Measure-free viewpoint on p-adic and adelic wavelets

Recently there were presented new examples of MRAs based on p-adic and adelic wavelets. This is a note of the methodological nature. Its aim is embed all known examples MRAs into the measure-free scheme based on abstract Hilbert space formalism. On the one hand, such an embedding unifies the known examples of MRAs. On the other hand, the presented measure free scheme can be used for construction of new examples of MRAs, especially in the infinite dimensional case. The last section of this paper is devoted to another abstract MRAs scheme based on Riesz spaces. We start paper with a review on p-adic and adelic wavelets.     

An Application of the Theory of Open Quantum Systems to Model the Dynamics of Party Governance in the US Political System

The Gorini-Kossakowski-Sudarshan-Lindblad equation allows us to model the process of decision making in US elections. The crucial point we attempt to make is that the voter’s mental state can be represented as a superposition of two possible choices for either republicans or democrats. However, reality dictates a more complicated situation: typically a voter participates in two elections, i.e. the congress and the presidential elections. In both elections the voter has to decide between two choices. This very feature of the US election system requires that the mental state is represented by a 2-qubit state corresponding to the superposition of 4 different choices. The main issue is to describe the dynamics of the voters’ mental states taking into account the mental and political environment. What is novel in this paper is that we apply the theory of open quantum systems to social science. The quantum master equation describes the resolution of uncertainty (represented in the form of superposition) to a definite choice.     

p-Adic wavelets and their applications

The theory of p-adic wavelets is presented. One-dimensional and multidimensional wavelet bases and their relation to the spectral theory of pseudodifferential operators are discussed. For the first time, bases of compactly supported eigenvectors for p-adic pseudodifferential operators were considered by V.S. Vladimirov. In contrast to real wavelets, p-adic wavelets are related to the group representation theory; namely, the frames of p-adic wavelets are the orbits of p-adic transformation groups (systems of coherent states). A p-adic multiresolution analysis is considered and is shown to be a particular case of the construction of a p-adic wavelet frame as an orbit of the action of the affine group.     

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

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Harmonic Analysis in the p-Adic Lizorkin Spaces: Fractional Operators, Pseudo-Differential Equations, p-Adic Wavelets, Tauberian Theorems

In this article the p-adic Lizorkin spaces of test functions and distributions are introduced. Multi-dimensional Vladimirov’s and Taibleson’s fractional operators, and a class of p-adic pseudo-differential operators are studied on these spaces. Since the p-adic Lizorkin spaces are invariant under these operators, they can play a key role in considerations related to fractional operator problems. Solutions of pseudo-differential equations are also constructed. Some problems of spectral analysis of pseudo-differential operators are studied. p-Adic multidimensional Tauberian theorems connected with these pseudo-differential operators for the Lizorkin distributions are proved.