We find the solution of a first-order autonomous differential equation defined on the entire real line. This problem is solved using elements of mathematical analysis and topology.

To keep pace with the development in the exchange of information through an insecure transmission medium, it is necessary to provide secure encryption methods. In this paper, we present a high-security encryption system ATTRU based on a new algebra AT.

In this article, we evaluate an interesting integral from the famous book of integrals by Gradshteyn and Ryzhik, which, in its seventh edition, includes an incorrect result. We derive the correct value. Moreover, we highlight the application of the hypergeometric functions formalism in the evaluation of this integral.

Cardiovascular diseases (CVDs) remain a leading cause of morbidity and mortality worldwide. Early detection of heart rhythm abnormalities and stress levels can significantly improve patient outcomes. This study leverages continuous monitoring via smartwatches and an AI-powered mobile application to facilitate early detection of both cardiovascular anomalies and stress levels. We developed advanced data science and forecasting algorithms utilizing machine learning and statistical analysis to identify potential deviations from normative physiological patterns. Over a 28-day period, continuous data were collected from wearable devices, analyzing heart rate (HR), stress levels, and additional physiological parameters in a participant cohort. Our findings demonstrate significant correlations (the correlation coefficient is greater than 0.83) between heart rate and stress levels, with Poincaré plot analysis providing valuable insights into heart rhythm disturbances (SD1/SD2 ratios). Furthermore, we present an innovative application of this technology for predicting opioid cravings, highlighting its broader health monitoring capabilities. The Behaivior AI Recovery™ application, compatible with various sensor devices, including smartwatches, offers a practical solution for real-world health monitoring in clinical and home settings, enabling early intervention and improved patient outcomes. This research underscores the transformative potential of integrating AI and wearable technology in healthcare, emphasizing the importance of continuous and remote patient monitoring.

In this paper, we propose a new digital signature scheme constructed over integral group rings, relying on the hardness of the Conjugacy Search Problem (CSP) and incorporating a cryptographic hash function to map arbitrary messages into the integral group ring. The construction is described in detail and supported by a concrete example. A brief security analysis is also provided, showing resistance to several potential attack scenarios.

In this work, we study the injective chromatic number of the graph C^k_n, which is a graph obtained from C_n by adding an edge to any two distinct vertices with distance 2 to k on C_n. An injective coloring of a graph G is a vertex coloring in which any pair of vertices with a common neighbor are assigned different colors. The injective chromatic number \chi_i (G) of a graph G is the least positive integer k such that G has an injective k-coloring. In this work, we determine \chi_i (C^k_n) for all n and k. When n \leq 2k+1, the graph C_n^k becomes a complete graph K_n, with injective chromatic number n. For larger values of n, we obtain the following main results: if r=0, then \chi_i (C^k_n) = 2k+1, while if 1 \leq r \leq 2k, then \chi_i (C^k_n)=2k+\tilde{x}+1, where \tilde{x} is the largest natural number such that 1+(2k+1)\lceil \frac{n}{2k+x}\rceil > n. Additionally, we find \chi_i (P^k_n) for all n and k where P^k_n is a graph obtained similarly from a path P_n.

In this work, we establish interrelations between the clique polynomials, clique neighborhood polynomials, and convex neighborhood polynomials of graphs over clique-preserving and distance-reducing operations of graphs.

In this article, we establish the moment's properties of a novel family of distribution are via generalized order statistics (GOS). We take into consideration the cases for order statistics and record values.

In this paper, we develop a new method of encryption, using DNA and polynomials as public and private keys, which gives a very high level of security for both the private keys and the original message.

In this paper, we propose integral representations of the one-parameter k-Pell and k-Pell-Lucas numbers. Our results are also deduced with the Pell and Pell-Lucas numbers.