Band 73, Heft 6

ABSTRACT In this paper, we provide a new characterization of ℒ-fuzzy ideals of residuated lattices, which allows us to describe ℒ-fuzzy ideals generated by ℒ-fuzzy sets. Thanks to the latter, we endow the lattice of ℒ-fuzzy ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of ℒ-fuzzy annihilator of an ℒ-fuzzy subset of a residuated lattice with respect to an ℒ-fuzzy ideal and investigate some of its properties. To this extent, we show that the set of all ℒ-fuzzy ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define some types of ℒ-fuzzy ideals of residuated lattices, namely stable ℒ-fuzzy ideals relative to an ℒ-fuzzy set, and involutory ℒ-fuzzy ideals relative to an ℒ-fuzzy ideal. Finally, we prove that the set of all stable ℒ-fuzzy ideals relative to an ℒ-fuzzy set is also a complete Heyting algebra, and that the set of involutory ℒ-fuzzy ideals relative to an ℒ-fuzzy ideal is a complete Boolean algebra.

ABSTRACT In 1973, Katriňák proved that regular double p -algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety ℝDℙℂℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties ℝDBLℙ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] , ℝDℙℂℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝℙℂℍd \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝDBLℍ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] of regular double p -algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝDMℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{P}\] , ℝDMℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , ℝDMDBLℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] , ℝDMDBLℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] of regular De Morgan p -algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p -algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p -algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2ℵ0 \[{{2}^{{{\aleph }_{0}}}}\] . We then define new logics, ℛDPCℋ \[\mathcal{R}\mathcal{D}\mathcal{P}\mathcal{C}\mathcal{H}\] , ℛPCℋd \[\mathcal{R}\mathcal{P}\mathcal{C}{{\mathcal{H}}^{d}}\] and ℛDℳℋ \[\mathcal{R}\mathcal{D}\mathcal{M}\mathcal{H}\] , and show that they are algebraizable with ℝDℙℂℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝℙℂℍd \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝDMℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2ℵ0 \[{{2}^{{{\aleph }_{0}}}}\] .

ABSTRACT We extend the notion of local connectedness and uniform local connectedness to the category M σ Frm of metric σ -frames. We provide the construction of the uniformly locally connected reflection of a locally connected metric σ -frame. We also discuss complete metric σ -frames and show the existence of a completion for a metric σ -frame in the category M σ Frm .

ABSTRACT We establish explicit constructions of real transcendental numbers that are not U -numbers with respect to Mahler’s classification by using regular continued fraction expansions of irrational real algebraic numbers.

ABSTRACT Let k ≥ 2. A generalization of the well-known Pell sequence is the k -Pell sequence whose first k terms are 0,…, 0, 1 and each term afterwards is given by the linear recurrence pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k). The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k -Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [ Repdigits in generalized Pell sequences , Arch. Math. (Brno) 56 (4) (2020), 249–262].

ABSTRACT We address Peirce decompositions for evolution algebras at idempotent elements contained in the associative nucleus of the evolution algebras. If the idempotent elements are natural vectors, the requirement of being nuclear is then proved to be equivalent to the evolution algebra to be baric. A description of baric evolution algebras is also provided.

ABSTRACT In this paper, we introduce the notion of generalized anti-symmetry laws in groupoids, and we apply this concept to several algebraic structures. Moreover, we show that every Fibonacci sequence on ( C , *) is periodic.

ABSTRACT We generalize cyclic Jensen’s inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ 2 -divergence.

ABSTRACT Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind, which is an elementary transform of the hypergeometric function and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz et al. [ Differential subordinations involving generalized Bessel functions , Bull. Malays. Math. Sci. Soc. 38 (3) (2015), 1255–1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.

ABSTRACT In this paper, we consider a first-order iterative differential inclusion. We study the existence of solutions and some topological proprieties of the attainable set, where the right hand side is an upper semi-continuous multifunction with convex values. Then, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity.

