Abstract:
We show that the Poincar\'e lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincar\'e lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction, and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical (m=0), not all closed forms are invariant. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

Abstract:
Using ideas from Boukerrioua and Guezane-Lakoud (2008), some nonlinear integral inequalities are established. 1. Introduction Integral inequalities provide a very useful and handy device for the study of qualitative as well as quantitative properties of solutions of differential equations. The Gronwall-Bellman type (see, e.g., [1–4]) is particularly useful in that they provide explicit bounds for the unknown functions. One of the most useful inequalities in the development of the theory of diferential equations is given in the following theorem. Theorem 1.1 (see [3]). If and are non-nonnegative continuous functions on satisfying for some constant , then The importance of this inequality lies in its successful utilization of the situation for which the other available inequalities do not apply directly. It has been frequently used to obtain global existence, uniqueness, stability, boundedness, and other properties of the solution for wide classes of nonlinear differential equations. The aim of this paper is to give other results on nonlinear integral inequalities and their applications. 2. Main Results In this section, we begin by giving some material necessary for our study. We denote by , the set of real numbers and the nonnegative real numbers. Lemma 2.1. For , one has Lemma 2.2 (see [1]). Let and be continuous functions for , let be a differentiable function for and suppose Then for , Now we state the main results of this work Theorem 2.3. Let be real-valued nonnegative continuous functions and there exists a series of positive real numbers and satisfy the following integral inequality, for then for . Proof. Define a function by then and (2.4) can be written as By (2.7) and Lemma 2.1, we get Differentiating (2.6), we get Using (2.8) and (2.9), it yields where By Lemma 2.2, we have Using (2.7) and (2.12), we get This achieves the proof of the theorem. Remark 2.4. if we take , then the inequality established in Theorem 2.3 become the inequality given in [4, Theorem ]. Theorem 2.5. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function is nondecreasing and for then for and where Proof. For By (2.7) and the fact that , one gets: Differentiating (2.6) and using (2.17), we obtain then Since the function is nondecreasing, for then, where Consequently For , we can see that then the function can be estimated as Let Now we estimate the expression by using (2.24) to get Remarking that we integrate (2.27) from 0 to to get replacing by its value in (2.28), we obtain then Using (2.7), (2.23), and (2.30) we have, This achieves the proof of the

Abstract:
We prove moving lemma for additive higher Chow groups of smooth projective varieties. As applications, we prove the very general contravariance property of additive higher Chow groups. Using the moving lemma, we establish the structure of graded-commutative differential graded algebra (CDGA) on these groups.

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.

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.

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 $[a, b]$.

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.

Abstract:
The investigation of initial value problems of differential equations where the initial time differs with each solution is initiated in this paper. Basic preliminary results in qualitative theory are discussed to understand possible ramifications.

Abstract:
In this article, we study linear differential equations of higher-order whose coefficients are square matrices. The combinatorial method for computing the matrix powers and exponential is adopted. New formulas representing auxiliary results are obtained. This allows us to prove properties of a large class of linear matrix differential equations of higher-order, in particular results of Apostol and Kolodner are recovered. Also illustrative examples and applications are presented.