In this work, we analyze the relation between reparametrizations of gradient flow and the induced implicit bias in linear models, which encompass various basic regression tasks. In particular, we study how reparametrization, loss function, and link function influence the convergence behavior and implicit bias of gradient flow. Our results provide conditions under which the implicit bias can be well-described and convergence of the flow is guaranteed. We furthermore show how to use these insights for designing reparametrization functions that lead to specific implicit biases which are closely connected to $\ell_p$- or trigonometric regularizers.

In this note, inspired partly by the works on the analytic trans-series of free-energies in 2D integrable QFTs in the UV limit, we study the corresponding asymptotic expansions in the IR limit of the integrable XXX spin chains, at the anti-ferromagnetic critical point. Starting from the linear TBA integral equation, we generate the perturbative series in the minimal-scheme coupling constants, up to the first order where the $ζ_3$ appears. To all perturbative orders, at spin one-half, the perturbative series relates to that of the $β^2=8π^-$ massive sine-Gordon by a sign flip of the coupling constant, and this relation generalizes for an arbitrary spin, to certain massive deformation of $su(2)_k$ WZW. This relation partially supports the conjectured field-theoretical interpretations of such critical points, at the leading-power $\&\&$ all-logarithmic accuracy. We also compute the perturbative coefficients attached to the first exponentially-small corrections up to the order where $ζ_3$ appears, in case $s>1$. An interesting feature is that the largest exponentially-small corrections at the locations $h^{2+\frac{2k}{s}}$ are non-vanishing, but the corresponding Borel singularities in the perturbative series are canceled for generic $s>1$, not necessarily integers or half-integers. This becomes most dramatic in the a 't Hooft type large $s$ limit, in which the perturbative series has a finite convergence radius, while these exponentially small corrections still survive.

We invent a new cohomology theory for Lie triple algebras. Using this cohomology, we introduce the notions of 2-term $L_\infty$-triple algebras and Lie triple 2-algebras. We prove that the category of 2-term $L_\infty$-triple algebras is equivalent to the category of Lie triple 2-algebras. Crossed modules of Lie triple algebras are studied in detail.

The trapping probability, $ψ$, as defined in Kovacevic and Pflug (2011), is modelled by assuming proportional capital losses, both in the case where there is no insurance and in the case where insurance is purchased by the household. Insurance coverage is likewise proportional, mirroring the structure of quota-share contracts, which are both prevalent in practice and analytically convenient. New closed formulae for $ψ$ are obtained in the case of no insurance when the distribution of the remaining proportion of capital is a power law, extending the results in Kovacevic and Pflug (2011). When proportional insurance is acquired and the remaining proportion of capital is uniformly distributed on $[0,1]$, $ψ$ satisfies a non-local differential equation whose analysis is based on the properties of diffusion processes. The non-local nature of the equation can be addressed using iterative solution methods, leading to a constructive determination of the trapping probability. Constraints on the parameters governing the capital process are derived in both the uninsured and insured cases to prevent the certainty of trapping. Numerical calculations are used to determine the trapping probability for the insured process and to illustrate the impact of different parameters. Consequences on the trapping probability for vulnerable non-poor populations with initial capital slightly above the poverty line are discussed.

In this paper, we compute the exact values of regularity of the quotient rings of the edge ideals associated to multi triangular snake and multi triangular ouroboros snake graphs. Also we compute the exact values of depth, Stanley depth, regularity and projective dimension of the quotient rings of the edge ideals associated to $q$-fold bristled graphs of multi triangular snake and multi triangular ouroboros snake graphs.

In this paper, we prove Deligne's conjecture on the algebraicity of the critical values of symmetric power $L$-functions associated with modular forms of weight at least 5. We also establish new cases of Blasius' conjecture on the algebraicity of the critical values of tensor product $L$-functions associated with modular forms. Additionally, we prove an algebraicity result for the critical values of Rankin--Selberg $L$-functions for $\GL_n \times \GL_2$ in the unbalanced case, which extends the previous results of Furusawa and Morimoto for ${\rm SO}(V) \times \GL_2$. These results are applications of our main theorem on the algebraicity of cross ratios of Rankin--Selberg $L$-functions at critical points.

