This paper studies nonlinear constitutive equations for gravitoelectromagnetism. Eventually, the problem is solved of finding, for a given particular solution of the gravity-Maxwell equations, the exact form of the corresponding nonlinear constitutive equations.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
-type and -type four dimensional plane wave solutions of Einstein's field eq...inventy
In the present paper, we have studied - type and -type plane wave solutions of Einstein's field equations in general theory of relativity in the case where the zero mass scalar field coupled with gravitational field and zero mass scalar field coupled with gravitational & electromagnetic field and established the existence of these two types of plane wave solutions in . Furthermore we have considered the case of massive scalar field and shown that the non-existence of these two types of plane wave solutions in GR theory.
ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOL...ijoejournal
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an amplifier
due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by optimized
annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account redistribution
of radiation defects (after implantation of ions of dopant) for optimization of the above annealing
have been done by using recently introduced analytical approach. The approach gives a possibility
to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
Solutions of Maxwell Equation for a Lattice System with Meissner EffectQiang LI
We show that Maxwell equation of a lattice system may have Meissner effect solutions when all carriers are surface state electrons. Some limitations on the wave function distributions of electrons in the system are identified.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
-type and -type four dimensional plane wave solutions of Einstein's field eq...inventy
In the present paper, we have studied - type and -type plane wave solutions of Einstein's field equations in general theory of relativity in the case where the zero mass scalar field coupled with gravitational field and zero mass scalar field coupled with gravitational & electromagnetic field and established the existence of these two types of plane wave solutions in . Furthermore we have considered the case of massive scalar field and shown that the non-existence of these two types of plane wave solutions in GR theory.
ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOL...ijoejournal
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an amplifier
due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by optimized
annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account redistribution
of radiation defects (after implantation of ions of dopant) for optimization of the above annealing
have been done by using recently introduced analytical approach. The approach gives a possibility
to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
Solutions of Maxwell Equation for a Lattice System with Meissner EffectQiang LI
We show that Maxwell equation of a lattice system may have Meissner effect solutions when all carriers are surface state electrons. Some limitations on the wave function distributions of electrons in the system are identified.
Weighted Analogue of Inverse Maxwell Distribution with ApplicationsPremier Publishers
In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit.
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS ...ijoejournal
We introduce an approach for increasing density of voltage restore elements. The approach based on
manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special configuration.
Several required sections of the layer should be doped by diffusion or ion implantation. After
that dopants and/or radiation defects should be annealed.
On prognozisys of manufacturing doublebaseijaceeejournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this
heterostructure by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects.
The approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher
compactness of the considered bipolar transistor.
The i(G)-graph is defined as a graph whose vertex set correspond 1 to 1 with the i(G)-sets of G . Two i(G)- sets say
S1 and 2 S are adjacent in i(G) if there exists a vertex S1
v , and a vertex wS2 such that v is adjacent to w and = { } { } 1 2 S S w v or equivalently = { } { } S2 S1 v w . In this paper we obtain i(G)-graph of some special graphs.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Weighted Analogue of Inverse Maxwell Distribution with ApplicationsPremier Publishers
In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit.
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS ...ijoejournal
We introduce an approach for increasing density of voltage restore elements. The approach based on
manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special configuration.
Several required sections of the layer should be doped by diffusion or ion implantation. After
that dopants and/or radiation defects should be annealed.
On prognozisys of manufacturing doublebaseijaceeejournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this
heterostructure by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects.
The approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher
compactness of the considered bipolar transistor.
The i(G)-graph is defined as a graph whose vertex set correspond 1 to 1 with the i(G)-sets of G . Two i(G)- sets say
S1 and 2 S are adjacent in i(G) if there exists a vertex S1
v , and a vertex wS2 such that v is adjacent to w and = { } { } 1 2 S S w v or equivalently = { } { } S2 S1 v w . In this paper we obtain i(G)-graph of some special graphs.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Derivation of Schrodinger and Einstein Energy Equations from Maxwell's Electr...iosrjce
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Transient Numerical Analysis of Induction Heating of Graphite Cruciable at Di...ijeljournal
Mathematical modeling of Induction heating process is done by using 2D axisymmetric geometry.
