First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants.     

Regular left-continuous t-norms

A left-continuous (l.-c.) t-norm ⊙ is called regular if there is an n<ω such that the map x ↦ x⊙a has, for any a∈[0,1], at most n discontinuity points, and if the function mapping every a∈[0,1] to the set behaves in a specifically simple way. The t-norm algebras based on regular l.-c. t-norms generate the variety of MTL-algebras. With each regular l.-c. t-norm, we associate certain characteristic data, which in particular specifies a finite number of constituents, each of which belongs to one out of six different types. The characteristic data determines the t-norm to a high extent; we focus on those t-norms which are actually completely determined by it. Most of the commonly known l.-c. t-norms are included in the discussion. Our main tool of analysis is the translation semigroup of the totally ordered monoid ([0,1];≤,⊙,0,1), which consists of commuting functions from the real unit interval to itself.     

Idempotent uninorms

Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. Two broad classes of idempotent uninorms are fully characterized: the class of left-continuous ones and the class of right-continuous ones. In particular, the important subclasses of conjunctive left-continuous idempotent uninorms and of disjunctive right-continuous idempotent uninorms are characterized by means of super-involutive and sub-involutive decreasing unary operators. As a consequence, it is shown that any involutive negator gives rise to a conjunctive left-continuous idempotent uninorm and to a disjunctive right-continuous idempotent uninorm.     

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

In this paper we study extensions of the arithmetic operators +, -, ·, ÷ to the lattice of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on which can be generated using these operators; these classes provide natural extensions of the Łukasiewicz, product, Frank, Schweizer–Sklar and Yager t-norms to .     

Counting measures,monotone random set functions

Summary This paper arose from work on random processes whose values are measures or more general set functions. Secs. 1–3, which have nothing specifically random, discuss two topologies for certain sigma-finite measures. One, applicable only to counting measures, is a quotient topology which is useful in the finite case but excessively weak in the infinite case. Making use of a well-known result of P. Hall on sets of representatives, we describe this topology and show that it can be enlarged to the stronger one generated by a modification of the Lévy-Prohorov (L-P) metric. Sec. 4 gives a property of the L-P metric for finite integer valued counting measures. The rest of the paper deals with a random monotone non-negative set function in a separable metric space X. If X is complete and if is subadditive and right continuous¹ in probability on certain classes of sets, we show the existence of a version of with right-continuous sample functions. If X is locally compact and is left continuous in probability on a certain class of open sets, there is a left-continuous version. With appropriate additional assumptions, we obtain versions that are measures or capacities. In the latter case, a 0–1 valued set function represents a random closed or compact set. The form of integer-valued strongly subadditive set functions is described for certain cases.Supported in part by National Science Foundation Grant GP-6216     

Tensor-product approximation to operators and functions in high dimensions

In recent papers tensor-product structured Nyström and Galerkin-type approximations of certain multi-dimensional integral operators have been introduced and analysed. In the present paper, we focus on the analysis of the collocation-type schemes with respect to the tensor-product basis in a high spatial dimension d. Approximations up to an accuracy are proven to have the storage complexity with q independent of d, where N is the discrete problem size. In particular, we apply the theory to a collocation discretisation of the Newton potential with the kernel , , d3. Numerical illustrations are given in the case of d=3.     

Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it.     

Gorenstein FP-Injective Dimension for Complexes

This article is a continuous work of [17 Hu , J. , Zhang , D. ( 2013 ). Weak AB-context for FP-injective modules with respect to semidualizing modules . J. Algebra Appl. 12 ( 7 ): 1350039 .[Crossref], [Web of Science ®] , [Google Scholar]], where the coauthors introduced the notion of 𝒢-FP-injective R-modules. In this article, we define a notion of 𝒢-FP-injective dimension for complexes over left coherent rings. To investigate the relationships between 𝒢-FP-injective dimension and FP-injective dimension for complexes, the complete cohomology group bases on FP-injectives is given.     

Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in [7] ofG. H. Meisters and improves the result [5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result [10] ofG. A. Willis.     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

An Infinite Analogue of Rings with Stable Rank One

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings) and include examples like End (V), the ring of endomorphisms of a vector space V over some field , and ( ), the ring of all row- and column-finite matrices over . We show that the category of QB-rings is stable under the formation of corners, ideals, and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e., a quasi-invertible element. Finally we show that the C*-algebras that are QB-rings are exactly the extremally rich C*-algebras studied by L. G. Brown and the second author.     

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^?1/4?ε; in the smoothness direction, a sufficient condition in Hölder classes isq∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^?1/4 and belong toC^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.     

