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We study Einstein's gravity with negative cosmological constant coupled to nonlinear electrodynamics proposed earlier. The metric and mass functions and corrections to the Reissner--Nordstr\"{o}m solution are obtained. Black hole solutions can have one or two horizons. Thermodynamics and phase transitions of magnetically charged black holes in Anti-de Sitter spacetime are investigated. The first law of black hole thermodynamics is formulated and the generalized Smarr relation is proofed. By calculating the Gibbs free energy and heat capacity we study the black hole stability. Zero-order (reentrant), first-order, and second-order phase transitions are analysed. The Joule--Thomson expansion is considered showing the cooling and heating phase transitions. It was shown that the principles of causality and unitarity are satisfied in the model under consideration., Comment: 32 pages, 12 figures. Corrected typo in published paper (Universe, Vol.9(10), 456 (2023))

Einstein's gravity in AdS space coupled to nonlinear electrodynamics (NED) with two parameters is studied. We investigate magnetically charged black holes. The metric and mass functions and their asymptotic are obtained showing that black holes may have one or two horizons. Thermodynamics in extended phase space was studied and it was proven that the first law of black hole thermodynamics and the generalized Smarr relation hold. The magnetic potential and the vacuum polarization conjugated to coupling (NED parameter), are computed and depicted. We calculate the Gibbs free energy and heat capacity., Comment: 14 pages, 5 figures. Extended version of published paper in MPLA. Added figures 3-5

We study Einstein's gravity coupled to nonlinear electrodynamics with two parameters in Anti-de Sitter spacetime. Magnetically charged black holes in an extended phase space is investigated. We obtain the mass and metric functions, their asymptotic and corrections to the Reissner--Nordstr\"{o}m metric function when the cosmological constant vanishes. The first law of black hole thermodynamics in extended phase space is formulated and the magnetic potential and the thermodynamic conjugate to the coupling are obtained. We proved the generalized Smarr relation. The heat capacity and the Gibbs free energy are computed and phase transitions are studied. It was shown that the electric field of charged objects at the origin and electrostatic self-energy are finite within the nonlinear electrodynamics proposed., Comment: 17 pages, 6 figures

WE analyse the universe inflation when the source of gravity is electromagnetic fields obeying nonlinear electrodynamics with two parameters and without singularities. The cosmology of the universe with stochastic magnetic fields is considered. The condition for the universe inflation is obtained. It is demonstrated that singularities of the energy density and pressure are absent as the scale factor approaches to zero. When the scale factor goes to infinity one has equation of state for ultra-relativistic case. The curvature invariants do not possess singularities. The evolution of universe is described showing that at large time the scale factor corresponds to the radiation era. The duration of universe inflation is analysed. We study the classical stability and causality by computing the speed of the sound. Cosmological parameters such as the spectral index $n_s$, the tensor-to-scalar ratio $r$ and the running of the spectral index $\alpha_s$ are evaluated., Comment: 19 pages, 8 figures. Corrected typo in equation (34) in published paper Eur.Phys.J.C 84 (2024) 2, 205

We investigate Einstein-Gauss-Bonnet (EGB) 4D massive gravity coupled to nonlinear electrodynamics (NED) in an Anti-de-Sitter (AdS) background and find an exact magnetically charged black hole solution. The metric function was analyzed for different values of massive gravity parameters. We verified the first law of black hole thermodynamics and the generalized Smarr formula, treating the cosmological constant as thermodynamic pressure. Vacuum polarization is defined as the conjugate to the NED parameter. To analyze the local stability of the black hole, we compute specific heat. We investigated the Van der Waals-like/reentrant phase transition of the black holes and estimated the critical points. We observed small/large black hole (SBH/LBH) and small/intermediate/large black hole (SBH/IBH/LBH) phase transitions. The Joule-Thomson coefficient, inversion, and isenthalpic curves are discussed. Finally, the minimum inversion temperature and the corresponding event horizon radius are obtained using numerical techniques., Comment: 79 pages

Einstein's gravity in AdS space coupled to nonlinear electrodynamics is studied. We analyse the metric, mass functions and corrections to the Reissner-Nordstrom solution. Magnetic black holes thermodynamics in extended phase space is investigated. We formulate the first law of black hole thermodynamics showing that the generalized Smarr relation holds. The black hole stability is studied by evaluating the Gibbs free energy and heat capacity. To study the cooling and heating phase transitions of black holes we consider the Joule--Thomson isenthalpic expansion. The Joule--Thomson coefficient and the inversion temperature are calculated., Comment: 28 pages, 10 figures

