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 Mathematics , 2015, Abstract: We prove weighted $q$-variation inequalities with $2  Mathematics , 2010, Abstract: We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective manifolds. Using Ahlfors currents in higher dimension, we obtain several strong degeneracy statements.  Xi-Liang Fan Mathematics , 2012, Abstract: In the paper, Harnack inequalities are established for stochastic differential equations driven by fractional Brownian motion with Hurst parameter$H<1/2$. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.  Saroj Panigrahi Electronic Journal of Differential Equations , 2009, Abstract: In this paper, we obtain Liapunov-type integral inequalities for certain nonlinear, nonhomogeneous differential equations of higher order with without any restriction on the zeros of their higher-order derivatives of the solutions by using elementary analysis. As an applications of our results, we show that oscillatory solutions of the equation converge to zero as$t o infty$. Using these inequalities, it is also shown that$(t_{m+ k} - t_{m}) o infty $as$m o infty$, where$1 le k le n-1$and$langle t_m angle $is an increasing sequence of zeros of an oscillatory solution of$ D^n y + y f(t, y)|y|^{p-2} = 0$,$t ge 0$, provided that$W(., lambda) in L^{sigma}([0, infty), mathbb{R}^{+})$,$1 le sigma le infty$and for all$lambda > 0$. A criterion for disconjugacy of nonlinear homogeneous equation is obtained in an interval$[a, b]$.  Mathematics , 2011, Abstract: This paper is devoted to the study of$L_{p}$Lyapunov-type inequalities ($ \ 1 \leq p \leq +\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in$\real^{N}$. It is proved that the relation between the quantities$p$and$N/2\$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.