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21 pages, 359 KiB

Open AccessArticle

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

by Viktor Abramov

Axioms 2024, 13(10), 687; https://doi.org/10.3390/axioms13100687 - 3 Oct 2024

Viewed by 910

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear [...] Read more.

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group

G A (1, 5) \subset S_{5}

. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed. Full article

(This article belongs to the Special Issue Recent Advances in Representation Theory with Applications)

17 pages, 342 KiB

Open AccessArticle

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

by Viktor Abramov and Olga Liivapuu

Universe 2024, 10(1), 2; https://doi.org/10.3390/universe10010002 - 21 Dec 2023

Cited by 1 | Viewed by 1521

Abstract

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a

Z_{3}

-skew-symmetric covariant

SO (3)

-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry [...] Read more.

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a

Z_{3}

-skew-symmetric covariant

SO (3)

-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group

SO (3) ↪ SU (5)

. We find two

SO (3)

-invariants of

Z_{3}

-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by

K (z, z)

. We study the invariant properties of

K (z, z)

and find its stabilizer. Making use of these invariant properties, we define an

SO (3)

-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an

SO (3)

-irreducible geometric structure and find its curvature and torsion. Full article

(This article belongs to the Special Issue The Languages of Physics—A Themed Issue in Honor of Professor Richard Kerner on the Occasion of His 80th Birthday)

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11 pages, 2434 KiB

Open AccessArticle

Random Laser Based on Ytterbium-Doped Fiber with a Bragg Grating Array as the Source of Continuous-Wave 976 nm Wavelength Radiation

by Andrey Rybaltovsky, Sergei Popov, Dmitry Ryakhovskiy, Alexey Abramov, Andrey Umnikov, Oleg Medvedkov, Viktor Voloshin, Alexander Kolosovskii, Igor Vorob’ev, Yuriy Chamorovskiy and Denis Lipatov

Photonics 2022, 9(11), 840; https://doi.org/10.3390/photonics9110840 - 8 Nov 2022

Cited by 7 | Viewed by 2727

Abstract

A random narrow-linewidth lasing at a wavelength of 976 nm was obtained in an ytterbium-doped germanophosphosilicate fiber with an array of weakly reflecting fiber Bragg gratings (FBGs). A random laser cavity was formed by implementing the standard phase mask method of FBG inscription [...] Read more.

A random narrow-linewidth lasing at a wavelength of 976 nm was obtained in an ytterbium-doped germanophosphosilicate fiber with an array of weakly reflecting fiber Bragg gratings (FBGs). A random laser cavity was formed by implementing the standard phase mask method of FBG inscription directly during the fiber drawing process. The UV radiation pulses of a KrF excimer laser (248 nm wavelength) synchronized with the fiber drawing speed were used to fabricate the in-fiber array of hundreds of similar FBGs. The developed laser’s slope efficiency in the backward-pumping scheme was measured as high as 33%. The stable continuous-wave operation mode of the laser was detected. The magnitude of the laser power fluctuations depends linearly on the cavity length. The random laser cavity modified with a single highlyreflected (90%) FBG demonstrates significantly better power stability and higher slope efficiency than the same one without an FBG. Full article

(This article belongs to the Special Issue Optical Fiber Lasers and Amplifiers. Practical Applications of Fiber-Based Laser Sources)

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17 pages, 1389 KiB

Open AccessReview

Interaction of Mitochondrial Calcium and ROS in Neurodegeneration

by Artyom Y. Baev, Andrey Y. Vinokurov, Irina N. Novikova, Viktor V. Dremin, Elena V. Potapova and Andrey Y. Abramov

Cells 2022, 11(4), 706; https://doi.org/10.3390/cells11040706 - 17 Feb 2022

Cited by 105 | Viewed by 9122

Abstract

Neurodegenerative disorders are currently incurable devastating diseases which are characterized by the slow and progressive loss of neurons in specific brain regions. Progress in the investigation of the mechanisms of these disorders helped to identify a number of genes associated with familial forms [...] Read more.

