Abstract We study the irreducible representations of the Burnside groups $$B(m,3)$$ . We relate their nontrivial images to $$3$$ -periodic groups with center of order $$3$$ and investigate such groups. We prove a result on the structure of these groups and use it to determine the dimensions of irreducible representations of $$B(5,3)$$ and $$B(6,3)$$ .
This paper investigates the existence and general structure of transcendental meromorphic solutions for the FitzHugh–Nagumo model, a fundamental system in neuroscience describing neural membrane excitation. Utilizing Nevanlinna theory and Clunie’s lemma, we significantly extend the recent results of Lü et al. [Bull. Malays. Math. Sci. Soc. (2021)]. While previous studies focused on models with rational coefficients, our work considers a more generalized FitzHugh–Nagumo system featuring arbitrary meromorphic coefficients. We provide a complete characterization of the solutions for the generalized equation: g_1V^2V_t+g_2V^3+g_3V_tt+g_4V_t+g_5V+g_6=0 where g_j are meromorphic functions. We prove that any such solution V(t) must take a specific form involving an entire function and either a finite or infinite product of its poles and zeros, depending on the nature of the ratio g_2/g_1 . Furthermore, we generalize results for higher-order differential equations within this framework. To validate our theoretical findings, we exhibit several explicit examples that demonstrate the necessity of our conditions.
We study the irreducible representations of the Burnside groups B(m,3) . We relate their nontrivial images to 3 -periodic groups with center of order 3 and investigate such groups. We prove a result on the structure of these groups and use it to determine the dimensions of irreducible representations of B(5,3) and B(6,3) .
In this article, we have explored transcendental entire solutions of several nonlinear quadratic trinomial equations governed by the q -shift operator. While some of these equations involve the ordinary q -shift and the function itself, others incorporate combinations such as the q -shift together with a differential operator or the q -differential operator together with the function. Our findings are illustrated through several examples. These results refine and extend earlier work in a new direction, providing deeper insights into the nature of such equations and their transcendental entire solutions.
In this paper, within the framework of comparison by weights of polynomials, we establish necessary and sufficient conditions of hypoellipticity.
