Generalized Kripke semantics for Nelson’s logic

A completeness theorem for logics N4 ^N and N3⁰ is proved. A characterization by classes of N4 ^N - and N3⁰-models is presented, and it is proved that all logics of four types η(L), η ³(L), η ⁿ (L), and η ⁰(L) are Kripke complete iff so are their respective intuitionistic fragments L. A generalized Kripke semantics is introduced, and it is stated that such is equivalent to an algebraic semantics. The concept of a p-morphism between generalized frames is defined and basic statements on p-morphisms are proved.     

Natural Deduction for Fitting’s Four-Valued Generalizations of Kleene’s Logics

In this paper, we present sound and complete natural deduction systems for Fitting’s four-valued generalizations of Kleene’s three-valued regular logics.     

Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

Natural Deduction for Post’s Logics and their Duals

In this paper, we introduce the notion of dual Post’s negation and an infinite class of Dual Post’s finitely-valued logics which differ from Post’s ones with respect to the definitions of negation and the sets of designated truth values. We present adequate natural deduction systems for all Post’s k-valued (\(k\geqslant 3\)) logics as well as for all Dual Post’s k-valued logics.     

Extensions of Hałkowska–Zajac's three-valued paraconsistent logic

 As it was proved in [4, Sect. 3], the poset of extensions of the propositional logic defined by a class of logical matrices with equationally-definable set of distinguished values is a retract, under a Galois connection, of the poset of subprevarieties of the prevariety generated by the class of the underlying algebras of the defining matrices. In the present paper we apply this general result to the three-valued paraconsistent logic proposed by Hałkowska–Zajac [2]. Studying corresponding prevarieties, we prove that extensions of the logic involved form a four-element chain, the only proper consistent extensions being the least non-paraconsistent extension of it and the classical logic. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5> 03B50, 03B53, 03G10 RID="ID=" <E5>Key words or phrases:</E5> Many-valued logic &ndash; Paraconsistent logic &ndash; Extension &ndash; Prevariety &ndash; Distributive lattice Received 12 August 2000 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5> 03B50, 03B53, 03G10 RID=" ID=" <E5>Key words or phrases:</E5> Many-valued logic &ndash; Paraconsistent logic &ndash; Extension &ndash; Prevariety &ndash; Distributive lattice     

An axiom system for Lingenberg’s metric-Euclidean planes

We show that Lingenbergs metric-Euclidean planes are the rectangular planes of Karzel and Stanik which satisfy the axiom If two of the perpendicular bisectors of a triangle exist, then so does the third.This paper was written while the author was at the Institute of Mathematics of University of Biaystok with a Fulbright grant. I thank the Polish-U.S. Fulbright Commission for the grant, Professor Krzysztof Pramowski for the hospitality, and Ewa Walecka for drawing the figures.     

A uniqueness theorem for series by Stromberg’s system

The paper is devoted to the Stromberg’s polynomial wavelets. A uniqueness theorem is proved for series by this system. For Paley function of this system a weak estimate of type (1,1) is obtained. Counterexamples are also discussed.     

Pontryagin’s principle for state-constrained evolutionary differential complementarity system

This paper is devoted to the state-constrained optimal control problem of evolutionary variational inequality. In this paper, the control domain is not necessarily convex. Moreover, since our state constraint is quite general and, in many cases, it requires pointwise behavior of the state, the framework of the partial differential equation (instead of the abstract framework) is used. Some optimality conditions (in the form of Pontryagin’s principle) for optimal controls are established.     

Galilei’s relativity principle for a system of reaction-convection-diffusion equations

The representations of the Galilean algebra and its extensions relative to which the system of nonlinear reaction-convection-diffusion equations can be invariant are investigated. The kinds of nonlinearities at which this system is invariant relative to those algebras are determined to within continuous equivalence transformations.     

Note on Halphen’s system

The Ramanujan Journal - In this paper, we construct extensions of the differential field obtained from Halphen’s system of ordinary differential equations (ODEs) using quasi-modular forms of...     

A scenario for the creation of chaotic attractors in Chua’s system

We investigate a scenario for the creation of irregular chaotic attractors in Chua’s system. We show that irregular attractors in Chua’s system are created by those and only those mechanisms that characterize Lorenz, Rössler, and other dissipative nonlinear systems described by ordinary differential equations. These mechanisms include cascades of Feigenbaum period doubling bifurcations, subharmonic cascades of cycle bifurcations in Sharkovskii’s order, and homoclinic cascades of bifurcations.     

Kantorovich’s majorants principle for Newton’s method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA. The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).     

Devaney’s chaos of a relay system

We address the differential equation with a pulse function, whose moments of discontinuity depend on the initial moment. The existence of a chaotic attractor, and the complex behavior of all solutions are investigated. An appropriate simulations are presented.     

Functionally invariant solutions to Maxwell’s system

We consider the problem of the existence of functionally invariant solutions to Maxwell’s system. The solutions found contain functional arbitrariness, which is used for determining the parameters of Maxwell’s system (the dielectric and magnetic constants).     

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.     

Harnack’s principle for quasiminimizers

Abstract We study Harnack type properties of quasiminimizers of the -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Keywords: Metric space, doubling measure, Poincaré inequality, Newtonian space, Harnack inequality, Harnack convergence theorem Mathematics Subject Classification (2000): 49J52, 35J60, 49J27