Induction heating is coupled field problem that includes electromagnetism and heat transfer. Mathematical
modeling of electromagnetism and heat transfer is done by using maxwell equations and classical heat
transfer equation respectively. Temperature dependent material properties are used for this analysis. This
analysis includes coil voltage distribution, crucible electromagnetic power, and coil equivalent impedance
at different frequency. Induction coil geometry effect on supply voltage is also analyzed. This analysis is
useful for designing of induction coil for melting of nonferrous metal such as gold, silver, uranium etc.
Transient numerical analysis of induction heating of graphite cruciable at di...ijeljournal
Mathematical modeling of Induction heating process is done by using 2D axisymmetric geometry. Induction heating is coupled field problem that includes electromagnetism and heat transfer. Mathematical
modeling of electromagnetism and heat transfer is done by using maxwell equations and classical heat
transfer equation respectively. Temperature dependent material properties are used for this analysis. This
analysis includes coil voltage distribution, crucible electromagnetic power, and coil equivalent impedance
at different frequency. Induction coil geometry effect on supply voltage is also analyzed. This analysis is useful for designing of induction coil for melting of nonferrous metal such as gold, silver, uranium etc.
Transient Numerical Analysis of Induction Heating of Graphite Cruciable at Di...ijeljournal
Mathematical modeling of Induction heating process is done by using 2D axisymmetric geometry.
Induction heating is coupled field problem that includes electromagnetism and heat transfer. Mathematical
modeling of electromagnetism and heat transfer is done by using maxwell equations and classical heat
transfer equation respectively. Temperature dependent material properties are used for this analysis. This
analysis includes coil voltage distribution, crucible electromagnetic power, and coil equivalent impedance
at different frequency. Induction coil geometry effect on supply voltage is also analyzed. This analysis is
useful for designing of induction coil for melting of nonferrous metal such as gold, silver, uranium etc.
I am Baddie K. I am a Magnetic Materials Assignment Expert at eduassignmenthelp.com. I hold a Masters's Degree in Electro-Magnetics, from The University of Malaya, Malaysia. I have been helping students with their assignments for the past 12 years. I solve assignments related to Magnetic Materials.
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Analytical Solution Of Schrodinger Equation With Mie-Type Potential Using Fac...ijrap
we have obtained the analytical solution of Schrödinger wave equation with Mie – type potential
using factorization method. We have also obtained energy eigenvalues of our potential and the
corresponding wave function using an ansatz and then compare the result to standard Laguerre’s
differential equation. Under special cases our potential model reduces two well known potentials such as
Coulomb and the Kratzer Feus potentials.
Similar to Nonlinear constitutive equations for gravitoelectromagnetism (20)
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU(2) using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary Σ-matrices introduced here play a role similar to ordinary matrix units, and their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The presentation of n-ary SU(2) in terms of full Σ-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q>4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4q(n-1)+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σ^het-matrices of order (4q(n-1))^4. Some examples of the lowest arities are presented.
https://arxiv.org/abs/2403.19361. *) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures, https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the unitless ternary division algebra of imaginary "half-octonions" we have introduced is ternary alternative.
https://arxiv.org/abs/2312.01366
https://www.amazon.com/s?k=duplij
https://arxiv.org/abs/2312.01366.
We introduce a new class of division algebras, hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the proposed earlier matrix polyadization procedure which increases the algebra dimension. The obtained algebras obey the binary addition and nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative and the corresponding map is n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. Then we obtain another series of n-ary algebras corresponding to the binary division algebras which have more dimension, that is proportional to intermediate arities, and they are not isomorphic to those obtained by the previous constructions. Second, we propose a new iterative process (we call it "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half dimension, we call them "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the introduced unitless ternary division algebra of imaginary "half-octonions" is ternary alternative.
