A Modal Logic Based on Linearly Ordered f-Spaces

A modal logic associated with the -spaces introduced by Ershov is examined. We construct a modal calculus that is complete w.r.t. the class of all strictly linearly ordered -frames, and the class of all strictly linearly ordered -frames.     

Modal Multilattice Logic

A modal extension of multilattice logic, called modal multilattice logic, is introduced as a Gentzen-type sequent calculus \(\hbox {MML}_n\). Theorems for embedding \(\hbox {MML}_n\) into a Gentzen-type sequent calculus S4C (an extended S4-modal logic) and vice versa are proved. The cut-elimination theorem for \(\hbox {MML}_n\) is shown. A Kripke semantics for \(\hbox {MML}_n\) is introduced, and the completeness theorem with respect to this semantics is proved. Moreover, the duality principle is proved as a characteristic property of \(\hbox {MML}_n\).     

Paraconsistent double negation as a modal operator

A paraconsistent modal‐like logic, , is defined as a Gentzen‐type sequent calculus. The modal operator in the modal logic can be simulated by the paraconsistent double negation in . Some theorems for embedding into a Gentzen‐type sequent calculus for and vice versa are proved. The cut‐elimination and completeness theorems for are also proved.     

Schauder decompositions and functional calculus

We establish a functional calculus, with nice properties, for one and several continuous operators on some non-normable locally convex spaces, more specifically, for operators on the space of entire functions and on other power series spaces. In particular we obtain spectral mapping theorems. The calculus rests on Schauder decompositions for the spaces under consideration, which are of independent interest.     

Ockham’s Calculus of Strict Implication

In his main work Summa Logicae written around 1323, William of Ockham developed a system of propositional modal logic which contains almost all theorems of a modern calculus of strict implication. This calculus is formally reconstructed here with the help of modern symbols for the operators of conjunction, disjunction, implication, negation, possibility, and necessity.     

Some set-theoretical aspects of differentiation theory

We study the behaviour under uncountable products of some structures of interest in the Frölicher-Kriegel calculus of differentiation theory: the Mackey closure topology, the compactly and sequentially generated topologies and bornologies. Some properties turn out to be independent from ZFC+(CH).     

One Calculus of Nonderivable Formulas of Propositional Modal Logic

We obtain the upper bound for the number of applications of rules in the deduction tree of modal formulas in some variants of the S4 calculus. We use this estimate to construct the calculus where the only derivable formulas are those whose counterparts in S4 are unprovable.     

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.     

Tableaus with Invertible Rules for Many‐Valued Modal Propositional Logics

We investigate tableaubased theorem proving for some families of manyvalued modal propositional logics. We consider manyvalued counterparts of wellknown twovalued modal logics obtained from various combinations of four basic properties of the accessibility relation, i.e., reflexivity, transitivity, seriality, and Euclideaness. For each of these logics we introduce tableau systems with certain invertible rules. We prove soundness and completeness of the presented tableau systems.     

On the Existence of a Generalized Solution of One Partial Differential System

We propose a method for the construction of generalized solutions for some nondivergent partial differential systems using set-valued analogs of the generalized statement of the problem based on subdifferential calculus. We establish new sufficient conditions for the existence of solutions of a variational inequality with set-valued operator under weakened coerciveness conditions. We consider examples of a weighted p-Laplacian in the Sobolev spaces , p 2.     

Limiting subhessians, limiting subjets and their calculus

We study calculus rules for limiting subjets of order two. These subjets are obtained as limits of sequences of subjets, a subjet of a function at some point being the Taylor expansion of a twice differentiable function which minorizes and coincides with at . These calculus rules are deduced from approximate (or fuzzy) calculus rules for subjets of order two. In turn, these rules are consequences of delicate results of Crandall-Ishii-Lions. We point out the similarities and the differences with the case of first order limiting subdifferentials.

     

Quantum groups, differential calculi and the eigenvalues of the Laplacian

We study -differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz' first calculus, the calculation of the eigenvalues of the Laplacian.

     

Effective Replaceability of the Omega—Rule for Restricted Sequents of the First–Order Linear Temporal Logic

We consider an effective replaceability of the omega–rule for the always operator in a restricted first–order linear temporal logic. This work is a continuation of two previous papers of the author where an infinitary (with the omega–rule) calculus without loop rules was constructed and founded. In the present paper, we use this calculus to construct the so–called saturated calculus consisting of four decidable deductive procedures replacing the omega–rule for the always operator.     

Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables

By using some basic calculus of multiple integration, we provide an alternative expression of the integral

     

Discrete wick calculus and stochastic functional equations

We develop Wick calculus over finite probability spaces and prove that there is a one-to-one correspondence between the solutions of Wick stochastic functional equations and the solutions of the deterministic functional equations obtained by turning off the noise. We also point out some possible applications to ordinary and partial stochastic differential equations.This research is supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap a.s. (STATOIL).     

A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that -sectorial operators generally do not admit a bounded H^∞ calculus over the right half-plane. In contrast to this, we prove that the H^∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as as and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that -sectorial operators admit a bounded calculus over the Besov algebra of the right half-plane. We also discuss the link between -sectorial operators and bounded Tadmor-Ritt operators.     

Upper bound on the lengthening of proofs by cut elimination

In this paper there are constructed sequential calculi KGL and IGL. The calculus KGI is a version of the classical predicate calculus, and IGL is a version of constructive calculus. KGL and IGL do not contain structural rules and there are no rules in them for which in some premise more than one lateral formula would be contained. A procedure for eliminating cuts from proofs in these calculi is described. It is shown that the height of a derivation obtained by this procedure exceeds 2 ^h, where h is the height of the original derivation, is the number of sequences in the original derivation,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 87–98, 1984.     

Rational functions and real Schubert calculus

We single out some problems of Schubert calculus of subspaces of codimension that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension and rational functions of one variable.

     

Everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions

We consider the integral functional of the calculus of variations

     

Loop-free calculus for modal logic S4. II

In [J. Andrikonis, Loop-free calculus for modal logic S4. I, Lith. Math. J., 52(1):1–12, 2012], loop-free calculus for modal logic S4 is presented. The calculus uses several types of indexes to avoid loops and obtain termination of derivation search. Although the mentioned article proves that derivation search in the calculus is finite, the proof of soundness and completeness is omitted and, therefore, is presented in this paper. Moreover, this paper presents loop-free calculus for modal logics K4, which is obtained by modifying the calculus for S4. Finally, some remarks for programming the given calculi are offered.