Abstract:
In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.

Abstract:
In the first part of this paper we prove some new Poincar\'e inequalities, with explicit constants, for domains of any hypersurfaces of Riemannian manifolds with sectional curvature bounded from above, involving the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurfaces. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as consequences we derive some monotonicity results involving the integral of the mean curvature.

Abstract:
We consider the integrals of $r$-mean curvatures $S_r$ of a complete hypersurface $M$ in space forms $\mathcal{Q}_c^{n+1}$ which generalize volume $(r=0)$, total mean curvature $(r=1)$, total scalar curvature $(r=2)$ and total curvature $(r=n)$. Among other results we prove that a complete properly immersed hypersurface of a space form with $S_r\geq 0$, $S_r\not\equiv 0$ and $S_{r+1}\equiv 0$ for some $r\le n-1$ has $\int_MS_rdM=\infty.$

Abstract:
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int_M|A|^3<+\infty$).

Abstract:
The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type formula for such a kind of operators and as applications we obtain some sharp estimates for the first nonzero eigenvalues in two special cases. These results can be considered as generalizations of the Lichnerowicz-Obata Theorem.

Abstract:
In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the extrinsic diameter of the domain.

Abstract:
objective:to analyze the relationship between the acute phase reactants and the disease activity of juvenile idiopathic arthritis (jia) and to evaluate the agreement between erythrocyte sedimentation rate and c-reactive protein during the acute phase of the disease. methods: a cohort retrospective study has been conducted based on the analysis of 30 children and adolescents who fulfilled the diagnostic criteria of jia. all of them were in current follow-up at the pediatric rheumatology outpatient clinic and had acute phase reactants blood tests performed. results: studied population comprised 30 patients: 21 (70%) of them were females and 19 (63.3%) presented oligoarticular subtype. the mean age at disease onset was 65.6 months; the age at diagnosis was 85.3 months and the follow-up had 57.2 months of duration. the acute phase reactants showed positive association with the disease activity. anemia was not associated with disease activity. during the acute phase of the disease, agreement between erythrocyte sedimentation rate and c-reactive protein was greater than 80%. conclusions: the acute phase reactants have a positive association with the activity of the disease and using both tests simultaneously increases their specificity.

Abstract:
It has long been argued that spatial aspects of language influence people’s conception of time. However, what spatial aspect of language is the most influential in this regard? To test this, two experiments were conducted in Hong Kong and Macau with literate Cantonese speakers. The results suggest that the crucial factor in literate Cantonese people’s spatial conceptualization of time is their experience with writing and reading Chinese script. In Hong Kong and Macau, Chinese script is written either in the traditional vertical orientation, which is still used, or the newer horizontal orientation, which is more common these days. Before the 1950s, the dominant horizontal direction was right-to-left. However, by the 1970s, the dominant horizontal direction had become left-to-right. In both experiments, the older participants predominately demonstrated time in a right-to-left direction, whereas younger participants predominately demonstrated time in a left-to-right direction, consistent with the horizontal direction that was prevalent when they first became literate.

Abstract:
In this paper we use Dirichlet's theorem in order to elementally prove two theorems. The first says that since a polynomial ax+b generates one prime, it also generates infinites. The second theorem (which is proved in a very simillar way to the first) says that x^2+1 generates infinitely many primes.