178 pages, 6 Chapters. DOI: 10.1088/978-0-7503-5281-9. This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers. In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way. Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.Part of IOP Series in Coherent Sources, Quantum Fundamentals, and Applications. Key features • Provides new mathematical methods for quantum computing. • Presents material in a widely accessible way. • Contains methods for unconventional computing where there is computational complexity. • Provides methods to increase speed and efficiency. For the light paperback version use MyPrint service here: https://iopscience.iop.org/book/mono/978-0-7503-5281-9, also PDF, ePub and Kindle. For the libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Innovative-Quantum-Computing/?K=9780750352796. Amazon ordering: https://www.amazon.de/gp/product/0750352795
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.
https://www.mdpi.com/books/book/6455
We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.
https://arxiv.org/abs/2212.08606
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Книга «Поэфизика души» представляет собой полное, на момент издания 2022 г., собрание прозаических произведений автора. Как рассказы, так и миниатюры на полстраницы, пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга поэтическими образами, воплощенными в прозе. Также включены юмористические путевые заметки о поездке в Китай.
Книга "Поэфизика души", Степан Дуплий – полное собрание прозы 2022, 230 стр. вышла в Ridero: https://ridero.ru/books/poefizika_dushi и Kindle Edition file на Амазоне: https://amazon.com/dp/B0B9Y4X4VJ . "Бумажную" книгу можно заказать на Озоне https://ozon.ru/product/poefizika-dushi-682515885/?sh=XPu-9Sb42Q и на ЛитРес: https://litres.ru/stepan-dupliy/poefizika-dushi-emocionalnaya-proza-kitayskiy-shtrih-punktir . Google books: https://books.google.com/books?id=9w2DEAAAQBAJ .
Книгу можно заказать из-за рубежа на AliExpress: https://aliexpress.com/item/1005004660613179.html .
Книга «Гравитация страсти» представляет собой полное собрание стихотворений автора на момент издания (август, 2022). Стихотворения пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга необычными поэтическими образами.
Книга "Гравитация страсти", Степан Дуплий - полное собрание стихотворений 2022, 338 стр. вышла в Ridero: https://ridero.ru/books/gravitaciya_strasti
. Книга в мягкой обложке доступна для заказа на Ozon.ru: https://ozon.ru/product/gravitatsiya-strasti-707068219/?oos_search=false&sh=XPu-9TbW9Q
, на Litres.ru: https://www.litres.ru/stepan-dupliy/gravitaciya-strasti-stihotvoreniya , за рубежом на AliExpress: https://aliexpress.com/item/1005004722134442.html , и в электронном виде Kindle file на Amazon.com: https://amazon.com/dp/B0BDFTT33W .
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures. We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.
In book: S. Duplij, "Polyadic Algebraic Structures", 2022, IOP Publishing (Bristol), Section 1.5. See https://iopscience.iop.org/book/978-0-7503-2648-3
https://arxiv.org/abs/2206.14840
The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories.
Suitable for university students of advanced level algebra courses and mathematical physics courses.
Key features
• Provides a general, unified approach
• Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
https://arxiv.org/abs/2201.08479
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be "entangled" such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields, which can lead to a new category of polyadic fields. Many illustrative concrete examples are presented.
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/ fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and ?-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" ?-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and ?-commutativity are studied.
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
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This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
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(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Nonlinear constitutive equations for gravitoelectromagnetism
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International Journal of Geometric Methods in Modern Physics
Vol. 11, No. 1 (2014) 1450004 (10 pages)
c World Scientific Publishing Company
DOI: 10.1142/S0219887814500042
NONLINEAR CONSTITUTIVE EQUATIONS
FOR GRAVITOELECTROMAGNETISM
STEVEN DUPLIJ
Theory Group, Nuclear Physics Laboratory
V. N. Karazin Kharkov National University
Svoboda Sq. 4, Kharkov 61077, Ukraine
sduplij@gmail.com
ELISABETTA DI GREZIA∗ and GIAMPIERO ESPOSITO†
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli
Complesso Universitario di Monte S. Angelo
Via Cintia Edificio 6, Napoli 80126, Italy
∗digrezia@na.infn.it
†gesposit@na.infn.it
ALBERT KOTVYTSKIY
Department of Physics
V. N. Karazin Kharkov National University
Svoboda Sq. 4, Kharkov 61077, Ukraine
kotvytskiy@gmail.com
Received 14 April 2013
Accepted 26 May 2013
Published 16 July 2013
This paper studies nonlinear constitutive equations for gravitoelectromagnetism. Even-
tually, the problem is solved of finding, for a given particular solution of the gravity-
Maxwell equations, the exact form of the corresponding nonlinear constitutive equations.
Keywords: General relativity; gravitoelectromagnetism.
