Papers

In 1948, Erdös and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \begin{equation}\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3},\nonumber\end{equation} which is still open. It is known that the conjecture holds if one can prove it for any prime $n\equiv 1\;(\mbox{mod}\;24)$. If $n=24m+1$ and $n_1\leq n_2,n_3$, then $n_1=6m+k$ with $1\leq k\leq 12m$. A solution $(n_1,n_2,n_3)$ of the above equation is called a tame solution if $n_2$ and $n_3$ are factors of $(6m+k)(24m+1)$. We call $n=24m+1$ wild if it does not have any tame solution. Based on the information in our earlier work on tame solutions posed in arXiv, Howerton found that there are only fourteen wild primes of the form $n=24m+1\leq 2.4\times 10^{11}$. In this paper, we derive thirty-four families of wild solutions of the above equation, which contain the solvability of the fourteen wild primes. Together with our earlier tame polynomial solutions, numeric test shows that they cover all the primes of the form $24m+1$.

We study large-overtone quasi-normal modes of a massive scalar field in a four-dimensional Schwarzschild de-Sitter black hole in the presence of a wall placed at the static sphere. Dirichlet boundary condition is imposed at the wall. At leading order the quasi-normal modes with large overtones scale as $n$ for $n\gg 1$ and are inversely proportional to the bouncing time. We analyze correction terms up to $n^{-1}$ order in the asymptotic expansion.

Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.

We derive an explicit Faber-Walsh representation for the planar conductivity problem with finitely many disjoint inclusions of general shape and independently chosen positive conductivities. Walsh's lemniscatic map gives a conformal coordinate without smallness or wide-separation assumptions. The field expansion is expressed through Faber-Walsh polarization tensors, which we relate to the complex contracted generalized polarization tensors by a triangular change of basis. For analytic interfaces, we prove conformal continuation and absolute and uniform convergence of the Walsh-Grunsky series near the boundary to obtain explicit factorizations of the Neumann-Poincaré operator and the Faber-Walsh polarization tensors. These factorizations separate canonical interaction, which can remain nonzero even when all Grunsky coefficients vanish, from conformal deformation, while retaining the componentwise conductivity dependence in a coupled resolvent. For harmonic polynomial incident fields, we prove geometric convergence of outgoing truncations with exact coefficients on compact subsets of fixed exterior branch domains. Numerical comparisons with independent boundary-integral solutions illustrate the representation for heterogeneous, asymmetric, and nearly touching configurations.

In this work, we study the one-dimensional regression problem under random design and Gaussian errors. Our framework is very general: we make no prior assumptions about the design (which may be nonstationary and exhibit short or long-range dependence), nor do we assume that the errors are homoscedastic. We examine in detail the cases where the error process exhibits short or long-range dependence. We adopt a least-squares penalized strategy using piecewise polynomials to estimate the regression function, following the framework of Baron, Birgé and Massart [1999]. We derive explicit penalties, up to calibration constants, to obtain adaptive estimators for which we establish risk bounds. Since these penalties depend on the dependence properties of the error process, which are unknown in practice, we propose several adaptations of the dimension jump calibration algorithm to make our procedures fully data-driven.

This data article provides the datasets, symbolic derivations, and scripts used to reproduce master diagrams, stationary-phase spectra, windowed first-order coherence functions, and quantum-noise stability maps for strongly chirped dissipative solitons of the cubic-quintic complex Ginzburg-Landau equation in normal and anomalous group-delay dispersion regimes. The repository includes node-regularized normal-dispersion spectra and energies; small-parameter expansions of the branch roots; cavity-map gain-loss update relations; Airy uniformization at the normal-dispersion spectral edge; anomalous-dispersion spectra and coherence calculations; and processed tables for plotting and stability analysis. OriginLab projects are accompanied by open-format .csv/.txt numerical tables to support reuse without proprietary plotting software. Data and code repository: https://doi.org/10.5281/zenodo.22690899.

