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We define the notion of a generalized Sasaki mapping on a d 0 -algebra. We also introduce d 0 -algebras with the Sasaki property and, for such d 0 -algebras, we construct the generalized Sasaki projection, which turns out to be a generalized Sasaki mapping.

In 1974, M. B. Nathanson proved that every irrational number α represented by a simple continued fraction with infinitely many elements greater than or equal to k , is approximable by an infinite number of rational numbers p/q satisfying |α−p/q|<1/(k2+4q2). $|\alpha-p/q|<1/(\sqrt{k^2+4}q^2).$ In this paper, we refine this result.

In this paper, for any rational prime p and for a fixed positive integer ν p , we provide infinite families of number fields defined by quintinomials satisfying ν p ( i ( K )) = ν p . We illustrate our results by some computational examples.

Let ℤ q be a complete residue system modulo an odd integer q ≥ 3. The main purpose of this paper is to give an asymptotic formula for the number of representations of an element n ∈ ℤ q as sum of two classical Lehmer numbers in ℤ q . In addition, we make explicit the known expression for the number of the classical Lehmer numbers.

The article deals with the issue of the existence of fundamental systems for solving some types of homogeneous linear differential equations with constant coefficients, which are created using higher derivatives of the so-called fundamental function. The first part of the article also partially expands on new information from the article [4]. Some interesting results are presented: Let f ( x ) = x n− 1 sin βx be the fundamental function of the solutions to the differential equation (D2+β2)ny=0, $(D^2+\beta^2)^ny=0,$ then the sequence of derivatives Dkif(x)i=12n $\left\{D^{k_i}f(x)\right\}_{i=1}^{2n}$ forms a fundamental system of solutions to this differential equation if the number of distinct even-order derivatives of f is equal to the number of distinct odd-order derivatives. Another result is that the set of n arbitrary distinct derivatives of the function f(x)=xn−1eλx(λ≠0) $f(x)=x^{n-1}{\text{e}}^{\lambda x} (\lambda \neq 0)$ forms the fundamental system of solutions to the differential equation ( D − λ ) n y = 0. While the article does not provide a comprehensive solution to the issue, it offers the potential for generalizing the presented results.

Let k be a nonzero complex number. In this paper, we consider a k -circulant matrix whose first row is ( L 2 , L 4 , … , L 2 n ), where L n is the n th Lucas number. The formulae for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from Theorem 2 of [Bueno, A.C.F.: On r-circulant matrices with Fibonacci and Lucas numbers having arithmetic indices , AIP Conf. Proc. 1905 (2017), 030010] is improved because this result can not be applied in some cases. Since Theorem 4 in the same work relies on Theorem 2, the obtained formulae for the eigenvalues of a k -circulant matrix involving the Lucas numbers of even index show that there are cases when the result of Theorem 4 can also not be applied. Additionally, we present a more precise formula for the Euclidean norm of such matrix compared to the formula presented in Theorem 5 in the aforementioned reference. The upper and lower bounds for the spectral norm of a k -circulant matrix whose first row is (L2−1,L4−1,…,L2n−1) $(L_{2}^{-1},L_{4}^{-1},\dots,L_{2n}^{-1})$ are also investigated. The obtained results are illustrated by examples.

In this article, we construct a new explicit formula for the Drazin inverse of a sum of two matrices P, Q ∈ ℂ n×n under conditions weaker than those used in some recent papers, and modify an invalid formula in [Dopazo et al.: Block representations for the Drazin inverse of anti-triangular matrices , Filomat 30 (2016), 3897–3906]. Furthermore, we apply our results to obtain some new representations for the Drazin inverse of a 2 × 2 block matrix. Some numerical examples are given to illustrate our results.

The paper deals with the nonoscillatory solutions of arbitrary noninteger-order neutral equations with distributed delays. Through the use of the LFD (Liouville Fractional Derivative) of order α ≥ 0 on the half-axis and BCP (Banach Contraction Principle), we are able to get the nonoscillation criteria. The obtained results are emphasized with some appropriate examples.

Inequalities involving fractional operators have also been an active area of research. These inequalities play a crucial role in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, firstly we established two new identities for the case of differentiable convex functions using generalized tempered fractional integral operators. By utilizing these identities, some novel inequalities like Simpson-type, Bullen-type, and trapezoidal-type are proved for differentiable s -convex functions. Additionally, from the obtained results, several special cases of the known results for different choices of parameters are recaptured. Finally, some applications to q -digamma and modified Bessel functions are given.

This work offers a generalization of the α -Bernstein-Durrmeyer operators Dm,Zα $\mathfrak{D}^{\alpha}_{m,\mathbf{Z}}$ about a collection Z of arbitrary measures, with the goal of building a bridge between approximation, probability, and measure theory. It has been discussed how to achieve pointwise convergence of Dm,Zα $\mathfrak{D}^{\alpha}_{m,\mathbf{Z}}$ and constraints on measure collection have been developed for these operators’ convergence. With certain values of the measures considered, the generalization yields a range of Bernstein-like operators with their α variation. In order to handle a wider variety of p.l.o. and investigate their convergence, the paper comprises numerous operators, making it a useful tool.

