RCAAP Repository

In this work, we will do some applications of the Moving Spheres method, a variant of the method of Moving Planes, in order to obtain some Liouville-type theorems and some Harnack-type inequalities in Rn, as well as in the Euclidian half space Rn +. Our study focuses on, mostly, in the article written by Yan Yan Li and Lei Zhan [32], as well as some references of the same article. We concentrate in studying some properties of positive solutions of some semilinear elliptic partial differential equations with critical exponent and giving different proofs, improvements, and extensions of some previously established Liouville-type theorems and Harnack-type inequalities.

In this work, we study the semisimple group algebras FqCn of the finite abelian groups Cn over a finite field Fq and give conditions so that the number of its simple components is minimal; i.e. equal to the number of simple components of the rational group algebra of the same group. Under such conditions, we compute the set of primitive idempotents of FqCn and from there, we study the abelian codes as minimal ideals of the group algebra, which are generated by the primitive idempotents, computing their dimension and minimum distances.

In this dissertation, a way to build dense STBCs with full diversity of maximal order in central simple algebra will be presented. We constructed a retriculated ST code with a nonzero determinant for a quad antenna MISO transmission. Also, we will present a general algorithm to test the limit of a given order, since by the use of a maximum order instead of just the algebraic integer ring, we can increase the capacity of the code without a loss in the minimum determinant. Furthermore, by using the ideal of a maximum order we can further improve the code, as we increase the minimum determinant.

We consider a dynamical one-dimensional nonlinear Von Kármán model for beams depending on the parameter " > 0 and we study their asymptotic behavior for t large, when " ! 0. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponential uniform with respect to the parameter ". In order for this to be true the damping mechanism has to have the appropriate scale with respect to ". In the limit as " ! 0 we obtain damped Berger- Timoshenko beam model for which the energy tends exponentially to zero. This is done both in the case of internal and boundary damping .

In this work, we study existence, multiplicity and nonexistence of positive solutions, with respect to a positive parameter , for a class of quasilinear elliptic problems in bounded domains of RN, N 2, involving the N-laplacian operator and a nonlinearity f(t) which behaves as t, for some 2 (0;N1), when t ! 0+ and has critical exponential growth of Trudinger-Moser type at +1. In order to obtain the results, we have used minimax theorems, sub and supersolution methods and a refinement of the Trudinger- Moser inequality due to P.-L. Lions.

In this work, using variational methods and the sub and super solutions method we study the existence and multiplicity of positive solutions for some classes of problems involving the p-Laplacian operator in bounded domains of RN. Initially, we study a result of existence of positive solution for a problem where the nonlinearity does not satisfy the classical Ambrosetti-Rabinowitz condition, and then we study the existence and multiplicity result of positive solutions for a class of problems where the considered nonlinearity can change sign.

This word is concerned with the study of bounded solutions of semilinear elliptic equations u − F0(u) = 0 in the whole space Rn, under the assumption that u is monotone in one direction, say, @u/@xn > 0 em Rn. The goal is to establish the one-dimensional character or symmetry of u, namely, that u only depends on one variable or, equivalently, that the level sets of u are hyperplanos. This type of symmetry question was raised by de Giorgi in 1978 (see [6]), who made the folowing conjecture: Conjecture Suppose that u 2 C2(Rn) is solution of the equation u + u − u3 = 0 satisfying |u(x)| 1 and @u @xn > 0 in the whole Rn. Then the level sets of u must be hyperplanes. We show a stronger version of De Giorgi s conjecture is indeed true in dimension 2 and 3 using some techniques in the linear theory developed by Berestychi, Caffarelli and Nirenberg [5] in one of their papers on qualitative properties of solutions of semilinear elliptic equations.

This work is a study of linear systems from the perspective of linear algebra. We will use the concepts of matrix, vector, linear combination, linear dependence and independence, vector space, basis and dimension. We will also calculate the determinants and implications. Our aim is to present the rudiments of Linear Algebra as helper tool in solving linear systems and display its geometry. We want it to manufacture a auxiliary text that can be explored by students and high school teachers, and so gently introducing this powerful mathematical tool. Throughout the text will be covered also some historical aspects.