ABSTRACT In this work, we study the integration of the differential-difference sine-Gordon equation with an integral type source. We deduce the time performance of the scattering data of the spectral problem which is associated with the discrete sine-Gordon equation. Using the inverse scattering method, we integrate the Cauchy problem for the differential-difference sine-Gordon equation with the integral type source in the class of the rapidly decreasing functions.

ABSTRACT This paper is concerned with the existence of nontrivial weak solutions for a class of p ( x )-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.

ABSTRACT The issue of distributivity of aggregation operators is crucial for many different areas such as fuzzy sets and fuzzy logic, pseudo-analysis and measure theory, and particulary in the decision making theory. The problem of distributivity of an operator form a special class of uni-nullnorms over a general uninorm is being addressed through this paper. The class in question consists of uni-nullnorms with continuous and Archimedean underlying t-norms and t-conorms, and the results presented here are a natural continuation and extension of some previous works with an emphasis on a much wider class of uninorms as inner operators.

ABSTRACT This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

ABSTRACT Let X , Y be two Banach spaces over K=ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F + C be a weakly coercive operator from X onto Y , where F is a Fredholm proper operator, and C is a C 1 -compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1 -diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f ( x ) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

ABSTRACT The almost contact metric structure that we have on a real hypersurface M in the complex quadric Q m = SO m +2 / SO m SO 2 allows us to define, for any nonnull real number k , the k -th generalized Tanaka-Webster connection on M , ∇^(k) . Associated to this connection, we have Cho and torsion operators FX(k) and TX(k) , respectively, for any vector field X tangent to M . From them and for any symmetric operator B on M , we can consider two tensor fields of type (1,2) on M that we denote by BF(k) and BT(k) , respectively. We classify real hypersurfaces M in Q m for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator L ξ on M .

ABSTRACT The countably paracompact normal spaces were characterised by Dowker and Katětov in terms of an insertion property. Dowker also described them by normality of their product with the closed unit interval. Michael used the Dowker-Katětov insertion property to motivate his selection characterisation of these spaces. Morita extended in a natural way Dowker’s product characterisation to all τ -paracompact normal spaces. In this paper, we look at these results from the point of view of open relations. Insertions and selections are equivalent for such relations. Furthermore, we obtain a natural characterisation of τ -paracompact normal spaces in terms of selections for convex-valued open relations. Based on this, we give simple alternative proofs of the above mentioned results. Other applications are obtained as well.

ABSTRACT We introduce the classes of (strongly) Θ-discrete homogeneous spaces. We discuss the relationships of these classes to other classes of spaces possessing homogeneity-related properties, such as (strongly) n -homogeneous spaces. Many examples are given distinguishing discrete homogeneity and other types of homogeneity.

ABSTRACT The generalization of the family of distributions that could provide a simple, and efficient algorithm for parameter estimation and study of the behavior of datasets from various fields has received significant interest. Such a model has enormous advantages, such as its flexible nature, and the regression form can easily be derived. In the literature, various generalized families of distributions have been introduced. Despite the merits of these distributions, they still have some limitations due to many parameters in the model. Thus, the estimation of parameters often becomes cumbersome. Therefore, this study introduced the alpha power Muth or Teissier-G family of continuous distributions with well-defined parameters, and obtained the joint progressive type-II censoring scheme and their reliability measures. Furthermore, we obtained the global and local influences of the APTG model. We used real-life and simulated data to evaluate the numerical applications of the introduced model. The results show that the alpha power Muth or Teissier-G family of distributions gave the best fits to both datasets than some existing models.

ABSTRACT It is known that the Golden space form may not be an Einstein manifold. In this paper, it is shown that the condition to be Einstein for a Golden space form is equivalent to being weakly Einstein. In addition, the partial geodesic and cyclic parallelism of the CR-submanifolds of a Golden space are examined, and the case of the constant Golden sectional curvature is determined. Moreover, the CR-submanifolds with semi-flat normal connection are studied and an inequality is obtained. The equality case of this inequality is also checked. We also consider the totally umbilical CR-submanifold of Golden Riemannian manifolds and show that such submanifolds are totally geodesic under certain conditions. Furthermore, we obtain an inequality involving the scalar curvature of CR-submanifold and check the existence of extrinsic spheres in Golden space forms.