Mathematics Subject Classification 2010: 83C05
1. Introduction
Over the past decade, a description of nonlinear classical electrodynamics and
Yang–Mills theory has been considered in the literature [1–3], with the hope of
being able to extend it to a broader framework, including gauge theories of grav-
ity [4] and quantum gravity [5].
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However, no explicit calculation had been performed, and the formulation
remained too general for the physics community to be able to appreciate its poten-
tialities. For this purpose, as a first step, we here consider the gravitoelectromag-
netism in the weak-field approximation (following, e.g. [6]). Recall the standard
Maxwell equations in SI units [7]
curl E = −
∂B
∂t
, div B = 0,
curl H =
∂D
∂t
+ j, div D = ρ,
(1.1)
where E is the electric field, B is the magnetic field, ρ is charge density, j is electric
current density. In the linear case
B = µ0H, D = ε0E. (1.2)
In the nonlinear case these equations can be presented in the form [8]
D = M(I1, I2)B +
1
c2
N(I1, I2)E,
H = N(I1, I2)B − M(I1, I2)E,
(1.3)
where the invariants are (Fµν being the electromagnetic field tensor, with Hodge
dual ∗
Fµν
)
I1 =
1
2
FµνFµν
= B2
−
1
c2
E2
, I2 = −
c
4
Fµν
∗
Fµν
= B · E. (1.4)
Their gravitational analogues in SI are
curl Eg = −
∂Bg
∂t
, div Bg = 0, (1.5)
curl Bg =
1
c2
∂Eg
∂t
+
1
εgc2
jg, div Eg =
1
εg
ρg, (1.6)
where Eg is the static gravitational field (conventional gravity, also called grav-
itoelectric for the sake of analogy), Bg is the gravitomagnetic field, ρg is mass
density, jg is mass current density, G is the gravitational constant, εg is the gravity
permittivity (analog of ε0). Here
εg = −
1
4πG
, µg = −
4πG
c2
, (1.7)
are the gravitational permittivity and permeability, respectively.
The main idea is to introduce analogues of H and D to write (1.5) and (1.6) in
the Maxwell form for four fields in SI as
curl Eg = −
∂Bg
∂t
, div Bg = 0, (1.8)
curl Hg =
∂Dg
∂t
+ jg, div Dg = ρg. (1.9)
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Nonlinear Constitutive Equations for Gravitoelectromagnetism
In the linear-gravity case
Dg = εgEg, (1.10)
Bg = µgHg, (1.11)
εgµg =
1
c2
. (1.12)
Note now that the linear-gravity case (1.10)–(1.12) corresponds to weak approxi-
mation and some special case of gravitational field configuration. We generalize it
to nonlinear case which can describe other configurations and non-weak fields, as
in (1.3), by
Dg = Mg(Ig1, Ig2)Bg +
1
c2
Ng(Ig1, Ig2)Eg, (1.13)
Hg = Ng(Ig1, Ig2)Bg − Mg(Ig1, Ig2)Eg, (1.14)
where the invariants are
Ig1 = B2
g −
1
c2
E2
g, Ig2 = Bg · Eg. (1.15)
The gravity-Maxwell equations (1.8) and (1.9) together with the nonlinear
gravity-constitutive equations (1.13) and (1.14) can give a nonlinear electrodynam-
ics formulation of gravity (or at least some particular instances of this construction).
2. Linear Gravitoelectromagnetic Waves
The gravity-Maxwell equations for gravitoelectromagnetic waves (far from sources)
are
curl Eg = −
∂Bg
∂t
, div Bg = 0, (2.1)
curl Hg =
∂Dg
∂t
, div Dg = 0, (2.2)
with generic values of permittivity and permeability (1.7). Then
curl Eg = −µg
∂Hg
∂t
, div Hg = 0,
curl Hg = εg
∂Eg
∂t
, div Eg = 0.
(2.3)
We differentiate the first equation with respect to time: curl ∂
∂t Eg = −µg
∂2
Hg
∂t2 ⇒
1
εg
curl(curl Hg) = −µg
∂2
Hg
∂t2 . Since curl(curl Hg) = grad(div Hg) − ∆Hg = −∆Hg,
then
∆Hg = εgµg
∂2
Hg
∂t2
. (2.4)
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S. Duplij et al.