For binary matroids \(M,N\) representable over a common field \(\F\), the Kronecker product of their \(\F\)-representations defines a matroid \(T_{\F}(M,N)\) independent of the chosen representations. We classify when this represented tensor product is regular, cographic, graphic, or binary. For simple nonfree factors, regularity holds exactly when, up to interchange, one factor is a cactus matroid and the other is outerplanar, or one is a triangular cactus matroid and the other is series–parallel. Cographicity holds exactly in the first case. For simple factors with nonempty ground sets, graphicity holds exactly when one factor is free and the other is graphic. For fixed factors, regularity, cographicity, and graphicity are independent of the common representation field, although the isomorphism type may vary with its characteristic. Products of at least three simple nonfree factors are nonregular.

This paper investigates entire classical separable variable radial solutions to a class of parabolic k-Hessian equations -u_t (sigma_k(lambda(D^2u)))^alpha = f(|x|)g(t), where the right-hand side consists of positive continuous functions. Without assuming that -u_t is positively bounded, we extend previous work for the equations whose right-hand side is 1 to more general cases, including cases where the right-hand side terms are bounded and periodic. By employing the method of Euler's broken line, we obtain the existence and nonexistence results. Furthermore, we obtain the precise asymptotic power exponent.

Computing spaces of modular and automorphic forms is an important problem in algorithmic number theory, with in particular Diophantine applications to generalised Fermat and other equations. The case of Shimura curves was studied by Greenberg and Voight by cohomological methods, allowing them to reduce the problem to linear algebra. Relying on work of Imbert, we leverage topological techniques to obtain linear systems so structured that they can be solved in time quasi-linear in the genus.

We study the Smoluchowski-Kramers approximation for a stochastic wave equation with state-dependent damping on a bounded domain, driven by both a $Q$-Wiener process and a Lévy process with finite second moment. As $\varepsilon\to0$, we prove that $u^\varepsilon$ converges in distribution, in the Meyer-Zheng topology, to the unique weak solution of an overdamped stochastic parabolic equation. The proof relies on a nonlinear transformation associated with the damping coefficient, uniform energy estimates, and compactness arguments in the pseudo-path topology. We identify the limiting equation, which contains both the classical Gaussian noise-induced drift caused by state-dependent damping and an explicit jump correction generated by the Lévy noise. We show that the latter coincides exactly with the Marcus-to-Itô correction associated with the canonical jump flow induced by the nonlinear damping transformation.

Variable-weight geometric orthogonal codes (GOCs) were introduced by Doty and Winslow for designing the macrobonds in 3D DNA origami. Recently, Wang et al. introduced the concept of geometric difference families (GDFs) and established the equivalence between GOCs and a certain type of GDFs. Based on the relationship, Su et al. gave two classes of variable-weight GOCs. In this paper, we first prove the existence of (n*m,K,1)-GDFs for K ={3,5} and {3,6}. For further research, we completely determine the existence of all the remaining parameters for an (n*m,{3,4,5},1)-GDF by Su et al. As consequences, the perfect (n*m,K,1)-GOCs for K = {3,5}, {3,6} and {3,4,5} are obtained.

Full-duplex (FD) communication theoretically doubles spectral efficiency. Despite this potential, its practical performance is primarily constrained by severe self-interference (SI) and co-channel interference. Moreover, conventional fixed antenna arrays suffer from limited spatial flexibility, resulting in insufficient spatial isolation for SI suppression. To address this issue, this letter proposes an FD architecture assisted by pinching antenna systems (PASS). By dynamically adjusting transmit and receive antenna positions, the system enables large-scale channel reconfiguration, thereby synergizing with the base station (BS) beamforming to suppress SI and enhance the desired signal reception. To demonstrate the potential of PASS for FD communication, we formulate a weighted sum-rate maximization problem that jointly optimizes antenna positions, BS beamforming, and power allocation. To tackle this non-convex problem, we reformulate it using the weighted minimum mean square error (WMMSE) framework and develop an efficient alternating optimization (AO) algorithm to iteratively update the optimization variables. Simulation results reveal that the proposed PASS-assisted FD architecture significantly outperforms conventional fixed antenna arrays, achieving substantial sum-rate gains while effectively mitigating SI to enable FD operation.

Let H^{n+1} be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^{beta}r; we treat both regimes: beta>1+k{alpha} and beta=1+k{alpha}. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_{infty} respectively in two different cases.