Neurodegenerative disorders are currently incurable devastating diseases which are characterized by the slow and progressive loss of neurons in specific brain regions. Progress in the investigation of the mechanisms of these disorders helped to identify a number of genes associated with familial forms of these diseases and a number of toxins and risk factors which trigger sporadic and toxic forms of these diseases. Recently, some similarities in the mechanisms of neurodegenerative diseases were identified, including the involvement of mitochondria, oxidative stress, and the abnormality of Ca²⁺ signaling in neurons and astrocytes. Thus, mitochondria produce reactive oxygen species during metabolism which play a further role in redox signaling, but this may also act as an additional trigger for abnormal mitochondrial calcium handling, resulting in mitochondrial calcium overload. Combinations of these factors can be the trigger of neuronal cell death in some pathologies. Here, we review the latest literature on the crosstalk of reactive oxygen species and Ca²⁺ in brain mitochondria in physiology and beyond, considering how changes in mitochondrial metabolism or redox signaling can convert this interaction into a pathological event. Full article

(This article belongs to the Section Mitochondria)

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9 pages, 1188 KiB

Open AccessArticle

Photosensitive Yb-Doped Germanophosphosilicate Artificial Rayleigh Fibers as a Base of Random Lasers

by Andrey Rybaltovsky, Sergei Popov, Denis Lipatov, Andrey Umnikov, Alexey Abramov, Oleg Morozov, Dmitry Ryakhovskiy, Viktor Voloshin, Alexander Kolosovskii, Igor Vorob’ev, Oleg Butov and Yuriy Chamorovskiy

Fibers 2021, 9(9), 53; https://doi.org/10.3390/fib9090053 - 1 Sep 2021

Cited by 5 | Viewed by 2580

Abstract

Asingle-mode Yb-doped germanophosphosilicate fiber with ultra-low optical losses (less than 2 dB/km) was fabricated by means of the MCVD method utilizing an all-gas-phase deposition technique developed “in house”. The absorption and luminescent spectral properties of the fiber were thoroughly studied. The photosensitivity of [...] Read more.

Asingle-mode Yb-doped germanophosphosilicate fiber with ultra-low optical losses (less than 2 dB/km) was fabricated by means of the MCVD method utilizing an all-gas-phase deposition technique developed “in house”. The absorption and luminescent spectral properties of the fiber were thoroughly studied. The photosensitivity of the pristine (non-hydrogenated) fiber to 248 nm-laser radiation was confirmed by means of fiber Bragg grating (FBG) inscription directly during the drawing process. The random single-frequency lasing at the 1060-nm-wavelength obtained in the 21-m-long fiber with an array of weak FBG was reported. The developed laser slope efficiency in the backward-pumping scheme was measured as high as 32%. Full article

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14 pages, 1079 KiB

Open AccessEditor’s ChoiceReview

Interaction of Oxidative Stress and Misfolded Proteins in the Mechanism of Neurodegeneration

by Andrey Y. Abramov, Elena V. Potapova, Viktor V. Dremin and Andrey V. Dunaev

Life 2020, 10(7), 101; https://doi.org/10.3390/life10070101 - 30 Jun 2020

Cited by 92 | Viewed by 7415

Abstract

Aggregation of the misfolded proteins β-amyloid, tau, huntingtin, and α-synuclein is one of the most important steps in the pathology underlying a wide spectrum of neurodegenerative disorders, including the two most common ones—Alzheimer’s and Parkinson’s disease. Activity and toxicity of these proteins depends [...] Read more.

Aggregation of the misfolded proteins β-amyloid, tau, huntingtin, and α-synuclein is one of the most important steps in the pathology underlying a wide spectrum of neurodegenerative disorders, including the two most common ones—Alzheimer’s and Parkinson’s disease. Activity and toxicity of these proteins depends on the stage and form of aggregates. Excessive production of free radicals, including reactive oxygen species which lead to oxidative stress, is proven to be involved in the mechanism of pathology in most of neurodegenerative disorders. Both reactive oxygen species and misfolded proteins play a physiological role in the brain, and only deregulation in redox state and aggregation of the proteins leads to pathology. Here, we review the role of misfolded proteins in the activation of ROS production from various sources in neurons and glia. We discuss if free radicals can influence structural changes of the key toxic intermediates and describe the putative mechanisms by which oxidative stress and oligomers may cause neuronal death. Full article

(This article belongs to the Collection Function, Regulation, and Dysfunction of Intrinsically Disordered Proteins)

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8 pages, 238 KiB

Open AccessArticle

3-Lie Superalgebras Induced by Lie Superalgebras

by Viktor Abramov

Axioms 2019, 8(1), 21; https://doi.org/10.3390/axioms8010021 - 6 Feb 2019

Cited by 6 | Viewed by 3083

Abstract

We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of

(m, n)

-block matrices taking a supertrace of a matrix as the [...] Read more.