By analogy, from the second equation curl ∂
∂t Hg = εg
∂2
Eg
∂t2 ⇒ −1
µg
curl(curl Eg) =
εg
∂2
Eg
∂t2 . Hence we get the wave equation for Eg,
∆Eg = εgµg
∂2
Eg
∂t2
. (2.5)
3. Nonlinear Gravitoelectromagnetic Waves
The differences begin with the constitutive equations (1.13) and (1.14). For sim-
plicity put first Mg = 0. Then
Dg =
N
c2
Eg, (3.1)
Bg =
1
N
Hg, (3.2)
where N ≡ Ng(Ig1, Ig2). The Maxwell equations become (hereafter the dots denote
time derivatives)
curl Eg = −
1
N
·
Hg −
1
N
∂Hg
∂t
, (3.3)
div
1
N
Hg = Hg grad
1
N
+
1
N
div(Hg) = 0, (3.4)
curl Hg =
˙N
c2
Eg +
N
c2
∂Eg
∂t
, (3.5)
div
N
c2
Eg = Eg grad
N
c2
+
N
c2
div(Eg) = 0. (3.6)
Take derivative of (3.3) with respect to time and get
curl
∂
∂t
Eg = −
1
N
··
Hg − 2
1
N
·
∂Hg
∂t
−
1
N
∂2
Hg
∂t2
. (3.7)
From (3.5), it follows
∂Eg
∂t = c2
N curl Hg −
˙N
N Eg. Then we get
curl
c2
N
curl Hg −
˙N
N
Eg = −
1
N
··
Hg − 2
1
N
·
∂Hg
∂t
−
1
N
∂2
Hg
∂t2
. (3.8)
The left-hand side here is
curl
c2
N
curl Hg −
˙N
N
Eg = grad
c2
N
× curl Hg +
c2
N
grad div Hg −
c2
N
∆Hg
−
˙N
N
curl Eg − grad
˙N
N
× Eg. (3.9)
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Nonlinear Constitutive Equations for Gravitoelectromagnetism
From (3.4), we get div(Hg) = −NHg grad( 1
N ) = 0. Thus, the nonlinear analogue
of the wave equation is
grad
c2
N
× curl Hg +
c2
N
grad div Hg −
c2
N
∆Hg −
˙N
N
curl Eg − grad
˙N
N
× Eg
= −
1
N
··
Hg − 2
1
N
·
∂Hg
∂t
−
1
N
∂2
Hg
∂t2
. (3.10)
Note that if N = const., then we obtain the usual wave equation
∆Hg =
1
c2
∂2
Hg
∂t2
. (3.11)
Take now the constitutive equations in the form
Dg = MBg +
N
c2
Eg, (3.12)
Hg = NBg − MEg, (3.13)
where N, M are constants. In absence of sources, the Maxwell equations become
curl Eg = −
∂Bg
∂t
, div Bg = 0, (3.14)
curl Hg =
∂Dg
∂t
, div Dg = 0. (3.15)
If we express the Maxwell equations through Eg and Bg, the second pair of equa-
tions become
curl Hg =
∂Dg
∂t
⇒ N curl Bg − M curl Eg = M
∂Bg
∂t
+
N
c2
∂Eg
∂t
. (3.16)
Since curl Eg = −
∂Bg
∂t , we get
curl Bg =
1
c2
∂Eg
∂t
. (3.17)
The second equation, div Dg = 0, reduces to M div Bg + N
c2 div Eg = 0. Since
div Bg = 0, we get
div Eg = 0. (3.18)
Thus, using constitutive equations with constant M and N we have Maxwell equa-
tions in terms of Bg and Eg, i.e.
curl Eg = −
∂Bg
∂t
, div Bg = 0, (3.19)
curl Bg =
1
c2
∂Eg
∂t
, div Eg = 0. (3.20)
At this stage, we get the wave equations in the standard way. The time derivative
of the first equation yields curl ∂
∂t Eg = −
∂2
Bg
∂t2 ⇒ c2
curl(curl Bg) = −
∂2
Bg
∂t2 . Since
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curl(curl Bg) = grad(div Bg) − ∆Bg = −∆Bg, then
∆Bg =
1
c2
∂2
Bg
∂t2
. (3.21)
By analogy curl ∂
∂t Bg = 1
c2
∂2
Eg
∂t2 ⇒ −curl(curl Eg) = 1
c2
∂2
Eg
∂t2 , and we get the wave
equation for Eg,
∆Eg =
1
c2
∂2
Eg
∂t2
. (3.22)
Thus, the gravitoelectromagnetic waves Eg and Bg have speed c and do not depend
on the constants M and N.