Droplet impact on surfaces is ubiquitous in many natural and industrial processes. While the impact dynamics of droplets composed of simple fluids have been studied extensively, droplets containing particles are less explored, but are more application relevant. The non-Newtonian behavior of particle suspension introduces new physics affecting the impact dynamics. Here, we investigated the impact dynamics of droplets containing cornstarch particles on a deep water pool and systematically characterized the impact outcomes with various Weber number and particle volume fractions. Distinctive phenomena compared to Newtonian droplet impact have been observed. A regime map of the impact outcomes is unveiled and the transition boundaries are quantified with scaling analysis. Rheology of the suspension is found to play a pivotal role in giving rise to distinct impact outcomes. The results lay the foundation for further characterization of the dynamics of suspension droplet impacting on liquid surfaces and can be translated to other suspension fluids.

We prove that if a subgroup $H$ of the automorphism group $\Aut(Σ^\Z)$ of a non-trivial full shift acts on points of finite support (= points bi-asymptotic to a fixed point) with a free orbit, then for every finitely-generated abelian group $A$, the abstract group $A \wr H$ also embeds in $\Aut(Σ^\Z)$. The groups admitting an action with such a free orbit include $A \wr \Z$ for $A$ a finite abelian group, and finitely-generated free groups. The class of such groups is also closed under commensurability and direct products. We obtain for example that $\Z \wr \Z$, $\Z_2 \wr (\Z_2 \wr \Z)$ and $\Z \wr (\Z_2 \wr \Z)$ embed in $\Aut(Σ^\Z)$. The group $\Z \wr \Z$ is the first example of a finitely-generated torsion-free subgroup of $\Aut(Σ^\Z)$ with infinite cohomological dimension, answering an implicit question of Kim and Roush and an explicit question of the author. We also explore a simpler variant of the construction that gives embeddings of certain Neumann groups, as well as some near-misses to higher iterated wreath products.

We construct a weak categorification of the quantum toroidal algebra action on the Grothendieck group of moduli space of stable (or framed) sheaves over an algebraic surface, which is constructed by Schiffmann-Vasserot and Neguţ. The new ingredient is two intersection-theoretic descriptions of the quadruple moduli space of stable sheaves.

We determine the scaling limit for permutations conditioned to have longest decreasing subsequence of length at most $d$. These permutations are also said to avoid the pattern $(d+1)d \cdots 2 1$ and they can be written as a union of $d$ increasing subsequences. We show that these increasing subsequences can be chosen so that, after proper scaling, and centering, they converge in distribution. As the size of the permutations tends to infinity, the distribution of functions generated by the permutations converges to the eigenvalue process of a traceless $d\times d$ Hermitian Brownian bridge.

Graphical functions are special position space Feynman integrals, which can be used to calculate Feynman periods and one- or two-scale processes at high loop orders. With graphical functions, renormalization constants have been calculated to loop orders seven and eight in four-dimensional $ϕ^4$ theory and to order five in six-dimensional $ϕ^3$ theory. In this article we present the theory of graphical functions in even dimensions $\geq4$ with detailed reviews of known properties and full proofs whenever possible.

Let $ρ>0$ be a constant, let $j\ge0$ be an integer, and let $Γ(z)$ denote the Euler gamma function. With the aid of the integral representation for the Riemann zeta function $ζ(z)$, by virtue of a monotonicity rule, and by means of some properties of the function $\frac{1}{e^t-1}$ and its derivatives, the authors discuss the increasing monotonicity of the function $t\mapsto\binom{t+ρ+j}ρ\frac{ζ(t+ρ)}{ζ(t)}$, study the absolute convexity and logarithmic convexity of the function $t\mapstoΓ(t+j)ζ(t)$, and derive the increasing monotonicity and inequalities of some sequences involving the ratios $\bigl|\frac{B_{2n+2}} {B_{2n}}\bigr|$ of the Bernoulli numbers $B_{2n}$, where $\binom{z}{z}$ denotes the extended binomial coefficient.

We introduce a classical analog of the algebraic Heun operator associated with a classical Leonard pair. Given two observables $X$ and $Y$ satisfying the classical counterpart of the Askey--Wilson relations, we define a \emph{classical Heun observable} $W$ as the most general bilinear combination of $X$, $Y$, and their Poisson bracket. We prove that, when $W$ is taken as Hamiltonian, the dynamics of X and Y is governed by quartic differential equations and, generically, by elliptic functions of second order. This result provides a universal algebraic mechanism transforming the elementary dynamics associated with classical Leonard pairs into elliptic dynamics, and yields an algebraic explanation of a classical observation of Manning on the connection between the Heun equation and elliptic solvability. The construction is illustrated on three examples: an extension of the Pöschl--Teller system, the Zhukovsky--Volterra gyrostat, and a relativistic $A_1$ model related to the classical Askey--Wilson algebra.

