p-adic Gibbs measures and Markov random fields on countable graphs

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures     

The p-adic Potts model on the Cayley tree of order three

We study a phase transition problem for the q-state p-adic Potts model on the Cayley tree of order three. We find certain conditions for the existence of p-adic Gibbs measures and then establish the existence of a phase transition.     

p-adic Gibbs measures for the hard core model with three states on the Cayley tree

We study p-adic hard core models with three states on the Cayley tree. It is known that there are four types of such models. We find conditions that must be imposed on the order k of the Cayley tree and on the prime p for a translation-invariant p-adic Gibbs measure to exist.     

Construction of a set of <Emphasis Type="Italic">p</Emphasis>-adic distributions

Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ?_p of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions).     

A p-Adic Generalized Gibbs Measure for the Ising Model on a Cayley Tree

Theoretical and Mathematical Physics - We consider a p-adic Ising model on the Cayley tree of order k ≥ 2. We completely describe all p-adic-translation-invariant generalized Gibbs measures...     

Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L₂-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).     

The local index and the Swan conductor

In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.     

<Emphasis Type="Italic">p</Emphasis>-Adic solid-on-solid model on a Cayley tree

We consider a p-adic solid-on-solid (SOS) model with a nearest-neighbor coupling, m+1 spins, and a coupling constant J ∈ Q _p on a Cayley tree. We find conditions under which a phase transition does not occur in the model. We show that if p | m + 1 for some J, then a phase transition occurs. Moreover, we formulate a criterion for the boundedness of p-adic Gibbs measures for the (m+1)-state SOS model.     

Characterizations by Automorphism Groups of Some Rank 3 Buildings, III. Moufang-Like Conditions

In this paper, we introduce the root-Moufang condition and the p-adic Moufang condition. We show that affine buildings of type Ã2 satisfying the root-Moufang condition are Bruhat–Tits buildings. Also, every rank 3 affine building satisfying the p-adic Moufang condition is a Bruhat-Tits building. We motivate the introduction of the new conditions by showing that all Bruhat– Tits Ã2-buildings satisfy the root-Moufang condition, and that the Ã2-buildings over a p-adic field also satisfy the p-adic Moufang condition. Another application of the p-adic Moufang condition is given in Part IV of this paper.     

单个生成元Walsh p-进制平移不变空间伸缩的交与并

p-进制MRA与GMRA是构造L~2(R_+)中小波框架的重要工具. L~2(R+)中嵌套子空间序列交集为{0},并集为L~2(R_+)是其构成p-进制MRA与GMRA的基本要求.本文研究单个生成元Walsh p-进制平移不变子空间伸缩的交与并,证明了:对任意单个生成元Walsh p-进制平移不变子空间,其p-进制伸缩的交是{0};若生成元分为Walsh p-细分函数,则其p-进制伸缩的并是L~2(R_+)中一个Walshp-进制约化子空间.特别地,其伸缩构成L~2(R_+)中p-进制GMRA当且仅当∪_(j∈z)p~j supp(■φ)=R+,其中■为定义在L~2(R_+)上的Walsh p-进制傅里叶变换.值得注意的是:形式上,我们的结果类似于通常L~2(R)的情形,然而其证明不是平凡的.这是因为定义在R_+上的p-进制加法"⊕"不同于定义在R上的通常加法"+".     

The <Emphasis Type="Italic">p</Emphasis>-adic sector of the adelic string

We study the construction of Lagrangians that can be considered the Lagrangians of the p-adic sector of an adelic open scalar string. Such Lagrangians are closely related to the Lagrangian for a single p-adic string and contain the Riemann zeta function with the d’Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach that combines all p-adic Lagrangians. This new Lagrangian is attractive because it is an analytic function of the d’Alembertian. Investigating the field theory with the Riemann zeta function is also interesting in itself.     

