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Том 10, № 3 (2018)
- Год: 2018
- Статей: 6
- URL: https://journals.rcsi.science/2070-0466/issue/view/12525
Review Articles
Holography and Local Fields
Аннотация
This paper, which is a summary (in which considerable creative license has been taken) of the author’s talk at the sixth international conference on p-adic mathematical physics and its applications (CINVESTAV, Mexico City, October 2017), reviews some recent work connecting field theories defined on the p-adic numbers and ideas from the AdS/CFT correspondence. Some results are included, along with general discussion of the utility and interest of p-adic analogues of Lagrangian field theories, at least from the author’s perspective. A few challenges, shortcomings, and ideas for future work are also discussed.
151-165
Research Articles
On the Classification of p-Adic UHF Banach Algebras
Аннотация
For any prime number p let Ωp be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (Mn) is an increasing sequence of p-adic Banach subalgebras of M such that each Mn is generated over Ωp by an algebraic system of matrix units {eij( n) | 1 ≤ i, j ≤ pn }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.
166-178
A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field
Аннотация
We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [a, b] ⊂ ℝ, f and its analytic continuation f̅∞ satisfy the same properties that can be expressed in the language of real closed ordered fields. If f is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.
179-191
Measure-Preservation and the Existence of a Root of p-Adic 1-Lipschitz Functions in Mahler’s Expansion
Аннотация
In this paper, we provide necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable mod p on ℤp to be measure-preserving, in Mahler’s expansion, and show that these conditions can be modified to guarantee the existence of a root of p-adic Lipschitz functions.
192-208
209-221
Active Beam Splitting Attack Applied to Differential Phase Shift Quantum Key Distribution Protocol
Аннотация
The differential phase shift quantum key distribution protocol is of high interest due to its relatively simple practical implementation. This protocol uses trains of coherent pulses and allows the legitimate users to resist individual attacks. In this paper, a new attack on this protocol is proposed which is based on the idea of information extraction from the part of each coherent state and then making decision about blocking the rest part depending on the amount of extracted information.
222-232