This work aims to explore, under certain conditions, the existence of solutions for the non-linear Dirichlet problem, encompassing the linear case, involving measures. We will also include the scenario in which the measure is absolutely continuous with respect to the Radon measure, that is, a measure defined by an integrable function. The adopted approximation method is to consider a sequence of Radon measures that converge in the weak-star topology, known as vague convergence. As an application, the study of the existence of a solution for the elliptic problem with exponential semilinearity with domain in the plane will be presented.

In this work, we study the tidy subgroup theory developed by G. Willis in the paper willis 1992, about the structure of locally compact and totally disconnected groups, we also show that the integer valued function, s, called the scale function, is well defined and continuous. Finally, we use these results to show the conjecture proposed by Hoffman and Mukherjea in their paper, Concentration Funtions and a Class of Non-Compact Groups, that says that a locally compact group is a neat group.

This dissertation deals with the unification problem regarding the associative-commutative equational theory with
homomorphism (ACh). This unification problem seeks to solve equations of the type , for first-order terms and , which
consists of finding a substitution that makes both terms when instantiated by this substitution identical, i.e., such that
. The unification modulo ACh problem consists of solving a similar problem but seeks substitutions such that both
instantiated terms are in the same ACh modulo equivalence class.
Past research established the undecidability of ACh Unification, defying the viability of finding solutions for this case.
However, recent work has introduced a new approach known as ‘Bounded ACh-Unification’. This new concept
strategically imposes a constraint by bounding the number of homomorphism function symbols occurring in a term.
With that, it was possible to define a set of inference rules and allegedly prove the correctness of the algorithm
defined by such rules. Our goal is to provide a comprehensive understanding of the procedure for solving the
bounded ACh-Unification problem by carefully examining its inference rules and validating the proof of termination,
soundness and completeness.

In this dissertation, we present the basic concepts of imaginary quadratic fields, with an emphasis on the ideal class group. Following that, we introduce the necessary analytical tools for proving the class number formula.

We will conduct a study on the mean curvature flow, seeking self-similar solutions. We will show the possible self-similar evolutions for surfaces immersed in Euclidean space. Next, we will classify non-trivial solutions of ruled, rotational, and helicoidal surfaces.

IIn this work we prove some results concerning to Caffarelli-Kohn-Nirenberg's type problems in
$\mathbb{R}^N$.
In the first chapter we prove the existence of nontrivial solutions with Berestycki-Lions type nonlinearities. More
precisely, we study the following classes of problems
$$
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)+ |x|^{-bp^{*}}|u|^{p-2}u= |x|^{-bp^{*}} h(u) \ \
\mbox{in} \ \mathbb{R}^{N},
\leqno{(PM)}
$$
and
$$
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)= |x|^{-bp^{*}}f(u) \ \
\mbox{in} \ \mathbb{R}^{N},
\leqno{(ZM)}
$$
where $1<p<N$, $0\leq a< \frac{N-p}{p}$, $a<b\leq a+1$, $p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}$ and $d=1+a-b$.
In the second chapter we prove the existence and concentration of ground state solutions for a class of subcritical,
critical or supercritical Caffarelli-Kohn-Nirenberg type problems. More precisely, we are going to study the following
class of quasilinear problems
\begin{equation*}\tag{$P_{\mu,\varrho,\sigma}$}
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)+|x|^{-bp^{*}}[1+\mu V(z)]|u|^{p-2}u = |x|^{-
bp^{*}}[f(u)+\varrho|u|^{\sigma-2}u]
\end{equation*}
in $\mathbb{R}^{N}$, where $1<p<N$, $0\leq a< \frac{N-p}{p}$, $a<b\leq a+1$, $p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}$,
$d=1+a-b$ and $\mu>0$.
In the third chapter we prove the existence of a positive and a nodal ground state solutions to the following class of
Caffarelli-Kohn-Nirenberg type problems
\begin{equation*}
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)+|x|^{-bp^{*}} V(x)|u|^{p-2}u = |x|^{-
bp^{*}}K(x)f(u) \quad\hbox{in $\mathbb{R}^{N}$}, \quad\hbox{(P)}\
\end{equation*}

\noindent where
$1<p<N$, $0\leq a< \frac{N-p}{p}$, $a<b\leq a+1$, $p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}$ and $d=1+a-b$.
In Chapter 3 is concerned with the existence of a ground state solution for the problem $(P_{3})$. Here $V$ and $f$
are continuous functions with $V$ being either periodic or asymptotic at infinity to a periodic function. The function
$f$ has subcritical growth and behaves like
$|u|^{q-2}u$ with $p<q< 2_{**}$. Using variational methods, we prove the existence of a ground state solution in the
subcritical case, i.e, $\beta=0$ and the critical case, i.e, $\beta=1$.