Qualitative theory of p-adic dynamical systems

We consider a class of dynamical systems over the p-adic number field: hierarchical dynamical systems. We prove a strong variant of the Poincaré theorem on the number of returns for such systems and show that hierarchical systems do not admit mixing. We describe hierarchical dynamical systems over the projective line and present an example of a nonhierarchical p-adic system that admits mixing: the p-adic baker’s transformation.     

p-adic Transcendence and p-adic Transcendence Measures for the Values of Mahler Type Functions

We prove the p-adic transcendence and p-adic transcendence measures for the values of some Mahler type functions.     

On <Emphasis Type="Italic">p</Emphasis>-adic quasi Gibbs measures for <Emphasis Type="Italic">q</Emphasis> + 1-state Potts model on the Cayley tree

In the present paper we introduce a new kind of p-adic measures, associated with q + 1-state Potts model, called p-adic quasi Gibbs measure, which is totally different from the p-adic Gibbs measure. We establish the existence of p-adic quasi Gibbs measures for the model on a Cayley tree. If q is divisible by p, then we prove the occurrence of a strong phase transition. If q and p are relatively prime, then there is a quasi phase transition. These results are totally different from the results of [F. M. Mukhamedov and U. A. Rozikov, Indag. Math. N. S. 15, 85–100 (2005)], since when q is divisible by p, which means that q + 1 is not divided by p, so according to a main result of the mentioned paper, there is a unique and bounded p-adic Gibbs measure (different from p-adic quasi Gibbs measure)     

On the simultaneous rational approximations to real and p-adic numbers

Lithuanian Mathematical Journal - We consider simultaneous rational approximations to real and p-adic numbers. We prove that for any irrational number α0 and p-adic number α, there are...     

Applications of <Emphasis Type="Italic">p</Emphasis>-adics to geophysics: Linear and quasilinear diffusion of water-in-oil and oil-in-water emulsions

In a very general setting, we discuss possibilities of applying p-adics to geophysics using a p-adic diffusion representation of the master equations for the dynamics of a fluid in capillaries in porous media and formulate several mathematical problems motivated by such applications. We stress that p-adic wavelets are a powerful tool for obtaining analytic solutions of diffusion equations. Because p-adic diffusion is a special case of fractional diffusion, which is closely related to the fractal structure of the configuration space, p-adic geophysics can be regarded as a new approach to fractal modeling of geophysical processes.     

Functional Fourier transformation and renormalization group transformation in bosonic field theory models

We introduce the notion of a functional Fourier transformation in bosonic p-adic and Euclidean models of statistical physics. We prove a commutation relation between the Fourier transformation and the Wilson renormalization group transformation and discuss the similarity of renormalization group formalisms in p-adic and Euclidean models.     

Sparse <Emphasis Type="Italic">p</Emphasis>-adic data coding for computationally efficient and effective big data analytics

We develop the theory and practical implementation of p-adic sparse coding of data. Rather than the standard, sparsifying criterion that uses the L ₀ pseudo-norm, we use the p-adic norm.We require that the hierarchy or tree be node-ranked, as is standard practice in agglomerative and other hierarchical clustering, but not necessarily with decision trees. In order to structure the data, all computational processing operations are direct reading of the data, or are bounded by a constant number of direct readings of the data, implying linear computational time. Through p-adic sparse data coding, efficient storage results, and for bounded p-adic norm stored data, search and retrieval are constant time operations. Examples show the effectiveness of this new approach to content-driven encoding and displaying of data.     

Characterizations by Automorphism Groups of Some Rank 3 Buildings – IV: Hyperbolic p-adic Moufang Buildings of Rank 3

In this paper, we introduce the p-adic Moufang condition for hyperbolic buildings of rank 3. It is the most obvious and simplest generalization of the p-adic Moufang condition for affine buildings, introduced in Part III of this sequence of papers. We show that p is very restricted, which confirms (but does not prove) the conjecture that no p-adic analogue is possible for the construction of Moufang (hyperbolic) buildings by Ronan and Tits.     

p-ADIC亚纯函数在截断重级下的一个定理

本文研究p-adic亚纯函数在截断重级下分担两个公共值集的唯一性问题，得到了一个相应的定理．定理的证明主要利用p-adic亚纯函数论中的两个基本定理和一个辅助函数．