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1. Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

   Casale, Guy; Freitag, James; Nagloo, Joel (November 2020, Annals of mathematics)

   null (Ed.)

   We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups. 

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2. Generic differential equations are strongly minimal

   https://doi.org/10.1112/S0010437X23007212  

   DeVilbiss, Matthew; Freitag, James (July 2023, Compositio Mathematica)

   In this paper we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of non-minimality of Freitag and Moosa. Our techniques are sufficient to show that generic order $$h$$ differential equations with non-constant coefficients are strongly minimal, answering a question of Poizat (1980). 

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3. A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

   https://doi.org/10.5488/CMP.25.33203  

   Mathews, W. N.; Esrick, M. A.; Teoh, Z. Y.; Freericks, J. K. (January 2022, Condensed Matter Physics)

   The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation. 

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4. A Tutorial on Matrix Perturbation Theory (using compact matrix notation)

   https://doi.org/10.48550/arXiv.2002.05001  

   Bamieh, Bassam (October 2020, ArXivorg)

   Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the non-Hermitian and degenerate cases, as well as for higher order terms. 

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5. Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients

   https://doi.org/10.1016/j.padiff.2024.100753  

   Adjibi, Kolade; Martinez, Allan; Mascorro, Miguel; Montes, Carlos; Oraby, Tamer; Sandoval, Rita; Suazo, Erwin (September 2024, Partial Differential Equations in Applied Mathematics)

   We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are timedependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations. 

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