Volume 74, Issue 2

Let 𝔔 be a reduced left-sided and left-stable quantale without zero divisors. This paper shows how the systematic use of the «quantization» of 2 viewed as a regular and strict right 𝔔-subalgebra of 𝔔 can both strengthen and simplify the rather complicated construction of quantale-enriched topologies given for semi-unital and semi-integral quantales in the original paper [3]. The underlying structure for this new construction is an adjoint situation between the dual category of semi-unital, semi-integral quantales and quantized topological spaces.

In this paper, we first give a topological representation of some algebraic lattices of ideals of C ( X ). Next, we apply these results and prove that a space X is normal if and only if the lattice of closed fixed ideals of C ( X ) is a sublattice of the lattice of ideals of C ( X ). It is proved that if two rings C ( X ) and C ( Y ) are isomorphic, then two lattices Z ∘ [ X ] and Z ∘ [ Y ] are isomorphic. We conclude that two rings C * ( X ) and C * ( Y ) are isomorphic if and only if two lattices Z [ βX ] and Z [ βY ] are isomorphic.

We say that an idempotent term t is an exact - m - majority term if t evaluates to a , whenever the element a occurs exactly m times in the arguments of t , and all the other arguments are equal. If m < n and some variety 𝓥 has an n -ary exact- m -majority term, then 𝓥 is congruence modular. For certain values of n and m , for example, n = 5 and m = 3, the existence of an n -ary exact- m -majority term neither implies congruence distributivity, nor congruence permutability.

Let a n be defined recursively by a n = a n –1 + a n –2 + pn + q if n ≥ 2, with a 0 = a 1 = 1, where p and q are constants. The p = 0 case of a n corresponds to what have been termed the generalized Leonardo numbers , the q = 1 case of which are the classical Leonardo numbers. Let Ln(x)=∑i=0naixn−i,$\begin{array}{}\displaystyleL_n(x)=\sum_{i=0}^na_ix^{n-i},\end{array}$ which we refer to as a generalized Leonardo coefficient polynomial. Here, it is shown, under some mild assumptions on p and q , that L n ( x ) has exactly one real zero if n is odd and no real zeros if n is even. This yields a previous result for the Fibonacci coefficient polynomials when p = q = 0 and a new analogous result for the Leonardo coefficient polynomials when p = 0, q = 1. Further, when p = 0 and –1 < q ≤ 1 + ρ , we show that the sequence consisting of the real zeros of the L n ( x ) for n odd is strictly decreasing with limit – ρ , where ρ=1+52.$\begin{array}{}\displaystyle\rho=\frac{1+\sqrt{5}}{2}.\end{array}$ Finally, under the same assumptions on p and q , we are able to show the convergence in modulus of all the zeros of L n ( x ).

The authors of the paper suggest a novel approach in order to examine an integral equality using conformable fractional operators. By using this identity, some Newton-type inequalities are proved for differentiable convex functions by taking the modulus of the newly established equality. Moreover, we prove some Newton-type inequalities by using the Hölder and power-mean inequality. Furthermore, some new results are presented by using special choices of obtained inequalities. Finally, we give some conformable fractional Newton-type inequalities for functions of bounded variation.

We present a variety of refined conditions for σ -algebras 𝓐 (on a set X ), 𝓕, 𝓖 (on a set U ) such that the distributivity equation ( A ⊗ F ) ∩ ( A ⊗ G ) = A ⊗ F ∩ G , $$\begin{array}{}\displaystyle(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),\end{array}$$ holds – or is violated. The article generalizes the results in an article of Steinicke (2021) and includes a positive result for σ -algebras generated by at most countable partitions, which was not covered before. We also present a proof that counterexamples may be constructed whenever X is uncountable and there exist two σ -algebras on X which are both countably separated, but their intersection is not. We present examples of such structures. In the last section, we extend Theorem 2 in Steinicke (2021) from analytic to the setting of Blackwell spaces.