4. Waves and Constitutive Equations for Linear
Constitutive Functions
Let us consider the constitutive equations (1.13) and (1.14) as linear functions of
the invariants, i.e.
M = Mg(Ig1, Ig2) = amIg1 + bmIg2, (4.1)
N = Ng(Ig1, Ig2) = c2
εg + anIg1 + bnIg2, (4.2)
am, bm, an, bn being some constants. From all the Maxwell equations in material
media, and in the absence of sources one finds curl Hg =
∂Dg
∂t , curl(NBg −MEg) =
∂
∂t (MBg + N
c2 Eg), and N curl Bg − M curl Eg = M
∂Bg
∂t + N
c2
∂Eg
∂t . Since curl Eg =
−
∂Bg
∂t , from the last equation one gets
curl Bg =
1
c2
∂Eg
∂t
. (4.3)
The second equation, div Dg = 0, reduces to div(MBg + N
c2 Eg) = 0, or M div Bg +
N
c2 div Eg = 0. Since div Bg = 0, one gets
div Eg = 0. (4.4)
5. Inverse Problem of Nonlinear Gravitoelectromagnetism
In electrodynamics the direct solution of the Maxwell equations together with the
nonlinear constitutive equations is a non-trivial and complicated task even for sim-
ple systems [1, 2]. In previous sections we presented some very special cases of the
nonlinear functions N and M. Here we formulate the following inverse problem: if
we have some particular solution of the gravity-Maxwell equations (1.8) and (1.9),
can we then find the exact form of the corresponding nonlinear gravity-constitutive
equations (1.13) and (1.14)?
It is natural to consider the case of plane gravitational waves, when the fields
have only one space coordinate. We will show that even in this case one can have
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a non-trivial nonlinearity. Let us choose Eg and Bg mutually orthogonal and per-
pendicular to the direction of motion
Eg =
E
0
0
, Bg =
0
0
B
, (5.1)
where E ≡ E(t, y), B ≡ B(t, y). Now the invariants (1.15) become
Ig1 = B2
−
1
c2
E2
≡ I, (5.2)
Ig2 = 0. (5.3)
The use of the nonlinear gravity-constitutive equations (1.13) and (1.14) gives for
the other fields
Dg =
1
c2
NE
0
MB
, Hg =
−ME
0
NB
, (5.4)
where N ≡ N(I), M ≡ M(I) are the sought for gravity-constitutive functions. They
depend on I only, because of Lorentz invariance (see [1, 2]). Inserting the fields (5.1)
and (5.4) into the gravity-Maxwell equations (1.8) and (1.9) without sources gives
us three equations (hereafter, a prime with the corresponding subscript denotes
the first partial derivative with respect to the variable in the subscript, while dot
denotes time derivative)
Ey = ˙B, (5.5)
(NB)y =
1
c2
(NE)·
, (5.6)
(ME)y = (MB)·
. (5.7)
Now we take into account that the gravity-constitutive functions N, M depend
only on the invariant I and present (5.6) and (5.7) as the differential equations for
them
NI BI y −
1
c2
E ˙I + N By −
1
c2
˙E = 0, (5.8)
MI EI y − B ˙I = 0, (5.9)
where we have exploited the identities
Ny = NIIy, My = MIIy,
˙N = NI
˙I, ˙M = MI
˙I.
(5.10)
The Eq. (5.9) can be immediately solved by
M(I) =
M0 = const., if EI y = B ˙I,
arbitrary, if EI y = B ˙I.
(5.11)
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The Eq. (5.8) can be solved if
λ ≡
(By −
˙E
c2 )
(BI y − E ˙I
c2 )
, (5.12)
depends only on I, which is a very special case. One then has the differential
equation
NI + λ(I)N = 0 (5.13)
and its solution is
N(I) = N0e−
R
λ(I)dI
. (5.14)
Otherwise, by using the expressions for Iy and ˙I from (5.2), i.e.