The covering number of a finite noncyclic group G, denoted sigma(G), is the smallest positive integer k such that G is a union of k proper subgroups. If G is 2-generated, let omega(G) be the maximal size of a subset S of G with the property that any two distinct elements of S generate G. Since any proper subgroup of G can contain at most one element of such a set S, omega(G) is at most sigma(G). For a family of primitive groups G with a unique minimal normal subgroup N isomorphic to a direct power of the alternating group A_n and G/N cyclic, we calculate sigma(G) for n divisible by 6 and m at least 2. This is a generalization of a result of E. Swartz concerning the symmetric groups, which corresponds to the case m=1. For the above family of primitive groups G, we also prove a result concerning pairwise generation: for fixed m at least 2 and n even, we calculate asymptotically the value of omega(G) when n goes to infinity and show that omega(G)/sigma(G) tends to 1 as n tends to infinity.

We studied the nonautonomous and non periodic Schrödinger equation with asymptotic growth $-\div(\xi(x) u) +
V(x)u$, in R^N. We demonstrated the existence of positive solutions and solutions that change sign when the
potentials $\xi$ and $V$ are positive. When the potential $V$ changes sign and $\xi$ is positive, we showed the
existence of positive solutions. We used the Mountain Pass Theorem and the Linking Theorem to obtain such
solutions.

The elliptic problem that describes the stationary logistic equation in a bounded domain is studied using variational methods and the Nehari manifold. The existence of a unique positive solution is obtained through the minimization of the functional associated with the Euler-Lagrange equation of the problem, restricted to the Nehari manifold. A second solution, which changes sign, is presented by applying the Mountain Pass Theorem on Nehari. Two cases are addressed: superlinear and subcritical nonlinearity, or asymptotically linear.

Let $(M^{n},g,\nabla f,\lambda)$ be a compact gradient Ricci
almost soliton with boundary. In this thesis, we obtain
rigidity theorems for $(M^{n},g,\nabla f,\lambda)$ so that
we can show if it is isometric to a closed hemisphere of an
Euclidean sphere, or a closed Euclidean ball, or a domain in
$\mathbb{H}^{n}$. Furthermore, we apply such theorems to
characterize gradient Ricci almost solitons on warped
product $M=B\times_{h}F$, where $B$ is a compact Riemannian
manifold with boundary.

Consider two magnetic dipoles fixed in the plane, free to spin, separated by a distance r, subjected to a homogenous external magnetic field applied with a certain orientation. This system is a non-linear dynamical system and the goal of this work is to determine and classify its equilibrium points and the bifurcations suffered by the system caused by the changes of the applied fields. The equations of the motion of the dipoles are obtained from Newton’s second law in angular terms considering the torques that each of the dipoles undergoes due to the presence of the other and due to rotational friction. We show that only two of the eight equilibrium points, obtained in the absence of an external magnetic field, are stable. Their basins of attraction were built using the Runge-Kutta method. As the intensity and orientation of the applied external field are varied, the system can undergo five different types of bifurcations that can destroy, create and change the stability of these equilibrium points. For high intensities, we observe that only four equilibrium points remain, and only one is stable. The results of this analysis were obtained from the application of a combination of the Continuation Method, the Newton-Raphson Method and the Runge-Kutta Method.

In this work, we provide a study of complete gradient shrinking Ricci solitons of dimension 4. We present in detail the proofs (originally exposed in an article by Huai-Dong Cao, Ernani Ribeiro Jr, and Detang Zhou) of two theorems that guarantee geometrical classifications and controls on the Ricci or Riemannian curvature, provided that pointwise estimates on the self-dual or anti-self-dual parts of the Weyl tensor or a certain control on the scalar curvature in terms of the soliton's potential function are satisfied.

The goal of this work is to investigate/describe structural properties of groups (finite or infinite) under some restrictions on their subgroups.

