Three constructions of Frobenius manifolds: a comparative study

The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the B--side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``non-linear cohomology''. The comparison of thesestructures sheds some light on the general Mirror Problem: establishing isomorphisms between Frobenius manifolds of different classes. Another theme is the study of tensor products of Frobenius manifolds, corresponding respectively to the K\"unneth formula in Quantum Cohomology, direct sum of singularities in Saito's theory, and presumably, the tensor product of the differential Gerstenhaber-Batalin-Vilkovisky algebras. We extend the initial Gepner's construction of mirrors to the context of Frobenius manifolds and formulate the relevant mathematical conjecture.Comment: 46 pages, AMSTe

Strategic decision-making processes as a mediator of the effect of board characteristics on company innovation: A study of publicly-listed firms in Greece

Based on the Upper Echelons Theory that suggests the demographic characteristics of executives are linked to organisational processes and outcomes, the paper proposes that strategic decision-making processes mediate the relationship between board members’ demographic characteristics and corporate innovation relating to product, process and organization. Based on questionnaires completed by 101 CEOs of Greek listed firms, the findings confirm that reporting and formalization as decision processes mediate the effect of board characteristics on innovation. Sound financial and formal mechanisms encourage Greek executives to take risks and invest in product or service innovation. Findings show that the executives’ educational level is positively associated with financial reporting and rule formalization activities due to the changes that have been occurred in the Greek education system over recent decades. Functional background is found to influence only financial reporting activities. Finally, the managerial implications of this study are discussed

Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization

In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the *-representation theory of such *-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C^*-algebras. Throughout this paper the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C^*-algebras, we show that the GNS construction of *-representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two *-algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate *-representations of the two *-algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over *-algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules.Comment: LaTeX2e, 51pages, minor typos corrected and Note/references adde

Exponential renormalization

Moving beyond the classical additive and multiplicative approaches, we present an "exponential" method for perturbative renormalization. Using Dyson's identity for Green's functions as well as the link between the Faa di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota-Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counterfactors and of order n bare coupling constants).Comment: revised version; accepted for publication in Annales Henri Poincar

A proof of strong normalisation using domain theory

Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-L\"{o}f dependent type theory extended with a program for Spector double negation shift.Comment: 16 page

Niceness theorems

Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, ... . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.Comment: 52 page