Band 70, Heft 6

The theory of fuzzy deductive systems in RM algebras is developed. Various characterizations of fuzzy deductive systems are given. It is proved that the set of all fuzzy deductive systems of a RM algebra 𝒜 is a complete lattice (it is distributive if 𝒜 is a pre-BBBCC algebra). Some characterizations of Noetherian RM algebras by fuzzy deductive systems are obtained. In pre-BBBZ algebras, the fuzzy deductive system generated by a fuzzy set is constructed. Finally, closed fuzzy deductive systems are defined and studied. It is showed that in finite CI and pre-BBBZ algebras, every fuzzy deductive system is closed. Moreover, the homomorphic properties of (closed) fuzzy deductive systems are provided.

The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K 2 -algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L °° and D ( L ) of L that were associated to L in [4]. An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant. As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra L S . It is shown that this is exactly when de Morgan subalgebra L °° of L is a perfect extension of the Boolean algebra B ( L ). Two examples illustrating when this special case happens and when it does not are presented.

In the paper, we introduce 𝔏-fuzzy state filters in state residuated lattices and investigate their related properties, where 𝔏 is a complete Heyting algebra. Moreover, we study the 𝔏-fuzzy state co-annihilator of an 𝔏-fuzzy set with respect to an 𝔏-fuzzy state filter. Finally, using the 𝔏-fuzzy state co-annihilator, we investigate lattice structures of the set of some types of 𝔏-fuzzy state filters in state residuated lattices. In particular, we prove that: (1) the set FSF [ L ] of all 𝔏-fuzzy state filters is a complete Heyting algebra; (2) the set S ν FSF [ L ] of all stable state filters relative to an 𝔏-fuzzy set ν is also a complete Heyting algebra; (3) the set I μ FSF [ L ] of all involutory 𝔏-fuzzy state filters relative to an 𝔏-fuzzy state filter μ is a complete Boolean algebra.

We introduce and investigate central lifting property (CLP) for orthomodular lattices as a property whereby all central elements can be lifted modulo every p-ideal. It is shown that prime ideals, maximal ideals and finite p-ideals have CLP. Also Boolean algebras, simple chain finite orthomodular lattices, subalgebras of an orthomodular lattices generated by two elements and finite orthomodular lattices have CLP. The main results of the present paper include the investigation of CLP for principal p-ideals and finite direct products of orthomodular lattices.

In this paper, we apply the module theory to EQ-algebras and we introduce EQ-modules, multiplication EQ-modules and investigate some properties about them. Then we construct the fraction of EQ-algebras, the fraction of EQ-modules, and prove some related results.

In this paper, we find all the solutions of the title Diophantine equation in positive integers ( m , n , k , x ), where P i is the i th term of the Pell sequence.

We consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ -ideal of classic microscopic sets.

The disjoint properties of finitely many composition operators acting on the weighted Banach spaces of holomorphic functions in the unit disk were investigated in this paper.

In this paper we prove existence and uniqueness of a common fixed point for non-self coincidentally commuting mappings with nonlinear, generalized contractive condition defined on strictly convex Menger PM-spaces proved.

Fundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

We outline the transition from classical probability space (Ω, A , p ) to its "divisible" extension, where (as proposed by L. A. Zadeh) the σ -field A of Boolean random events is extended to the class 𝓜( A ) of all measurable functions into [0,1] and the σ -additive probability measure p on A is extended to the probability integral ∫( · ) d p on 𝓜( A ). The resulting extension of (Ω, A , p ) can be described as an epireflection reflecting A to 𝓜( A ) and p to ∫( · ) d p . The transition from A to 𝓜( A ), resembling the transition from whole numbers to real numbers, is characterized by the extension of two-valued Boolean logic on A to multivalued Łukasiewicz logic on 𝓜( A ) and the divisibility of random events: for each random event u ∈ 𝓜( A ) and each positive natural number n we have u/n ∈ 𝓜( A ) and ∫( u/n ) d p = (1 /n ) ∫u d p . From the viewpoint of category theory, objects are of the form 𝓜( A ), morphisms are observables from one object into another one and serve as channels through which stochastic information is conveyed. We study joint random experiments and asymmetrical stochastic dependence/independence of one constituent experiment on the other one. We present a canonical construction of conditional probability so that observables can be viewed as conditional probabilities. In the present paper we utilize various published results related to "quantum and fuzzy" generalizations of the classical theory, but our ultimate goal is to stress mathematical (categorical) aspects of the transition from classical to what we call divisible probability.

In this paper, we investigate the asymptotic behavior of the multivariate record values by using the Reduced Ordering Principle (R-ordering). Necessary and sufficient conditions for weak convergence of the multivariate record values based on sup-norm are determined. Some illustrative examples are given.

In this paper, we investigate the asymptotic properties of a nonparametric estimator of the relative error regression given a functional explanatory variable, in the case of a scalar censored response, we use the mean squared relative error as a loss function to construct a nonparametric estimator of the regression operator of these functional censored data. We establish the strong almost complete convergence rate and asymptotic normality of these estimators. A simulation study is performed to illustrate and compare the higher predictive performances of our proposed method to those obtained with standard estimators.

The Kumaraswamy generalized family of distributions proposed by Cordeiro and de-Castro (2011), has received increased attention in modern distribution theory with 624 google citations, and more than 50 special models have been studied so far. We define another generator, and then propose a new Kumaraswamy generalized family of distributions by inducting this new generator. Some useful properties of the proposed family are obtained such as quantiles, linear representation of the density, moments and generating function. The method of maximum likelihood is used for estimating family parameters. The properties of a special model of the family, called new Kumaraswamy-Burr XII distribution, are reported. A simulation study is conducted to assess the performance of maximum likelihood estimates of the proposed model. Two real-life data sets are analyzed to illustrate the flexibility of proposed model.

In this paper, we suggest a novel public key scheme by incorporating the twisted Edwards model of elliptic curves. The security of the proposed encryption scheme depends on the hardness of solving elliptic curve version of discrete logarithm problem and Diffie-Hellman problem. It then ensures secure message transmission by having the property of one-wayness, indistinguishability under chosen-plaintext attack (IND-CPA) and indistinguishability under chosen-ciphertext attack (IND-CCA). Moreover, we introduce a variant of Nyberg-Rueppel digital signature algorithm with message recovery using the proposed encryption scheme and give some countermeasures to resist some wellknown forgery attacks.

In the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.