The purpose of this monograph is to succinctly present the theory of E-functions and prerequisites for the first fundamental theorem test. From this result, we will prove the Second Fundamental Theorem from which we will obtain as corollary the famous LidemannWeierstrass theorem.

The present thesis aims to answer the following questions. Even in a scenario where an effective vaccine is not developed in the coming years, does the strategy of social isolation and repeated reopening reduce the number of deaths? And why did SARS (2002) and MERS (2012) not cause as many problems as the COVID-19 pandemic? We use ideas from control theory and the classic SIR model to answer the first question, while to answer the second question, it is necessary to introduce an extension of this model, which we call SECIAR, and describe its global dynamics.

This work aims to present results on the non-abelian tensor square G ⊗ G of a group
G, for the class of powerful finite p-groups. Some properties and results about finite
p-groups and powerful p-groups that will be used in the context will also be presented,
as well as the main properties of the group ν(G), a certain extension of G ⊗ G by
G × G. In addition, we will address some bounds for the order, the exponent and the
rank of G ⊗ G and of the non-abelian exterior square G ∧ G for finite p-groups G.

In this work, we study spacelike surfaces in Minkowski space $\mathbb{E}^3_1$ and that satisfying the Weingarten linear equation of the type $aH+bK=c$, where $a,b$ and $c$ are constants and $H$ e $K$ denotes, respectively, the mean curvature and $K$ the Gaussian curvature of the surface. We show that if these surfaces are foliated by circles in parallel planes and ($H\neq0$ and $K\neq0$), then these surfaces must be surfaces of revolution. Furthermore, we show that if a spacelike surface satisfies the Weingarten linear equation and is foliated by circles in planes that are not parallel, then this surface is pseudohyperbolic.

The aim of this work is to investigate the existence and uniqueness of solutions for a class of
nonlinear reaction-diffusion equations with non-local boundary conditions, as well as to analyze the dynamics of the
problem. To achieve this, we employ the method of sub- and supersolution for elliptic and parabolic equations.

In this work, we state and prove a result of global bifurcation due to Rabinowitz. Subsequently, we
apply this theory to obtain positive solutions of elliptic Kirchhoff-type problems in bounded domains under varying
assumptions regarding the nonlinearity.

Based on the work of Armando V. Corro, Antonio Martínez, and Keti Tenenblat, in this dissertation,
we will apply Ribaucour transformations to rotational flat surfaces in the three-dimensional hyperbolic space, H^3,
providing new explicit families of flat surfaces in H^3 that are determined by various parameters. By choosing certain
parameters in a special way, it is possible to obtain surfaces that exhibit periodicity with respect to one variable and
also surfaces that have an arbitrary even number of embedded ends of horosphere type, or even an infinite number
of such ends.

This dissertation aims to study the global attractivity for nonautonomous Nicholson’s blowfly equation with a pair of time-varying delays. More precisely, we will investigate the permanence, local stability and global attractivity of its positive equilibrium K.

In this dissertation we study the asymptotic properties of an estimator, presented by Goegebeur et al. (2022), for the risk measure known as conditional tail moment. The situation considered corresponds to extrapolation outside the data range and requires arguments from extreme value theory for the construction of the appropriate estimator. We performed a brief analysis of the main risk measures found in the literature, as well as their relationships with conditional tail moment. The results obtained by Goegebeur et al. (2022), which establish
under suitable conditions, the limit distribution of the properly normalised estimator are presented in detail.

We prove that an oriented hypersurface in a space form, whose mean cur-
vature does not vanish at any point, is an initial condition for a solution

to the inverse mean curvature flow (IMCF) by parallel hypersurfaces if and
only if it is isoparametric. Considering the isoparametric hypersurfaces in

space forms, we obtain the solutions to the IMCF by parallel hypersurfa-
ces explicitly. Moreover, we study these solutions in detail, describing their

behavior in the maximal interval where they are defined. For the isopara-
metric hypersurfaces in the hyperbolic space and in the sphere, with two

or four distinct principal curvatures, we consider the additional assumption
that these curvatures have the same multiplicity.

