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The small parameter method allows one to construct solutions of differential equations in the form of power series and has become widespread in mathematical physics. In most cases, these series are asymptotically convergent. The aim of this work is to find conditions for the ordinary convergence of series in powers of a small parameter representing solutions of perturbation theory problems.

(1) Background: This paper is devoted to the study of a class of semilinear initial boundary value problems of parabolic type. (2) Methods: We make use of fractional powers of analytic semigroups and the interpolation theory of compact linear operators due to Lions–Peetre. (3) Results: We give a functional analytic proof of the C2 compactness of a bounded regular solution orbit for semilinear parabolic problems with Dirichlet, Neumann and Robin boundary conditions. (4) Conclusions: As an application, we study the dynamics of a population inhabiting a strongly heterogeneous environment that is modeled by a class of diffusive logistic equations with Dirichlet and Neumann boundary conditions.

Nuclear bodies are RNA and protein-rich, membraneless organelles that play important roles in gene regulation. The largest and most well-known nuclear body is the nucleolus, an organelle whose primary function in ribosome biogenesis makes it key for cell growth and size homeostasis. The nucleolus and other nuclear bodies behave like liquid-phase droplets and appear to condense from the nucleoplasm by concentration-dependent phase separation. However, nucleoli actively consume chemical energy, and it is unclear how such nonequilibrium activity might impact classical liquid–liquid phase separation. Here, we combine in vivo and in vitro experiments with theory and simulation to characterize the assembly and disassembly dynamics of nucleoli in earlyCaenorhabditis elegansembryos. In addition to classical nucleoli that assemble at the transcriptionally active nucleolar organizing regions, we observe dozens of “extranucleolar droplets” (ENDs) that condense in the nucleoplasm in a transcription-independent manner. We show that growth of nucleoli and ENDs is consistent with a first-order phase transition in which late-stage coarsening dynamics are mediated by Brownian coalescence and, to a lesser degree, Ostwald ripening. By manipulatingC. eleganscell size, we change nucleolar component concentration and confirm several key model predictions. Our results show that rRNA transcription and other nonequilibrium biological activity can modulate the effective thermodynamic parameters governing nucleolar and END assembly, but do not appear to fundamentally alter the passive phase separation mechanism.

We are concerned with the inviscid limit of the Navier–Stokes equations to the Euler equations for barotropic compressible fluids in R3.When the viscosity coefficients obey a lower power law of the density (i.e., ρδ with 0<δ<1), we identify a quasi-symmetric hyperbolic–singular elliptic coupled structure of the Navier–Stokes equations to control the behavior of the velocity of the fluids near a vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. Some uniform estimates on both the local sound speed and the velocity in H3(R3)with respect to the viscosity coefficients are also obtained, which lead to the strong convergence of the regular solutions of the Navier–Stokes equations with finite mass and energy to the corresponding regular solutions of the Euler equations in L∞([0,T];Hlocs​(R3)) for any s∈[2,3). As a consequence, we show that, for both viscous and inviscid flows, it is impossible that the L∞ norm of any global regular solution with vacuum decays to zero asymptotically, as t tends to infinity. Our framework developed here is applicable to the same problem for other physical dimensions via some minor modifications.

The partitioning behavior of Cl and F between apatite and sediment melt has been investigated by performing piston-cylinder experiments at 2.5 GPa, 800 °C using a hydrous experimental pelite staring material (EPSM) with ∼7 wt% H2O and variable Cl (∼0, 500, 1000∼0, 500, 2000, or 3000 ppm) and F (∼0, 700, or 1500 ppm) contents, relevant for subduction zone conditions. Cl and F partitioning between apatite and melt is non-Nernstian, with DClAp-melt varying from 1.9-10.6 and DFAp-melt varying from 16-72. In contrast, Cl and F partition coefficients between phengite/biotite and melt (DClPhen-melt, DClBi-melt, DFPhen-melt, and DFBi-melt) were determined to be 0.24 ± 0.01, 0.86 ± 0.05, 1.4 ± 0.1, and 3.7 ± 0.4, respectively. The Nernstian partitioning of Cl and F between phengite/biotite and melt suggests ideal mixing of F, Cl, and OH in phengite, biotite, and melt. Exchange coefficients for F, Cl, and OH partitioning between apatite and melt were determined, with KdCl-OHAp-melt = 19-49, KdF-OHAp-melt = 164-512, and KdF-ClAp-melt = 7-21. The evident variation of Kd values was attributed to non-ideal mixing of F, Cl, and OH in apatite. A regular ternary solution model for apatite was developed by modeling the variation of Kd values for experiments from this study and those from Webster et al. (2009) and Doherty et al. (2014) Positive values (∼15 to ∼25 kJ/mol) obtained for Margules parameters WApCl-OH, WApF-Cl, and WApF-OH at low-pressure conditions (0.2 GPa, 0.05 GPa, and 900 °C) are in contrast to zero or negative values at 2.5 GPa, 800 °C. Based on a thermodynamic framework for F, Cl, and OH exchange between apatite and melt, using values for -ΔrG°Cl-OH(P,T), -ΔrG°F-OH(P,T), -ΔrG°F-Cl(P,T), WApCl-OH, WApF-Cl, and WApF-OH obtained through regression, F and Cl contents in melt can be derived from apatite compositions.

