-noise ratio (SNR)-adaptive channel encoder dynamically encodes the attribute information of the compacted features, while a Low-Density Parity-Check (LDPC) encoder ensures reliable transmission of geometric information. At the cloud side, an SNR-adaptive channel decoder guides the decoding of attribute information, and the LDPC decoder corrects geometry errors. Finally, a feature decompaction module restores the channel-wise dimensionality, and a diffusion-based feature upsampling module reconstructs shallow-layer features, enabling multi-scale feature reconstruction. On the KITTI dataset, our method achieved a 172-fold reduction in feature size with 3D average precision scores of 93.17%, 86.96%, and 77.25% for easy, moderate, and hard objects, respectively, over a 0 dB SNR wireless channel. Our source code will be released on GitHub at: https://github.com/yuanhui0325/T-PCFC.

Let \( \ell_n(x) \) denote the maximal run-length among the first \( n \) digits of the Lüroth expansion of \( x\in(0,1] \). While \( \ell_n(x) \) grows logarithmically, we investigate the finer multifractal properties of the exceptional set where $\ell_n(x)$ exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set \[ \left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = α, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = β\right\}, \] for all \( 0 \le α\le β\le 1 \).

We present a time-domain scheme for computing Casimir forces within the Maxwell stress tensor formalism, together with a specific realization using the finite-element-based discontinuous Galerkin time-domain method. The approach enables accurate evaluation of Casimir--Lifshitz interactions for a wide range of geometries and material properties at finite temperature. At the core of the method, the electromagnetic Green's tensor is expressed as the system's response to dipolar excitations, thereby recasting the Maxwell stress tensor into a set of classical scattering problems driven by electric and magnetic dipoles. We validate the approach against reference calculations of the Casimir interaction between parallel half-spaces at both zero and nonzero temperature. We further demonstrate its applicability to finite, cylindrically symmetric geometries for which closed-form solutions are unavailable, obtaining accurate agreement with asymptotic predictions based on physical considerations. These findings illustrate the method's potential for studying Casimir interactions in realistic micro- and nanoscale structures, relevant to nanodevice design and experimental settings.

We resolve a question posed by Benedetti and Sagan by constructing a signreversing involution on Takeuchi's expansion that yields the antipode for the ring of symmetric functions in terms of the Schur basis.

Linear response theory asserts that sufficiently small external biases produce currents proportional to the applied force and forms the theoretical foundation of nonequilibrium transport. Here we demonstrate that linear response can break down even in uniformly hyperbolic deterministic systems when hierarchical asymmetry is present. Using a minimal class of uniformly expanding chaotic maps with hierarchical multiscale structure, we show that progressively finer transport channels become dynamically active as the applied bias decreases. The resulting force current relation is monotone and exhibits a hierarchical, fractal-like organization of activation thresholds. As a consequence, the effective mobility diverges as F to 0, demonstrating breakdown of linear response despite strong chaos and uniform hyperbolicity. The effect arises from deterministic multiscale activation rather than intermittency, stochastic noise, or singular invariant measures. These results identify hierarchy as an independent deterministic mechanism for nonperturbative transport response and demonstrate that uniform hyperbolicity alone does not guarantee the validity of linear response.

The exponential growth of AI has created unprecedented demand for computational resources, pushing chip designs to the limit while simultaneously escalating the environmental footprint of computing. As the industry transitions toward heterogeneous integration (HI) to address the yield and cost challenges of monolithic scaling, minimizing the carbon cost of these complex HI systems becomes critical. To fully exploit HI, a co-design approach spanning application, architecture, chip, and packaging is essential. However, this creates a vast design space with competing objectives, specifically the trade-offs between performance, cost, and carbon footprint (CFP) for sustainability. CarbonPATH is an early-stage pathfinding framework designed to address this multi-objective challenge. It identifies optimized HI systems by co-designing workload mapping, architectural parameters, and packaging technologies, while treating sustainability as a first-class design constraint. The framework accounts for a wide range of factors, including compute and memory sizes, chiplet technology nodes, communication protocols, integration style (2D, 2.5D, 3D), operational CFP, embodied CFP, and interconnect type. Using simulated annealing, CarbonPATH explores this high-dimensional space to identify solutions that balance traditional metrics against environmental impact. By capturing interactions across applications, architectures, chiplets, and packaging, CarbonPATH uncovers system-level solutions that traditional methods often miss due to restrictive assumptions or limited scope.