In this work, we study isoparametric hypersurfaces in product manifolds of
dimension 4. First of all, we characterize and classify the isoparametric hypersurfaces with
constant principal curvatures in the product spaces Q c1 x Q c2 , where Q ci is a space form with
constant sectional curvature ci, for ci {-1,0,1} e c1 c2. We show that such hypersurfaces are
given as open subsets of either a product hypersurface, where one factor is a curve of
constant curvature, or a diagonal structure in H 2 x R 2 , constructed from horocycles in H 2 and
straight lines in R 2 .
Next, we classify the hypersurfaces in Q 3 x R with the three distinct constant principal
curvatures, where in this case . We show that such hypersurfaces are cylinders over

isoparametric surfaces of Q 3 with two non-null distinct principal curvatures. We also prove
that the hypersurfaces with constant principal curvatures in Q 3 x R are isoparametric.
Furthermore, we provide a necessary and sufficient condition for an isoparametric
hypersurface on Q 3 x R to have constant principal curvatures.
Finally, we describe the evolution by the mean curvature flow of isoparametric hypersurfaces
in product manifolds of dimension 4. We show that the evolution of isoparametric
hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a
reparametrization of the flow by parallel hypersurfaces in a short time, as long as the
uniqueness of the mean curvature flow holds for the initial data and the corresponding
ambient space. Through this result, we describe the evolution of the hypersurfaces classified
in the first and second parts of the work. We also describe the evolutions of isoparametric
hypersurfaces in S 2 x S 2 and H 2 x H 2 , classified by Urbano (2019) and Dong Gao, Hui Ma and
Zeke Yao (2022), respectively, and of isoparametric hypersurfaces in Q 3 x R with g distinct
constant principal curvatures, g {1,2}, classified by Chaves and Santos (2019).

The discrete element method (DEM) is a numerical technique widely used to sim ulate granular materials. The temporal evolution of these simulations is often performed using a Verlet-type algorithm, because of its second order and its desirable property of energy conservation. However, when dissipative forces are considered in the model, such as the nonlinear Kuwabara-Kono model, the Verlet method no longer behaves as a second order method, but instead its order decreases to 1.5. This is caused by the singular behavior of the damping force in the Kuwabara-Kono model at the beginning and in the end of particle collisions. In this work, we introduce a simplified problem which reproduces the singularity of the Kuwabara-Kono model and prove that the order of the method decreases from 2 to 1 + q, where 0 < q < 1 is the exponent of the nonlinear singular term. Furthermore, we propose a regularized normal force model based on the concept of mollifiers. We show numerically that the Verlet method combined with this regularized force model can integrate collisions with second order accuracy and that the coefficient of restitution of the system tends to increase as a function of the regularization parameter. Furthermore, using the DEM algorithm, we construct a granular Taylor-Couette computer simulation to generate coarse-grained data that will be fed into a SINDy machine learning algorithm in order to infer constitutive laws for granular flows based on the mu(I) rheology.

In this thesis, we study splittings of profinite groups as HNN-extensions and amalgamated free products. In fact, these constructions can be considered as particular cases of a profinite fundamental group of a graph of groups, which we denote by $\Pi_1(\GA,\G)$. Hence, if a given profinite group $G$ has a splitting $G=\Pi_1(\GA,\G)$ for some profinite graph of groups $(\GA,\G)$, we obtain not only properties of the group $G$ but also properties of the graph of groups $(\GA,\G)$. In the first part, given an abstract group $G$, that splits as the fundamental group of an infinite graph of groups, we construct a profinite graph of groups $(\overline{\GA},\overline{\G})$ such that $\G$ embeds in $\overline{\G}$ and the profinite completion of $G$ splits as $\widehat{G}=\Pi_1(\overline{\GA},\overline{\G})$. This answers an Open Question of Ribes. With this construction in hand, we answer two more Open Questions of Ribes. The first concerns the closure of normalizers, which generalizes the main Theorem of a paper by Ribes and Zalesski. The second is related to subgroup conjugacy separability of virtually free groups, generalizing the main Theorem of a paper by Chagas and Zalesski. Our strategy for solving the problems above is to describe the profinite fundamental group of a graph of groups in the language of paths. Since it behaves very well via inverse limits, it facilitates the interrelation between the abstract and the profinite settings. We continue our journey by investigating the celebrated Stallings' decomposition Theorem. It states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. The pro-$p$ version of this result was obtained by Weigel and Zalesskii in 2017. We proved that, in the category of pro-$p$ groups, splitting theorems hold beyond splittings over finite groups. In fact, if $G$ is a finitely generated pro-$p$ group having an open normal subgroup $H$ that splits as $H=\Pi_1(\HA,\D)$, and we suppose conjugacy classes of vertex groups are $G$-invariant then $G$ also splits as $G=\Pi_1(\GA,\G)$. If $H$ is a non- trivial free pro-$p$ product we obtain, as a particular case, the aforementioned Weigel-Zalesski Theorem. The main tool behind the proof is our Limitation Theorem, which establishes that $|E(\G)| \leq |E(\D)|$. With this construction in hand, we provide three applications. First, we show that if $G$ is a finitely generated pro-$p$ group having an open normal subgroup $H$ acting on a pro-$p$ tree $T$, with $\{H_v \mid v \in V(T)\}$ being $G$-invariant, then $G$ splits as $G=\Pi_1(\GA,\G)$. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability. We finish the thesis by showing that our Theorem 11 holds even for Wilkes' example of a pro-$p$ inaccessible group.

