Volume 74, Issue 1

Related to generalized arithmetic triangle, we introduce the hyper ( r , q )-Fibonacci polynomials as the sum of these elements along a finite ray starting from a specific point, which generalize the hyper-Fibonacci polynomials. We give generating function, recurrence relations and we show some properties whose application allows us to extend the notion of Cassini determinant and to study some ratios. Moreover, we derive a connection between these polynomials and the incomplete ( r , q )-Fibonacci polynomials defined in this paper.

Let R α := { r ∈ ℝ : coz( α − r ) ≠ → p } for every α ∈ 𝓡( L ). The ring 𝓒 c ( L ) is introduced as a pointfree version of the subring 𝓒 c ( X ) of C ( X ) by R α . In this paper, we show that 𝓒 c ( X ) is a z -good ring and every radical ideal in it is an absolutely convex ideal. Also, we study this result which for any frame L , there exists a zero-dimensional frame M , which is a continuous image of L and 𝓒 c ( L ) ≅ 𝓒 c ( M ).

Let ( P m ) m ≥0 be the sequence of Pell numbers given by P 0 = 0, P 1 = 1, and P m +2 = 2 P m +1 + P m for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ± 1, which is a product of two Pell numbers.

In this study, a novel family of analytical functions connected to convex functions in the open unit is introduced and investigated. Additionally, relationships between this class and other subclasses of analytic functions are deduced. Further, different results for the mentioned class and several new interesting properties are obtained.

In Geometric function theory, the Ma-Minda class of starlike functions has a unique place as it unifies various subclasses of starlike functions. There has been an vivid interplay between special functions and their geometric properties, like starlikeness. In this article, we establish certain special function’s radius of Ma-Minda starlikness. As an application, we obtain conditions on parameters for these special functions to be in the Ma-Minda class. Further, we focus on certain convolution properties for the Ma-Minda class that are not done so far, and study their applications in radius problem. Finally, we prove a variational problem of Goluzin, namely, the region of variability for the Ma-Minda class. Our results simplify and generalize the already-known ones.

This article is devoted to the study of mappings defined in the plane domain. Under certain conditions, the upper estimate of the distortion of the modulus of families of paths is obtained. Similarly, the upper estimate of the modulus of the families of paths in the pre-image under the mapping is also obtained.

In the present paper, new oscillation criteria are established for fourth-order delay differential equations of the form (a3(t)(a2(t)a1(t)x′(t)′)′)′+b(t)xα (σ (t))=0 $$\begin{array}{} \displaystyle \Big(a_3(t)\Big(a_2(t) \left( a_1(t) x'(t)\right)' \Big)' \Big)'+b(t)x^\alpha(\sigma(t))=0 \end{array}$$ under the assumption (noncanonical) ∫t0∞ 1aj(t)dt<∞ ,j=1, 2, 3. $$\begin{array}{} \displaystyle \int\limits_{t_0}^\infty \frac{1}{a_j(t)}\text{d} t \lt \infty,\qquad j=1,~2,~3. \end{array}$$ We convert the equation into a canonical type, utilize the comparison method, and the Riccati transformation to find sufficient conditions for oscillation of all solutions to the aforementioned problem. This approach greatly simplifies the examination analysis, and provides a substantial improvement of the current results and this is documented by several evidences and illustrated by numerical examples.

We study Hermite interpolation for the space of polynomials of total degree in ℝ N and the space of homogeneous polynomials in ℝ N +1 . We investigate the relations between the two types of Hermite interpolation. We show that they have the same regularity and continuity property.

It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.

In this paper, we introduce the concept of Hadamard 𝔇-hom-der on Banach algebras. Further, we apply the fixed point method to prove the stability of the Hadamard 𝔇-hom-der on Banach algebras.

The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in C λ , u ( X ) for a locally- λ space X when λ ⊇ 𝓕( X ), for a hemi- λ λ f -space X when λ ⊆ 𝓟 𝓢( X ), and for a k -space X when λ ⊇ 𝓚( X ). This paper also studies that every bounded subset of C λ , u ( X ) has compact closure for some classes of topological spaces X .

In this paper, we define some combinatorial principles to characterize spaces X whose hyperspace satisfies some variation of some classical star selection principle. Specifically, the variations characterized are the selective and absolute versions of the star selection principles for the Menger and Rothberger cases; also, the hyperspaces considered in these characterizations are CL ( X ), 𝕂( X ), 𝔽( X ) and ℂ𝕊( X ) in both cases, endowed with either the Fell topology or the Vietoris topology.

The classical density topology is an extension of the natural topology on the real line, as the interior of arbitrary Lebesgue measurable set A is contained in the set of density points of A . Also each density point of A belongs to the closure of A for arbitrary measurable set A . In this paper, we concentrate on lower density operators for which the inclusions mentioned above are not fulfilled. In the first part, examples of such lower density operators generated by measure-preserving bijections are given. There are introduced three conditions to investigate lower density operators for which only the second inclusion holds. In the second part, the concept of operator D introduced by K. Kuratowski is applied to the characterization of such operators.

For n a power of two, we give a complete description of the cohomology algebra H * ( G͠ n ,3 ; ℤ 2 ) of the Grassmann manifold G͠ n ,3 of oriented 3-planes in ℝ n . We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for H * ( G͠ n ,3 ; ℤ 2 ).

In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [ The von Bahr–Esseen moment inequality for pairwise independent random variables and applications , J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [ Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications , J. Math. Inequal. 10 (3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

In this paper, let { X n , n ≥ 1} be a sequence of random variables with heavy tailed distributions and define S n = X 1 + X 2 + ⋯ + X n , max1≤ k≤ n $\begin{array} \displaystyle \max\limits_{1\le k\le n} \end{array}$ S k = max{ S 1 , S 2 , …, S n } and M n = max{ X 1 , X 2 , …, X n }. We consider the large deviation behaviors and establish the upper bounds of large deviations for the above three quantities based on some dependent random variables { X n , n ≥ 1}, such as acceptable random variables, widely acceptable random variables and a class of random variables that satisfies the Marcinkiewicz-Zygmund type inequality and Rosenthal type inequality. Furthermore, the lower bounds of large deviations for some non-negative and identically distributed dependent sequences are obtained.

In this paper, we study the space SPGn $\begin{array} \displaystyle SP_{G}^{n} \end{array}$ X of G -permutation degree. Especially, new properties of this space are investigated. We study its metrazibility, semi-metrazibility, stratifiability, semi-stratifiability and uniformizability. Moreover, we study the uniform space (SPGnX,SPGnΥ), $\begin{array} \displaystyle (SP_{G}^{n}X, SP_{G}^n\Upsilon), \end{array}$ investigating when this space is totally bounded ( ω -bounded). Finally, a study of universal elements in classes of spaces that are presented in this article, and related open questions complete this paper.

Paolo de Lucia passed away on March 18, 2022 in Nyon, Switzerland. He was undoubtedly one of the most respected experts on Real Analysis and Measure Theory, but most of us miss him first of all as a scientific role model, colleague and fatherly friend. Last year, some of his coauthors, students and friends attended the ‘Abstract Measure Theory and Applications: a symposium remembering Paolo de Lucia’s contributions’, held in Naples on March 30–31, 2023, and contributed orally with lectures and speeches in the section ‘Memories’. For this reason, this year we wish to commemorate him by transcribing our two contributions and keeping them unchanged, albeit without the familiar context of the symposium, because unchanged is the sorrow for his loss and the weight of his absence for those who knew him. So, despite an unbridgeable distance, we can imagine him blushing slightly, smiling and saying ironically “At last, for me even a jurist agrees to publish in a mathematics journal”.