Volume 74, Issue 5

In this paper, we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids wEnd C n and End C n of all weak endomorphisms and all endomorphisms of an undirected cycle graph C n with n vertices. We also describe Green’s relations and regularity of these monoids and calculate their cardinalities.

The crossing number cr( H ) of a graph H is the minimum number of edge crossings over all drawings of H in the plane. Let H ∗ be the connected graph of order five isomorphic to K 5 ∖ e obtained by removing one edge from the complete graph K 5 . The main aim of the paper is to give the crossing numbers of the join products H ∗ + P n and H ∗ + C n , where P n and C n are the path and the cycle on n vertices, respectively. The proofs are done with the help of a suitable classification of a large number of drawings of the graph H ∗ in view of the existence of a separating cycle of two possible types.

An irreducible complete atomic oml of infinite height cannot be algebraic and have the covering property. However, modest departure from these conditions allows infinite-height examples. We use an extension of Kalmbach’s construction to the setting of infinite chains to provide an example of an infinite-height, irreducible, algebraic oml with the 2-covering property, and Keller’s construction provides an example of an infinite-height, irreducible, complete oml that has the covering property and is completely hereditarily atomic. Completely hereditarily atomic omls generalize algebraic oml s suitably to quantum predicate logic.

Let d ( n ) and φ ( n ) denote the number of positive divisors of n and the Euler’s phi function of n , respectively. In this paper, we prove various (conditional and unconditional) results about the solvability of the Diophantine equation d ( n ) = d ( φ ( n )) and a related inequality. For further research, we present some open problems.

We say a monic polynomial g ( x ) ∈ ℤ[ x ] of degree n is monogenic if g ( x ) is irreducible over ℚ and {1, θ , …, θ n −1 } is a basis for the ring ℤ K of integers of number field K = ℚ( θ ), where θ is a root of g ( x ). Let f(x)=xn+c∑ i=1n(ax)n− i∈ Z[x]andF(x)=xn+c∑ i=1nai− 1xn− i∈ Z[x] $\begin{array}{} \displaystyle f(x)=x^n+c\sum_{i=1}^{n}(ax)^{n-i} \in \mathbb{Z}[x] \,\text{and}\, F(x)=x^n+c\sum_{i=1}^{n}a^{i-1}x^{n-i} \in \mathbb{Z}[x] \end{array}$ be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a , c , n for the polynomials f ( x ) and F ( x ) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group S n , the symmetric group on n letters.

This paper generalizes the results of the first part of the paper [Fecenko, J.—Diekema, E.: On the linear (in)dependence of sequences of derivatives of the functions x n sin x and x n cos x , http://arxiv.org/abs/2305.11184]. The main goal is to prove that the sequence of functions f ( x ), Df ( x ), …, D 2 n –1 f ( x ), where f ( x ) is x n −1 e αx sin βx or x n −1 e αx cos βx for β ≠ 0, forms a fundamental system of solutions of the differential equation ( D 2 – 2 αD + α 2 + β 2 ) n y = 0. Or more generally, that the sequence of functions D k f ( x ), D k +1 f ( x ), …, D 2 n + k –1 f ( x ), k ∈ ℕ also forms a fundamental system of solutions of the aforementioned differential equation. The problem is solved using a suitable transformation of the Wronskian matrix, by solving a nonlinear differential equation and then calculating the determinant of the modified Wronskian matrix by applying Laplace’s generalized expansion.

Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K ) = {0}. We prove that a convex compact-valued map F : ℝ → 2 Y ∖ {∅} is K -continuous and K -additive if and only if there are non-empty convex compact sets A , B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal “modulo K ” to the continuous set-valued map G(t)=tA,t≥ 0,tB,t< 0. $$\begin{array}{} \displaystyle G(t)=\begin{cases} tA,&t\geq0,\\ tB,& t \lt 0. \end{cases} \end{array}$$ Next, we use this result to characterize convex compact-valued maps F : ℝ N → 2 Y ∖ {∅}.

In this paper, we prove some new inequalities of Hermite–Hadamard type for differentiable functions with h -convex derivatives. It is also shown that the newly established inequalities are extension of the existing inequalities in the literature. Finally, we give applications of the new results and outline some future problems.

In this paper, we present the existence of global mild solutions for the Hilfer fractional neutral evolution equations (HFNEEs), regardless of whether the semigroups are compact or noncompact. We achieve our main results by utilizing the generalization Ascoli-Arzelà theorem, Krasnoselskii’s fixed point theorem, Laplace transform, and measures of noncompactness. To demonstrate the feasibility of our method, we provide an example.

We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

In the present study, we determine a Schurer type of complex generalized Stancu operators and examine its some approximation properties. Firstly, we obtain upper quantitative estimates for the convergence and then we get lower estimates from a qualitative Voronovskaja-type result for these specified operators attached to the analytic functions in an open disk centered at the origin. Hence, we present the exact order of simultaneous approximation by using obtained upper and lower estimates.

