Volume 75, Issue 4

Some years ago, Shitov proved that the chromatic number of the power graph of semi-groups is at most countable, thus answering a question raised by Aalipour et al. about whether this statement holds for groups. Later, Dalal and Kumar proved that the chromatic number of the enhanced power graph of a group is also countable. In the line of these recent works, we first generalize the concept of power graphs to coprime power graph, and then we prove the same type of result for this generalization. Furthermore, we state a conjecture about their common relationship as graphs.

DRl-semigroups are a generalization of lattice ordered groups (l-groups) containing a.o. Boolean algebras, Brouwerian algebras and MV-algebras. Since their introduction in the 1960s their axiomatic system has been several times reduced. In this paper, we show that it can be simplified even further. Specifically, we show that the axiom ensuring compatibility of the semigroup operation + and the lattice operations is equivalent to a significantly weaker condition of monotonicity of + with respect to the implied order ≤. Because the same equivalence holds for l-groups, we observe that the axiomatic systems of l-groups and DRl-semigroups are more aligned than originally thought.

M. Sambasiva Rao recently investigated some sorts of special filters in distributive pseudocomplemented lattices. In our paper, we extend this study to lattices which need neither be distributive nor pseudocomplemented. For this sake, we define a certain modification of the notion of a pseudocomplement as the set of all maximal elements belonging to the annihilator of the corresponding element. We prove several basic properties of this notion and then define coherent, closed and median filters as well as D -filters. In order to be able to obtain valuable results, we often must add some additional assumptions on the underlying lattice, e.g. that this lattice is Stonean or D -Stonean. Our results relate properties of lattices and to those of corresponding filters. We show how the structure of a lattice influences the form of its filters and vice versa.

Reichenbach’s common cause principle (RCCP) is a metaphysical claim about the causal structure of the world. It entails that all correlations can be explained causally either by pointing at the causal connection between the correlated entities or by displaying a common cause of the correlation. We contribute to its mathematical side. Firstly, after adopting the RCCP axioms, we indicate the importance of positive covariance of events in connection with RCCP. In fact, RCCP is meaningful only for positively correlated events. Secondly, we find explicit requirements for a state to have a common cause in a quantum logic. Afterwards, we compare RCCP in the standard (Boolean) case and in the quantum setup that admits new properties of correlation, impossible in the classical case. We show that the notion of maximal correlation differs considerably in these two cases. We answer an open question by providing a counterexample based on a quantum logic given by a free orthomodular lattice. Subsequently, we find a relation between the Darboux property and RCCP. We invent a new technique for obtaining the central result that atomless σ -complete quantum logics are common cause complete. A variety of new examples is presented. Finally, we contribute to a fundamental question whether a common cause incomplete quantum logic can be embedded into one that is common cause complete . We contribute by an advanced construction that allows for a positive answer to this question for quantum logics with finitely many atoms and even for some quantum logics with countably many atoms.

Given a magma G = ( G, · ) and a biunary algebra I = ( I, λ, ρ ) (an algebra with two unary operations), we define a new magma G I = ( G I ,· ) as follows: ( x · y )( i ) = x ( λi ) · y ( ρi ) for any x, y ∈ G I and any i ∈ I . The main goal of the paper is to characterize the variety generated by all the magmas G I obtained in this way if G and I are taken from arbitrary (but fixed) varieties of magmas and biunary algebras, respectively.

In this paper, we prove that there does not exist a set of four positive integers {3, 6, c, d }, such that a product of any two of them increased by 3 is a triangular number.

In this paper, we first introduce the fractional integral operators of a function with respect to another function. Then, we prove a new version of Hermite-Hadamard for newly introduced fractional integral operators. For this aim, we use the h -convex function in the case the function h is a B -function. Moreover, we also establish several corresponding trapezoid and midpoint type inequalities by using h -convex function and Hölder inequality.

In this paper, we give some generalizations for the filtering of regular operator mappings through a (potentially nonunital) completely positive linear mapping. We find a general formulation of the Jensen operator inequality for convex regular mappings and a unital field of completely positive linear mappings. We introduce the notion of non-commutative F -divergence operators for regular operator mappings and continuous fields of Hilbert space operators and obtain some relations between the perspective of a convex regular mapping and a non-commutative F -divergence operator. We provide a refinement for the general formulation of the Jensen operator inequality and a refinement for the extension of the significant Choi-Davis-Jensen inequality.

The Gaussian hypergeometric function is an important class of special functions, whose mathematical properties have long been of interest to the academic community. In this paper, by utilizing monotonicity rules, we establish the necessary and sufficient conditions for the monotonicity of the ratio of two Gaussian hypergeometric functions x↦Fa1,b1;c1;xFa2,b2;c2;x $x \mapsto \frac{F\left(a_1, b_1 ; c_1 ; x\right)}{F\left(a_2, b_2 ; c_2 ; x\right)}$ for positive parameters a i , b i , c i ( i = 1, 2). Subsequently, we also derive some corollaries and provide some conclusions related to the complete elliptic integral function and its variants.

This paper examines the oscillation and asymptotic properties of a class of third-order neutral differential equations with distributed deviating arguments by employing comparison principles. A series of oscillation criteria are derived, which essentially extend and improve the existing related results and generalize to more general third-order differential equations by removing several constraints. Some examples are also provided to demonstrate our main results.