In this work are introduced the concepts of involutive affine and projective varieties. Taking into account that every projective variety in P2n-1 has dimension greater than or equal to n-1 and that every hypersurface is involutive, we put our focus on the study of involutive curves in P3, noting that a curve in P3 contained in a plane will be involutive if and only if it is a union of lines passing through the point associated to the suported by plane the correspondence between points and planes determined by the standard symplectic form in P3. We started using the Poisson bracelete invariance of the definition ideal of a varity criterion to determine the involutive lines and conics in P3. Moreover, we exhibit a family of involutive twisted curves. Finally, having in mind that the parameters spaces for involutive lines and conics are 3 and 5 dimensional spaces, respectively. We find how many involutive lines and conic meet 3 and 5 given lines in P3, respectively.

In this paper, we study a version of the Multiplicative Ergodic Theorem of Oseledets for diffeomorphisms of class C1 on a compact Riemannian manifold of finite dimension which ensures the existence of Lyapunov exponents at almost every point with respect to a Borel probability measure invariant by diffeomorphism. In fact, we demonstrate the theorem in a more general version, namely in the context of linear cocycles. The theorem of Oseledets for diffeomorphisms will be established as a special case of this version.

In this work we will study the definitions, examples and basic properties of Ore extensions. In particular, we will present a special case of Ore extensions, the Weyl algebras An(K) over a field K. We will see that An(K) is a simple noetherian domain. We will study also the dimension d(M) of a finitely generated An(K)-module and we will prove the Bernstein's inequality, n d(M) 2n. Finally we will study the holonomic An(K)- modules, that is, the finitely generated An(K)-modules such that d(M) = n:

We study the deformation theorem using the condition introduced by Cerami [8]. Furthermore, we study the following Dirichlet problem: ( u = f(x; u); x 2 u = 0; x 2 @ where is a smooth and bounded domain in RN and f : R ! R is a Caratheodory function with subcritical growth. In the above problem, we use the condition of Cerami [8] again, to ensure the existence of non-trivial solution. For this purpose, we use General Minimax Theorem proved by Bartolo in [12].

The goal this work is to understand the local cohomology functor, and some of its properties. We show that this functor has a relation with the functor Ext. Furthermore, we show the followings theorems: Grothendieck's Vanishing Theorem, Hartshorne's Vanishing Theorem, Grothendieck's Non-Vanishing Theorem and Hartshorne-Linchenbaum's Vanishing Theorem.

In this work, we present the notion of Rees algebra of an ideal and some of its basic properties. Such concept is related to the normality of rings and ideals, and to reductions of ideals as well. Finally, we shall exhibit the Rees algebra of a module, proving some generalizations of results in the case of ideals.

We consider a transmission wave equation in two embedded domains in R2, where the speed is a1 > 0 in the inner domain and a2 > 0 in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strongly convex and a1 > a2. As a consequence of this inequality, uniqueness and Lipschitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coeficient from a single time dependent Neumann boundary measurement.

We study two types of Neumann problem related to Capillary problem and to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. In particular, we prove the existence of Killing graphs with prescribed mean curvature and prescribed boundary conditions.

Set a generalization of Möbius transformation and build a theory of inductive that may be an n-dimensional hyperbolic space. This theory allows for the inductive starting with n = 1, together with the extension notion of the Poincaré build a chain groups GM(n) transformation Möbius and spaces hyperbolic H2 members. We will see explicit formulas for the Poincaré bisectors in size 2. And may on models of hiperbolic space ball these bisectors coincide with the isometric spheres of isometries. We will be using explicit formulas of bissectors, to ge youself an algorithm, the DAFC, to obtain generators for Fuchsianos groups, which will be our study group.

In this work we present a recent result, due to D. Pellegrino and E. V. Teixeira, that characterizes the continuous linear operators between lpspaces which attain their norms. To this end, we Örstly explore some topics from the Banach space theory, such as Banachís Theorem for basis, Bessaga-Pe ̃czynski Selection Principle and Pittís Theorem.

The text on Limits, Continuity, Derivability and Applications is aimed at students of undergraduate courses in mathematics and basic education teachers, especially those active in high school as a way to aid in their studies in subjects that address the differential calculus. Whenever possible, approach the issues presented with content of basic education seeking a link between some content. The examples are as clear as possible, because the goal here is not to present challenges but show direct applications of key settings.

Cook's local infuence approach based on normal curvature is an important diagnostic tool for assessing local infuence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of infuence measures for objective functions at a point with a nonzero rst derivative. The aim of this paper is to develop a diferential-geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and ane curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model.