In this article, we introduce novel variations on several well-known model-theoretic tree properties (TPs), and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the $\mathcal {I}$-tree property ($ \mathcal {I}$-TP), for an arbitrary Ramsey index structure $\mathcal {I}$. We focus attention on the colored linear order index structure c, showing that c-TP is equivalent to instability. After introducing c-$\text {TP}_1$ and c-$ \text {TP}_2$, we prove that c-$ \text {TP}_1$ is equivalent to $\text {TP}_1$, and that c-$ \text {TP}_2$ is equivalent to IP. We see that these three TPs give a dichotomy theorem, just as with TP, $\text {TP}_1$, and $\text {TP}_2$. Along the way, we observe that appropriately generalized tree index structures $\mathcal {I}^{<\omega }$ are Ramsey, allowing for the use of generalized tree indiscernibles.

Globally, macadamia nut production generates substantial amounts of by-products such as macadamia husk, which is rich in phenolic compounds known for their antioxidant and health benefits(1). In this study, we encapsulated a phenolic-rich extract from macadamia green husk into soy-lecithin liposomes and applied a chitosan biopolymer coating to protect phenolic integrity, enhance physical stability(2), and improve delivery for food applications(3), as the phenolic compounds are often unstable and susceptible to environmental conditions such as heat, light, enzymes, and pH. Liposomes were prepared in aqueous media by high-pressure homogenization (2,000 MPa) and subsequently coated with 0.4% (w/v) chitosan. Particle size, ζ-potential, and polydispersity index (PDI) were measured by dynamic light scattering, morphology was assessed by transmission electron microscopy (TEM), and encapsulation efficiency (EE) was determined by quantifying free total phenols (unencapsulated phenolics). In vitro bioaccessibility was evaluated by measuring total phenolic content during an in vitro simulated gastrointestinal digestion. Empty and extract-loaded liposomes had mean diameters ranging between ~77 to 112 nm. Chitosan coating increased vesicle size (empty: 523 nm; loaded: 501 nm) and produced a broader size distribution (PDI = 0.41). Uncoated liposomes were strongly anionic (ζ = −62 mV empty; −43 mV loaded); chitosan coating reversed the surface charge to strongly positive values of +48 mV for empty and +50 mV for loaded liposomes, indicating successful adsorption of the cationic polymer. TEM confirmed largely spherical vesicles with sizes consistent with those measured by dynamic light scattering. Coated liposomes achieved higher encapsulation efficiency (91.15%) than uncoated liposomes (81.50%). Simulated digestion showed that liposomal encapsulation provided good protection under gastric (low pH) conditions but lost membrane integrity under intestinal conditions, releasing the phenolic compounds. The release of phenolic compound from coated liposomes was lower than uncoated liposomes. These results demonstrate that chitosan-coated liposomes improve the stability and in vitro bioaccessibility of macadamia husk-derived phenolics and provide practical design guidance for the development of liposomal delivery systems in food and pharmaceutical applications.

We study the semi-infinite Neumann problem, which models the variation law of a cline in a semi-infinite habitat. Using the bifurcation analysis method, we find that there is a unique solution curve emanating from $(\arctan \alpha ,0)$ with $\alpha> 0$, which is strictly increasing and approaches $1$ in $C[0,+\infty )$. Furthermore, we show that any cline (bifurcation solution) is stable, thereby providing a confirmed answer to a conjecture. Moreover, we also establish the stability of the trivial solution. Our conclusions are consistent with the related numerical results and biological reality.

Compared with superhydrophobic coatings, superamphiphobic coatings offer broader application potential in various fields. However, their practical applications are often hindered by insufficient repellency towards low-surface-tension, high-viscosity and complex multicomponent liquids, as well as by their stability. Herein, a palygorskite-based stable superamphiphobic coating with a simple fabrication process and low cost is reported. The coating was fabricated through sequential spray deposition of a fluorosilicone resin (FSR) bonding layer and a superamphiphobic functional layer composed of fluorinated palygorskite (F-PAL), fluorinated carbon black (F-CB) and FSR. In this coating, F-PAL nanorods serve as the building blocks for constructing the primary nanostructure, while F-CB nanoparticles occupy the interstitial spaces between the PAL nanorods, forming the secondary nanostructure. Together, these components generate a multiscale hierarchical micro-/nanostructure that is critical for achieving superamphiphobicity. By systematically optimizing the mass ratio of F-CB to F-PAL and the FSR content, a low-surface-energy multiscale hierarchical micro-/nanostructure was successfully established, imparting the coating with excellent superamphiphobicity towards water, hydroxyl-terminated polybutadiene/dioctyl sebacate mixtures (HTPB-H) and their aluminium powder-containing suspensions (HTPB-H/Al). The coating exhibits outstanding mechanical, chemical and thermal stability. Furthermore, the coating demonstrates remarkable anti-adhesion performance against both HTPB-H and HTPB-H/Al, together with superior passive anti-icing performance, as evidenced by a significantly prolonged freezing delay time (1291 s, ∼30-fold increase) and a substantially reduced ice adhesion strength (from 224.3 to 65.8 kPa). Moreover, efficient photothermal de-icing performance was achieved under low-temperature, high-humidity and weak-light conditions. These results demonstrate that the proposed superamphiphobic coating holds significant promise for practical applications in anti-adhesion and anti-icing under harsh conditions.

