AppliedMath

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.

In Algebraic Statistics, M.A. Cueto, J. Morton and B. Sturmfels introduced a statistical model, the Restricted Boltzmann Machine, which introduced the Hadamard product of two or more vectors of an affine or projective space, i.e., the componentwise product of their entries, forcing Algebraic Geometry to enter. The Hadamard product $X\star Y$ of two subvarieties is defined as the Zariski closure of the Hadamard product of its elements. Recently, D. Antolini and A. Oneto introduced and studied the definition of Hadamard rank, and we prove some results on it. Moreover, we prove some theorems on the dimension and shape of the Hadamard powers of X. The aim is to describe the images of the Hadamard products without taking the Zariski closure. We also discuss several scenarios describing the case in which some of the data, i.e., the variety X, is wrong or it is not possible to recover it.