Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension $n\geqslant3$. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space $\mathbb{R}^{n}$. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear equations corresponding to the Boussinesq system. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear equations on the half-line by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear equations to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.Comment: 18 page

On stability problem for the stationary Boussinesq system in Morrey-type spaces

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on $\mathbb{R}^n$, where $n \geqslant 3$. In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from $[0, \infty)$ to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates , Holder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces. Our results are generalized the previous ones of the steady Boussinesq systems in weak-$L^p$ spaces obtained by Hishida [T. Hishida, {\it On a class of Stable Steady flow to the Exterior Convection Problem}, Journal of Differential Equations, Vol. 141, Iss. 1 (1997), pages 54-85] and Ferreira et al. [L.C.F. Ferreira and E.J. Villamizar-Roa, {\it On the stability problem for the Boussinesq equations in weak-$L^p$ spaces}, Commun. Pure Appl. Anal. (2010), Vol. 9, No. 3, pages 667-684] and of the Navier-Stokes equations in Morrey spaces obtained by Kozono et al. [H. Kozono and M. Yamazaki, {\it The stability of small stationary solutions in Morrey Spaces of the Navier-Stokes equation}, Indiana University Mathematics Journal, Vol. 44, No. 4 (1995), pages 1307-1336].Comment: 17 page

Dynamics of evoked and spontaneous calcium transients in synaptic boutons of neocortical pyramidal neurons

In response to an action potential (AP), a transient rise in the intracellular calcium concentration ([Ca2+]i) causes transmitter release from nerve terminals. As the spatiotemporal dynamics of this calcium rise can affect the efficacy and plasticity of synaptic connections, it is essential to understand their determinants. To characterise factors that shape calcium transients in neocortical synaptic boutons, layer 5 pyramidal cells in the rat somatosensory cortex were filled through the patch pipette with a fluorescent calcium indicator for the measurement of [Ca2+]i. For accurate calculation of [Ca2+]i from the fluorescence intensity, the calcium binding affinities (Kd) of the indicators were measured in vitro, in solutions that were similar to the patch-clamp internal solution. These solutions were made with various concentrations of calcium chloride, but a constant concentration of a calcium buffer. The resultant free [Ca2+] was measured with a calcium-selective macroelectrode. It was found that the Kd values of the calcium indicators were considerably different from those previously published or provided by the manufacturers. Two main determinants of the intracellular calcium dynamics are the capacity of endogenous calcium buffers and the activity of calcium sequestration mechanisms. By measuring the peak amplitude of single AP-evoked calcium transients with different concentrations of OGB-1 or OGB-6F, a value of 7 was estimated for the calcium-binding ratio of endogenous buffers. Thus, in response to a single AP and in the absence of exogenous buffers, [Ca2+]i was raised by 5.3 microM, with a total change of approximately 50 microM. The rate constant of calcium sequestration (0.60 per s) was estimated from the slow decay time constant of the measured transients. The initial fast decay did not prolong when intracellular calcium uptake was inhibited, or speed up during repetitive stimulation. These findings suggest that calcium-induced calcium release (CICR), buffer saturation, and a non-linear calcium transporter were not the main cause of the bi-exponential decay. A 3D model of a bouton en passant showed that diffusion of calcium into the axon was likely the underlying mechanism. During high-frequency stimulation, CICR contributed to a supralinear summation of [Ca2+]i. Spontaneous increases in [Ca2+]i have been observed in several nerve terminals. They have been implicated in a number of cellular processes, including calcium homeostasis and spontaneous transmitter release. Here, the high-affinity calcium indicator OGB-1 was used to monitor small changes in [Ca2+]i. Spontaneous calcium transients (sCaTs) were observed at a frequency of around 0.2 per min. The increase in [Ca2+]i associated with each sCaT was 1.4–2.3 microM, in the absence of exogenous buffers. It was hypothesised that sCaTs arose from calcium release from presynaptic stores. In support of this, caffeine increased the average frequency of sCaTs by approximately 90%. The amplitude and kinetics of sCaTs identified in caffeine and in the control condition were not different from each other, suggesting that the majority of sCaTs might have been a result of calcium release through ryanodine receptors. The functional consequence(s) of sCaTs in neocortical synaptic boutons remains to be determined