In this paper, we study a nonlinear free boundary problem modelling the growth of radially symmetric tumours. The tumour consists of a central necrotic core, an intermediate quiescent layer and an outer proliferating shell. The evolution of tumour layers and the movement of the tumour boundary are totally governed by external nutrient supply and conservation of mass. The three-layer structure generates three free boundaries with discontinuous nutrient consumption rates and cell growth rates. We develop a nonlinear analysis method to clarify the interactive relationships among free boundaries. By carefully studying the dependence of the critical-state tumour growth rate on the external nutrient concentration, we reveal the evolutionary mechanism in tumour growth and the mutual transformation of its internal structures. The existence and uniqueness of the radial stationary solution is proved, and its globally asymptotic stability towards different dormant tumour states is established.

In this study, we develop epidemic reaction-diffusion models by incorporating the dependency of the diffusion rate of susceptible individuals on new infection cases, employing both Fickian and Fokker–Planck-type diffusion laws. As the first part of a two-part series, we focus on epidemics driven by frequency-dependent incidence. We explore linear, exponential and algebraic relationships between diffusion rate of the susceptible population and new infection cases to provide deeper biological insights. Our analysis establishes the global existence of solutions and characterizes the threshold dynamics using basic reproduction numbers. We find that in quasilinear parabolic systems, the Fokker–Planck-type diffusion law tends to induce spatial segregation of susceptible and infected individuals, while the Fickian law favours spatial homogenization of susceptible individuals. Additionally, the Fokker–Planck-type model, where the diffusion rate of infected individuals depends on new recovery cases, more accurately captures the cognitive diffusion behaviour of individuals.

In this article, we first investigate weighted Minkowski-type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of Glaudo (2022, Adv. Math. 408, 108595) by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov–Fenchel-type inequalities in $\mathbb {R}^{n+1}$ and $\mathbb {H}^{n+1}$, extending the earlier results of VanBlargan and Wang (2024, Commun. Contemp. Math. 26, 2350026) and Zhou and Zhou (2024, J. Geom. Anal. 34, 376). These inequalities hold for nearly spherical sets that are $W^{2,\infty }$-close to geodesic spheres coupled with general convex weights.

In this paper, we show the existence, uniqueness and stability of nontrivial solutions to the following Minkowski-curvature problems on unbounded domains:

$$ \begin{align*} \bigg(\frac{x'}{\sqrt{1-x^{\prime2}}}\bigg)'=f(t,\ x),\quad t\geq t_0, \quad \lim_{t\to\infty}x(t)=\psi_0,\quad \lim_{t\to\infty}x'(t)e^{t}=0, \end{align*} $$

$f:\ [t_0, \infty )\times \mathbb {R}\rightarrow \mathbb {R}$

$t_0>0$

$\psi _0\in \mathbb {R}$

Chrysoperla carnea (Stephens) (Neuroptera: Chrysopidae) is an important agricultural biocontrol agent that preys on various soft-bodied insect pests. This study was carried out to evaluate the fitness parameters of bifenthrin-selected (Bifen-Sel) and unselected (Unsel) strains of C. carnea and their reciprocal crosses (C1 and C2) by using an age-stage and two-sex life table approach. After continuous selection with bifenthrin (44 generations), the Bifen-Sel strain of C. carnea developed a 9.55-fold resistance level when compared to that of the Unsel. Biological parameters, such as the pupal period, pre-adult duration, male and female longevity, adult pre-oviposition period, and oviposition period of the Bifen-Sel strain, all were significantly different from that of the Unsel strain and their crosses C1 and C2. The larval period, total pre-oviposition period, female ratio, reproductive female ratio, and fecundity of the Bifen-Sel strain were significantly different from those of the Unsel strain. Demographic parameters, including intrinsic rate (rm), finite rate (λ), net reproductive rate (R0), mean generation rate (T), and gross reproductive rate of the Bifen-Sel strain, were similar to those of the Unsel strain but significantly higher than those of the C1 and C2 crosses of C. carnea. Bifenthrin resistance was stable in the Bifen-Sel strain of C. carnea. These findings provide valuable insights into bifenthrin resistance dynamics in C. carnea as it resulted in no fitness cost and support its selective use under controlled conditions, while highlighting the need for future research on predation potential of this resistant strain under field conditions with bifenthrin applications.

We study a size-structured tree growth model from [4–6], described by the nonlinear renewal equation $\phi(t) = \mathfrak{F} \phi_t, \ \phi_t \in L^1_\rho(\mathbb{R}_{-}),$ with reproduction, death, and growth rates $\beta$, $\mu$, and $g$. We prove that, under mild conditions on these rates, the equation generates a semiflow in $L^1_\rho(\mathbb{R}_{-})$ that is permanent and possesses a compact global attractor $\mathcal{A}$. If $\beta$ is monotone, $\mathcal{A}$ reduces to a single asymptotically stable equilibrium attracting all compact sets with positive initial data. Adapting an approach from [21], originally developed for simpler renewal equations, we investigate stability and persistence in this more complex setting via the one-dimensional recurrence $b_{n+1} = \mathfrak{F} b_n,$ thereby complementing the functional-analytic framework of [13].