In this paper, we investigate the compactness problem for the super-Liouville equation under Dirichlet-chiral boundary conditions. We prove that no blow-up occurs in this setting.

For $0<\kappa\leq4$, we construct a closed response calculus for $\mathrm{SLE}_\kappa$ weldings and a canonical Dirichlet form. The divergence covariance combines the Weil–Petersson and Velling–Kirillov forms. The integrated response gives exact changes of measure and canonical Liouville–capacity increments on conformally removable weldings.

Let $(M,g)$ be a closed negatively curved surface. If $g$ is strictly $\tfrac 19$-pinched and has the same unmarked length spectrum as a hyperbolic metric, we show that $g$ is hyperbolic. As a consequence, we show that any hyperbolic metric on a surface admits a $C^2$-neighborhood in the space of metrics in which it is characterized by its unmarked length spectrum, up to isometry.

We propose $p$ and $rp$-spectral element methods for elliptic boundary layer problems on two dimensional rectangular domains. To resolve boundary layers, we use $p$-version with a fixed mesh and an $rp$-version with a boundary layer mesh consisting of thin needle-like elements near the boundary layer and coarse elements away from the layer. Stability estimates are derived using non-conforming spectral element functions. The numerical scheme for both the methods is based on minimising a least-squares functional in appropriate Sobolev norms. We construct robust preconditioners to manage the condition number of the normal equations derived from the least-squares formulation. The method is able to approximate boundary layers at a rate $O\left(\frac{\log W}{W^2}\right)$ for the $p$-version and at the rate $O({\epsilon}\alpha^{2W})$, uniformly in $\epsilon$, for the $rp$-version, where $0<\epsilon\leq1$ is the boundary layer parameter, $\alpha<1$ is a constant and $W$ denotes the degree of the approximating polynomial. Numerical results are provided for model elliptic boundary layer problems for a range of boundary layer parameters. Simulation results demonstrate the efficiency and robustness of the method in capturing boundary layers on rectangular domains.

We present a new proof of a classical theorem of Dieudonné: if a linear space of $n\times n$ matrices consists entirely of singular matrices, then its dimension is at most $n^2-n$. Our proof is based on a surprising “duality” argument: we prove this universal upper bound by exhibiting a single matrix space that serves as a lower bound for a related problem. Interestingly, this approach only works for certain fields, but we use model-theoretic arguments to obtain the same result for all fields. We hope that this approach can be generalized to provide new duality-based proofs of other classical theorems on matrix spaces, and give some preliminary results in this direction.

By using spherical harmonicas decomposition and Gaussian-type Poincaré inequalities, we establish several sharp stability estimates for the following second-order weighted Heisenberg Uncertainty Principle \begin{equation*} \int_{\mathbb{R}^N} \!\frac{|\Delta u|^2} {|x|^{2\alpha}} \mathrm{d}x \int_{\mathbb{R}^N} |x|^{2\alpha+2}|\nabla u|^2\mathrm{d}x \geq \frac{(N+4\alpha+2)^2}{4} \left(\int_{\mathbb{R}^N} |\nabla u|^2 \mathrm{d}x\right)^2, \end{equation*} and \begin{equation*} \int_{\mathbb{R}^{N}} \frac{|\Delta u|^{2}} {|x|^{2\alpha}} \mathrm{d}x +\int_{\mathbb{R}^{N}} \left|x\right|^{2\alpha+2} |\nabla u|^{2}\mathrm{d}x \ge\left(N+4\alpha+2\right) \int_{\mathbb{R}^{N}} |\nabla u|^{2}\mathrm{d}x. \end{equation*} We also provide the explicit value and the necessary and sufficient condition for attainability of the sharp stability constants. Moreover, when $\alpha=0$, our results reduce into those of [Calc. Var. Partial Differential Equations 64 (2025), Paper No. 129], [J. Funct. Anal. 290 (2026), Paper No. 111321] and [arXiv:2510.00453].

Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its extensions have found widespread use in many other research areas such as molecular dynamics, climate science, engineering, finance, and neuroscience. Applications include dimensionality reduction, forecasting, system identification, control, and spectral clustering. In order to apply DMD to partial differential equations, the spatial domain is typically first discretized using finite difference or finite element techniques, thus implicitly rendering the problem finite-dimensional. We extend projected and exact DMD to infinite-dimensional systems. Rather than estimating matrices from vector-valued observations, our DMD variants learn finite-rank operators from functional data such as observables, densities, or wavefunctions. We show that conventional DMD algorithms can be regarded as special cases of their functional DMD counterparts. All results will be illustrated with the aid of guiding examples. We focus in particular on Koopman, Perron-Frobenius, and Koopman-von Neumann operators associated with graphons, ordinary differential equations, and stochastic differential equations.

We prove that, for vector-valued convolution Calderón–Zygmund operators, boundedness on a single nontrivial Morrey space is equivalent to the corresponding global $L^p$ boundedness. Thus one Morrey scale already contains the full finite-$p$ boundedness information. The implication from Morrey to $L^p$ is obtained by a separated-copy amplification argument that reconstructs the global norm from a single scale-local estimate; the converse is proved in the same vector-valued framework by a local/far-field decomposition. As a consequence, boundedness of the vector-valued Hilbert transform on one nontrivial Morrey space is equivalent to the UMD property. The result shows that Morrey estimates do not bypass the classical Banach-space obstruction: they detect it exactly.

For every integer $n \ge 2$, we construct explicit tetrahedral counterexamples showing that repeated longest-edge (LE) n-section refinement does not, in general, preserve shape regularity, contrary to long-standing conjectural expectations. For $n = 2$, the construction disproves the finite-family conjecture put forward by Adler in 1983 and the non-degeneracy conjecture for tetrahedral longest-edge bisection formulated by Rivara and Levin in 1992. It also shows that the sharper higher-dimensional diameter decay suggested by Stynes in 1983,following his 1980 planar finite-similarity result, cannot hold for arbitrary tetrahedra. For $n = 3$, it disproves the published non-degeneracy conjecture for tetrahedral longest-edge trisection formulated in 2011 by Suárez, Abellón, Abad and Plaza on the basis of numerical experiments. The resulting infinite descendant sequences violate both the minimum and maximum angle conditions. For $n \ge 3$, the selected edge is always the unique longest edge at every refinement step. The two-step recurrence used in the construction exhibits two distinct degeneration behaviors: a flat-tetrahedron regime for $2 \le n \le 5$ and a skinny-tetrahedron regime for $n \ge 6$. The degenerating sequences can also be realized within conforming partitions generated by the conforming LE n-section algorithm. For both the classical and the conforming LE n-section algorithms, the maximal element diameter tends to zero as the number of refinement steps tends to infinity, regardless of which longest edge is chosen. Thus diameter convergence, and even conformity, do not prevent shape degeneration. These algorithms should therefore be used with care in applications requiring uniform shape regularity.

We develop compactness and defect-of-compactness results for Banach-valued evolution classes with Morrey control in time. For \[ \begin{aligned} \mathbb W_M^{p,\lambda}(0,T;E_0,E_1) =\{u\in\mathcal M^{p,\lambda}(0,T;E_0):\;& u'\in\mathcal M^{p,\lambda}(0,T;E_1)\},\\[-1mm] &0<\lambda<1. \end{aligned} \] the exact trace exponent \[ \theta=1-\frac{1-\lambda}{p} \] governs both continuity and compactness. If $E_0\hookrightarrow\!\hookrightarrow E\hookrightarrow E_1$, bounded sets are compact in the same Morrey space $\mathcal M^{p,\lambda}(0,T;E)$, not merely in $L^p(0,T;E)$. In contrast, compactness in $C([0,T];E)$ holds if and only if the exact trace space $(E_1,E_0)_{\theta,\infty}$ embeds compactly into $E$. We also obtain compact lower-order Hölder embeddings and a sharp Hilbert-triple threshold. On unbounded domains we prove a tightness criterion for global Morrey compactness. In the Hilbert-valued case we establish, by a direct time-averaging argument, cocompactness modulo spatial translations and a translation profile decomposition whose remainder vanishes in every strictly subcritical time-Morrey–Sobolev target; a uniform little-Morrey condition removes the loss in the time exponent. At the doubly critical endpoint, heat and whole-space Stokes dynamics force balanced parabolic profiles, and the remainder vanishes in $\mathcal M_t^{2,\lambda}L_x^{2^*}$. Finally, in three-dimensional Navier–Stokes we show that the classical nonlinear profile decomposition has a scale-sensitive Morrey refinement: the remainder is small in $\mathcal M_t^{p,1-p/4}L_x^6$ for every $2\le p<4$, and orthogonal profiles have vanishing interaction in the corresponding Morrey forcing space.