We present a generative framework for inverse design of five-layer transmissive Huygens' metasurfaces (HMSs), addressing a longstanding challenge in achieving full-phase, high-efficiency unit cell designs with minimal full-wave simulations. The key to achieving this is our reliance on the field-based semianalytical (SA) scheme developed in Part I of this paper, which allows rapid and highly effective synthesis of such multilayer composites, however with limited accuracy. To overcome the prohibitive data demands of traditional pipelines, we employ Mamba, a selective state space model well suited for long-range sequence modeling as the backbone of our learning framework. A bidirectional Mamba (Bi-Mamba) forward surrogate is first trained on SA-generated data and subsequently fine-tuned with full-wave CST samples. An ablation over a 1080-sample CST pool shows that as few as 270 full-wave calibration samples suffice to reach near-CST-level agreement at a fraction of the simulation cost. An autoregressive Mamba inverse generator is subsequently trained on surrogate-augmented data, treating unit-cell synthesis as a sequential generation task. The resulting one-to-many generative model produces diverse unit cell geometries conditioned on target scattering responses. It achieves CST-validated designs with field transmission magnitude 0.9 across the full 0-$2π$ phase range at 20 GHz. Moreover, a CST-calibrated surrogate trained to accurately predict frequency responses (18-22 GHz) enables functional post-selection of inverse generated designs. Together, the hybrid SA-generative methodology in this two-part compilation establishes a scalable and data-efficient solution for multilayer HMS synthesis, with natural extensions toward broadband, oblique-incidence, and higher-dimensional electromagnetic inverse-design problems.

Magnetic resonance imaging (MRI) is central to the diagnosis of multiple sclerosis, where the identification of biomarkers such as the central vein sign benefits from high-resolution images. However, most clinical brain MRI scans are performed using 1.5 T scanners, which provide lower sensitivity compared to higher-field systems. We propose a blind super-resolution framework to enhance real 1.5 T MRI images acquired in clinical settings, where only post-processed data are available and the degradation model is not fully known. The problem is formulated as a non-convex blind inverse problem involving the joint estimation of the high-resolution image and the blur kernel. Image regularization is handled through a Plug-and-Play strategy based on a pretrained denoiser, while suitable constraints are imposed on the blur kernel. To solve the resulting model, we design a heterogeneous alternating block-coordinate method in which the two variables are updated using different types of algorithms. Convergence properties are rigorously established. Experiments on FLAIR and SWI sequences acquired at 1.5 T show improved structural definition and enhanced visibility of clinically relevant features, with visual comparison against 3 T images.

Empirical research in reverse engineering and software protection is crucial for evaluating the efficacy of methods designed to protect software against unauthorized access and tampering. However, conducting such studies with professional reverse engineers presents significant challenges, including access to professionals and affordability. This paper explores the use of students as participants in empirical reverse engineering experiments, examining their suitability and the necessary training; the design of appropriate challenges; strategies for ensuring the rigor and validity of the research and its results; ways to maintain students' privacy, motivation, and voluntary participation; and data collection methods. We present a systematic literature review of existing reverse engineering experiments and user studies, a discussion of related work from the broader domain of software engineering that applies to reverse engineering experiments, an extensive discussion of our own experience running experiments ourselves in the context of a master-level software hacking and protection course, and recommendations based on this experience. Our findings aim to guide future empirical studies in RE, balancing practical constraints with the need for meaningful, reproducible results.

Leptogenesis from low-energy CP violation provides a vital link between neutrino physics and the observed baryon asymmetry of the universe. However, this connection is typically obscured by unknown high-energy parameters. In this work, we investigate thermal leptogenesis in the Minimal Left-Right Symmetric Model with generalized parity as the left-right symmetry, where the hermiticity of the Dirac neutrino coupling allows the right-handed mixing matrix $V_\mathrm{R}$ to be determined with minimal assumptions. We show that for a real $V_\mathrm{R}$, these conditions favor CP-conserving Majorana phases, leaving the Dirac CP-violating phase ($δ$) as the sole source of asymmetry. By numerically exploring all four leptogenesis scenarios, we demonstrate that $δ$ alone can generate the observed baryon asymmetry with the correct sign within specific regions of the parameter space. The results exhibit a high sensitivity to the neutrino mass ordering and the lightest neutrino mass, providing a stringent, testable framework for future experimental measurements of the CP phase and neutrino mass scale.

We introduce a two-phase quadratic integrate-and-fire (QIF) neuron whose membrane potential evolves according to two alternating Riccati equations within finite bounds. This simple extension removes the unphysical voltage divergence of the standard QIF model while producing realistic spike waveforms. Despite this modification, the system retains an exact low-dimensional description in the thermodynamic limit, governed by a single complex Riccati equation. Expressions for collective quantities such as the firing rate and mean voltage remain compact and analytically tractable. Because the formalism preserves the mathematical structure of the standard QIF ensemble, it inherits its many generalizations and can serve as a drop-in replacement in existing mean-field frameworks, providing a more biologically plausible yet still exactly solvable neuronal model.