In this paper, we consider the class 𝓚 𝓢 ( ψ 0 ) of close-to-convex functions associated to a rational function ψ 0 ( z ) = ( k 2 + z 2 )/( k 2 – kz ), where k = 2$\begin{array}{}\displaystyle\sqrt{2}\end{array}$ + 1 and the rational function ψ 0 is related to cardioid shaped bounded domain. We compute the radius of convexity, growth, distortion and certain coefficient inequalities of such functions. We determine bounds on second and third order Toeplitz determinants. Moreover, we also compute bounds on second order Hankel determinants and second and third order Vandermonde determinants.

In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper “ Monotonicity properties for a ratio of finite many gamma functions ” published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value.

In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

This paper deals with some qualitative properties of solutions to nonlinear neutral hybrid differential equations connected to ψ -Caputo fractional derivative with time-varying lags. First, we demonstrate the problem possesses a mild solution uniquely where the source function may have temporal singularities. Second, in some cases, we indicate that the problem possesses a unique mild solution under some weaker conditions than the previous one. Third, we also obtain a result on a global mild solution for the problem. Finally, the results are further enriched by studying a new type of Ulam–Hyers stability for the main equation. The main results are obtained by applying the nice inequality, first proposed and proven in this paper. Some befit examples are given to justify the applicability of the main results.

In this paper the authors prove the existence of a positive solution to the second order boundary value problem with a fully nonlinear term − u ″ + λ u = a ( t ) f ( t , u ( t ) , u ′ ( t ) ) , t ∈ I , u ( 0 ) = u ∞ = 0 , $$\begin{array}{}\displaystyle\left\{\begin{array}{ll}-u''+\lambda u=a(t)f(t,u(t),u'(t) ), &t\in I,\\u(0)=u\left(\infty \right)=0,\end{array}\right.\end{array}$$ where I = (0, ∞), f : ℝ + × ℝ + × ℝ → ℝ + and a : I → ℝ + are continuous, and λ > 0. An example illustrates the main result.

In this work we investigate a nonlocal problem for the Fredholm integro-differential equation involving integral condition. The main tool used in our considerations is Dzhumabaev parametrization method. We make use of the numerical implementation of the Dzhumabaev parametrization method to obtain the desired result, which is well-supported with numerical examples.

In this paper, bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation are construed. Inequalities involving Shannon entropy, Kullback–Leibler discrimination, triangle distance and Jeffrey distance, are studied as particular instances by using various types of convex functions. Several new bounds of certain divergence measures for some specified time scales are also discussed.

The main aim of this paper is to extend the concepts of the 1MP and MP1 inverses defined for rectangular complex matrices. We present the weighted 1MP and MP1 inverses for a bounded linear operator between two Hilbert spaces as two new kinds of generalized inverses. The notions of the weighted 1MP and MP1 inverses are new in the context of rectangular complex matrices too. We establish a number of characterizations and some representations of the weighted 1MP and MP1 inverses. Several operator equations are solved applying the weighted 1MP and MP1 inverses. A special case of one of these equations is the normal equation which is related with the least-squares solution. As consequences of our results, we obtain new properties of the 1MP and MP1 inverses.

GPE-algebras were introduced by Dvurečenskij and Vetterlein as unbounded pseudo-effect algebras. Recently, they have been characterized as partial L -algebras with local duality. In the present paper, GPE-algebras with an everywhere defined L -algebra operation are investigated. For example, linearly ordered GPE-algebra are of that type. They are characterized by their self-similar closures which are represented as negative cones of totally ordered groups. More generally, GPE-algebras with an everywhere defined multiplication are identified as negative cones of directed groups. If their partial L -algebra structure is globally defined, the enveloping group is lattice-ordered. For any self-similar L -algebra A , exponent maps are introduced, generalizing conjugation in the structure group. It is proved that the exponent maps are L -algebra automorphisms of A if and only if A is a GPE-algebra. As an application, a new characterization of cone algebras is obtained. Lattice GPE-algebras are shown to be equivalent to ∧-closed L -algebras with local duality.

