A_{n-1} singularities and nKdV hierarchies

According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the {\em total descendent potential} in the theory of Gromov -- Witten invariants of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive: math.DG/0108160 contain two equivalent constructions, motivated by some results in Gromov -- Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito's Frobenius structure on the miniversal deformation of the $A_{n-1}$-singularity, the total descendent potential is a tau-function of the $n$KdV hierarchy. We derive this result from a more general construction for solutions of the $n$KdV hierarchy from $n-1$ solutions of the KdV hierarchy.Comment: 29 pages, to appear in Moscow Mathematical Journa

Tax Reforms in an Endogenous Growth Model with Pollution

This paper discusses the effects of a green tax reform in an AK growth model without abatement activities and with a negative environmental externality in utility function. There is also a non-optimal level of public spending. The results depend on the financing source of public spending. When there is not public debt, a revenue-neutral green tax reform has not any effect on pollution, growth and welfare. On the contrary, when short-run deficits are financed by debt issuing, a variety of green tax reforms increase welfare. Nevertheless, in this framework, non-green tax reforms are also welfare improving.Environmental externalities, Economic growth, Pollution taxes, Laffer Curve.

Moyal Planes are Spectral Triples

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces $\R^{2N}$ endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes--Lott functional action, are given for these noncommutative hyperplanes.Comment: Latex, 54 pages. Version 3 with Moyal-Wick section update

The unexpected resurgence of Weyl geometry in late 20-th century physics

Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn by its author from physical theorizing in the early 1920s. It had a comeback in the last third of the 20th century in different contexts: scalar tensor theories of gravity, foundations of gravity, foundations of quantum mechanics, elementary particle physics, and cosmology. It seems that Weyl geometry continues to offer an open research potential for the foundations of physics even after the turn to the new millennium.Comment: Completely rewritten conference paper 'Beyond Einstein', Mainz Sep 2008. Preprint ELHC (Epistemology of the LHC) 2017-02, 92 pages, 1 figur

ACUOS: A System for Order-Sorted Modular ACU Generalization

[ES] La generalización, también denominada anti-unificación, es la operación dual de la unificación. Dados dos términos t y t' , un generalizador es un término t'' del cual t y t' son instancias de sustitución. El concepto dual del unificador más general (mgu) es el de generalizador menos general (lgg). En esta tesina extendemos el conocido algoritmo de generalización sin tipos a, primero, una configuración order-sorted con sorts, subsorts y polimorfismo de subtipado; en segundo lugar, la extendemos para soportar generalización módulo teorías ecuacionales, donde los símbolos de función pueden obedecer cualquier combinación de axiomas de asociatividad, conmutatividad e identidad (incluyendo el conjunto vacío de dichos axiomas); y, en tercer lugar, a la combinación de ambos, que resulta en un algoritmo modular de generalización order-sorted ecuacional. A diferencia de las configuraciones sin tipos, en nuestro marco teórico en general el lgg no es único, lo que se debe tanto al tipado como a los axiomas ecuacionales. En su lugar, existe un conjunto finito y mínimo de lggs, tales que cualquier otra generalización tiene a alguno de ellos como instancia. Nuestros algoritmos de generalización se expresan mediante reglas de inferencia para las cuales damos demostraciones de corrección. Ello abre la puerta a nuevas aplicaciones en campos como la evaluación parcial, la síntesis de programas, la minería de datos y la demostración de teoremas para sistemas de razonamiento ecuacional y lenguajes tipados basados en reglas tales como ASD+SDF, Elan, OBJ, CafeOBJ y Maude. Esta tesis también describe una herramienta para el cómputo automatizado de los generalizadores de un conjunto dado de estructuras en un lenguaje tipado módulo un conjunto de axiomas dado. Al soportar la combinación modular de atributos ecuacionales de asociatividad, conmutatividad y existencia de elemento neutro (ACU) para símbolos de función arbitrarios, la generalización ACU modular aporta suficiente poder expresivo a la generalización ordinaria para razonar sobre estructuras de datos tipadas tales como listas, conjuntos y multiconjuntos. La técnica ha sido implementada con generalidad y eficiencia en el sistema ACUOS y puede ser fácilmente integrada con software de terceros.[EN] Generalization, also called anti-uni cation, is the dual of uni cation. Given terms t and t 0 , a generalization is a term t 00 of which t and t 0 are substitution instances. The dual of a most general uni er (mgu) is that of least general generalization (lgg). In this thesis, we extend the known untyped generalization algorithm to, rst, an order-sorted typed setting with sorts, subsorts, and subtype polymorphism; second, we extend it to work modulo equational theories, where function symbols can obey any combination of associativity, commutativity, and identity axioms (includ- ing the empty set of such axioms); and third, to the combination of both, which results in a modular, order-sorted equational generalization algo- rithm. Unlike the untyped case, there is in general no single lgg in our framework, due to order-sortedness or to the equational axioms. Instead, there is a nite, minimal set of lggs, so that any other generalization has at least one of them as an instance. Our generalization algorithms are expressed by means of inference systems for which we give proofs of cor- rectness. This opens up new applications to partial evaluation, program synthesis, data mining, and theorem proving for typed equational rea- soning systems and typed rule-based languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude. This thesis also describes a tool for automatically computing the gen- eralizers of a given set of structures in a typed language modulo a set of axioms. By supporting the modular combination of associative, com- mutative and unity (ACU) equational attributes for arbitrary function symbols, modular ACU generalization adds enough expressive power to ordinary generalization to reason about typed data structures such as lists, sets and multisets. The ACU generalization technique has been generally and e ciently implemented in the ACUOS system and can be easily integrated with third-party software.Espert Real, J. (2012). ACUOS: A System for Order-Sorted Modular ACU Generalization. http://hdl.handle.net/10251/1921

Motivations and Physical Aims of Algebraic QFT

We present illustrations which show the usefulness of algebraic QFT. In particular in low-dimensional QFT, when Lagrangian quantization does not exist or is useless (e.g. in chiral conformal theories), the algebraic method is beginning to reveal its strength.Comment: 40 pages of LateX, additional remarks resulting from conversations and mail contents, removal of typographical error