This paper investigates the joint optimization of the transmit precoder and the transmission/reflection coefficients of a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) to maximize the weighted sum rate (WSR) in a multi-user downlink. We propose a particle-swarm-assisted gradient meta-learning (PSA-GML) algorithm for this non-convex problem. The original problem is first equivalently transformed via an amplitude-split parameterization and a collapsed precoder representation, which automatically satisfy the energy-conservation constraint and reduce the search dimension. Particle swarm optimization (PSO) then performs a global search over the STAR-RIS coefficients to yield a high-quality, initialization-robust warm start, with the transmit precoder obtained in closed form. Departing from conventional alternating optimization (AO), a coordinate-wise long short-term memory (LSTM) meta-optimizer trained by first-order gradient meta-learning further refines the coefficients and precoder jointly, learning per-coordinate adaptive update rules from data. The meta-optimizer is trained offline and applied to unseen channels without further adaptation. Numerical results show that PSA-GML attains an 11.06 bits/s/Hz WSR at 10 dB with N=32 elements and K=4 users, exceeding AO by 13.1% (and by 6.2% even with multiple random restarts) and the random-phase scheme by 35.1%. In the interference-limited regime it reaches 83.9% of the hand-designed Adam refinement without manual hyper-parameter tuning, and it transfers zero-shot across regimes, indicating that the learned update rule captures the intrinsic WSR landscape structure.

This paper investigates a fundamental question in stripeless erasure coding based on symmetric balanced incomplete block designs (SBIBDs): how many replicas per object are precisely required to guarantee recovery from any set of at most $p$ node failures? We refine the known sufficient recovery guarantee for the generalized SBIBD $(v,k,\lambda)$ construction. One replica is necessary and sufficient for $p=1$, and $q=\lambda(p-1)+2$ replicas ($q\le k$) guarantee recovery for $p\ge2$. Recovery takes one round for $p=2$ and at most two rounds in general. To study the tightness of this count, we define the universal replica threshold $q^\ast(A,p)$ for a fixed zero-diagonal SBIBD representative~$A$. We determine $q^\ast(A,p)$ for $p=1$ and $p=2$. For $p\ge3$, we identify pairwise separated sets and private target permutations as the structures determining whether the sufficient count is tight or can be further reduced. These conditions give exact thresholds for all but the case where $\lambda>1$ and $A$ contains no pairwise separated set of size~$p$. We improve the bounds for this remaining case and leave its exact threshold open. Finally, we give sufficient conditions for pairwise separated sets and show how affinity relabeling can realize private target permutations.

Trace spaces determine exactly which initial values are compatible with an evolution class. We identify the exact trace generated by vector-valued Morrey control in time and show that it differs essentially from the classical $L^p$ theory. For a Banach couple $X_1\hookrightarrow X_0$, the trace of the natural Morrey evolution class is the weak real-interpolation space $(X_0,X_1)_{\theta,\infty}$, where $\theta=1-(1-\lambda)/p$. Every element of this space occurs as a trace through a bounded extension operator. The result is sharp in both interpolation parameters: in general the smoothness exponent cannot be increased and the fine index $\infty$ cannot be replaced by any finite index. Thus Morrey control changes the exact trace mechanism rather than merely strengthening an integrability estimate. At the limiting endpoint, bounded mean oscillation (BMO) control yields finite-index interpolation traces together with a logarithmic modulus of continuity. The result isolates the precise initial-data space naturally associated with local, scale-sensitive time regularity and provides a trace framework suited to evolution equations with Morrey-type maximal regularity. \keywords{Morrey spaces \and trace spaces \and real interpolation \and evolution equations \and bounded mean oscillation}

