This paper develops a Stepanov framework for almost periodicity over p -adic fields. We begin by defining Stepanov spaces using p -adic balls and the normalized Haar measure on the locally compact abelian additive group ℚ_p . Stepanov almost periodicity is then defined through relatively dense sets of translation numbers. To bridge Stepanov and Bohr almost periodicity, we introduce the class of uniformly regular Stepanov almost periodic functions, which enables the derivation of Bohr almost periodic solutions under weaker assumptions. To illustrate the theory, we establish existence results for translation-difference equations, convolution equations, and semilinear evolution equations involving the Vladimirov pseudodifferential operator. The existence of almost periodic mild solutions is obtained by means of the Schauder fixed-point theorem.
This paper investigates the hypothesis of ultrametric organization of abstract energy landscapes defined on random Erdös-Rényi graphs. Within the proposed model, each vertex of the graph is assigned a random free energy uniformly distributed over a given interval, and the kinetics of the system is described by a Markov process with a Kramers transition rate matrix. Based on an exact macroscopic coarse-graining procedure and the spectral decomposition of the effective generator, a kinetic metric between basins of attraction is constructed. Computational experiments for graphs with V=5000 vertices and E=15000 edges (average degree ⟨ k⟩=6.0 ) demonstrate that this metric reveals a high degree of nontrivial ultrametricity, which monotonically increases from ≈21% at an energy interval width of Δ F=10 kJ/mol to ≈92% at Δ F=1000 kJ/mol. For a rigorous mathematical justification of this phenomenon, a limit theorem is formulated and proved, establishing that as the width of this interval tends to infinity, the logarithmic asymptotics of the proposed metric converges pointwise to the classical single-linkage ultrametric. Analysis of computational experiments shows that at finite values of the interval width, suboptimal paths and thermal fluctuations introduce corrections that violate the strong triangle inequality; however, in the regime of large energy fluctuations, their contribution is exponentially suppressed, due to which the metric asymptotically acquires ultrametric properties. The results obtained indicate that the ultrametric organization of energy landscapes is a universal asymptotic property of networks of arbitrary topology with a highly rugged landscape, which for graphs with moderate edge density manifests itself already at sufficiently large energy spreads.
Small functions were defined in complex analysis and next in ultrametric analysis. The order of growth was also defined in complex analysis and has a similar definition in ultrametric analysis. It has already been examined in a previous article. Here we study additional properties designed to compute the growth order using the distribution of the zeros and the application to very small functions. We can then apply a theorem on entire functions sharing 3 small functions.
Let 𝕂 be an algebraically closed field of characteristic zero, complete for a nontrivial non-Archimedean absolute value. Let a,b∈𝕂∖{0} and let n,k,p be positive integers. We study non-constant functions f and g whose differential polynomials f^n+a(f^(k))^p and g^n+a(g^(k))^p share the value b CM . For entire functions, if n≥ p+2 and (n,p)=1 , then f=g unless f and g are proportional polynomials of degree at most k . For meromorphic functions, if n≥ 2p(k+1)+8 , then f and g satisfy one of three explicit algebraic relations. These results extend earlier work of Tuan and Quang and, for p=1 , improve their bounds on n .
We study the norm convergence of sequences of the general matrix transform means σ_n^T(f) of Vilenkin-Fourier series with respect to a bounded Vilenkin system. We give a sufficient condition for convergence for each integrable function f , namely ∑_k=1^nt_k,n→1 and ∑_k=1^nt_k,n^2=O(1/n). In comparison with previous results in this topic, we do not assume monotonicity of the defining sequence (t_k,n,1≤ k≤ n, k∈ℙ) of the matrix T . The T summation is a common generalisation of several well-known summation methods, such as Fejér, Weierstrass, Riesz, Picard, Bessel, and Nörlund summation methods. The method also includes Cesàro means; the present sufficient condition covers the range α>1/2 .
