Abstract:
O artigo aponta algumas permanências e rupturas na obra A República dos Bugres, do escritor Ruy Tapioca, em rela o ao padr o lukacsiano de romance histórico bem como estabelece algumas rela es da mesma obra com a metafic o historiográfica.

Abstract:
We deal with a class of Lipschitz vector functions $U=(u_1,...,u_h)$ whose components are non negative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the Poho\u{z}aev identity, we prove that the nodal set is a collection of $C^{1,\alpha}$ hyper-surfaces (for every $0<\alpha<1$), up to a residual set with small Hausdorff dimension. This result applies to the asymptotic limits of reaction-diffusion systems with strong competition interactions, to optimal partition problems involving eigenvalues, as well as to segregated standing waves for Bose-Einstein condensates in multiple hyperfine spin states.

Abstract:
In this paper we focus on existence and symmetry properties of solutions to the cubic Schr\"odinger system \[ -\Delta u_i +\lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i \quad \text{in $\Omega \subset \mathbb{R}^N$},\qquad i=1,\dots d \] where $d\geq 2$, $\lambda_i,\beta_{ii}>0$, $\beta_{ij}=\beta_{ji}\in \mathbb{R}$ for $j\neq i$, $N=2,3$. The underlying domain $\Omega$ is either bounded or the whole space, and $u_i\in H^1_0(\Omega)$ or $u_i\in H^1_{rad}(\mathbb{R}^N)$ respectively. We establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients, as well as in the purely cooperative case.

Abstract:
We study the existence of ground states for the coupled Schr\"odinger system \begin{equation} \left\{\begin{array}{lll} \displaystyle -\Delta u_i+\lambda_i u_i= \mu_i |u_i|^{2q-2}u_i+\sum_{j\neq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i \\ u_i\in H^1(\mathbb{R}^n), \quad i=1,\ldots, d, \end{array}\right. \end{equation} $n\geq 1$, for $\lambda_i,\mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called "symmetric attractive case") and $1

Abstract:
In this paper we consider a class of gradient systems of type $$ -c_i \Delta u_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}\Omega, \qquad u_1=...=u_k=0 \text{on} \partial \Omega, $$ in a bounded domain $\Omega\subseteq \R^N$. Under suitable assumptions on $V_i$ and $P$, we prove the existence of ground-state solutions for this problem. Moreover, for $k=2$, assuming that the domain $\Omega$ and the potentials $V_i$ are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.

Abstract:
In this paper we prove the existence of infinitely many sign-changing solutions for the system of $m$ Schr\"odinger equations with competition interactions $$ -\Delta u_i+a_i u_i^3+\beta u_i \sum_{j\neq i} u_j^2 =\lambda_{i,\beta} u_i \quad u_i\in H^1_0(\Omega), \quad i=1,...,m $$ where $\Omega$ is a bounded domain, $\beta>0$ and $a_i\geq 0\ \forall i.$ Moreover, for $a_i=0$, we show a relation between critical energies associated with this system and the optimal partition problem $$ \mathop{\inf_{\omega_i\subset \Omega \text{open}}}_{\omega_i\cap \omega_j=\emptyset\forall i\neq j} \sum_{i=1}^{m} \lambda_{k_i}(\omega_i), $$ where $\lambda_{k_i}(\omega)$ denotes the $k_i$--th eigenvalue of $-\Delta$ in $H^1_0(\omega)$. In the case $k_i\leq 2$ we show that the optimal partition problem appears as a limiting critical value, as the competition parameter $\beta$ diverges to $+\infty$.

Abstract:
There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed. The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the $n$-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all. This analysis is, as far as we know, original.

Abstract:
The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$ \partial_t u-\Delta u=a u-b(x) u^p \text{in} \Omega\times \R^+, u(0)=u_0, u(t)|_{\partial \Omega}=0 $$ as $p\to +\infty$, where $\Omega$ is a bounded domain and $b(x)$ is a nonnegative function. We deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards we fully describe its long time behavior.

Abstract:
based on the contributions of lovisolo (1995), using the lévi-strauss bricoleur to think the docent praxis, this article tries to resignify the contributions of the docent formation branch denominated reflexive researcher professor (ppr). its main authors, specially stenhouse, are analyzed in a perspective that interpretation dominates and emphasizes autonomy, creativeness, art, putting in opposition with the position delineated by lovisolo based on understandings of lévi-straus about primitive and scientific thinking forms. it is questioned the education functioning that the romantic formation field of ppr seems to operate refusing both this kind of question as its possible answers.

Abstract:
Let $\Omega\subset \mathbb{R}^N$ be an open bounded domain and $m\in \mathbb{N}$. Given $k_1,\ldots,k_m\in \mathbb{N}$, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form \[ \inf\left\{F(\lambda_{k_1}(\omega_1),\ldots, \lambda_{k_m}(\omega_m)):\ (\omega_1,\ldots, \omega_m)\in \mathcal{P}_m(\Omega)\right\}, \] where $\lambda_{k_i}(\omega_i)$ denotes the $k_i$--th eigenvalue of $(-\Delta,H^1_0(\omega_i))$ counting multiplicities, and $\mathcal{P}_m(\Omega)$ is the set of all open partitions of $\Omega$, namely \[ \mathcal{P}_m(\Omega)=\left\{(\omega_1,\ldots,\omega_m):\ \omega_i\subset \Omega \text{ open},\ \omega_i\cap \omega_j=\emptyset\ \forall i\neq j\right\}. \] While existence of a quasi-open optimal partition $(\omega_1,\ldots, \omega_m)$ follows from a general result by Bucur, Buttazzo and Henrot [Adv. Math. Sci. Appl. 8, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and regularity of the partition in the sense that the free boundary $\cup_{i=1}^m \partial \omega_i\cap \Omega$ is, up to a residual set, locally a $C^{1,\alpha}$ hypersurface. This last result extend the ones in the paper by Caffarelli and Lin [J. Sci. Comput. 31, 2007] to the case of higher eigenvalues.