Volume 72, Issue 4

This paper concerns notions of Booleanness in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, σ - and κ -frames. We provide four distinct conditions which all reduce in the case of full frames to each element having a complement. These concepts depend heavily on the free frame over a partial frame, the congruence frame of a partial frame and the relations between them. We also examine the adjoint situation between the congruences of a partial frame and those of its free frame.

We discuss exchangeability and independence in the setting of σ -complete Riesz MV-algebras. We define and link to each other the notions of exchangeability and distribution law for a sequence of observables (i.e. non classical random variables), as well as the notion of independence for a sequence of algebras. We obtain two categorical dualities for σ -complete Riesz MV-algebras endowed with states and we define a “canonical” state on the coproduct of a sequence of probability Riesz tribes, giving a weak version of de Finetti’s result. Finally, we discuss statistical models.

By means of the telescoping method, several summation formulae are established for the arctangent function with its argument being Pell and Pell–Lucas polynomials. Numerous infinite series identities involving Fibonacci and Lucas numbers are included as particular cases.

The aim of this paper is to introduce a new two-variable polynomials defined via Hermite polynomials. In order to construct some fundamental properties of these polynomials, we first derive a generating function relation. By using definition and this generating relation, we arrive at several recurrence relations, an integral representation, some implicit summation formulae, a symmetry identity for these new two-variable polynomials. Furthermore, we obtain some results which give various classes of multilinear and multilateral generating functions. Then, some special cases are presented. Finally, we also give a general class of these new polynomials and prove explicit closed-form formulae of them.

With the aid of convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the author presents decreasing property of a ratio constituted via three derivatives of a sum involving trigamma function and discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic.

In this paper, we investigate the solvability of a fourth-order differential evolution equation on singular cylindrical domain containing a cuspidal point. Some regularity results are obtained for the classical solutions by using the Dunford operational calculus.

Anew class of implicit Hadamard fractional differential equations with Riemann-Stieltjes integral boundary conditions is studied in this research paper. The existence and uniqueness results of the aforesaid problem are investigated using Schauder’s fixed point theorem and Banach’s contraction mapping principle. A simulative example is given to highlight the acquired outcomes.

The Riccati equation method is used to establish stability criteria for systems of two first-order linear ordinary differential equations. Examples are presented in which the obtained result is compared with the results obtained by the Lyapunov and Bogdanov methods, by a method involving estimates of solutions in the Lozinskii logarithmic norms and by the freezing method.

Let Ω ⊂ R N ( N ≥ 2) be a simply connected bounded domain, containing the origin, with C 2 boundary denoted by ∂ Ω. Denote by Ω ext:=RN∖ Ω ¯ $\Omega^{\mathrm{ext}}:=\mathbb{R}^{N} \backslash \bar{\Omega}$the exterior of Ω. We consider the perturbed eigenvalue problem − Δ p u − Δ q u = μ K ( x ) | u | p − 2 u  for  x ∈ Ω ext  u ( x ) = 0  for  x ∈ ∂ Ω u ( x ) → 0 ,  as  | x | → ∞ , $$\left\{\begin{array}{lcl}-\Delta_{p} u-\Delta_{q} u=\mu K(x)|u|^{p-2} u & \text { for } & x \in \Omega^{\text {ext }} \\u(x)=0 & \text { for } & x \in \partial \Omega \\u(x) \rightarrow 0, & \text { as } & |x| \rightarrow \infty,\end{array}\right.$$ where p, q ∈ (1 ,N ), p≠q$p \neq q$and K is a positive weight function defined on Ω ext having the property that K ∈ L ∞ (Ω ext ) ∩ L N/p (Ω ext ) . We show that the set of parameters μ for which the above eigenvalue problem possesses nontrivial weak solutions is exactly an unbounded open interval.

The paper deals with extension of bounded bilinear maps. It gives a necessary and sufficient condition for extending a bounded bilinear map on the Cartesian product of subspaces of Banach spaces. This leads to a full characterization for extension of bounded bilinear maps on the Cartesian product of arbitrary subspaces of Hilbert spaces. Applications concerning projective tensor products are also investigated.