Consider the finite simple group $\mathbf{G}:=\mathrm{PSL}(2,\mathbf{F}_{11})$ of order 660, which has an irreducible representation $V_{10}$ of dimension 10. In this note, we study a special trivector ${σ_0}\in \bigwedge^3V_{10}^\vee$ that is $\mathbf{G}$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperkähler fourfold $X_6^{σ_0}\subset\mathrm{Gr}(6,V_{10})$ with many symmetries. We will also look at the associated Peskine variety $X_1^{σ_0}\subset \mathbf{P}(V_{10})$, which is highly symmetric as well and admits 55 isolated singular points. It will help us to better understand the geometry of the special Debarre-Voisin fourfold $X_6^{σ_0}$. We also discuss an application of this example to the global geometry of the moduli space of Debarre-Voisin fourfolds.

In the context of the massive secondary object recently observed in the compact-star merger GW190814, we investigate the possibility of producing massive neutron stars from a few different equation of state models that contain exotic degrees of freedom, such as hyperons and quarks. Our work shows that state-of-the-art relativistic mean field models can generate massive stars reaching $\gtrsim 2.05\,\Msun$, while being in good agreement with gravitational-wave events and x-ray pulsar observations, when quark vector interactions and non-standard self-vector interactions are introduced. In particular, we present a new version of the Chiral Mean Field (CMF) model in which a different quark-deconfinement potential allows for stable stars with a pure quark core. When rapid rotation is considered, our models generate stellar masses that approach, and in some cases surpass $2.5\,\Msun$. We find that in such cases fast rotation does not necessarily suppress exotic degrees of freedom due to changes in stellar central density, but require a larger amount of baryons than what is allowed in the non-rotating stars. This is not the case for pure quark stars, which can easily reach $2.5\,\Msun$ and still possess approximately the same amount of baryons as stable non-rotating stars. We also briefly discuss possible origins for fast rotating stars with a large amount of baryons and their stability, showing how the event GW190814 can be associated with a star containing quarks as one of its progenitors.

This article is devoted to inverse problems for nonlinear equations of the modified transfer, which can be regarded as a manageable problem. Various productions such problems for normal (unmodified) of the transport equation studied earlier by A.I. Prilepko and most complete form in the dissertation of his follower N. Volkov. If management is a factor in the coefficient of absorption or scattering indicatrix, even in the case of conventional linear transfer equation inverse problem become nonlinear (in the thesis N. P. Volkov overcome this difficulty). The paper deals with modification of the transport equation (for Similar to the way it is done Ozeen for the Navier - Stokes equations)

In this note, we give a geometric realization of the wheel conditions initiated by Feigin-Odesskii through the K-theoretic Hall algebra on the affine plane.

A fermionic three-site generalization of the popular bosonic Swanson model is studied as providing an exactly solvable five-parametric example of the quantum-mechanical unitary-evolution process leading to an ultimate loss of the observability and fall in an exceptional-point singularity (EP3). The instant of degeneracy is found to have an explicit one-parametric form. Its unitarity-compatible vicinity (i.e., the corridor of access to EP3) is also specified in closed form. The exact, numerical-error-independent solvability is found essential due to another, avoided, false energy-level crossing which is found to occur not too far from the true EP3 singularity.

While recent JWST observations of mature super-Earths and sub-Neptunes have frequently revealed featureless transmission spectra, their inflated progenitors offer a unique window into understanding their primordial compositions. As part of the KRONOS (Keys to Revealing the Origin and Nature Of sub-neptune Systems) JWST program, we present the NIRISS/SOSS transmission spectrum of V1298 Tau c, a $\sim$23 Myr super-Earth progenitor orbiting a young Solar analog. We detect H$_2$O in V1298 Tau c's atmosphere with a $\log_{10}$ volume mixing ratio of $-1.83^{+0.68}_{-0.77}$, but no additional molecules from these data alone. We find consistent results for the planetary atmospheric properties in both retrievals with and without informed priors on stellar heterogeneities based on the observed stellar spectrum. We infer an atmospheric metallicity [O/H] of $14.8^{+56.0}_{-12.28}\times$ the Solar value. This metallicity is similar to literature measurements for other young planets, including its massive outer companion V1298~Tau~b. In contrast, this measured metallicity is systematically lower than the metallicities of mature planets of similar mass and temperature. Altogether, these results provide tentative but growing evidence that the exoplanet mass--metallicity relation evolves with planetary age.

