For a class of locally scaling functions on ℤ_2 with bijective restrictions we derive a representation similar to Mahler expansion. Using this representation we characterize functions uniformly differentiable modulo 2. We derive a representation of locally scaling functions with bijective restrictions via coordinate functions.
In this paper, we consider the Cauchy problem for a system of N∈ℕ coupled nonlinear wave equations in three-dimensional space. The system involves nonlinear terms that are monotone and satisfy concavity conditions. We first formulate the problem with prescribed initial data and then investigate the existence and uniqueness of positive, bounded, smooth solutions. To establish existence, we construct an iterative scheme that generates a sequence of approximate solutions. We prove that this sequence converges uniformly to a limit, which coincides with the solution of the original system. Furthermore, we show that under suitable structural assumptions on the nonlinearities, the solution obtained in this way is unique within the class under consideration. The results provide a constructive framework for analyzing coupled nonlinear wave equations and demonstrate applicability to a broad class of models arising in physics and engineering.
We generalize the classical p -adic norm on ℚ by introducing the (p_1,…,p_s) -adic norm, where p_1,…,p_s are pairwise distinct prime numbers, and study the associated completion ℚ_p_1,…,p_s . We establish its main properties, including non-Archimedeanity, sub-multiplicativity, and an Ostrowski-type classification of non-trivial ultrametric sub-multiplicative norms on ℚ . We also prove a unique expansion theorem and an analogue of Hensel’s lemma.
Using the canonical Fourier harmonic analysis on the field ℚ_p of p -adic numbers, we prove an analogue of one classical Titchmarsh theorem on description of the image under the Fourier transform of the class of function satisfying the Lipschitz condition in L^2 .
This paper is devoted to studying the behavior of Fourier coefficients and Fourier series in the multiplicative systems of corrected functions. The article also constructs an integrable function and a set that possess the property of strong universality with respect to the Vilenkin system.