Solutions of drugs may behave as ideal solutions, real solutions, or irregular solutions. It is necessary to understand the behaviour of these solutions before attempting to handle them. Various theories/models are reported in the literature to explain their behaviour. The importance of models in predicting the solubility of aripiprazole is demonstrated using its solubility in dioxane-water blends. The method utilizes theoretical and semiempirical approaches to predict solubility. The experimental solubility data for aripiprazole are validated using both ideal and nonideal solutions, focusing on the Scatchard-Hildebrand equation for regular solutions. Furthermore, the Extended Hildebrand Solubility approach is employed to identify the most suitable equation that yields calculated solubility data in agreement with experimental results. Interestingly, a method that directly correlates the solubility parameter of solvent combinations with the logarithm of the mole fraction solubility produces findings comparable to those obtained with the Extended Hildebrand Solubility approach. The results imply that aripiprazole solutions behave as irregular solutions. The solubility profile of aripiprazole may be precisely determined using a quartic equation developed based on regression of activity coefficient versus solubility parameter of the solvent blends. This method saves time and money compared to experimental methods.

Pharmaceutical, food, and cosmetic formulations often contain binary or ternary surfactant mixtures with synergistic interactions amongst micellar building blocks. Here, a ternary mixture of the surfactants hexadecyltrimethylammonium bromide, dodecyltrimethylammonium bromide, and sodium deoxycholate is examined to see if the molar fractions of the surfactants in the ternary mixed micellar pseudophase are determined by the interaction coefficients between various pairs of the surfactants or by their propensity to self-associate. Critical micelle concentrations (CMC) of the analyzed ternary mixtures are determined experimentally (spectrofluorimetrically using pyrene as the probe molecule). Thermodynamic parameters of ternary mixtures are calculated from CMC values using the Regular Solution protocol. The tendency for monocomponent surfactants to self-associate (lower value of CMC) determines the molar fractions of surfactant in the mixed micelle if there is no issue with the packing of the micelle building units of the ternary mixed micelle. If a more hydrophobic surfactant is incorporated into the mixed micelle, the system (an aqueous solution of surfactants) is then the most thermodynamically stabilized.

We study the solvability of the Ionkin problem for some differential equations with one space variable. These equations include parabolic and quasiparabolic, hyperbolic and quasihyperbolic, pseudoparabolic and pseudohyperbolic, elliptic and quasielliptic equations and equations of many other types. For the above equations, the following theorems are proved with the use of the splitting method: the existence of regular solutions—solutions that all have weak derivatives in the sense of S. L. Sobolev and occur in the corresponding equation.

The existence of regular and singular bound state solutions to △ p u + f ( u ) = 0 , r ∈ R n ∖ { 0 } is considered. Our result concerns the solution according to its behavior as r → 0 and r → ∞ . Under the assumption that f is supercritical for small u > 0 and is subcritical for large u > 0 , we show the existence of various types of solutions. The Pohozaev identity plays a crucial role in our investigation.

Silicate liquid immiscibility leading to formation of mixtures of distinct iron-rich and silica-rich liquids is common in basaltic and andesitic magmas at advanced stages of magma evolution. Experimental modeling of the immiscibility has been hampered by kinetic problems and attainment of chemical equilibrium between immiscible liquids in some experimental studies has been questioned. On the basis of symmetric regular solutions model and regression analysis of experimental data on compositions of immiscible liquid pairs, we show that liquid–liquid distribution of network-modifying elements K and Fe is linked to the distribution of network-forming oxides SiO2, Al2O3 and P2O5 by equation: logKdK/Fe=3.796ΔXSiO2sf+4.85ΔXAl2O3sf+7.235ΔXP2O5sf-0.108,where KdK/Fe is a ratio of K and Fe mole fractions in the silica-rich (s) and Fe-rich (f) immiscible liquids: KdK/Fe=XKs/XKf/XFes/XFef and ΔXisf is a difference in mole fractions of a network-forming oxide i between the liquids (s) and (f): ΔXisf=Xis-Xif. We use the equation for testing chemical equilibrium in experiments not included in the regression analysis and compositions of natural immiscible melts found as glasses in volcanic rocks. Departures from equilibrium that the test revealed in crystal-rich multiphase experimental products and in natural volcanic rocks imply kinetic competition between liquid–liquid and crystal–liquid element partitioning. Immiscible liquid droplets in volcanic rocks appear to evolve along a metastable trend due to rapid crystallization. Immiscible liquids may be closer to chemical equilibrium in large intrusions where cooling rates are lower and crystals may be spatially separated from liquids.