The development of radiation-tolerant materials capable of maintaining structural, electrical, and thermal stability in extreme, radiation-rich environments remains a critical challenge in materials science. In this work, the effects of 60 MeV 35Cl ion irradiation on highly oriented pyrolytic graphite (HOPG) and multilayer reduced graphene oxide (ML-rGO) were investigated. The samples were exposed to fluences of 5.11 x 10^9 and 1.3 x 10^10 ions/cm^2 and characterized by X-ray diffraction (XRD), Raman spectroscopy, scanning electron microscopy (SEM), atomic force microscopy (AFM), and electrical transport measurements. The results show that the irradiation response is strongly influenced by the initial structural organization of the material. In HOPG, ion exposure leads to a progressive loss of crystalline order, evidenced by XRD peak broadening and an increase in the Raman ID/IG ratio, accompanied by a reduction in electrical transport performance. In contrast, ML-rGO exhibits distinct behavior at higher fluences, suggesting partial structural reorganization. The appearance of more defined graphitic features in XRD and Raman analyses, along with changes in surface morphology and electrical response, suggests the formation of more ordered sp2 domains. These findings indicate that irradiation effects vary with the initial degree of order, providing useful insights for selecting carbon-based materials for devices operating under severe radiation conditions.

In this paper, we would like to investigate whether a generalized Einsteinian cubic gravity, in which three possible cubic interactions ${\cal P}$, ${\cal C}$, and ${\cal C}'$ are treated on an equal footing, admits a de Sitter solution as its stable cosmological solution. As a result, we are able to confirm the existence of the corresponding de Sitter solution for this gravity by solving analytically its field equations. Remarkably, only the cubic interaction ${\cal P}$ gives rise to the existence of the de Sitter solution. Then, we convert the field equations into the corresponding dynamical system for a stability analysis purpose. A fixed point of this dynamical system is found and shown to be equivalent to the obtained de Sitter solution. However, the perturbed dynamical system turns out to be incomplete, leaving undetermined information of the stability of the fixed point (or equivalently the de Sitter solution). Fortunately, we show that this loophole can be cured once the well-known Starobinsky term $R^2$ is introduced into the action of the generalized Einsteinian cubic gravity, despite the fact that it contributes nothing to the value of the de Sitter solution.

Developing Reynolds-averaged Navier-Stokes (RANS) turbulence models that remain accurate across diverse flow regimes is a long-standing challenge. In this work, we propose a novel framework, termed the progressive mixture-of-experts (PMoE), designed to enable continual learning for RANS turbulence modelling. The framework employs a modular autoencoder-based router to associate each flow scenario with a specialised turbulence model, referred to as an expert. When a new flow regime cannot be adequately represented by the existing router and expert set, a new expert together with its routing component can be introduced at low cost, without modifying or degrading previously trained ones, thereby naturally avoiding catastrophic forgetting. The framework is applied to a range of flows with distinct physical characteristics, including airfoil wake, channel, periodic hill, and square duct flows. The resulting PMoE model effectively integrates multiple experts and achieves improved predictive accuracy across both seen and unseen test cases that differ in operating conditions or configurations. Owing to sparse activation, model expansion does not incur additional computational cost during inference. The proposed framework therefore provides a scalable pathway towards lifelong-learning turbulence models for industrial computational fluid dynamics.

Recent work highlighted the importance of higher-order correlations in quantum dynamics for a deeper understanding of quantum chaos and thermalization. The full Eigenstate Thermalization Hypothesis, the framework encompassing correlations, can be formalized using the language of Free Probability theory. In this context, chaotic dynamics at long times are proposed to lead to free independence or "freeness" of observables. In this work, we investigate these issues in a paradigmatic semiclassical model - the kicked top - which exhibits a transition from integrability to chaos. Despite its simplicity, we identify several non-trivial features. By numerically studying 2n-point out-of-time-order correlators, we show that in the fully chaotic regime, long-time freeness is reached exponentially fast. These considerations lead us to introduce a large deviation theory for freeness that enables us to define and analyze the associated time scale. The numerical results confirm the existence of a hierarchy of different time scales, indicating a multifractal approach to freeness in this model. Our findings provide novel insights into the long-time behavior of chaotic dynamics and may have broader implications for the study of many-body quantum dynamics.