Generalized Bernstein polynomials with Pollaczek weight

Bernstein polynomials are a useful tool for approximating functions. In this paper, we extend the applicability of this operator to a certain class of locally continuous functions. To do so, we consider the Pollaczek weight

which is rapidly decaying at the endpoints of the interval considered. In order to establish convergence theorems and error estimates, we need to introduce corresponding moduli of smoothness and K-functionals. Because of the unusual nature of this weight, we have to overcome a number of technical difficulties, but the equivalence of the moduli and K-functionals is a benefit interesting in itself. Similar investigations have been made in [B. Della Vecchia, G. Mastroianni, J. Szabados, Weighted approximation of functions with endpoint or inner singularities by Bernstein operators, Acta Math. Hungar. 103 (2004) 19–41] in connection with Jacobi weights.     

Continuous Dependence of Renormalized Solution for Nonlinear Degenerate Parabolic Problems in the Whole Space

We consider the general degenerate parabolic equation: $$u_t - \Delta b(u) + div\ \tilde{F}(u) = f \qquad{\rm in} \ \ Q\ =\ ]0,T[ \, \times \, =\mathbb{R}^N , \ \ T > 0. $$ We suppose that the flux \({\tilde{F}}\) is continuous, b is nondecreasing continuous, and both are not necessarily Lipschitz continuous functions. The well-posedness (existence and uniqueness) of the renormalized solution of the associated Cauchy problem for L ¹ initial data and source term is studied in Maliki and Ouédraigo (Ann. Fac, Sci. Toulouse Math. (6) 17(3):597–611, 2008) under a structure condition \({\tilde{F}(r)=F(b(r))}\) and an assumption on the modulus of continuity of b. In the same framework, our aim is here to establish the continuous dependence of this renormalized solution with respect to the data. The novelty is the fact that we are working in the whole space \({\Omega=\mathbb{R}^{N}}\) with unbounded data (u ₀, f) and b, \({\tilde{F}}\) are not Lipschitz functions.     

Characterizing n-Isoclinism Classes of Lie Algebras

In this article, we introduce the notion of the equivalence relation, n-isoclinism, between Lie algebras, and obtain some criterions under which Lie algebras are n-isoclinic. In particular, we show that n-isoclinic Lie algebras can be isoclinically embedded into one Lie algebra. Also, we present the notion of an n-stem Lie algebra and prove its existence within an arbitrary n-isoclinism class. In addition, similar to a result of Hekster [6 Hekster , N. S. ( 1986 ). On the structure of n-isoclinam classes of groups . J. Pure Appl. Algebra 40 : 63 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]] in the group case, we characterize the n-stem Lie algebras in the n-isoclinism classes which contains at least one finitely generated Lie algebra L with dim (L ⁿ⁺¹) finite.     

The canonical projection associated with certain possibly infinite generalized iterated function systems as a fixed point

In this paper, influenced by the ideas from Mihail (Fixed Point Theory Appl 2015:15, 2015), we associate to every generalized iterated function system \(\mathcal {F}\) (of order m) an operator \(H_{\mathcal {F}}:\mathcal {C} ^{m}\rightarrow \mathcal {C}\), where \(\mathcal {C}\) stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of \(\mathcal {F}\)) for the operator \(H_{\mathcal {F}}\) to be continuous, contraction, \(\varphi \)-contraction, Meir–Keeler or contractive. We also give sufficient condition under which \(H_{\mathcal {F}}\) has a unique fixed point \(\pi _{0}\). Moreover, we prove that, under these circumstances, the closure of the imagine of \(\pi _{0}\) is the attractor of \(\mathcal {F}\) and that \(\pi _{0}\) is the canonical projection associated with \(\mathcal {F}\). In this way we give a partial answer to the open problem raised on the last paragraph of the above-mentioned Mihail’s paper.     

Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients

In this paper we extend the results of Caffarelli, Jerison, and Kenig [Ann. of Math. (2) 155 (2002)] and Caffarelli and Kenig [Amer. J. Math. 120 (1998)] by establishing an almost monotonicity estimate for pairs of continuous functions satisfying in an infinite strip (global version) or a finite parabolic cylinder (localized version), where ${\cal L}$ is a uniformly parabolic operator with double Dini continuous ${\cal A}$ and uniformly bounded b and c. We also prove the elliptic counterpart of this estimate. This closes the gap between the known conditions in the literature (both in the elliptic and parabolic case) imposed on u_± in order to obtain an almost monotonicity estimate. At the end of the paper, we demonstrate how to use this new almost monotonicity formula to prove the optimal C^1,1‐regularity in a fairly general class of quasi‐linear obstacle‐type free boundary problems. © 2010 Wiley Periodicals, Inc.     

Regularized minimization under weaker hypotheses

LetV andW be two Banach spaces, withV reflexive, a bounded convex set ofV, A a linear mapping fromV intoW, and letF be a convex functional onW. We minimizeJ(v)=F(Av) on using hypotheses about particular sequences in IfV is uniformly convex, we prove existence and uniqueness of a solution of minimal norm minimizingJ. In the Hilbert space case, withF defined byF(w)=w–f ²,f given inW, we get existence and uniqueness of the projection off on A(), which generalizes the case where A() is a closed set ofW (taking closed andA continuous). Finally, we give examples, and we study an unbounded operator case.