We prove global well-posedness of the one-dimensional defocusing cubic nonlinear Schrödinger equation in the modulation space $M^{\infty,1}(\mathbb{R})$. This space imposes no spatial decay and contains $C_b^2(\mathbb{R})$ as well as all absolutely convergent sums of plane waves. The result applies to arbitrary data in this space, including large smooth quasiperiodic profiles and their localized perturbations. The proof constructs a nonnegative density satisfying a local conservation law from forward Weyl ratios, which are defined through half-line square-integrable solutions of the associated spectral problem. A suitable nonlinear combination of localized integrals of this density controls the modulation norm. Finally, choosing the spatial localization scale and NLS scaling in a coordinated way makes the accumulated boundary flux small enough to continue every mild solution globally.

Repeated evaluation can estimate a benchmark score accurately while still requiring replication to certify narrow uncertainty. We characterize that requirement on a fixed grid of $M$ tasks with $L$ binary paths per task under the hard budget $(M+t)K$, where each path costs at most $K$ responses or episodes. For fixed $L \ge 3$ and $0 < \alpha \le 1/12$, the optimal expected width on the worst pure cohort is $\Theta_{\alpha,L}([M(t+1)]^{-1/2})$ when every task is observed and $\Theta_{\alpha,L}([M(t+\sqrt{M})]^{-1/2})$ when omission is allowed. The lower bounds cover adaptive hard-budget policies, and fixed random-subset designs attain both rates through disagreement certificates. A joint mean/disagreement interval turns the task-covering law into practical finite-budget inference. In an equal-budget LiveCodeBench replay with 16 models, 880 tasks, and five outputs per task, the task-covering design reduces median point-estimation MSE by 87.0% relative to pooled uniform sampling, while the Joint certificate produces narrower confidence intervals in 15/16 panels and reduces median interval width by 30.6%. Finite-regime analyses identify task coverage as the effective choice at the evaluated scale and characterize how cohort size and within-task agreement determine the useful operating region. Together, the sharp laws and fixed-budget evidence make replication and task coverage explicit design variables for information-efficient repeated evaluation.

This paper investigates the following May–Nowak type model for viral infection in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 2$: $\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \kappa - u - uw - \mu \dfrac{u^{1+\alpha}}{\ln^k(u+e)}, \\[1mm] \gamma v_t = \Delta v - v + uw, \\[1mm] w_t = \Delta w - w + v, \end{cases}$ where $\chi \in \mathbb{R}$, $\mu > 0$, $k \in [0,1)$, $\alpha > 0$, and $\gamma \in \{0,1\}$. We establish the global existence and uniform-in-time boundedness of classical solutions for suitably regular initial data under one of the following conditions: (A) $\gamma = 1$, $n = 2$, $k \in [0,1)$, $\alpha = 1$, and $\mu > 0$; (B) $\gamma = 1$, $3 \le n \le 5$, $k = 0$, $\alpha = 1$, and $\mu $ is sufficiently large; (C) $\gamma = 1$, $n \ge 6$, $k = 0$, $\alpha > \frac{n-2}{4}$, and $\mu > 0$; (D) $\gamma = 0$, $n \ge 2$, $k = 0$, $\alpha > \frac{n+2}{2}$, and $\mu>0$. In particular, in the physically relevant dimensions $n = 2,3$, both subquadratic and quadratic damping are sufficient to prevent blow-up. Moreover, in the fully parabolic case ($\gamma = 1$), our results improve upon recent findings by relaxing the condition $\alpha > \frac{n}{2}$ to weaker assumptions within the above parameter regimes.

In this paper, we conduct a systematic study of quandle colorings and quandle coloring quivers for pretzel links using the dihedral quandle $\mathbb{Z}_{n}$. First, we systematically investigate all possible colorings of 3-pretzel links, determining the number of distinct colorings in each case as well as the structure of their quandle coloring quivers. In order to obtain more general conclusions, we impose restrictions on $n$ based on the properties of the coefficient matrix of the system of congruence equations. So we examine the number of quandle colorings and the quandle coloring quivers for 4-pretzel links in the case where $n$ is prime. Finally, Combining the results of 4-pretzel links we rigorously derive both the coloring numbers and quandle coloring quivers for general $m$-pretzel links in the case where $n$ is prime, with full proofs provided.