This paper deals with a class of fourth-order elliptic equations of p ( x )-Kirchhoff type of the form Δp(x)2u−M∫Ω1p(x)|∇u|p(x)dxΔp(x)u=λW(x)|u|q(x)−2u in Ω,u=Δu=0 on ∂Ω, $$\begin{cases}\Delta_{p(x)}^2 u-M\left(\int\limits_{\Omega} \frac{1}{p(x)}|\nabla u|^{p(x)} \mathrm{d} x\right) \Delta_{p(x)} u=\lambda W(x)|u|^{q(x)-2} u & \text { in } \Omega, \\ u=\Delta u=0 & \text { on } \partial \Omega,\end{cases}$$ where Ω ⊂ ℝ N ( N ≥ 2) is a smooth bounded domain with boundary ∂Ω,p:Ω‾→R $\partial \Omega, p: \overline{\Omega} \rightarrow \mathbb{R}$ is a log-Hölder continuous function, M ( t ) = a + bt κ is a Kirchhoff function with a,κ>0,b≥0,Δp(x)2u= $a, \kappa>0, b \geq 0, \Delta_{p(x)}^2 u=$Δ( | Δ u| p ( x ) − 2 Δ u ) is the operator of fourth order called the p ( x )-biharmonic operator, Δ p ( x ) u = div ( |∇u| p ( x ) − 2 ∇u ) is the p ( x )-Laplacian, W : Ω → ℝ is a weighted function and λ is a positive parameter. Using variational techniques, we establish some multiplicity results to the problem in two cases when the function W is sign-changing or not and two examples are given to illustrate the main results.

This paper investigates singular discrete Dirac equations and establishes famous Weyls classification for these equations. First, the limit-circle and limit-point classifications are obtained. Then, the existence of boundary conditions is demonstrated. Finally, it is shown that the limit-circle case does not occur for these equations.

In the present paper, we introduce bivariate logarithmic weighted spaces of functions and study approximation properties of bivariate exponential sampling series in these spaces. We obtain pointwise and uniform convergence of the series and we determine rate of convergence by introducing a new modulus of continuity for functions belonging to bivariate logarithmic weighted spaces of continuous functions. Furthermore, in order to determine a rate of pointwise convergence, we estimate remainder of Mellin-Taylor formula thanks to the modulus of continuity and we present a quantitative Voronovskaja-type theorem. Finally, we give some graphical representation of approximation of continuous functions by EWϕf $E_\mathbf{W}^\phi f$ using kernels which satisfy certain assumptions.

Derived from the results in [Giang et al.: Convolutions for the Fourier transforms with geometric variables and applications , Math. Nachr. 283(12) (2010), 1758–1770], in this paper, we devoted to studying the boundedness properties for the Fourier-cosine convolution weighted by a Gaussian function of the form γ=exp(−12y2) $\gamma =\exp(-\frac{1}{2}y^2)$ via Young’s type theorem and Saitoh’s type inequality. New norm estimations in the weighted space are obtained, and the application of the corresponding class of convolutions in Fredholm’s second kind of integral equation is discussed. The conditions for the solvability of this equation on L 1 space are also found, along with the analysis of an illustrative numerical example, which exemplifies that the present object and method solve cases that are not under the conditions of previously known techniques.

The aim of this paper is to establish the existence and uniqueness of solutions to the κ -fractional iterative differential equations ( κ -FIDEs) using Schauder’s fixed point theorem. We also present the continuous dependence of the solution on the input data and a Hyers-Ulam stability analysis for this problem. Finally, some examples are provided to illustrate our main results.

We introduce and study almost cosymplectic manifolds whose characteristic vector field ξ belongs to a generalized ( k, μ ) ʹ -nullity distribution, which involves the tensor hʹ = φ ◦ h . If hʹ ≠ 0, we prove that the smooth functions k and μ are uniquely determined by k < 0 and μ = ξ (ln λ ), where λ=−k. $\lambda=\sqrt{-k}.$ Moreover, we show that the spectrum of hʹ is { 0 , λ,−λ} and that the considered manifolds cannot be Ricci symmetric. Under certain conditions, we further prove that such manifolds are locally a warped product. Examples are also provided.

In this paper, we investigate R-, H-, and M- nw -selective properties introduced in [Bonanzinga, M.—Giacopello, D.: A generalization of M-separability by networks , Atti Accad. Peloritana dei Pericolanti Cl. Sci. Fis. Mat. Nat. 101(2) (2023), A11]. In particular, we provide consistent uncountable examples of such spaces and we define trivial R-, H-, and M- nw -selective spaces the ones with countable net weight having, additionally, the cardinality and the weight strictly less then cov( 𝓜 ), 𝔟, and 𝔡, respectively. Since we establish that spaces having cardinalities more than cov( 𝓜 ), 𝔟, and 𝔡, fail to have the R-, H-, and M- nw -selective properties, respectively, non-trivial examples should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R- nw -selective and H- nw -selective spaces and we establish some limitations to constructions of non-trivial examples. Moreover, we consistently prove the existence of two H- nw -selective spaces whose product fails to be M- nw -selective. Finally, we study some relations between nw -selective properties and a strong version of the HFD property.

Let X be a Hausdorff topological space, MU ( X , ℝ) and MC ( X , ℝ) be the space of all minimal usco maps with values in ℝ and the space of all minimal cusco maps with values in ℝ, respectively. By τ p is denoted the topology of pointwise convergence. We characterize some cardinal invariants of ( MU ( X , ℝ) , τ p ) and ( MC (ℝ,ℝ) , τ p ). Let Q (ℝ,ℝ) be the space of all quasicontinuous functions on ℝ with values in ℝ and C (ℝ,ℝ) the space of all continuous functions on ℝ with values in ℝ. We compare cardinal invariants of ( MU (ℝ,ℝ) , τ p ), ( MC (ℝ,ℝ) , τ p ), ( Q (ℝ,ℝ) , τ p ) and ( C (ℝ,ℝ) , τ p ).