On the first eigenvalue of the linearized operator of ther-th mean curvature of a hypersurface

We generalize Reilly's inequality for the first eigenvalue of immersed submanifolds ofIR ^m +¹ and the total (squared) mean curvature, to hypersurfaces ofIR ^m +¹ and the first eigenvalue of the higher order curvatures. We apply this to stability problems. We also consider hypersurfaces in hyperbolic space.     

Essential Oils Extracted from Organic Propolis Residues: An Exploratory Analysis of Their Antibacterial and Antioxidant Properties and Volatile Profile

The industrial processing of crude propolis generates residues. Essential oils (EOs) from propolis residues could be a potential source of natural bioactive compounds to replace antibiotics and synthetic antioxidants in pig production. In this study, we determined the antibacterial/antioxidant activity of EOs from crude organic propolis (EOP) and from propolis residues, moist residue (EOMR), and dried residue (EODR), and further elucidated their chemical composition. The EOs were extracted by hydrodistillation, and their volatile profile was tentatively identified by GC-MS. All EOs had an antibacterial effect on Escherichia coli and Lactobacillus plantarum as they caused disturbances on the growth kinetics of both bacteria. However, EODR had more selective antibacterial activity, as it caused a higher reduction in the maximal culture density (D) of E. coli (86.7%) than L. plantarum (46.9%). EODR exhibited mild antioxidant activity, whereas EOMR showed the highest antioxidant activity (ABTS = 0.90 μmol TE/mg, FRAP = 463.97 μmol Fe²⁺/mg) and phenolic content (58.41 mg GAE/g). Each EO had a different chemical composition, but α-pinene and β-pinene were the major compounds detected in the samples. Interestingly, specific minor compounds were detected in a higher relative amount in EOMR and EODR as compared to EOP. Therefore, these minor compounds are most likely responsible for the biological properties of EODR and EOMR. Collectively, our findings suggest that the EOs from propolis residues could be resourcefully used as natural antibacterial/antioxidant additives in pig production.     

Surfaces of M_k^2\times \mathbb R invariant under a one-parameter group of isometries

We assume that an immersed constant mean curvature surface ${\varSigma } \looparrowright {M_k} \times \mathbb R $ satisfies a relation involving the mean curvature, the Gaussian curvature and the angle that the unit vector of the factor $\mathbb R $ makes with the normal to the surface. This relation, although given initially in its pointwise form, can be shown to be equivalent to an integral relation. From the assumed relation, it follows that $\varSigma $ is invariant under a one-parameter group of isometries of $M_k^2\times \mathbb{R }$ which are induced by the isometries of $M_k^2$  . An application is made to describe qualitatively those surfaces for which the Abresch-Rosenberg complex quadratic form vanishes.     

Crackling sound generation during the formation of liquid bridges: A lattice gas model

Due to abnormal mechanical instabilities, liquid bridges may form in the small airways blocking airflow. Liquid bridge ruptures during inhalation are the major cause of the crackling adventitious lung sound, which can be heard using a simple stethoscope. Recently, Vyshedskiy and colleagues (2009) [1] described and characterized a crackle sound originated during expiration. However, the mechanism and origin of the expiratory crackle are still controversial. Thus, in this paper, we propose a mechanism for expiratory crackles. We hypothesize that the expiratory crackle sound is a result of the energy released in the form of acoustic waves during the formation of the liquid bridge. The magnitude of the energy released is proportional to the difference in free energy prior and after the bridge formation. We use a lattice gas model to describe the liquid bridge formation between two parallel planes. Specifically, we determine the surface free energy and the conditions of the liquid bridge formation between two parallel planes separated by a distance 2h$2h$ by a liquid droplet of volume Ω$\Omega$ and contact angle Θ$\Theta$, using both Monte Carlo simulation of a lattice gas model and variational calculus based on minimization of the surface area with the volume and the contact angle constrained. We numerically and analytically determine the phase diagram of the system as a function of the dimensionless parameter hΩ^−1/3$h{\Omega }^{-1/3}$ and Θ$\Theta$. We can distinguish two different phases: one droplet and one liquid bridge. We observe a hysteresis curve for the energy changes between these two states, and a finite size effect in the bridge formation. We compute the release of free energy during the formation of the liquid bridge and discuss the results in terms of system size. We also calculate the force exerted from liquid bridge on the planes by studying the dependence of the free energy on the separation between the planes 2h$2h$. The simulation results are in agreement with the analytical solution.     

On the Gauss map of hypersurfaces with constant scalar curvature in spheres

In this work we consider connected, complete and orientable hypersurfaces of the sphere with constant nonnegative -mean curvature. We prove that under subsidiary conditions, if the Gauss image of is contained in a closed hemisphere, then is totally umbilic.

     

Relating airway diameter distributions to regular branching asymmetry in the lung

We study the distribution Pi(n)(D) of airway diameters D as a function of generation N in asymmetric airway trees of mammalian lungs. We find that the airway bifurcations are self-similar in four species studied. Specifically, the ratios of diameters of the major and minor daughters to their parent are constants independent of N until a cutoff diameter is reached. We derive closed form expressions for Pi(N)(D) and examine the flow resistance of the tree based on an asymmetric flow division model. Our findings suggest that the observed diameter heterogeneity is consistent with an underlying regular branching asymmetry.     

Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

For a normal variation of a hypersurface M ⁿ in a space form Q _c ⁿ⁺¹ by a normal vector field fN, R. Reilly proved:

Sedimentation rate inferred from 210Pb and 137Cs dating of three sediment cores at Itaipu reservoir (Paraná State,Brazil) the world’s second largest hydroelectricity producer

Journal of Radioanalytical and Nuclear Chemistry - This study provides the sedimentation rates (SR) of three drainage basins at Itaipu Reservoir (Paraná State, Brazil) by applying 210Pb and...     

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.