The classical Bohr inequality for scalars was extended to the non-commutative case of Hilbert space operators in the literature. The sole goal of this article is to discuss the operator Bohr inequality and present some of its new variants. This includes fresh reverses and refinements of this inequality with applications towards an operator’s real and imaginary parts, not to forget the new discussion of different domains of the parameters. One further application towards the operator Dunkl-Williams inequality will be presented too. While the new results are interesting, we emphasize that the approach used to explore these inequalities differs from the existing literature methods for this context.

We consider the weighted Bergman space Aα 2$\begin{array}{}\displaystyleA_{\alpha}^{2}\end{array}$(𝔻) of analytic functions f on the unit disc 𝔻 = { z ∈ ℂ : | z | < 1} such that f A α 2 2 := ∫ D f z 2 d A α z < + ∞ , $$\begin{array}{}\displaystyle\left\Vert f\right\Vert _{A_{\alpha}^{2}}^{2}:=\int\limits_{\mathbb{D}}\left\vert f\left( z\right) \right\vert ^{2}\text{d} A_{\alpha}\left( z\right) \lt +\infty,\end{array}$$ where d A α ( z ) = ( α +1) (1 − | z | 2 ) α d A ( z ) and dAz=dxdyπ $\begin{array}{}\displaystyle\text{d} A\left( z\right) =\frac{\text{d} x\text{d} y}{\pi}\end{array}$ (the normalized Lebesgue area measure). We investigate the generalized Riccati operator equation X A 1 X + A 2 X A 3 + A 4 Y A 5 + A 6 = 0 $$\begin{array}{}\displaystyleXA_{1}X+A_{2}XA_{3}+A_{4}YA_{5}+A_{6}=0\end{array}$$ with A i ∈ 𝓑(Aα 2$\begin{array}{}\displaystyleA_{\alpha}^{2}\end{array}$(𝔻) ), i = 1, 6 . Also we study the solvability of the operator equation X 1 T φ 1 + X 2 T φ 2 + ⋯ + X n T φ n = I A α 2 , $$\begin{array}{}\displaystyleX_{1}T_{\varphi_{1}}+X_{2}T_{\varphi_{2}}+\dots+X_{n}T_{\varphi_{n}%}=I_{A_{\alpha}^{2}},\end{array}$$ in the set of Toeplitz operators on the weighted Bergman space Aα 2$\begin{array}{}\displaystyleA_{\alpha}^{2}\end{array}$(𝔻). Moreover, we characterize compactness of the operator T φ T ψ − T η in terms of Berezin transforms.

Several naturally defined lower separation axioms for bitopological spaces obtained by modifying the axioms T 0 , T 1 , and R 0 appear in the literature. We introduce and study analogous separation axioms for bitopogenous spaces. In particular, we investigate relationships between the axioms and discuss the conditions under which the relationships are similar to those between the corresponding separation axioms for bitopological spaces.

The Fisher information matrix (FIM) relevant to order statistics (OSs) and their concomitants of the shape-parameters vector of the Cambanis bivariate distribution is investigated. Singly or multiply censored bivariate samples drawn from the Cambanis bivariate distribution are used to obtain the Fisher information (FI). In addition, the FI contained in the scale and shape parameters of generalized exponential distributions in the concomitants of OSs is obtained. The cumulative residual FI in the concomitant of OSs based on the Cambanis family is theoretically and numerically studied. Finally, a bivariate real-world data set has been analyzed for illustrative purposes, and the performance of the proposed method is quite satisfactory.

Given a continuum X , C n ( X ) denotes the hyperspace of nonempty closed subsets of X with at most n components. A strong size level is a subset of the form σ −1 ( t ), where σ is a strong size map for C n ( X ) and t ∈ (0, 1]. In this paper, answering a question by Capulín-Pérez, Fuentes-Montes de Oca, Lara-Mejía and Orozco-Zitli, we prove that for each n ≥ 2, no strong size level for C n ( X ) is irreducible.