In this work, we investigate the eigenvalue problem for an elliptic-hyperbolic equation with a non-smooth line of type changing within a special domain. The boundary conditions considered include Frankl’s condition, which is crucial for such problems involving mixed type equations. We derive the eigenvalues and corresponding eigenfunctions of the problem, paying special attention to the structural complexities introduced by the discontinuity in the equation’s type across the domain. Furthermore, we rigorously prove that the system of eigenfunctions obtained from the problem is not complete in the L 2 space, meaning that not every square-integrable function in the domain can be represented as a series expansion in terms of these eigenfunctions. This non-completeness is demonstrated by constructing a specific function orthogonal to the entire system of eigenfunctions. We utilize advanced techniques, including the transformation to polar coordinates and variable substitution, to express integrals involving Bessel functions, thereby providing a clear framework for proving the non-completeness. Our findings are applicable to both symmetric and anti-symmetric cases of the problem, with boundary conditions that either involve equal or opposite values at points reflected across a line of symmetry. This study highlights significant challenges in dealing with mixed-type equations and contributes to the broader understanding of how eigenvalue problems with Frankl’s conditions behave in non-standard geometries, particularly in domains with discontinuities.

In this paper, we investigate the orbital Hausdorff continuous dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses. The concepts of orbital Hausdorff continuous dependence and stability are used to characterize the relations of solution corresponding to the impulsive moments and the difference between the impulsive moments and the junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee the orbital Hausdorff continuous dependence and stability on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.

The time scales version of the Leighton oscillation theorem states that if (*) ∫t0∞1a(t)Δt=∫t0∞p(t)Δt=∞, $$ \int\limits_{t_0}^\infty \frac{1}{a(t)}\,{\Delta}t=\int\limits_{t_0}^\infty {p(t)}\,{\Delta}t=\infty,$$ where a and p are rd-continuous with a ( t ) > 0 for t ≥ t 0 , then every solution of the second-order self-adjoint dynamic equation (a(t)xΔ)Δ+p(t)xσ=0 $$ (a(t)x^\Delta)^\Delta+p(t)x^\sigma=0$$ is oscillatory. The theorem turns into the famous Leighton oscillation theorem when the time scale is taken as the set of real numbers, and its discrete version when the time scale is taken as the set of integers. The divergence of the first improper integral in ( ∗ ) means that the dynamic equation is in canonical form. The equation is called noncanonical when the integral is convergent. In this study, we establish an improved version of the Leighton oscillation theorem on time scales that can be applied to both canonical and noncanonical types of dynamic equations. Furthermore, we allow the second improper integral in ( ∗ ) to be convergent. In the special case, we derive completely new Leighton-type oscillation theorems for second-order self-adjoint difference equations Δ(akΔxk)+pkxk+1=0, $$ {\Delta}(a_k\Delta x_k)+p_kx_{k+1}=0,$$ where Δ is the forward difference operator, defined by Δ x k = x k +1 − x k (the derivative). Examples are given to illustrate the significance of these theorems.

Departing from a categorical concept of topogenous orders defined relative to the bifibration of subobjects, we introduce and discuss topogenous orders on forms, i.e., faithful and amnestic functors. These topogenous orders are shown to include both closure and interior operators on forms. We define and study two special morphisms relative to a topogenous order, namely strict and final morphisms. We give a characterization of the two morphisms by the help of their cartesian and cocartesian liftings. Some examples from topology and algebra demonstrating our results are included.

In order to make the fundamental group (one of the most well-known invariants in algebraic topology) more useful and powerful, some researchers have introduced and studied various topologies on the fundamental group from the beginning of the 21st century onwards. In this paper, first we recall and review these topologies and then using the concept of subgroup topology, we compare these topologies in order to present some results on topologized fundamental groups.

For one-dimensional semimartingales, the Itô formula can be extended from C 2 -functions to C 1 -functions with a locally absolutely continuous derivative. We propose a new, different proof of this result, which is simple, straightforward and quite elementary, avoiding in particular the extensive theory of local times for semimartingales.

We propose an estimator of the kernel-based conditional mean dependence measure obtained from an appropriate modification of a naive estimator based on usual empirical estimators. We then get asymptotic normality of this estimator both under conditional mean independence hypothesis and under the alternative hypothesis. A new test for conditional mean independence of random variables valued into Hilbert spaces is then introduced.

This comprehensive study delves into the examination and application of novel statistical distributions, namely the Ristić-Balakrishnan-Topp-Leone-Exponentiated half Logistic-G (RB-TL-EHL-G) family of distributions, emphasizing their paramount importance in reliability and medical data modeling. We meticulously explore a multitude of this family of novel distributions, accentuating their respective features, properties, and real-world applicability. The probability density, the cumulative distribution, the hazard rate, and the quantile functions are provided. The density functions of the RB-TL-EHL-G family are expanded, enabling a deeper understanding of their statistical properties, including various moments, generating functions, order statistics, stochastic orderings, probability weighted moments, and the Rényi entropy. A significant portion of the investigation is dedicated to the intensive analysis of various data sets, to which these distributions are fitted, unveiling noteworthy insights into their behavior and performance. Furthermore, the discussions extend to a comparative study, delineating the advantages and limitations of each distribution, fostering a deeper understanding and selection criteria for practitioners.