The arithmetic nature of a number given as an image of an algebraic number by a transcendental function is a subject studied by several mathematicians since the 19th century. One of the main interested in this type of problem was Mahler, who proposed questions of great interest in Transcendental Number Theory. One of these questions deals
with the existence of a transcendental function with integer and bounded coefficients that assumes algebraic values at algebraic points. The first goal of this work is to show the existence of such a function, but with almost all bounded coefficients.

We will also show the existence of a transcendental function f ∈ Z{z} with almost all bounded coefficients such that f and all its derivatives take algebraic values in algebraic points.

Another problem proposed by Mahler asks whether there are transcendental functions with a prescribed exceptional set. Related to this problem, we show that certain subsets of algebraic numbers are exceptional sets of some transcendental function f ∈ Z{z} with almost all bounded coefficients.

In this thesis we study the Z_p-splittings of a pro-p group G as a free pro-p product amalgamating infinite pro-cyclic subgroup or as an HNN-extension with infinite pro-cyclic associated subgroup, and prove the pro-p version of Theorem 2.1 and Theorem 3.6 of [1]. Furthermore, using the definition of a commensurator of a subgroup, we prove that when a procyclic group C acts freely on a p-tree T, the quotient of the commensurator of G over a normal subgroup contained in a G-edge stabilizer is pro-infinite cyclic or pro-p infinite dihedral, which is a more generalized version of Proposition 8.1 of [6]. Finally we show under some natural conditions that a pro-p group $G$ acting on a pro-p tree is equal to the commensurator of an edge G-stabilizer.

We study a general nonlinear elliptic problem in Rnand prove, by means of a variational structure of the problem, the existence of a ground state solution (of minimal energy), which is also the minimum of the functional constrained to the Pohozaev manifold. This minimum coincides the mountain pass level since the associated functional possesses the necessary geometry. We then propose and implement an algorithm for finding numerical ground state solutions for a wide class of elliptic problems in Rn, and provide several examples for which this new method can be applied.

This work presents a brief study on minimal isometric immersions in the hyperbolic space . Our goal is to demonstrate Simons' formula for the Laplacian of the second fundamental form norm and to use this formula to prove rigidity theorems that determine under what conditions a minimal submanifold of the hyperbolic space is totally geodesic. In addition, one also will define the concept of superstability on minimal submanifolds and use Simons' formula to prove estimates for the first eigenvalue of the stability operator of these minimal embeddings.

The aim of this work will be to study two kind of problems: obtain sufficient conditions to guarantee the existence of non-commutator elements in certain groups; and in a complementary perspective, describe criteria in which the derived subgroup coincides with the set of all commutators.

In this thesis, we study the functional Volterra--Stieltjes integral equations given by:
where the integral on the right--hand side is taken in the sense of Henstock--Kurzweil--Stieltjes.
In this work, we present sufficient conditions in order to guarantee the existence, uniqueness and
prolongation of solutions for this type of equations. We also prove the correspondence between
these equations and the functional Volterra delta integral equations on time scales, as well as with
the impulsive functional Volterra--Stieltjes integral equations. We present results concerning
stability, continuous dependence with respect on parameters and periodicity. The new results can
be found in \cite{GL, GLM2, GLM, LMS}.