Non-Abelian Hopf Cohomology II -- the General Case --

We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed by the authors in a previous article. We establish their functoriality and compute explicit examples. Further we classify Hopf torsors.Comment: This paper is conceived as the continuation of the article of the same authors entitled "Non-Abelian Hopf Cohomology", J. Algebra t. 312, (2007), no 2, p. 733 -- 75

Non-Abelian Hopf Cohomology

We introduce non-abelian cohomology sets of Hopf algebras with coefficients in Hopf modules. We prove that these sets generalize Serre's non-abelian group cohomology theory. Using descent techniques, we establish that our construction enables to classify as well twisted forms for modules over Hopf-Galois extensions as torsors over Hopf-modules.Comment: 19 page

Homogeneous algebras

Various concepts associated with quadratic algebras admit natural generalizations when the quadratic algebras are replaced by graded algebras which are finitely generated in degree 1 with homogeneous relations of degree N. Such algebras are referred to as {\sl homogeneous algebras of degree N}. In particular it is shown that the Koszul complexes of quadratic algebras generalize as N-complexes for homogeneous algebras of degree N.Comment: 24 page

Statistical Field Theory and Networks of Spiking Neurons

This paper models the dynamics of a large set of interacting neurons within the framework of statistical field theory. We use a method initially developed in the context of statistical field theory [44] and later adapted to complex systems in interaction [45][46]. Our model keeps track of individual interacting neurons dynamics but also preserves some of the features and goals of neural field dynamics, such as indexing a large number of neurons by a space variable. Thus, this paper bridges the scale of individual interacting neurons and the macro-scale modelling of neural field theory

A Path Integral Approach to Interacting Economic Systems with Multiple Heterogeneous Agents

This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems' interactions and agent's complexity. The formalism does not seek to aggregate agents: it rather replaces the standard optimization approach by a probabilistic description of the agents' behaviors and of the whole system. This is done in two distinct steps. A first step considers an interacting system involving an arbitrary number of agents, where each agent's utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around its utility optimum. The whole system of agents is thus defined by a composite probability depending on time, agents' interactions, relations of strategic dominations, agents' information sets and expectations. This setting allows for heterogeneous agents with different utility functions, strategic domination relations, heterogeneity of information, etc. This dynamic system is described by a path integral formalism in an abstract space -- the space of the agents' actions -- and is very similar to a statistical physics or quantum mechanics system. We show that this description, applied to the space of agents' actions, reduces to the usual optimization results in simple cases. Compared to the standard optimization, such a description markedly eases the treatment of a system with a small number of agents. It becomes however useless for a large number of agents. In a second step therefore, we show that, for a large number of agents, the previous description is equivalent to a more compact description in terms of field theory. This yields an analytical, although approximate, treatment of the system. This field theory does not model an aggregation of microeconomic systems in the usual sense, but rather describes an environment of a large number of interacting agents. From this description, various phases or equilibria may be retrieved, as well as the individual agents' behaviors, along with their interaction with the environment. This environment does not necessarily have a unique or stable equilibrium and allows to reconstruct aggregate quantities without reducing the system to mere relations between aggregates. For illustrative purposes, this paper studies several economic models with a large number of agents, some presenting various phases. These are models of consumer/producer agents facing binding constraints, business cycle models, and psycho-economic models of interacting and possibly strategic agents

A Path Integral Approach to Business Cycle Models with Large Number of Agents

This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems' interactions and agents' complexity. This formalism does not seek to aggregate agents. It rather replaces the standard optimization approach by a probabilistic description of both the entire system and agents' behaviors. This is done in two distinct steps. A first step considers an interacting system involving an arbitrary number of agents, where each agent's utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around his utility optimum. The entire system of agents is thus defined by a composite probability depending on time, agents' interactions and forward-looking behaviors. This dynamic system is described by a path integral formalism in an abstract space - the space of the agents' actions - and is very similar to a statistical physics or quantum mechanics system. We show that this description, applied to the space of agents' actions, reduces to the usual optimization results in simple cases. Compared to a standard optimization, such a description markedly eases the treatment of systems with small number of agents. It becomes however useless for a large number of agents. In a second step therefore, we show that for a large number of agents, the previous description is equivalent to a more compact description in terms of field theory. This yields an analytical though approximate treatment of the system. This field theory does not model the aggregation of a microeconomic system in the usual sense. It rather describes an environment of a large number of interacting agents. From this description, various phases or equilibria may be retrieved, along with individual agents' behaviors and their interactions with the environment. For illustrative purposes, this paper studies a Business Cycle model with a large number of agents