AppliedMath

This paper addresses the numerical integration of first-order ordinary differential equations by developing a continuous linear multistep block method. The method is constructed through the approximation of the exact solution using a linear combination of shifted Vieta–Lucas polynomials defined on the interval . The use of this polynomial basis extends traditional approximation approaches and provides improved stability while maintaining high-order accuracy. Theoretical analysis shows that the proposed method attains sixth-order convergence and possesses an extended stability interval of , ensuring reliable performance for moderately stiff problems. Numerical experiments confirm that the method achieves lower errors and higher computational efficiency than conventional methods. These results demonstrate the suitability of the proposed approach for scientific computing applications, including engineering simulations and mathematical modeling, where accurate numerical integration of first-order differential equation is required.

The paper retrieves quiescent dispersive solitons in dispersion-flattened optical fibers having nonlinear chromatic dispersion and the Kerr law of self-phase modulation. The platform model is the Schrödinger–Hirota equation. The enhanced direct algebraic method has made this retrieval possible. The intermediary functions are Jacobi’s elliptic function and Weierstrass’ elliptic function. The final results appear with parameter constraints for the existence of such solitons.

Modeling legal reasoning with artificial intelligence and machine learning presents formidable challenges. Legal decisions emerge from a complex interplay of factual circumstances, statutory interpretation, case precedent, jurisdictional variation, and human judgment—including the behavioral characteristics of judges and juries. This paper takes an exploratory approach to investigating how contemporary ML techniques might capture aspects of this complexity. Using pharmaceutical patent litigation as an illustrative domain, we develop a multi-layer analytical pipeline integrating text mining, clustering, topic modeling, and classification to analyze 698 U.S. federal district court decisions spanning January 2016 through December 2018, comprising substantive validity and infringement rulings under the Hatch-Waxman regulatory framework. Results demonstrate that the pipeline achieves 85–89% prediction accuracy—substantially exceeding the 42% baseline majority-class rate and comparing favorably with prior legal prediction studies—while producing interpretable intermediate outputs: clusters that correspond to recognized doctrinal categories (Abbreviated New Drug Application—ANDA litigation, obviousness, written description, claim construction) and topics that capture recurring legal themes. We discuss what these findings reveal about both the possibilities and limitations of computational approaches to legal reasoning, acknowledging the significant gap between statistical prediction and genuine legal understanding.