scholarly journals METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION

2020 ◽  
Vol 6 (334) ◽  
pp. 5-12
Author(s):  
А.Т. Ibrayev ◽  
◽  
Keyword(s):  
Field Theory ◽  
Commutative Algebra ◽  
Digital Processing ◽  
Real Numbers ◽  
Hypercomplex Numbers ◽  
Octonion Algebra ◽  
Technical Problems ◽  
Multidimensional Signals ◽  
Quaternion Space ◽  
Vector Part

In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

Prospective Digital Processing Methods of Multidimensional Signals with the Use of Irregular Meshes

Vestnik MEI ◽  
2019 ◽  
Vol 3 (3) ◽  
pp. 98-107
Author(s):  
Sergey V. Vishnyakov ◽  
◽  
Elizaveta A. Sokolova ◽  
Vitaliy V. Pekhterev ◽  
◽  
...  
Keyword(s):  
Digital Processing ◽  
Processing Methods ◽  
Multidimensional Signals ◽  
Irregular Meshes

scholarly journals On two-channel circuits of subband filtration of multidimensional signals

2021 ◽  
Vol 15 ◽  
pp. 9
Author(s):  
V.F. Babenko ◽  
A.A. Ligun ◽  
O.M. Shumeiko
Keyword(s):  
High Frequency ◽  
Low Frequency ◽  
Digital Processing ◽  
Multidimensional Signals

We propose the method to construct high-frequency and low-frequency analysing and restoring filters for digital processing of multidimensional signals.

THE CLASSICAL HARMONIC OSCILLATOR ON GALOIS AND P-ADIC FIELDS

1989 ◽  
Vol 04 (09) ◽  
pp. 2211-2233 ◽  
Author(s):  
YANNICK MEURICE
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Difference Equation ◽  
Harmonic Oscillator ◽  
Continuum Limit ◽  
The Other ◽  
Quantum Field ◽  
Real Numbers ◽  
Technical Tool ◽  
Classical Motion

Starting from a difference equation corresponding to the harmonic oscillator, we discuss various properties of the classical motion (cycles, conserved quantity, boundedness, continuum limit) when the dynamical variables take their values on Galois or p-adic fields. We show that these properties can be applied as a technical tool to calculate the motion on the real numbers. On the other hand, we also give an example where the motions over Galois and p-adic fields have a direct physical interpretation. Some perspectives for quantum field theory and strings are briefly discussed.

scholarly journals A counterexample to the containment I(3) ⊂ I2 over the reals

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Adam Czapliński ◽  
Agata Główka ◽  
Grzegorz Malara ◽  
Magdalena Lampa-Baczyńska ◽  
Patrycja Łuszcz-Swidecka ◽  
...  
Keyword(s):  
Commutative Algebra ◽  
Homogeneous Ideal ◽  
Real Numbers ◽  
The Real ◽  
Symbolic Powers

AbstractThe purpose of this note is to give counterexamples defined over the real numbers to a question relevant in commutative algebra, concerning a containment relation between algebraic and symbolic powers of a homogeneous ideal.

scholarly journals Image Processing for Technological Diagnostics of Metallurgical Facilities

2012 ◽  
Vol 12 (4) ◽  
pp. 66-76 ◽  
Author(s):  
Lyubka Doukovska ◽  
Venko Petkov ◽  
Emil Mihailov ◽  
Svetla Vassileva
Keyword(s):  
Image Processing ◽  
Wavelet Transforms ◽  
Digital Processing ◽  
Processing Technique ◽  
Image Processing Technique ◽  
One Dimensional ◽  
Image Processing Techniques ◽  
Digital Image Processing Technique ◽  
Multidimensional Signals ◽  
Processing Techniques

Abstract The paper presents an overview of the image-processing techniques. The set of basic theoretical instruments includes methods of mathematical analysis, linear algebra, probability theory and mathematical statistics, theory of digital processing of one-dimensional and multidimensional signals, wavelet-transforms and theory of information. This paper describes a methodology that aims to detect and diagnose faults, using thermographs approaches for the digital image processing technique.

scholarly journals Particle Creation and Annihilation: Two Bohmian Approaches

2018 ◽  
Vol 5 (1) ◽  
pp. 77-85 ◽  
Author(s):  
Andrea Oldofredi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Physics ◽  
Particle Creation ◽  
Bohmian Mechanics ◽  
Quantum Field ◽  
Technical Problems ◽  
Dirac Sea ◽  
Theory Of Fields ◽  
Particle Creation And Annihilation

This paper reviews and discusses two extensions of Bohmian Mechanics to the phenomena of particle creation and annihilation typically observed in Quantum Field Theory (QFT): the so-called Bell-type Quantum Field Theory and the Dirac Sea representation. These theories have a secure metaphysical basis as they postulate a particle ontology while satisfying the requirements imposed by the Primitive Ontology approach to quantum physics. Furthermore, their methodological perspective intentionally provides a set of rules to immunize physical theories to the conceptual and technical problems plaguing the standard formulation of Quantum Mechanics and QFT. A metaphysical analysis of both theories will be given, emphasizing the relevant features of each proposal. Finally, it will be acknowledged that, despite the metaphysical virtues and niceties of these frameworks, ultimately they do not provide definitive answers to other cogent foundational issues in QFT. Thus, these theories (as well as the other Bohmian extensions to QFT) should be considered as partial solutions to the problems raised by the quantum theory of fields. This situation can be considered incentive for further research.

scholarly journals Isoparametric hypersurfaces with four principal curvatures, II

2011 ◽  
Vol 204 ◽  
pp. 1-18 ◽  
Author(s):  
Quo-Shin Chi
Keyword(s):  
Commutative Algebra ◽  
Quaternion Algebra ◽  
Type A ◽  
Isoparametric Hypersurface ◽  
Principal Curvatures ◽  
Octonion Algebra ◽  
Isoparametric Hypersurfaces ◽  
New Approach ◽  
Condition B

AbstractIn this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) inS15is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively asof OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraintm2≥2m1-1 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5),(3,4),(7,8), and (6, 9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6, 9), and (7,8) remain open now.

scholarly journals Cosmology: Preprogrammed Evolution (The problem of Providence)

2021 ◽  
Vol 4 (2) ◽  
Keyword(s):  
Field Theory ◽  
Theoretical Physics ◽  
Real Numbers ◽  
Universal System ◽  
Physical Universe ◽  
Pure Mathematics ◽  
Infinite Set ◽  
Mathematical Machine ◽  
The Continuum

This is continued from the article Superunification: Pure Mathematics and Theoretical Physics published in this journal and intended to discuss the general logical and philosophical consequences of the universal mathematical machine described by the superunified field theory. At first was mathematical continuum, that is, uncountably infinite set of real numbers. The continuum is self-exited and selforganized into the universal system of mathematical harmony observed by the intelligent beings in the Cosmos as the physical Universe.

scholarly journals Isoparametric hypersurfaces with four principal curvatures, II

2011 ◽  
Vol 204 ◽  
pp. 1-18 ◽  
Author(s):  
Quo-Shin Chi
Keyword(s):  
Commutative Algebra ◽  
Quaternion Algebra ◽  
Type A ◽  
Isoparametric Hypersurface ◽  
Principal Curvatures ◽  
Octonion Algebra ◽  
Isoparametric Hypersurfaces ◽  
New Approach ◽  
Condition B

AbstractIn this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint m2≥ 2m1 -1 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5),(3,4),(7,8), and (6, 9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6, 9), and (7,8) remain open now.

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