Dispersion and nonlinearity properties of small solid-core photonic fibers with As2Se3 substrate

Characteristics of As2Se3 photonic crystal fibers (PCFs) with a solid-core and small-core diameter are numerically investigated in the long-wavelength range (from 2 to 10 μm). A full modal analysis and optical properties of designed photonic crystal fibers with lattice constant Λ and filling factor d/Λ are presented in terms of chromatic dispersion, effective refractive index, nonlinear coefficients, and confinement loss. The simulation results show that a high nonlinear coefficient of 4410.303 W–1·km–1 and a low confinement loss of 10−20 dB·km–1 can simultaneously be achieved in the proposed PCFs at a 4.5 μm wavelength. Chromatic dispersions are flat. The values of dispersion increase with increasing filling factor d/Λ and decrease with the increase in lattice constant Λ. In particular, some chromatic dispersion curves also cut the zero-dispersion line at two points. The flat dispersion feature, high nonlinearity, and small confinement loss of the proposed photonic crystal fiber structure make it suitable for supercontinuum

Chemical diversity of the Melaleuca cajuputi leaf oils from six locations in southern Vietnam

Melaleuca cajuputi Powell, a member of Myrtaceae family, is a popular and valuable plant in Vietnam, especially in the southern region. In this study, gas chromatography-mass spectrometry was used to determine the diversity in chemical composition of M. cajuputi leaf oil collected from six sites in the South of Vietnam. A total of sixty compounds has been identified from the essential oils of M. cajuputi leaf, such as α-thujene, α-pinene, benzaldehyde, β-pinene, β-myrcene, α-phellandrene, α-terpinene, benzene, 1-methyl-2-(1- methylethyl), 1,8-cineole, γ-terpinene, etc. Among those, five compounds were found in all six samples including α-thujene, α-pinene, α-phellandrene, γ-terpinene, and α-terpinolene, whereas other compounds were only present in some samples with varying amounts. Principal Component Analysis (PCA) has been performed to examine the similarities in the composition of leaf oils among the collection sites using XLSTAT software. The results from PCA suggested that six samples of leaf oil could be classified into 2 chemotypes with different chemical constituents. Note that 1,8-cineole only occurred in chemotype group 1 whereas it was omitted in chemotype group 2. This finding explains the diversity of bioactivity of M. cajuputi collected from different regions and provide more information on M. cajuputi for further application in medicine and food industry

Growth, survival and food utilization efficiency of longfin batfish (<em>Platax teira</em> Forsskål, 1775) larvae reared under different salinity levels

Salinity is crucial in fish larval rearing. In longfin batfish (Platax teira), little research has been conducted regarding the specific effects of salinity on growth, survival, deformity, and food utilization efficiency. This study aimed to determine the optimal salinity level for larval rearing of the longfin batfish by testing five different salinity levels (10, 15, 20, 25, and 30‰). Larvae of 1.5 cm in length and 0.2 g/fish in weight were stocked in cylindrical fiberglass tanks (300 L) at a density of 1 fish/L. The fish were fed to meet their dietary requirement and divided into four daily feedings. Each treatment was replicated three times over a 28-day period of rearing. The results revealed that salinity significantly influenced the growth (length, weight, biomass), and food utilization efficiency of the longfin batfish larvae. Overall, larvae reared at salinity levels of 15-20‰ exhibited superior performance compared to those exposed to salinity levels of 10, 25, and 30‰. However, salinity did not affect the coefficient of variation, survival, and deformity. From these findings, it is recommended to rear longfin batfish larvae at a salinity of 15-20‰ to achieve optimal growth and food utilization efficiency. This study provides valuable insights for longfin batfish larval rearing guidance, contributing to the aquaculture development of this economically valuable species