The gravitational wave event GW231123, with component black hole masses lying within or above the pair-instability mass gap, poses a significant challenge to current stellar evolution models. In this work, we describe how we investigated its origin by coupling the galaxy formation model GAMESH with the cluster population synthesis code RAPSTER and using two distinct binary population synthesis codes, SEVN and BSEEMP. This framework allowed us, for the first time, to reconstruct the life cycle of GW231123-like candidates within the same cosmological simulation, enabling a self-consistent comparison between different formation channels. Although both population synthesis codes can in principle produce black holes compatible with GW231123, we find that isolated binary evolution fails to reproduce the inferred merger redshift. In SEVN, massive black hole binaries form with semi-major axes > 10^3 Rsun , preventing coalescences within a Hubble time. In BSEEMP, candidates arise only at extremely low metallicities (Z = 10^{-10}), which contribute negligibly to the star formation rate density in our overdense simulated volume. Our results therefore strongly support a dynamical hierarchical origin. The observed black hole masses are naturally reproduced through successive mergers in dense globular clusters. The high dimensionless spins reported by the LIGO-Virgo-KAGRA Collaboration are consistent with this hierarchical population. We found a local merger rate density of 0.78 Gpc^{-3} yr^{-1}, with a peak at z = 4 - 6, tracing the maximum formation rate of globular clusters in metal-poor environments (Z = 0.006). Overall, GW231123 may represent a benchmark event for a robust population of hierarchical black holes formed in the early Universe.

Consecutive points in rotation curve measurements are correlated with each other, but this is usually ignored when constructing galaxy mass models. We apply a recently proposed data-driven approach to include the characteristic amplitude and scale length of such correlations as `nuisance parameters'. We construct mass models for $134$ galaxies from the SPARC rotation curve compilation with Navarro-Frenk-White (NFW) and pseudo-isothermal sphere (pISO) models for the dark halo. Allowing for correlations in the rotation curves generally improves the goodness of fit for both halo models, often yielding a formally good fit ($χ^2_\mathrm{r}\approx 1$) and model uncertainties that seem more representative of the constraining power of the data. For both halo models the inference on the typical correlation amplitude and scale length are very similar and physically plausible, $\sim 20\,\mathrm{km}\,\mathrm{s}^{-1}$ and $\sim 5\,\mathrm{kpc}$, respectively. The parametric form that we use to describe the correlations is intentionally simple, and our fitting approach makes the parameters describing possible correlations prone to `absorbing' other systematic errors, so we regard these estimates as upper limits. Without allowing for correlations we find a statistical preference for the pISO over the NFW model for $88$/$134$ galaxies; this preference essentially disappears when correlations are allowed for. Accounting for correlations in rotation curves when constructing mass models fundamentally affects how they are interpreted, highlighting an important systematic uncertainty that affects evidence for cusps or cores in dark matter haloes.

We experimentally investigate charge transport in In-CrSb and In-CrSb-In proximity devices, which are formed as junctions between superconducting indium leads and thick single crystal flakes of altermagnet CrSb. For double In-CrSb-In junctions, $dV/dI(B)$ curves are mirrored in respect to zero field for two magnetic field sweep directions, which is characteristic behavior of a Josephson spin valve. Also, we demonstrate Josephson diode effect by direct measurement of the critical current for two opposite directions in external magnetic field. We interpret these observations as a joint effect of the spin-polarized topological surface states and the altermagnetic spin splitting of the bulk bands in CrSb. For a single In-CrSb interface, the superconducting gap oscillates in magnetic field for both field orientations, which strongly resembles the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) behavior. FFLO is based on finite-momentum Cooper pairing, therefore, it is fully compatible with the requirements for the Josephson diode effect.

