Infinitely many monotone Lagrangian tori in del Pezzo surfaces

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $\mathbb{C}P^2 \# k \overline{\mathbb{C}P^2}$, for k=0,3,4,5,6,7,8. We name these tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ also have infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ . Finally, the Lagrangian tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$ inside a del Pezzo surface $X$ can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus $\Sigma$. We argue that $\Theta^{n_1,n_2,n_3}_{p,q,r}$ give rise to infinitely many exact Lagrangian tori in $X \setminus \Sigma$, even after attaching the positive end of a symplectization to the boundary of $X \setminus \Sigma$.Comment: 28 pages, 20 figure

Quantumness of correlations in indistinguishable particles

We discuss a general notion of quantum correlations in fermionic or bosonic indistinguishable particles. Our approach is mainly based on the identification of the algebra of single-particle observables, which allows us to devise an activation protocol in which the \textit{quantumness of correlations} in the system leads to a unavoidable creation of entanglement with the measurement apparatus. Using the distillable entanglement, or the relative entropy of entanglement, as entanglement measure, we show that our approach is equivalent to the notion of minimal disturbance in a single-particle von Neumann measurement, also leading to a geometrical approach for its quantification.Comment: 7 pages, 2 figure

O STF como legislador positivo: justiça constitucional pela efetividade dos direitos fundamentais

Esta dissertação tem por objeto o papel do STF no controle da omissão normativa inconstitucional e, por objetivo, demonstrar que, neste enfrentamento, o Tribunal Constitucional pode e deve desempenhar, como de fato desempenha, uma atuação normativa em sentido amplo e mesmo materialmente legislativa em sentido estrito, sobretudo a bem da eficácia e da efetividade dos direitos fundamentais, sem que isso implique ofensa à arquitetura constitucional de repartição e compartilhamento funcional do exercício do poder, bem como à democracia. Inicialmente, a partir de uma análise crítica da ideia de separação de poderes, aborda-se a inserção da Justiça Constitucional na repartição e no compartilhamento do exercício das funções do Estado, destacando-se a sua multifuncionalidade. Na sequência, enfoca-se a legitimidade democrática da Justiça Constitucional, com os principais fundamentos institucionais, procedimentais e substanciais que a legitimam para dialogar com os demais órgãos de soberania. Passa-se, então, ao exame da posição ocupada pelos direitos fundamentais no Estado Democrático de Direito e a omissão inconstitucional, especialmente a normativa, como grande óbice à sua eficácia e efetividade. Por fim, enfoca-se especificamente, com análise de jurisprudência, a atuação do STF como “legislador positivo”, no enfrentamento da omissão normativa inconstitucional, com seus pressupostos, características e limites.This dissertation concerns the role of the STF – Supreme Federal Court in regards to controlling the unconstitutional normative omission. It aims to demonstrate that, in this confrontation, the Constitutional Court can and must perform, as it does, a role that is normative in a wide sense and even materially legislative in a strict sense, above all, for the sake of efficiency and effectiveness of the fundamental rights, without incurring offense to the constitutional architecture of functional sharing and distribution of power, neither to democracy. In the first part, based on a critical analysis of the idea of separation of powers, we discuss the insertion of Constitutional Justice in the sharing and distribution of the exercise of the functions of the state, emphasizing its multifunctional aspect. Following that, we focus on the democratic legitimacy of Constitutional Justice, presenting the main substantial, procedural and institutional foundations, which legitimate it to dialogue with the other sovereign institutions. The second part begins by examining the position occupied by the fundamental rights in the Democratic Rule-of-Law State and the unconstitutional omission, in special, the normative one, as a great obstacle to its efficiency and effectiveness. Finally, with analysis of jurisprudence, the thesis specifically focuses on the performance of the STF as the “positive legislator” in confronting the unconstitutional normative omission, with its assumptions, characteristics and limitations