Method of collective variables with references system for the grand canonical ensemble

L'vov Branch of Statistical Physics, Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 2, pp. 282–296, May, 1989.     

Indistinguishability of particles or independence of the random variables?

After establishing the formula of information, we turn to two cases: to the independent case and to the dependent case. If the characteristics of the particles (e.g., the energy) as random variables are independent, we come in the discrete case to the Boltzmann distribution, while in the continuous case we come to the Gibbs formula-irrespectively of whether the particles are distinguishable or not. Our effort is nimed at clarity of the ideas, not at their mathematical completeness. Supported by the Hungarian National Foundation for Scientific Research (grant No. T 016384 and No. 4007-016237) Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part III.     

Renormalization group, causality, and nonpower perturbation expansion in QFT

The structure of the QFT expansion is studied in the framework of a new “invariant analytic” version of the perturbative QCD. Here, an invariant coupling constant α(Q ² /Λ ² ) = β ₁ α_s(Q ² )/(4π) becomes a Q ² -analytic invariant function α _an (Q²/Λ ² ) ≡A(x), which, by construction, is free of ghost singularities because it incorporates some nonperturbative structures. In the framework of the “analyticized” perturbation theory, an expansion for an observable F, instead of powers of the analytic invariant charge A(x), may contain specific functions A_n(x)=[aⁿ(x)] _an , the “nth power of a(x) analyticized as a whole.” Functions A _n>2(x) for small Q² ≤Λ ² oscillate, which results in weak loop and scheme dependences. Because of the analyticity requirement, the perturbation series for F(x) becomes an asymptotic expansion à la Erdélyi using a nonpower set {A _n (x)}. The probable ambiguities of the invariant analyticization procedure and the possible inconsistency of some of its versions with the renormalization group structure are also discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 55–66, April, 1999.     

Functional of the grand partition function in the method of collective variables with distinguished reference system. Multicomponent system

L'vov Branch of Statistical Physics, Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 1, pp. 72–82, April, 1990.     

Individual and collective opportunities

We consider the bargaining problem in the context of a variable number of agents. When new agents enter the scene but the opportunities open to the enlarged group do not expand, some solutions paradoxically may recommend that some of the agents originally present gain. We propose a quantitative measure of the extent to which a solution allows this phenomenon to occur and we rank the major solutions on that basis. The Kalai-Smorodinsky solution performs better than all weakly pareto-optimal and anonymous solution, and in particular strictly better than the Nash solution. However, the two solutions are equivalent when it is the opportunities for gains offered to initial groups, instead of individuals, that are being compared.     

Teaching and learning mathematics in the collective

In this paper we analyse and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented. We frame our discussion within enactivism—a theory of cognition that has helped us to understand classroom processes, particularly at the level of the group. We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective. Our analysis reveals teaching practices that value, capitalize upon, and promote group cognition, our discussion positions such work as teaching a way of being with mathematics, and we close by offering implications for teaching, educational policy, and further research.