Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

Козырев, С. В.; Khrennikov, Andrei

doi:10.1070/im2005v069n05abeh002284

Cited by 39 publications

(29 citation statements)

References 27 publications

(26 reference statements)

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“…Our class of orthonormal bases in L 2 (Q p ) is new, including Kozyrev' s orthonormal base [5][6][7] as special case.…”

Section: 1mentioning

confidence: 99%

The Cauchy problem for a class of pseudodifferential equations over p-adic field

Chuong

2008

Journal of Mathematical Analysis and Applications

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In this paper, some classes much more general than the one in [N.M. Chuong, Yu.V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (Eds.), Harmonic, Wavelet and p-Adic Analysis, World Scientific, Singapore, 2007] of Cauchy problems for an interesting class of pseudodifferential equations over p-adic fields are studied. The used functions belong to mixed classes of real and p-adic functions. Even for p-adic partial differential equations such problems in such function spaces have not been discussed yet. The established mathematical foundation requires very complicated and very difficult proofs. Days after days, these equations occur increasingly in mathematical physics, quantum mechanics. Explicit solutions of such problems are very needed for specialists on applied mathematics, physics, and engineering.

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“…Our class of orthonormal bases in L 2 (Q p ) is new, including Kozyrev' s orthonormal base [5][6][7] as special case.…”

Section: 1mentioning

confidence: 99%

The Cauchy problem for a class of pseudodifferential equations over p-adic field

Chuong

2008

Journal of Mathematical Analysis and Applications

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“…We describe the recursive procedure for constructing the solution of system (5). The initial condition for (1) as a function in D 0 (X) has the expansion over wavelets…”

Section: Proof Any Functionmentioning

confidence: 99%

“…We note that we have localization not only in the space but also on the tree T (X). (1) for v in D 0 (X) ⊗ C 1 ([0, ∞)) is equivalent to system of ordinary differential equations (5). System (5) is an example of the so-called cascade model.…”

Section: Remarkmentioning

confidence: 99%

“…In [4], p-adic wavelets were introduced, and the relation of wavelet analysis to the spectral theory of p-adic pseudodifferential operators was described. The analysis of pseudodifferential operators and wavelets on general (locally compact) ultrametric spaces was developed in [5]- [7]. The nonlinear theory of p-adic generalized functions was developed in [8], [9].…”

Section: Introductionmentioning

confidence: 99%

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Toward an ultrametric theory of turbulence

Козырев¹

2008

Theor Math Phys

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“…The analysis of wavelets and pseudodifferential operators on general locally compact ultrametric spaces was developed in [178,179,180,14]. A pseudodifferential operator on ultrametric space X is defined by Kozyrev as the following integral operator:…”

Section: Analysis On General Ultrametric Spacesmentioning

confidence: 99%

On p-adic mathematical physics

Драгович¹,

Khrennikov

Козырев

et al. 2009

P-Adic Num Ultrametr Anal Appl

278

167

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A brief review of some selected topics in p-adic mathematical physics is presented.1 Numbers: Rational, Real, p-AdicWe present a brief review of some selected topics in p-adic mathematical physics. More details can be found in the references below and the other references are mainly contained therein. We hope that this brief introduction to some aspects of p-adic mathematical physics could be helpful for the readers of the first issue of the journal p-Adic Numbers, Ultrametric Analysis and Applications.The notion of numbers is basic not only in mathematics but also in physics and entire science. Most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. p-Adic numbers (see, e.g. [1]), introduced by Hensel, are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography.The following view how to do science with numbers has been put forward by Volovich in [2,3]. Suppose we have a physical or any other system and we 1 make measurements. To describe results of the measurements, we can always use rationals. To do mathematical analysis, one needs a completion of the field Q of the rational numbers. According to the Ostrowski theorem there are only two kinds of completions of the rationals. They give real R or p-adic Q p number fields, where p is any prime number with corresponding p-adic norm |x| p , which is non-Archimedean. There is an adelic formula p |x| p = 1 valid for rational x which expresses the real norm |x| ∞ in terms of p-adic ones. Any p-adic number x can be represented as a series x = ∞ i=k a i p i , where k is an integer and a i ∈ {0, 1, 2, ..., p − 1} are digits. To build a mathematical model of the system we use real or p-adic numbers or both, depending on the properties of the system [2,3].Superanalysis over real and p-adic numbers has been considered by Vladimirov and Volovich [4,5]. An adelic approach was emphasized by Manin [6].One can argue that at the very small (Planck) scale the geometry of the spacetime should be non-Archimedean [2,3,7]. There should be quantum fluctuations not only of metrics and geometry but even of the number field. Therefore, it was suggested [2] the following number field invariance principle: Fundamental physical laws should be invariant under the change of the number field.One could start from the ring of integers or the Grothendieck schemes. Then rational, real or p-adic numbers should appear through a mechanism of number field symmetry breaking, similar to the Higgs mechanism [8, 9].Recently (for a review, see [10,11,12,13,14]) there have been exciting achievements exploring p-adic, adelic and ultrametric structures in various models of physics: from the spacetime geometry at small scale and strings, via spin glasses and other complex systems, to the universe as a whole. There has been al...

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Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

Cited by 39 publications

References 27 publications

The Cauchy problem for a class of pseudodifferential equations over p-adic field

The Cauchy problem for a class of pseudodifferential equations over p-adic field

Toward an ultrametric theory of turbulence

On p-adic mathematical physics

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