Continuation of Direct Products of Distributions

If, in some problems, one has to deal with the ``product'' of distributions $\rm f_i$ (also called generalized functions) $\rm\bar T = \Pi^m_{i=1} f_i$, this product has a priori no definite meaning as a functional $(\rm \bar T, \phi)$ for $\rm\phi \in S$. But if $\rm x^{\kappa +1} \Pi^m_{i=1} f_i$ exists, whatever the associativity is between some powers $\rm r_i$ of $\rm x$ ($\rm r_i \in \Bbb N, \sum_i r_i\leq \kappa +1, r_i \geq 0$) and the various $\rm f_i$, then a continuation of the linear functional $\rm \bar T$ from $\rm M$ onto $\rm S^{(N)}$ for some $\rm N$ is shown to exist in such a way that $\rm x^{\kappa +1} \bar T$ is defined unambiguously, and $\rm (\bar T, \phi), \phi \in S$, significant, though not unique.Comment: 3 pages, late

On the Four-Color-Map Theorem

Coloring planar Feynman diagrams in spinor quantum electrodynamics, is a non trivial model soluble without computer. Four colors are necessary and sufficient.Comment: 8 page

A search for exact superstring vacua

We investigate $2d$ sigma-models with a $2+N$ dimensional Minkowski signature target space metric and Killing symmetry, specifically supersymmetrized, and see under which conditions they might lead to corresponding exact string vacua. It appears that the issue relies heavily on the properties of the vector $M_{\mu}$, a reparametrization term, which needs to possess a definite form for the Weyl invariance to be satisfied. We give, in the $n = 1$ supersymmetric case, two non-renormalization theorems from which we can relate the $u$ component of $M_{\mu}$ to the $\beta^G_{uu}$ function. We work out this $(u,u)$ component of the $\beta^G$ function and find a non-vanishing contribution at four loops. Therefore, it turns out that at order $\alpha^{\prime 4}$, there are in general non-vanishing contributions to $M_u$ that prevent us from deducing superstring vacua in closed form.Comment: 9 pages, latex, CERN-TH.6946/9

Explicit supertring vacua in a background of gravitational waves and dilaton

We present an explicit solution of superstring effective equations, represented by gravitational waves and dilaton backgrounds. Particular solutions will be examined in a forthcoming note.Comment: 10 pages, latex, CERN-TH.7373/9

PIANTEY, Alain. Principes de diplomatie, nouvelle édition. Paris, Éditions Pedone, 2000, 458 p.

The article explores some problems present in the foundation of the great theories on law in German idealism by comparing Fichte’s perspective with that of Schelling in the period 1796-7. In spite of the obvious similarities between the two theoretical frameworks, attention can be drawn to certain differences in order to emphasize the rarely recognised originality of Schelling’s contribution to the debate on the status of law. Thus when he criticises natural law, like Fichte, Schelling proposes a new theory of the relationship between law and autonomy, opening the way for a view of the collective subjectivity founded on equality and liberty. By analysing the practical impact of the meeting with «the heteronomy of the autonomous», Schelling’s treatise brings about a decisive step towards a theory of the law as common law (transl. J. Dudley).L’article explore quelques problèmes présents dans la fondation des grandes théories sur le droit au sein de l’idéalisme allemand, en comparant la perspective fîchtéenne avec la perspective schellingienne dans la période 1796-1797. Malgré les similarités évidentes des deux cadres théoriques, certaines différences peuvent être mises en évidence afin de souligner l'originalité peu reconnue de l’apport schellingien au débat sur le statut du droit. Lorsqu’il critique ainsi le droit naturel, à l’instar de Fichte, Schelling propose une nouvelle théorie de la relation entre droit et autonomie, ouvrant la voie d’une pensée de la subjectivité collective fondée dans l’égalité et la liberté. En analysant la portée pratique de la rencontre de «l’hétéronomie de l’autonome», le traité de Schelling opère un pas décisif vers une pensée de la loi comme droit commun.Popa Délia. La liberté en communauté. Droit et résistance chez Fichte et Schelling en 1796-1797. In: Revue Philosophique de Louvain. Troisième série, tome 111, n°2, 2013. pp. 283-307