Iy = 2BBy −
2EEy
c2
, ˙I = 2B ˙B −
2E ˙E
c2
, (5.15)
we obtain
2NI B2
By +
1
c4
E2 ˙E −
2
c2
EBEy + N By −
1
c2
˙E = 0, (5.16)
where the sum of terms in brackets is not a function of I, in general.
Usually, in the wave solutions the dependence of fields on frequency ω and wave
number k is the same, and therefore we can consider the concrete choice
E(t, y) = f(εωt + ky) ≡ f(X(t, y)), B(t, y) = g(εωt + ky) ≡ g(X(t, y)), (5.17)
where ε ≡ ±1, with f and g arbitrary smooth nonvanishing functions. Bearing in
mind that
Ey = fXXy = kfX, By = gXXy = kgX,
˙E = fX
˙X = εωfX, ˙B = gX
˙X = εωgX,
our Eq. (5.5) yields
kf X = εωgX. (5.18)
Therefore,
g(X) =
k
εω
f(X) + α, (5.19)
where α is a constant, so that both E and B can be expressed through one function
only, i.e. f, and the invariant I reads eventually as
I =
1
ω2
k2
−
ω2
c2
f2
+ 2
k
εω
αf + α2
. (5.20)
The equations for the gravity-constitutive functions take therefore the form
NI 2I k2
−
ω2
c2
+ 2
ω2
c2
α2
+ N k2
−
ω2
c2
= 0, (5.21)
MIfX
2f
εω
k2
−
ω2
c2
+ 2kα α = 0, (5.22)
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having exploited the identities
gI y −
f ˙I
c2
= gf y −
f ˙f
c2
2f
ω2
k2
−
ω2
c2
+ 2
k
εω
α , (5.23)
gf y −
f ˙f
c2
=
fX
εω
f k2
−
ω2
c2
+ kεωα , (5.24)
and, after some cancellations,
f k2
−
ω2
c2
+ kεωα
2f
ω2
k2
−
ω2
c2
+ 2
k
εω
α
= 2 k2
−
ω2
c2
I + 2
ω2
c2
α2
, (5.25)
while
fI y − g ˙I = (ffy − g ˙f)
2f
ω2
k2
−
ω2
c2
+ 2
k
εω
α , (5.26)
ffy − g ˙f = fX(kf − εωg) = −εωfXα. (5.27)
The results of our analysis now depend on whether or not α vanishes. Indeed,
if α = 0, M is arbitrary and hence we obtain the equation
k2
−
ω2
c2
(2IN I + N) = 0, (5.28)
which implies that either the dispersion relation
k2
−
ω2
c2
= 0 (5.29)
holds, with N kept arbitrary, or such a dispersion relation is not fulfilled, while N
is found from the differential equation
2IN I + N = 0, (5.30)
which is solved by
N(I) =
N0
√
I
. (5.31)
By contrast, if α does not vanish, M equals a constant M0, while N solves the
more complicated Eq. (5.21). At this stage, to be consistent with the dependence
of N on I only, we have to require again that the dispersion relation (5.29) should
hold, jointly with NI = 0, which implies the constancy of N : N = N0.
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10. 3rd Reading
July 12, 2013 16:15 WSPC/S0219-8878 IJGMMP-J043 1450004
S. Duplij et al.
6. Concluding Remarks
We have brought “down to earth” the general program of considering nonlinear con-
stitutive equations for gravitoelectromagnetism, by solving the problem of finding,
for a given solution of the gravity-Maxwell equations, the exact form of nonlinear
constitutive equations. We look forward to being able to construct other relevant
examples, as well as being able to re-express our models in the language of differ-
ential forms, which turned out to be very powerful for general relativity [9–11].
Acknowledgments
S. Duplij thanks M. Bianchi, J. Gates, G. Goldin, A. Yu. Kirochkin, M. Shifman, V.
Shtelen, D. Sorokin, A. Schwarz, M. Tonin, A. Vainshtein, A. Vilenkin for fruitful
discussions. E. Di Grezia and G. Esposito are grateful to the Dipartimento di Fisica
of Federico II University, Naples, for hospitality and support.
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