The theory of {\vec Z}C(2)^2-lattices is decidable

For arbitrary finite group and countable Dedekind domain such that the residue field is finite for every maximal -ideal , we show that the localizations at every maximal ideal of two -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations of the above results to arbitrary -torsion-free -modules and we apply the previous results to show the decidability of the theory of -lattices. Eventually, we show that -lattices have undecidable theory. Received November 28, 1995     

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     

Ordinal arithmetic and \Sigma_{1}-elementarity

We will introduce a partial ordering on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use to provide a new characterization of the ubiquitous ordinal . Received: 18 August 1997     

The decision problem for {\vec Z}C(p^3)-lattices with p prime

We show undecidability for lattices over a group ring where has a cyclic subgroup of order for some odd prime . Then we discuss the decision problem for -lattices where is a cyclic group of order 8, and we point out that a positive answer implies – in some sense – the solution of the “wild undecidable” conjecture. Received November 15, 1995     

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to{\rm e}^{z}

Summary. With denoting the -th partial sum of ${\rm e}^{z}$, the exact rate of convergence of the zeros of the normalized partial sums, , to the Szeg\"o curve was recently studied by Carpenter et al. (1991), where is defined by Here, the above results are generalized to the convergence of the zeros and poles of certain sequences of normalized Pad\'{e} approximants to , where is the associated Pad\'{e} rational approximation to . Received February 2, 1994     

Around provability logic

We present some results on algebraic and modal analysis of polynomial (intrinsically definable) distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel-Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.     

Construction of universal modal worlds based on hyperset theory

Knowledge modal formulas are interpreted by a universal modal world in the hypersets universe [A]. This remedies the limitation of the interpretation of knowledge formulas by a tower of modal worlds in the well founded universe [F], where each world can interpret only a portion of knowledge modal formulas. Received: 21 August 1995     

Einstein metrics on S^{2}-bundles

New Einstein metrics are constructed on the associated , , and -bundles of principal circle bundles with base a product of K?hler-Einstein manifolds with positive first Chern class and with Euler class a rational linear combination of the first Chern classes. These Einstein metrics represent different generalizations of the well-known Einstein metrics found by Bérard Bergery, D. Page, C. Pope, N. Koiso, and Y. Sakane. Corresponding new Einstein-Weyl structures are also constructed. Received 25 October 1996 / Revised version 1 April 1997     

A characterization of the \Sigma_1-definable functions of KP\omega + (uniform\; AC)

The subject of this paper is a characterization of the -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and . This goal is achieved by a G?del Dialectica-style functional interpretation of and a computability proof for the involved functionals. Received October 9, 1996     

Optimal $\mathcal{L}^2$ error bounds for MITC3 type element

Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized MITC3 and MIN3 and show it is of order uniform in the plate thickness. Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001     

Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ${\mathbb R}^3$

Summary. In this paper we describe and analyse a class of spectral methods, based on spherical polynomial approximation, for second-kind weakly singular boundary integral equations arising from the Helmholtz equation on smooth closed 3D surfaces diffeomorphic to the sphere. Our methods are fully discrete Galerkin methods, based on the application of special quadrature rules for computing the outer and inner integrals arising in the Galerkin matrix entries. For the outer integrals we use, for example, product-Gauss rules. For the inner integrals, a variant of the classical product integration procedure is employed to remove the singularity arising in the kernel. The key to the analysis is a recent result of Sloan and Womersley on the norm of discrete orthogonal projection operators on the sphere. We prove that our methods are stable for continuous data and superalgebraically convergent for smooth data. Our theory includes as a special case a method closely related to one of those proposed by Wienert (1990) for the fast computation of direct and inverse acoustic scattering in 3D. Received May 29, 2000 / Revised version received March 26, 2001/ Published online December 18, 2001     

A test for expandability

A model of countable similarity type and cardinality is expandable if every consistent extension of its complete theory with is satisfiable in and it is compactly expandable if every such extension which additionally is finitely satisfiable in is satisfiable in . In the countable case and in the case of a model of cardinality of a superstable theory without the finite cover property the notions of saturation, expandability and compactness for expandability agree. The question of the existence of compactly expandable models which are not expandable is open. Here we present a test which serves to prove that a compactly expandable model of cardinality of a superstable theory is expandable. It is stated in terms of the existence of a certain elementary submodel whose corresponding theory of pairs of models satisfies a weak elimination of Ramsey quantifiers. Received May 20, 1996     

Leibniz filters and the strong version of a protoalgebraic logic

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?⁺ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?⁺ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?⁺. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. Received: 30 October 1998 /?Published online: 15 June 2001     

Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). Received: February 15, 1996     

Weak covering and the tree property

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if has the tree property then and is weakly compact in K. Received: 11 June 1997     

Ordinal diagrams for recursively Mahlo universes

In this paper we introduce a recursive notation system of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each in KPM proves that the initial segment of determined by is a well ordering. Proof theoretic study for such theories will be reported in [9]. Received: 13 January, 1998