A method of proving interpolation in paraconsistent extensions of the minimal logic

The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess _CIP, nor even the restricted interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 627–648, September–October, 2007.     

A new multiquadric quasi‐interpolation operator with interpolation property

In this article, we discuss a class of multiquadric quasi‐interpolation operator that is primarily on the basis of Wu–Schaback's quasi‐interpolation operator and radial basis function interpolation. The proposed operator possesses the advantages of linear polynomial reproducing property, interpolation property, and high accuracy. It can be applied to construct flexible function approximation and scattered data fitting from numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Automatic recognition of interpolation in modal calculi

We deal with some issues on automatic recognition of interpolation properties in modal calculi extending the logics S5 and S4.3. Supported by RFBR grant No. 06-01-00358 and by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 103–119, January–February, 2007.     

有理插值的一种递推算法

针对给定的数据型值点,利用多项式带余除法,给出有理函数插值的一种新型递推算法,可以求出满足插值条件的所有有理插值函数,并举例说明其有效性.     

Weak interpolation in extensions of the logics <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript> and <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>

Conditions are specified which are necessary and sufficient for a logic over K4 to possess the weak interpolation property. For this goal to be met, simple transitive modal algebras are described, and we establish a criterion for the class of such algebras to be amalgamable. For extensions of K4, the weak interpolation property is proved decidable. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 705–722, November–December, 2008.     

半群的模糊同余扩张

本文引入半群的模糊同余扩张的概念,给出了模糊同余扩张的同态性质,同时,本文研究了带有模糊同余扩张性质的半群类,证明了一个半群S有模糊同余扩张性质当且仅当S有同余扩张性质,最后进一步给出 有模糊同余张张性质的半群类的特征。     

Restricted interpolation property in superintuitionistic logics

The restricted interpolation property IPR in modal and superintuitionistic logics is investigated. It is proved that in superintuitionistic logics of finite slices and in finite-slice extensions of the Grzegorczyk logic, the property IPR is equivalent to the projective Beth property PB2. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). Translated from Algebra i Logika, Vol. 48, No. 1, pp. 54-89, January-February, 2009.     

Meshfree isoparametric finite point interpolation method (IFPIM) with weak and strong forms for evaporative laser drilling

An isoparametric finite point interpolation method (IFPIM) with weak and strong forms has been developed to analyze evaporative laser drilling. The method is based on isoparametric finite point representation of the unknowns in the influence domain. The local influence domains are mapped onto a master domain where the shape functions and their derivatives are known. The solution in the master domain is approximated by a linear combination of shape functions. The present method employs a simple strong form in the domain and a weak form on the boundary. Three different types of boundary conditions considered are of essential, convection, and laser irradiation type. The problem is geometrically nonlinear because the domain is not known a priori due to material removal in drilling. An iterative scheme is used to solve the nonlinear problem. The material removal is handled by redistributing points in the domain. This renders the point distribution non-uniform as in random distribution. The numerical results show excellent agreement with those by FEM and BEM in terms of groove shape, temperature and heat flux distributions, and amount of material removal. The results are superior to those from the isoparametric finite point interpolation methods with only strong forms.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

Conjugate interpolation problems in spaces of entire functions

A generalization of the Taylor expansions of entire functions with variable center is used to illustrate the notion of conjugate interpolation problem. The radii of existence, uniqueness, and convergence for such problems are found. A relationship between the interpolation functional systems of two mutually conjugate problems is revealed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 616–628, April, 2000.     

A problem about the weak Hilbert property

In this paper, we prove the main result: Let both (K, S) and (K ^*,S ^*) be preordered fields, and let (K ^*,S ^*) be a finitely generated extension of (K, S). IfK ^* is transcendental overK, then (K ^*,S ^*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover, some results on the weak Hilbert property are established. Project supported by National Natural Science Foundation of China     

The Beth property and interpolation in lattice-based algebras and logics

We deal with logics based on lattices with an additional unary operation. Interrelations of different versions of interpolation, the Beth property, and amalgamation, as they bear on modal logics and varieties of modal algebras, superintuitionistic logics and varieties of Heyting algebras, positive logics and varieties of implicative lattices, have been studied in many works. Sometimes these relations can and sometimes cannot be extended to the logics without implication considered in the paper. Supported by INTAS (grant No. 04-77-7080) and by RFBR (grant No. 06-01-00358). Supported by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 307–334, May–June, 2008.     

Restricted Interpolation and the Projective Beth Property in Equational Logic

Interconnections between syntactic and categorical properties of equational theories are established. The notions of restricted interpolation and of restricted amalgamation are introduced and their equivalence proved; interrelations of the above-mentioned properties and the projective Beth property, interpolation, and amalgamation are studied.     

A random weak ergodic property of infinite products of operators in metric spaces

ABSTRACT

We study the random weak ergodic property of infinite products of mappings acting on complete metric spaces. Our results describe an aspect of the asymptotic behaviour of random infinite products of such mappings. More precisely, we show that in appropriate spaces of sequences of operators there exists a subset, which is a countable intersection of open and everywhere dense sets, such that each sequence belonging to this subset has the random weak ergodic property. Then we show that several known results in the literature can be deduced from our general result.     

On triangular norm based axiomatic extensions of the weak nilpotent minimum logic

In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t‐norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM‐algebras (?????) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM‐chains, in particular varieties generated by standard WNM‐chains, or equivalently t‐norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM‐chains, i. e. those that generate the variety ?????, and we give finite axiomatizations for some t‐norm based extensions of WNM. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of some real interpolation spaces

We interrelate the real interpolation spaces associated with the couples , and , proving among others the identities

for all .

     

Alternate signs Banach-Saks property and real interpolation of operators

In the space of bounded linear operators acting between Banach spaces we define a seminorm vanishing on the subspace of operators having the alternate signs Banach-Saks property. We obtain logarithmically convex-type estimates of the seminorm for operators interpolated by the Lions-Peetre real method. In particular, the estimates show that the alternate signs Banach-Saks property is inherited from a space of an interpolation pair to the real interpolation spaces with respect to for all and .