We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of

(m, n)

-block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras. Full article

10 pages, 290 KiB

Open AccessArticle

(q, σ, τ)-Differential Graded Algebras

by Viktor Abramov, Olga Liivapuu and Abdenacer Makhlouf

Universe 2018, 4(12), 138; https://doi.org/10.3390/universe4120138 - 1 Dec 2018

Cited by 2 | Viewed by 2858

Abstract

We propose the notion of

(q, σ, τ)

-differential graded algebra, which generalizes the notions of

(σ, τ)

-differential graded algebra and q-differential graded algebra. We construct two examples of [...] Read more.

We propose the notion of

(q, σ, τ)

-differential graded algebra, which generalizes the notions of

(σ, τ)

-differential graded algebra and q-differential graded algebra. We construct two examples of

(q, σ, τ)

-differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a

(σ, τ)

-twisted graded q-commutator. In order to construct the second example, we introduce the notion of

(σ, τ)

-pre-cosimplicial algebra. Full article

(This article belongs to the Special Issue Estate Quantistica Conference - Recent Developments in Gravity, Cosmology, and Mathematical Physics)

11 pages, 263 KiB

Open AccessArticle

Generalization of Nambu–Hamilton Equation and Extension of Nambu–Poisson Bracket to Superspace

by Viktor Abramov

Universe 2018, 4(10), 106; https://doi.org/10.3390/universe4100106 - 15 Oct 2018

Cited by 2 | Viewed by 2786

Abstract

We propose a generalization of the Nambu–Hamilton equation in superspace

R^{3 | 2}

with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace

R^{3 | 2}

by means of the right-hand sides of the [...] Read more.

We propose a generalization of the Nambu–Hamilton equation in superspace

R^{3 | 2}

with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace

R^{3 | 2}

by means of the right-hand sides of the proposed generalization of the Nambu–Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of the Nambu–Hamilton equation in superspace leads to a family of ternary brackets of even degree functions defined with the help of a Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space

R^{3}

. We study the structure of the ternary bracket in a more general case of a superspace

R^{n | 2}

with n real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is the usual Nambu–Poisson bracket, extended in a natural way to even degree functions in a superspace

R^{n | 2}

, and the second is a new ternary bracket, which we call the

Ψ

-bracket, where

Ψ

can be identified with an invertible second order functional matrix. We prove that the ternary

Ψ

-bracket as well as the whole ternary bracket (the sum of the

Ψ

-bracket with the usual Nambu–Poisson bracket) is totally skew-symmetric, and satisfies the Leibniz rule and the Filippov–Jacobi identity ( Fundamental Identity). Full article

(This article belongs to the Special Issue Estate Quantistica Conference - Recent Developments in Gravity, Cosmology, and Mathematical Physics)

23 pages, 31349 KiB

Open AccessArticle

Optimization of Airborne Antenna Geometry for Ocean Surface Scatterometric Measurements

by Alexey Nekrasov, Alena Khachaturian, Evgeny Abramov, Dmitry Popov, Oleg Markelov, Viktor Obukhovets, Vladimir Veremyev and Mikhail Bogachev

Remote Sens. 2018, 10(10), 1501; https://doi.org/10.3390/rs10101501 - 20 Sep 2018

Cited by 12 | Viewed by 3697

Abstract

We consider different antenna configurations, ranging from simple X-configuration to multi-beam star geometries, for airborne scatterometric measurements of the wind vector near the ocean surface. For all geometries, track-stabilized antenna configurations, as well as horizontal transmitter and receiver polarizations, are considered. The wind [...] Read more.

We consider different antenna configurations, ranging from simple X-configuration to multi-beam star geometries, for airborne scatterometric measurements of the wind vector near the ocean surface. For all geometries, track-stabilized antenna configurations, as well as horizontal transmitter and receiver polarizations, are considered. The wind vector retrieval algorithm is generalized here for an arbitrary star geometry antenna configuration and tested using the Ku-Band geophysical model function. Using Monte Carlo simulations for the fixed total measurement time, we show explicitly that the relative wind speed estimation accuracy barely depends on the chosen antenna geometry, while the maximum wind direction retrieval error reduces moderately with increasing angular resolution, although at the cost of increased retrieval algorithm computational complexity, thus, limiting online analysis options with onboard equipment. Remarkably, the simplest X-configuration, while the simplest in terms of hardware implementation and computational time, appears an outlier, yielding considerably higher maximum retrieval errors when compared to all other configurations. We believe that our results are useful for the optimization of both hardware and software design for modern airborne scatterometric measurement systems based on tunable antenna arrays especially, those requiring online data processing. Full article

(This article belongs to the Special Issue Ocean Radar)

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