In this paper, we offer new technique for investigation of the third-order linear differential equation with mixed argument y‴(t)+p(t)y(τ (t))=0. $$\begin{array}{} \displaystyle y'''(t)+p(t)y(\tau(t))=0. \end{array}$$(E) We establish new criteria for property A of (E) which fulfill the gap in the oscillation theory.

This work is devoted to the study of the Cauchy problem for a loaded differential equation with the Bessel operator. When studying problems for loaded equations, the properties of Erdélyi-Kober operators are used as transformation operators concerning a relation. We obtain an explicit form of the solution to the Cauchy problem for a loaded one-dimensional differential equation. At the end of the work, we will show several examples on graphs.

In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u+h1vp≤ 0 in V,Δ v+h2uq≤ 0 in V, $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where ( V , E ) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1 , h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N -dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs , arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

The q -Durrmeyer polynomials are one of the popular q -versions of the classical operators of approximation theory. They have been studied from different points of view by a number of researchers. The aim of this work is to estimate the rate of convergence for the sequence of the q -Durrmeyer polynomials in the case 0 < q < 1. It is proved that for any compact set 𝓓 ⊂ ℂ, the rate of convergence is O ( q n ) as n → ∞. The sharpness of the obtained result is demonstrated.

The objective of this work is the construction of `Boyd-Wong fixed point theorem’ in the setting of generalized parametric metric space and discussion its application on existence criteria of solutions to a second order initial value problem. Also an analogue of `Banach type fixed point theorem’ of generalized parametric metric space is proved and its application on the existence of a solution of first order periodic boundary value differential equation is examined.

Kannan contraction is broadly investigated topic in Metric Fixed Point Theory due to its importance in omitting the continuity presumption. Of a great significance is also its role in characterizing completeness of a metric space through existence and uniqueness of a fixed point of arbitrary Kannan contraction in the observed setting. The concept of Kannan contraction has been adapted, extended and transferred to various types of spaces including cone metric spaces, quasi metric spaces, b -metric spaces, partial metric spaces, among others. The main aim of this article is to prove that for any Kannan contraction T on a complete metric space ( X , d ) there exists another metric on the set X in relation to which T is a Banach contraction while the completeness is preserved. In that way, all results on Kannan contraction may be derived as corollaries of the Banach contraction principle. The converse also holds since, by altering the metric, Banach contraction becomes a Kannan contraction. The obtained theoretical results are substantiated with adequate examples.

In this study, we disclose several statistical properties of concomitants of dual generalized order statistics (DGOSs) under a generic setting from the Sarmanov family of bivariate distributions. For this family, cross-moments and joint distributions of concomitants of DGOSs are explored. These results' implications for order statistics and record values are examined. Extropy and cumulative residual extropy (CRE), two well-known information measures, are also investigated. This study offers a few beautiful symmetrical relationships that these inaccuracy measurements satisfy. In addition, based on the concomitants of DGOSs, the issue of estimating the negative cumulative residual extropy (NCRE), empirical NCRE and NCE are investigated using the empirical technique. Lastly, a set of bivariate data from the real world is examined.

In this paper, we introduce a novel concept of a dynamic residual inaccuracy measure based on extropy. We extend the traditional residual inaccuracy measure to its dynamic version, which allows us to measure the discrepancy between two residual life distributions. We study the properties of the proposed measure, including its discrimination principle and the proportional hazard rate model. We also investigate a characterization problem related to the extropy inaccuracy measure and propose some alternative expressions of the dynamic residual measure of inaccuracy. Furthermore, we establish upper and lower bounds and some inequalities concerning dynamic residual inaccuracy measures based on extropy. We demonstrate that the defined measure of inaccuracy is invariant under scale but not under location transformation. The given findings have important implications for statistical inference, estimation, and modeling. The proposed extropy-based dynamic residual inaccuracy measure provides a powerful tool for quantifying the discrepancy between two residual life distributions over time. At the end of the paper, we provide two non-parametric estimators for the proposed extropy measure of inaccuracy for both the non-censored (complete sample) and the right-censored scheme. The performances of these estimators are compared numerically based on their bias and MSE.

W * is the category of the title. For G ∈ W * , we have the canonical compact space YG , and Yosida representation G ≤ C ( YG ), thus, for g ∈ G , one has the cozero-set coz( g ) in YG . The ideals at issue in G include the principal ideals and polars, G ( g ) and g ⊥⊥ , respectively, and the W * -kernels of W * -morphisms from G . The “coincidences of types” include these properties of G : (M) Each G ( g ) = g ⊥⊥ ; (Y) Each G ( g ) is a W * -kernel; (CR) Each g ⊥⊥ is a W * -kernel (iff each coz( g ) is regular open). For each of these, we give numerous “rephrasings” and examples, and note that (M) = (Y) ∩ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on W * brings topology to bear, and sharpens the view.