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.
We provide an explicit construction of a Gabor orthonormal bases for a local field K that provides maximal localization in both time and frequency. Such a localization is not true in case of ℝ due to the uncertainty principle. In particular, we construct examples of functions f ∈ L^2(K) such that the support of the ambiguity function of f is of minimum measure. Moreover, we establish a quantitative uncertainty principle for local fields, which follows as a consequence of Lieb's inequalities for general locally compact abelian group. In addition, we develop fundamental operator representations for Gabor systems defined over local fields.
Sufficient dyadic conditions are obtained for functions in the Lebesgue space L^p([0, +∞)) with 1≤ p≤ 2 , ensuring the integrability of their Fourier-Walsh transforms. These conditions are expressed in terms of moduli of smoothness and are shown to be sharp. As a result, a recent result established by Platonov in this framework is deduced.
This paper investigates a class of multidimensional p -adic Hardy-Hilbert-type integral operators with homogeneous kernels of degree -n . We establish the boundedness of these operators on various weighted function spaces, including weighted p -adic Lebesgue spaces, weighted p -adic Morrey spaces, and weighted p -adic mixed Morrey spaces.
We study the computational complexity of p -adic linear regression: given data (x_i,y_i) with x_i∈ℚ^n and y_i∈ℚ , find coefficients β∈ℚ^n minimising the p -adic residual sum L(β)=∑_i=1^r| y_i-x_i^⊤β|_p . Here r is the number of observations. Unlike least-squares and ℓ_1 regression over ℝ , the ultrametric inequality produces a discrete, hierarchical loss landscape in which small perturbations can change divisibility patterns abruptly. We show that this discretisation has worst-case computational consequences: computing an optimal p -adic regression solution is nondeterministic polynomial-time (NP)-hard. The proof is by a polynomial-time reduction from the maximum cut (Max-Cut) problem. We construct a regression instance in which the coefficients corresponding to vertices can be rounded to {0,1} without increasing loss, and each edge contributes 0 to the loss exactly when it crosses the induced cut. Hence minimising L(β) is equivalent to maximising the cut size. Our result complements existing tractable regimes for p -adic regression (e.g., polynomial-time solvability in fixed dimension by enumerating hyperplanes through n+1 data points) and motivates studying approximation, parameterised complexity, and alternative aggregations of p -adic residuals.
We consider a three-state solid-on-solid (SOS) model in the presence of a nonzero external field on a Cayley tree. A system of functional equations corresponding to this model is derived, where each solution defines a quasi Gibbs measure. Based on this system, we investigate the translation-invariant p -adic quasi Gibbs measures (TIpQGMs) of the model. In the case of a binary Cayley tree, we explicitly determine the TIpQGMs under the same conditions. Furthermore, by analyzing the boundedness of these measures, we establish the existence of a phase transition.
In this paper we study some problems of the canonical harmonic analysis on the field ℚ_p of p -adic numbers. The main elements of the canonical harmonic analysis on ℚ_p are canonical Fourier integral transforms, canonical generalized translation operators and canonical convolution products for functions on ℚ_p . We consider various results of the canonical harmonic analysis for functions from Lebesgue spaces L^ρ(ℚ_p) , 1≤ρ≤∞ . Basic concepts of the canonical harmonic analysis on ℚ_p are expand to generalized functions (or distributions), among them the canonical Fourier transforms on ℚ_p , the generalized translation operators on ℚ_p and others. The analogues of various results of classical harmonic analysis, including analogues of the Paley-Wiener-Schwartz theorems, are proved. We introduce a canonical convolution product on ℚ_p for usual and generalized functions and establish some of its properties.
We give sufficient conditions for a.e. convergence of Vilenkin-Fourier series on [0,1) , pointwise one on (0,1) and uniform convergence on [ε,1) , ε∈ (0,1) , in terms of behavior of some linear means of cited above series. Also, a condition for such means to converge a.e. on [0,1) is proved.
In this paper, we investigate spin systems on general infinite trees. The spins can take countably many values, and nearest-neighbor interactions are governed by a p -adic stochastic matrix. We establish sufficient conditions on the stochastic matrix that guarantee the uniqueness of the associated Markov chain. Furthermore, we identify a family of stochastic matrices that lead to the existence of at least two distinct p -adic Markov chains on an infinite tree, particularly a Cayley tree.
The aim of this paper is to provide some necessary and sufficient conditions for the boundedness of p -adic rough Hausdorff operators ℋ_Ψ,Ω^p on the grand Morrey-Herz spaces. In each case, we estimate the norm of the operators ℋ_Ψ,Ω^p .
We propose a new formulation of p -adic optimisation as the infinitesimal limit of the least squares method, and introduce several algorithms for p -adic optimisation. Since the optimisation problem includes the maximal feasible subsystem problem of linear equations over the finite field 𝔽_p , which is APX-complete, i.e. complete for the class of problems which allow constant-factor approximations, by E. Amaldi and V. Kann, we mainly deal with heuristic approaches to the p -adic optimisation under mild assumptions. In particular, we deal with p -adic polynomial regression under the assumption that noise occurs digitwise sparsely.
Let 𝕂 be an algebraically closed non-Archimedean field with characteristic zero such that it is complete for a nontrivial non-Archimedean absolute value. In this note, we construct examples of bi- URS for ℳ(𝕂) of the form {a_1, a_2, a_3, a_4}, {ω} .
We define a representation of isometric functions on the ring of p -adic integers ℤ_p in terms of permutations of the set {0,…, p-1}. We also provide new ergodicity criteria using these representations.
This article is dedicated to the study of dynamical systems over the field of p -adic numbers in higher dimension, defined by a specific monomial function in each component. We determine a condition on the Jacobian matrix for a point to be an attractor of the system. We also found basins of attraction and centers of Siegel disks in terms of basins of attraction and Siegel disks of its components, respectively.
We study nonlinear reaction-diffusion equations with non-local source terms, in which the results arise in population dynamics and epidemiology. Under mild conditions on the initial data, interaction kernel, and nonlinearity, we prove the existence, uniqueness, and positivity of bounded classical solutions. The analysis employs a Hammerstein-Volterra nonlinear integral formulation and a constructive monotone iteration scheme that preserves positivity and continuity. Special cases with vanishing reproduction rate and constant source intensity on finite time intervals are also investigated. The results extend known one-dimensional models to a general multidimensional framework.
