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15 pages, 393 KiB

Open AccessArticle

A p-Adic Model of Quantum States and the p-Adic Qubit

by Paolo Aniello, Stefano Mancini and Vincenzo Parisi

Entropy 2023, 25(1), 86; https://doi.org/10.3390/e25010086 - 31 Dec 2022

Cited by 5 | Viewed by 3210

We propose a model of a quantum

N

We propose a model of a quantum

N

-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an

N

-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where

N = 2

, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed. Full article

(This article belongs to the Special Issue Selected Feature Papers from Italian Quantum Information Science Conference 2022)

17 pages, 2265 KiB

Open AccessArticle

A New Pooling Approach Based on Zeckendorf’s Theorem for Texture Transfer Information

by Vincent Vigneron, Hichem Maaref and Tahir Q. Syed

Entropy 2021, 23(3), 279; https://doi.org/10.3390/e23030279 - 25 Feb 2021

Cited by 7 | Viewed by 2614

Abstract

The pooling layer is at the heart of every convolutional neural network (CNN) contributing to the invariance of data variation. This paper proposes a pooling method based on Zeckendorf’s number series. The maximum pooling layers are replaced with Z pooling layer, which capture texels from input images, convolution layers, etc. It is shown that Z pooling properties are better adapted to segmentation tasks than other pooling functions. The method was evaluated on a traditional image segmentation task and on a dense labeling task carried out with a series of deep learning architectures in which the usual maximum pooling layers were altered to use the proposed pooling mechanism. Not only does it arbitrarily increase the receptive field in a parameterless fashion but it can better tolerate rotations since the pooling layers are independent of the geometric arrangement or sizes of the image regions. Different combinations of pooling operations produce images capable of emphasizing low/high frequencies, extract ultrametric contours, etc. Full article

(This article belongs to the Special Issue Information Transfer in Multilayer/Deep Architectures)

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20 pages, 1395 KiB

Open AccessArticle

Solvability of the p-Adic Analogue of Navier–Stokes Equation via the Wavelet Theory

by Ehsan Pourhadi, Andrei Khrennikov, Reza Saadati, Klaudia Oleschko and María de Jesús Correa Lopez

Entropy 2019, 21(11), 1129; https://doi.org/10.3390/e21111129 - 17 Nov 2019

Cited by 17 | Viewed by 3344

Abstract

P-adic numbers serve as the simplest ultrametric model for the tree-like structures arising in various physical and biological phenomena. Recently p-adic dynamical equations started to be applied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oil emulsion) in capillary networks in porous random media. In particular, a p-adic analog of the Navier–Stokes equation was derived starting with a system of differential equations respecting the hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium. This equation describes propagation of fluid’s flow through Geo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillary networks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomian decomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equation in term of series in general form. This solution may help researchers to come closer and find more facts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from the analogous aspects of the real world phenomena. Full article

(This article belongs to the Section Multidisciplinary Applications)

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28 pages, 8337 KiB

Open AccessArticle

Modeling Fluid’s Dynamics with Master Equations in Ultrametric Spaces Representing the Treelike Structure of Capillary Networks

by Andrei Khrennikov, Klaudia Oleschko and María De Jesús Correa López

Entropy 2016, 18(7), 249; https://doi.org/10.3390/e18070249 - 7 Jul 2016

Cited by 43 | Viewed by 7136

Abstract

We present a new conceptual approach for modeling of fluid flows in random porous media based on explicit exploration of the treelike geometry of complex capillary networks. Such patterns can be represented mathematically as ultrametric spaces and the dynamics of fluids by ultrametric diffusion. The images of p-adic fields, extracted from the real multiscale rock samples and from some reference images, are depicted. In this model the porous background is treated as the environment contributing to the coefficients of evolutionary equations. For the simplest trees, these equations are essentially less complicated than those with fractional differential operators which are commonly applied in geological studies looking for some fractional analogs to conventional Euclidean space but with anomalous scaling and diffusion properties. It is possible to solve the former equation analytically and, in particular, to find stationary solutions. The main aim of this paper is to attract the attention of researchers working on modeling of geological processes to the novel utrametric approach and to show some examples from the petroleum reservoir static and dynamic characterization, able to integrate the p-adic approach with multifractals, thermodynamics and scaling. We also present a non-mathematician friendly review of trees and ultrametric spaces and pseudo-differential operators on such spaces. Full article

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