What can be done with PRA?

Let S⁺ denote system JR _° ² + AC _° ¹ in a classical second-order arithmetic, in which the induction rule is permitted to apply only to quantifier-free formulas and to _° ² -formulas not containing functional variables, while the convolution axiom is permitted to apply only to _° ¹ -formulas without functional variables. Also postulated is the closedness of the function class being examined, relative to primitive recursive operations. System, S⁺ turns out to be sufficiently rich: in it a theory of recursions and an elementary recursive analysis can be developed, a theorem on the continuity of effective operators and a theorem on cuteliminability from -deductions can be proved, and the usual analytic proofs of many number-theoretic theorems, including the prime distribution law, can be derived (with insignificant changes). (A formalization in S⁺ of the proof of Konig's lemma on paths in binary trees and of Godel's completeness theorem is described in the note.) On the other hand, the system admits of an interpretation in primitive recursive arithmetic (PRA). In particular, quantifier-free theorems in S⁺ are deducible in PRA, while theorems of form xyR(x,y) with a quantifier-free formula R have calculi R (x,(x)) with primitive recursive function , deducible in PRA. Thus, the suppressing part of operating constructive analysis can be developed already at the finite stages of the Shanin majorant hierarchy. In addition, a purely mechanical method exists for obtaining elementary number-theoretic proofs from many analytic proofs.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 93–102, 1976. Results announced September 4, 1975.     

New Andreev-type states in superconducting nanowires

Superconducting nanowires can exhibit a spatially inhomogeneous pair condensate that leads to the formation of new Andreev-type states. Such states are mainly located beyond the regions where the order parameter is enhanced, and no normal-superconducting contact or external magnetic field is needed for their formation. Our numerical self-consistent solutions of the Bogoliubov-de Gennes equations for cylindrical nanowires, in the clean limit, demonstrate that these new Andreev-type states decrease the ratio of the energy gap to the critical temperature as compared to its bulk value. The low-lying excitations in a clean superconducting nanowire are these new Andreev-type states induced by quantum confinement of the electrons in the transverse direction.     

Exoelectronic emission in active strain and creep conditions

It is established in the course of investigating exoelectronic emission of lead in a strain process with constant rate and under creep conditions, that the kinetics and emission current parameters are determined by the development of a thermal activation process of dynamic recovery.     

The influence of paramagnetic impurities on magnetic effects in radical reactions

The influence of paramagnetic impurities on magnetic and spin effects in radical reactions in liquid solutions is treated. The recombination probability of a radical pair is obtained. It is shown, that at high impurity concentration or in high-viscosity solvents the effect of the magnetic field vanishes.     

On dissipative structures in nonuniform media

It is found that in many physical, chemical and biological systems the nonuniform active media may be considered as a “matrix” controlling the selection of the dissipative structures of definite type.     

Normal deduction in the intuitionistic linear logic

We describe a natural deduction system NDIL for the second order intuitionistic linear logic which admits normalization and has a subformula property. NDIL is an extension of the system for !-free multiplicative linear logic constructed by the author and elaborated by A. Babaev. Main new feature here is the treatment of the modality !. It uses a device inspired by D. Prawitz' treatment of S4 combined with a construction introduced by the author to avoid cut-like constructions used in -elimination and global restrictions employed by Prawitz. Normal form for natural deduction is obtained by Prawitz translation of cut-free sequent derivations. Received: March 29, 1996     

Complexity of subclasses of the intuitionistic propositional calculus

We investigate the complexity of the decision problem for subclasses of the intuitionistic propositional calculus and present upper bounds for decision procedures locating these subclasses into lower complexity classes like co-NP or polynomial time.