Let 𝓗 be a complex Hilbert space and A be a non-zero positive bounded linear operator on 𝓗 . The main aim of this paper is to discuss a general method to develop A -operator seminorm and A -numerical radius inequalities of semi-Hilbertian space operators using the existing corresponding inequalities of bounded linear operators on 𝓗 . Among many other inequalities we prove that if S , T , X ∈ 𝓑 A ( 𝓗 ), i.e., if A -adjoint of S , T , X exist, then2∥S♯AXT∥A≤∥SS♯AX+XTT♯A∥A.$$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if T ∈ 𝓑 A ( 𝓗 ), then14∥T♯AT+TT♯A∥A≤18(∥T+T♯A∥A2+∥T−T♯A∥A2)≤18(∥T+T♯A∥A2+∥T−T♯A∥A2)+18cA2(T+T♯A)+18cA2(T−T♯A)≤wA2(T).$$\begin{align*} & \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A\\ \leq & \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg)\\ \leq & \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big)\leq w^2_A(T). \end{align*}$$ Here w A (⋅), c A (⋅) and ∥⋅∥ A denote A -numerical radius, A -Crawford number and A -operator seminorm, respectively.

Let φ be an analytic self-map of the open unit disk 𝔻 in the complex plane ℂ and u be an analytic function on 𝔻. The weighted composition operator is defined on the space H (𝔻) of analytic functions on 𝔻 by u C φ f = u ⋅ ( f ∘ φ ) ,   f ∈ H ( D ) . $$u{C_\varphi }f = u \cdot \left( {f \circ \varphi } \right),\quad f \in H\left(D\right).$$ The boundedness and the compactness of weighted composition operators from the Besov space into n th weighted type spaces are given in this work.

In this paper, we introduce the concept of fuzzy ideal topological vector spaces and study the basic properties of fuzzy-I-open and fuzzy-I-closed sets in fuzzy ideal topological vector spaces. Also, we study the properties of fuzzy-I-Hausdorff and fuzzy-I-compact in fuzzy ideal topological vector spaces. Furthermore, we introduce the concepts of fuzzy-I-homogenous space, fuzzy-I-monomorphism space, fuzzy-I-isomorphism space and fuzzy-I-automorphism space. Finally, we introduce the concepts of fuzzy-I-bounded set, fuzzy-I-balanced set, fuzzy-I-symmetric set and study their properties in fuzzy ideal topological vector spaces.

Continuous partial actions and continuous enveloping actions are investigated in the category of convergence spaces. Product and quotient constructions are considered. Further, it is shown that a continuous partial action on a convergence space can be extended to a continuous partial action on a compactification of the convergence space.

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by π1qs(X,x)$\pi_1^{qs} (X, x)$ for a pointed space ( X , x ). Then we prove that, unlike the small loop group, the quasi-small loop group π1qs(X,x)$\pi_1^{qs}(X, x)$ does not depend on the base point, and that it is a normal subgroup containing π1sg(X,x)$\pi_1^{sg}(X, x)$, the small generated subgroup of the fundamental group. Also, we show that a space X is homotopically path Hausdorff if and only if π1qs(X,x)$\pi_1^{qs} (X, x)$ is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.

In the paper we consider the families of symmetrically porouscontinuous functions. We find maximal additive classes for these families. Furthermore, we define new families of topologies generated by the symmetric porosity, which are useful to studying maximal multiplicative classes for symmetrically porouscontinuous functions. Some relevant properties of defined topologies are considered.

This paper introduces a two-parameters generator of continuous statistical probability distributions called the Alpha Power Rayleigh-G (APRAY-G) family, some statistical properties of the family of distributions were derived, and we introduced a two-submodels of the generator. We estimate the parameters of the models based on the method of maximum likelihood estimation and explored simulation studies based on the introduced submodels. We observed that the biasedness and root mean square errors decrease as the sample size becomes large. We examined the applications of the models based on real-life data sets. We compared the obtained results with some existing probability distribution models. The results showed that the proposed models gave a better fitness to the data under investigation.

In this paper, we first propose a new general method to introduce various lifetime distributions by choosing an appropriate kernel distribution. They have some characteristics in common with the well-known Birnbaum-Saunders distribution. Then, we choose the triangular distribution as a kernel model and construct the new distribution. This distribution has its support on the positive real axis and consists of two-pieces. We show that the newly defined distribution is in fact a generalized Birnbaum-Saunders distribution. It is mathematically tractable for studying its theoretical properties in detail. Different methods of estimation of parameters are proposed. The existence and uniqueness problem of the maximum likelihood estimation method is discussed. The performances of the estimators are evaluated through simulation studies. A real data fitting which compares it with the ordinary Birnbaum-Saunders, Laplace Birnbaum-Saunders and other some generalized Birnbaum-Saunders distributions is also given.

This paper is concerned with the existence of positive solutions for three point boundary value problems of Riemann-Liouville fractional differential equations with p -Laplacian operator. By means of the properties of the Green’s function and Avery-Peterson fixed point theorem, we establish a condition ensuring the existence of at least three positive solutions for the problem.