Annals of Communications in Mathematics

ABSTRACT.This article is devoted to five distinct integral inequalities of the Hardy-Hilbert  type, each possessing its own originality. In particular, we highlight new trigonometric variants that yield sharp upper bounds involving weighted integral norms of the underlying functions. Complete and rigorous proofs are provided in detail.

ABSTRACT. A dominating set S in G is called a pendant dominating set if ⟨S⟩ contains at least one pendant vertex. The minimum cardinality of a pendant dominating set is called the pendant domination number denoted by γpe(G). The pendant domination polynomial of G is denoted by Dpe(G, x) and is defined as Dpe(G, x) = Pn i=γpe(G) dpe(G, i)x i , where dpe(G, i)x i is the number of pendant dominating sets of size i. In this paper, we obtained the pendant domination number and pendant domination polynomial of the corona of some graphs, namely, Pm ◦ Kn, Cm ◦ Kn and Km ◦ Kn.

ABSTRACT. An injective function f : V (G) → {L1, L2, . . . , Ln}, where Lj is the jth Lucas number (j = 1, 2, . . . , n) is said to be Lucas cordial labeling if the induced function f ∗ : E(G) → {0, 1} defined by f ∗(uv) = (f(u) + f(v)) (mod 2) satisfies the condition |ef (0) − ef (1)| ≤ 1, where ef (0) is the number of edges labeled with 0 and ef (1) is the number of edges labeled with 1. A graph which admits Lucas cordial labeling is called Lucas cordial graph.

ABSTRACT. Let η ≥ 3 be an integer with primitive root π. For a simple connected graph G of order n, a bijective function f : V (G) → {1, 2, ..., n} is called a logarithmic cordial labeling to the base π modulo η if the induced function f ∗ π,η : E(G) → {0, 1}, defined by f ∗ π,η(ab) = 0 if indπ,η(f(a) + f(b)) ≡ 0(mod 2) or gcd(f(a) + f(b), η) ̸= 1, and f ∗ π,η(ab) = 1 if indπ,η(f(a) + f(b)) ≡ 1(mod 2), satisfies the condition |ef∗π,η (0) − ef∗π,η (1)| ≤ 1 where ef∗π,η (i) is the number of edges with label i(i = 0, 1). In this paper, we study the logarithmic cordial labeling of various classes of graphs, including path graphs, cycle graphs, star graphs, and complete graphs.

ABSTRACT. Probability distributions are essential tools for modeling, prediction, and statistical inference. In recent years, several generalized families of distributions have been proposed to extend classical models and increase their flexibility in capturing complex data behaviors. This paper reviews selected generalized families published between 2023 and 2025, focusing on their construction mechanisms, statistical properties, estimation methods, and real-world applications. The families discussed include trigonometric-based, inverse, Lomax-generated, Topp–Leone, and hybrid forms. To illustrate their performance, five families were combined with the exponential distribution and fitted to a real dataset. The comparison shows that all extended models provide an adequate fit, while the standard exponential model performs poorly. The findings confirm the practical value of generalized families in improving data modeling.

ABSTRACT. In this article, we present a new generalized version of the Hardy integral inequality. It has the property of depending on an auxiliary function. Thanks to this function, numerous variants are examined. The theory is complemented by two secondary results,  one showing that the main inequality can be improved under additional assumptions, and another giving a valuable lower bound for the main integral term. Several examples  are given for illustration.

ABSTRACT. Here we research the univariate multi-composite sigmoid activated quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued multi-composite sigmoid activated neural network operators. We perform also the related Banach space valued multi-composite sigmoid activated fractional approximation. These multi-composite sigmoid acti-vated approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are deÖned by using a multi-composite density function induced by a gen- eral multi-composite sigmoid function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We Önish with a convergence analysis.