This article considers a second-order difference equation with constant coefficients in its standard form and two different classes of two-point homogeneous boundary conditions. First, we construct the corresponding Green functions and derive some important properties for further analysis. Next, we propose adequate conditions for the existence of solutions to the considered boundary value problems. Finally, we offer two examples to show the applicability of the main results.

In this paper, we establish several new arctangent- and logarithmic-Hardy-Hilbert integral inequalities. The approach combines fundamental principles with refined techniques from the theory of integral inequalities, leading to a range of original results. Complete proofs are presented together with a discussion of their sharpness and potential applications.

This note introduces a \(1\)-parameter of cubic curves naturally associated to the sphere \(S^4\) considered in the unique \(5\)-dimensional irreducible representation space of \(SO(3)\). Eight examples are discussed with the last two being elliptic curves. Also, two conics are defined naturally in our setting by a special basis of the Lie algebra \(sl(3, \mathbb{R})\).

This paper introduces a unified framework for fixed point theorems involving asymptotically regular mappings in \(b\)-metric spaces through the concept of contractive families. We establish a general fixed point theorem that encompasses various existing results, including those of Kannan-type and generalized contractive conditions, as special cases. In particular, we demonstrate that the recent results of Nagac and Tas [1] emerge naturally as special cases of our main theorem through appropriate parameter choices. The main result employs coefficient functions and a general auxiliary function with strengthened continuity conditions, providing flexibility that allows the derivation of numerous particular cases. Several corollaries with complete proofs are presented to demonstrate that our results properly generalize and extend well-known theorems in the literature.

This paper introduces the concept of edge sum labeling in hypergraphs, where the edges of a hypergraph \(\mathcal{H}\) are assigned distinct positive integers such that the sum of the labels of all edges incident to any vertex is itself an edge label of \(\mathcal{H}\). Moreover, if the sum of the labels of any collection of edges equals the label of another edge in \(\mathcal{H}\), those edges must be incident to at least one common vertex. Additionally, we define and investigate zero edge sum hypergraphs, exploring their unique properties and presenting various results related to this new class of hypergraphs.

The basic building block of a modern block cipher is the substitution box (S-box), which provides the nonlinearity that is required when fending off advanced cryptanalytic methods. The novel approach introduced in the present work is computationally efficient, but it is still a robust algorithm for generating an 8 8 S-box through an operation of a specifically defined bijective mapping over the Galois Field \(GF(2^{8})\). The proposed S-box was strictly tested on a broad range of standard security criteria to prove its cryptographic integrity. A good performance has been identified in the analysis with a nonlinearity of 112 and linear approximation probability (LAP) of 0.0625, and the outstanding element of differential approximation probability (DAP) of 0.0156. Using a strong cryptographic construction, the new AES structure implements an S-box whose avalanche-like properties are best shown by a low value of the strict avalanche criterion (SAC) 0.4995 and strong bit independence (BIC) scores. The experimental results have supported the hypothesis that the proposed S-box has a much greater resistance to both the differential and the linear attacks compared with the state-of-the-art algebraic, heuristic, and chaos-based designs. In order to show how applicable the S-box can be in practical terms, a framework of image encryption incorporates the use of the S-box in it, whereby it operates as the basis element of a block cipher. The resulting cipher image achieved an entropy of 7.9978, which demonstrates a very high degree of randomness and strong resistance to statistical attacks. The feature article introduces a significant step towards systematizing cryptographic design through the introduction of a sound and carefully defined framework for the construction of high-security S-boxes.