Interactions with the host galaxy strip stars and dark matter from the outer regions of satellite galaxies. Meanwhile, some stars from the central regions can migrate outward due to dynamical heating, producing an excess in the outer surface brightness relative to the extrapolation of the inner Sérsic profile. Recently discovered dark matter deficient galaxies (DMDGs) appear to be representative examples of such tidally disturbed systems. In this work, we investigate how the break radius, defined as the radius beyond which this surface brightness excess emerges, forms and evolves, by performing $N$-body simulations of a satellite galaxy interacting with a host, where the satellite serves as a plausible progenitor of a DMDG. Our simulations naturally reproduce a break radius consistent with that observed in DMDGs. We find that the break radius grows over time and exhibits a characteristic evolutionary behaviour: during each pericentric passage it briefly contracts due to tidal compression, and then rapidly and strongly expands as the satellite undergoes dynamical relaxation. After the satellite reaches a quasi-equilibrium configuration, the break radius shows only mild variations until the next pericentre. Across our suite of simulations, the ratio of the break radius to the effective radius remains approximately constant, even when we change the orbital parameters and internal structure of the satellite. Based on these findings, we develop a prescription for predicting the time evolution of the break radius, which can be used to constrain the tidal interaction history of satellite galaxies, including DMDGs and splashback galaxies.

Higher-order interactions have recently emerged as a promising framework for describing new dynamical phenomena in heterogeneous contagion processes. However, a fundamental open question is how to understand their contribution from the perspective of the physics of critical phenomena. Based on a mesoscopic field-theoretic Langevin description, we show that: (i) pairwise mechanisms such as facilitation or thresholding are formally equivalent to higher-order ones, (ii) pairwise interactions at coarse-grained scales govern the higher-order contact process and, (iii) the interplay between noise and topology is determined by the network spectral dimension. In short, we demonstrate that classical field theories, rooted on model symmetries and/or network dimensionality, still capture the nature of the phase transition, also predicting finite-size effects in real and synthetic networks.

As of mid-2026, 11 objects have been discovered prior to impacting the Earth, with warning times between 2 - 20 hours. Using real metre-sized Earth impactors from the last decade, we ask the question: ``If the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) had been operating over the last decade, how many imminent impactors would it have observed and discovered pre-impact, and how early would these discoveries have been?'' We use the LSST Solar System Survey Simulator Sorcha and a population of real fireballs observed by orbital sensors over the last decade to investigate which events would have been observed pre-impact. We find that the LSST would have observed 30 (13.9%) of the 216 simulated objects, with most objects receiving 2 - 4 observations. Using the default linking algorithm, only two (0.9%) of these objects would have been `discovered' pre-impact. Using a modified linking algorithm better suited to fast moving objects, this increases to eight (3.7%). Based on this, we predict that the LSST will discover 8 +/- 2 imminent impactors over its nominal 10 year survey, at the low end of previous estimations. However, we predict these objects to be discovered 4 days pre-impact, substantially earlier than the current average. This will bring significant opportunities for telescopic follow-up, targeted fireball observations, planetary defence planning, and public engagement. There is also significant potential for precovery for impactors observed by the LSST but discovered by other surveys, instantly lengthening observation arcs and thereby reducing the orbital and impact location uncertainties. In some cases, these observations may also enable the linkage of telescopic observations with observed fireballs post-impact, providing valuable pre-impact astrometric and photometric data. This has significant implications for both asteroid research and planetary defence.

Context: The detection of the highest energy neutrino observed to date by KM3NeT, with an estimated energy of 220 PeV, opens up new possibilities for the study and identification of the astrophysical sources responsible for a diffuse flux of such ultra-high-energy neutrinos, among which gamma-ray bursts are longstanding candidates. Aims: Based on the event KM3-230213A, we derive constraints on the baryon loading and density of the surrounding environment in models of blastwaves in long-duration gamma-ray bursts. Methods: We compute the diffuse flux from gamma-ray burst blastwaves, either expanding in a constant density interstellar medium or developing in a radially decreasing density of a wind-like environment surrounding the gamma-ray burst progenitor star, by taking into account the expected neutrino spectra and luminosity function. We use a Poisson likelihood method to constrain the blastwave model parameters by calculating the expected number of neutrino events within the 90% confidence level energy range of KM3-230213A and by using the joint exposure of KM3NeT/ARCA, IceCube and Pierre Auger. Results: We constrain the baryon loading to be $\leq \{392, 131, 39, 13\}$ at 90% confidence level, which is inversely proportional to a varying interstellar medium particle density of $\{1, 3, 10, 30\}$ cm$^{-3}$. In the wind-like environment case, the baryon loading is $\leq \{20, 50, 100\}$ at 90% confidence level, which is proportional to the sixth power of a varying density parameter of $\{0.05, 0.06, 0.07\}$.