It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space $\mathcal{X}$, a continuous map $γ\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ γ$ is a function of bounded variation for every Lipschitz function $f\colon\mathcal{X} \to \mathbb{R}$. In this note, we show that the continuity assumption is of crucial importance: for many interesting examples of metric spaces there are no analogs of that characterization without the continuity assumption on $γ$. The interesting examples are: $\ell_2$, infinite metric trees, and Laakso-type spaces. However, for ultrametric spaces the said characterization holds without any continuity assumptions.

While EEG features differentiate Major Depressive Disorder (MDD) from healthy controls (HC), their clinical utility as biomarkers depends on a monotonic trajectory across the disease spectrum, from the acute (AC) phase to the maintenance (MA) phase and finally to the healthy baseline. However, the progression of the MA phase remains poorly understood in traditional marker analysis. Analyzing EEG data from 74 individuals (24 AC, 23 MA, and 27 HC), this study provides a comprehensive evaluation of classic ERP and resting-state indices across AC, MA, and HC groups. Our results demonstrate that almost no conventional metrics strictly satisfy the criterion of monotonic progression, likely due to profound inter-individual heterogeneity. These findings highlight the inherent limitations of group-level feature extraction and provide critical insights for developing future paradigms and algorithms to identify neurobiological markers with genuine clinical utility.

For a cardinal $κ$, denote by $\mathbf{H}^κ$ the algebraic real hyperbolic space of dimension $κ$. For a topological group $Γ$, we study the set of continuous representations $Γ\to \operatorname{Isom}(\mathbf{H}^κ)$ up to continuous self-representations $\operatorname{Isom}(\mathbf{H}^κ)\to \operatorname{Isom}(\mathbf{H}^κ)$. The novelty of this work relies in considering simultaneously all cardinals, finite or infinite. We will endow this set of classes of representations with a natural topology, and show that this character variety is compact. This will also enable us to recover all previous compactifications of actions on $\mathbf{H}^n$ by certain actions on real trees for the equivariant Gromov-Hausdorff topology. A class of representations recovers in particular the homothety class of its marked length spectrum. We will define the notion of algebraic cross-ratio and prove a GNS-embedding result, enabling us to generalize some rigidity properties of the marked length spectrum. We will also introduce a notion of abstract cross-ratio, and use it to show that a wide class of groups $Γ$ (characterized by the existence of what we call a $3$-full action on a $\operatorname{CAT}(-1)$-space) admit at most one class of irreducible representations into $\operatorname{Isom}(\mathbf{H}^κ)$ whose boundedness properties are controlled by those of $(X,d)$. We will apply this to topological groups $Γ$ such as the isometry group $\operatorname{Isom}(\mathbf{H}^κ)$ itself, the automorphism group $\operatorname{Aut}(T_ω)$ of the simplicial tree with countably infinite valency, and the automorphism group $\operatorname{PGL}_2(\mathbb{K}, \lvert\cdot \rvert)$ of the projective line over a non-Archimedean field.

Modeling inviscid compressible flows with shocks and vortex dominated dynamics remains challenging for particle grid methods due to moving discontinuities, cell crossing noise, and quadrature degradation under strong deformation. Building on a FLIP/APIC framework with vorticity aware tensor artificial viscosity, we identify a long time RTI failure mode: particle depletion at spike heads degrades quadrature and particle grid coupling, producing nonphysical, void-like dents. Standard mitigations (CPDI lite and subcell-jittered seeding) reduce but do not eliminate this artifact. We therefore add two sampling-aware controls: (i) conservative split resampling that replenishes depleted cells while exactly conserving mass, momentum, and internal energy; and (ii) a soft-switch that attenuates only the APIC affine term when local support is insufficient. Tests on the Sod shock tube and single/multi mode RTI show that the method removes spike head voids in long-time RTI while preserving vortex roll up, and matches reference Euler growth metrics

In accordance with the Cameron-Goethals-Seidel-Shult Classification Theorem, we extend the characterization of Hoffman colorability of line graphs from (Abiad, Bosma, Van Veluw, 2025) to all connected graphs with smallest eigenvalue at least $-2$; we give a characterization of Hoffman colorability of generalized line graphs, and we completely classify the Hoffman colorable exceptional graphs. The 245 Hoffman colorable exceptional graphs from this classification admit a natural partial ordering, and we determine the 29 graphs that are maximal in this respect, in a way similar to the classification of maximal ($E_8$-representable) exceptional graphs as described in (Cvetković, Rowlinson, Simić, 2004). Lastly, as a byproduct and also similarly as in (loc. cit.), we determine all 39 graphs that are maximal with respect to being representable in the $E_7$ root system.