Conflito, resistência e philía no Héracles furioso, de Eurípides

En la pieza Heracles Furioso, de Eurípides, el héroe se somete a la prueba definitiva: vencerse a sí mismo, aceptar seguir con vida después de cometer un error irremediable. La acción dramática tiene doble motivación, humana y divina. El conflicto humano requiere capacidad de resistencia de la familia y amigos de Heracles y la oposición de Lico es un elemento clave para la acción. En el enfrentamiento posterior entre Heracles y la diosa Hera, la victoria corresponde a la deidad, motivando a la catástrofe, pero se confirma la heroicidad de Heracles, que se resiste a la tentación de aniquilación tras el asesinato de sus niños. Su amigo, el rey Teseo, le proporciona el apoyo necesario para disuadirle de su propósito de aniquilamiento y fortalece al héroe decaído para la resistencia. La valoración de philia es un elemento importante en la constitución de la significación del texto de EurípidesIn Heracles Mainomenos, by Euripides, the hero is submited to the ultimate test: win himself, accepting stay alive after committing an irremediable error. The dramatic action has dual motivation, human and divine. The human conflict requires resilience of families and friends of Heracles and the opposition of Lico is a key element to the action. In the subsequent clash between Heracles and the goddess Hera, victory lies with the deity, motivating the disaster, but confirms the heroics of Heracles, that resists the urge to annihilation after the murder of children. His friend, King Theseus, provides him the support needed to dissuade him from his purpose of annihilation and strengthens him for endurance. The valuation of philia is an important element in shaping the sense of this euripidean textNa peça Héracles furioso, de Eurípides, o heroi é submetido ao teste supremo: vencer a si mesmo, aceitando continuar vivo depois de cometer um erro irremediável. A açao dramática tem dupla motivaçao, humana e divina. O conflito humano requer resistência dos familiares e amigos de Héracles e a oposiçao de Lico é um elemento fundamental para a açao. No confronto posterior, entre Héracles e a deusa Hera, a vitória cabe à divindade, motivadora do desastre, mas confirma-se a heroicidade de Héracles, que resiste ao desejo de aniquilamento após ao assassínio dos filhos. Seu amigo, o rei Teseu, fornece-lhe o amparo necessário para demovê-lo de seu intento de aniquilaçao e fortalece-o para a resistência. A valorizaçao da philía constitui um elemento importante na estruturaçao do sentido desse texto euripidian

STRUCTURE FOR REGULAR INCLUSIONS

We study pairs (C,D) of unital C∗-algebras where D is an abelian C∗-subalgebra of C which is regular in C in the sense that the span of {v 2 C : vDv∗ [ v∗Dv D} is dense in C. When D is a MASA in C, we prove the existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D whose restriction to D is the identity on D. We show that the left kernel of E, L(C,D), is the unique closed two-sided ideal of C maximal with respect to having trivial intersection with D. When L(C,D) = 0, we show the MASA D norms C in the sense of Pop-Sinclair-Smith. We apply these results to significantly extend existing results in the literature on isometric isomorphisms of norm-closed subalgebras which lie between D and C. The map E can be used as a substitute for a conditional expectation in the construction of coordinates for C relative to D. We show that coordinate constructions of Kumjian and Renault which relied upon the existence of a faithful conditional expectation may partially be extended to settings where no conditional expectation exists. As an example, we consider the situation in which C is the reduced crossed product of a unital abelian C∗-algebra D by an arbitrary discrete group acting as automorphisms of D. We charac- terize when the relative commutant Dc of D in C is abelian in terms of the dynamics of the action of and show that when Dc is abelian, L(C,Dc) = (0). This setting produces examples where no conditional expectation of C onto Dc exists. In general, pure states of D do not extend uniquely to states on C. However, when C is separable, and D is a regular MASA in C, we show the set of pure states on D with unique state extensions to C is dense in D. We introduce a new class of well behaved state extensions, the compatible states; we identify compatible states when D is a MASA in C in terms of groups constructed from local dynamics near an element 2 ˆD. A particularly nice class of regular inclusions is the class of C∗-diagonals; each pair in this class has the extension property, and Kumjian has shown that coordinate systems for C∗-diagonals are particularly well behaved. We show that the pair (C,D) regularly embeds into a C∗-diagonal precisely when the intersection of the left kernels of the compatible states is trivial

Uniquely D-colourable digraphs with large girth

Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring if for every arc $uv$ of D, either $f(u)f(v)$ is an arc of C or $f(u)=f(v)$, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of Mathematic

Unique Pseudo-Expectations for C∗C^*-Inclusions

Given an inclusion D $\subseteq$ C of unital C*-algebras, a unital completely positive linear map $\Phi$ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D $\subseteq$ C $\subseteq$ B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' $\cap$ C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D $\subseteq$ C has a unique pseudo-expectation $\Phi$ which is also faithful, then the C*-envelope of any operator space X with D $\subseteq$ X $\subseteq$ C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions.Comment: 26 page

Exploring a ΣcDˉ\Sigma_{c}\bar{D} state: with focus on Pc(4312)+P_{c}(4312)^{+}

Stimulated by the new discovery of $P_{c}(4312)^{+}$ by LHCb Collaboration, we endeavor to perform the study of $P_{c}(4312)^{+}$ as a $\Sigma_{c}\bar{D}$ state in the framework of QCD sum rules. Taking into account the results from two sum rules, a conservative mass range 4.07\sim4.97~\mbox{GeV} is presented for the $\Sigma_{c}\bar{D}$ hadronic system, which agrees with the experimental data of $P_{c}(4312)^{+}$ and could support its interpretation as a $\Sigma_{c}\bar{D}$ state.Comment: 9 pages, 6 figures. arXiv admin note: text overlap with arXiv:1801.0872