The modified $F(R)$ gravity theory with the function $F(R)=-(1/\beta)\ln(1-\beta R)$ is studied. The action at small coupling $\beta$ becomes Einstein--Hilbert action. The bound on the parameter $\beta$ from local tests is $\beta\leq 2\times 10^{-6}$ cm$^2$. We find the constant curvature solutions and it was shown that the de Sitter space is unstable but a solution with zero Ricci scalar is stable. The potential and the mass of the scalar field (scalaron) are obtained in the Einstein's frame. The slow-roll cosmological parameters are studied and e-folds number is evaluated. The critical points of autonomous equations are analyzed. The function $m(r)$ that describes the deviation from the $\Lambda$CDM model is calculated., Comment: 16 pages, 5 figures. Accepted for publication in IJMPD

The process of the Joule-Thomson adiabatic expansion within RNED-AdS spacetime is investigated. The isenthalpic P-T diagrams and the inversion temperature were depicted. The inversion temperature depends on the magnetic charge and RNED coupling constant of black holes. When the Joule-Thomson coefficient vanishes, cooling-heating phase transition occurs. We consider the cosmological constant as a thermodynamic pressure ant the black hole mass is treated as the chemical enthalpy., Comment: 10 pages, 3 figures, 1 table

We investigate electrically charged black holes in $(2+1)$ dimensional gravity coupled to nonlinear electrodynamics (NED). The metric function $f(r)$ is depicted, showing that there can be one or two horizons. We study black hole thermodynamics in extended phase space. The cosmological constant played the role of thermodynamics pressure. Electrostatic potential, vacuum polarization, internal energy, Helmholtz and Gibbs free energies as functions of black hole horizon and Hawking temperature were analysed. The global stability of the black hole is discussed. The specific heat and local thermodynamic stability of black holes are studied. Finally, the Van der Walls behaviour of the black holes is investigated., Comment: 12 pages, 14 figures, Published in Indian Journal of Physics

The principles of causality and unitarity are studied within rational nonlinear electrodynamics proposed earlier. We investigate dyonic and magnetized black holes and show that in the self-dual case, when the electric charge equals the magnetic charge, corrections to Coulomb's law and Reissner-Nordstr\"{o}m solutions are absent. In the case of the magnetic black hole, the Hawking temperature, the heat capacity and the Helmholtz free energy are calculated. It is shown that there are second-order phase transitions and it was demonstrated that at some range of parameters the black holes are stable., Comment: 15 pages, 9 figures. Published in: Future Relativity, Gravitation, Cosmology. Editor: V. V. Dvoeglazov et al. ISBN: 979-8-88697-455-3 (Nova Science Publishers, Inc., Hauppauge, NY, 11788, USA, p.99-110 (2023)). arXiv admin note: substantial text overlap with arXiv:2103.14087

Spherically symmetric solution in 4D Einstein--Gauss--Bonnet gravity coupled to modified logarithmic nonlinear electrodynamics (ModLogNED) is found. This solution at infinity possesses the charged black hole Reissner--Nordstr\"{o}m behavior. We study the black hole thermodynamics, entropy, shadow, energy emission rate and quasinormal modes. It was shown that black holes can possess the phase transitions and at some range of event horizon radii black holes are stable. The entropy has the logarithmic correction to the area law. The shadow radii were calculated for variety of parameters. We found that there is a peak of the black hole energy emission rate. The real and imaginary parts of the quasinormal modes frequencies were calculated. The energy conditions of ModLogNED are investigated., Comment: 19 pages, 5 figures

Thermodynamics of magnetically charged black holes in Anti-de Sitter space in an extended phase space is studied. The cosmological constant is considered as a pressure and the black hole mass is treated as the chemical entalpy. The black hole thermodynamics is similar to the Van der Waals liquid-gas thermodynamics. Quantities conjugated to the nonlinear electrodynamics parameter and a magnetic charge are obtained. The first law of thermodynamics and the generalized Smarr relation take place. We investigate critical behavior of black holes and Joule--Thomson expansion. The Gibbs free energy, the Joule--Thomson coefficient and the inversion temperature are calculated., Comment: 28 pages, 8 figures, 1 table, accepted for publication in IJGMMP

The critical behavior of magnetically charged by rational nonlinear electrodynamics (RNED) AdS black holes in an extended phase space is investigated. The cosmological constant is considered as a thermodynamic pressure and the black hole mass is identified with the chemical enthalpy. The analogy with the Van der Waals liquid-gas system is found and the critical exponents coincide with those of the Van der Waals system. The thermodynamics of RNED-AdS black holes and phase transitions are studied. A new thermodynamic quantities conjugated to the nonlinear parameter of RNED and a magnetic charge are defined. It is demonstrated the consistency of the first law of black hole thermodynamics and the Smarr formula., Comment: 24 pages, 10 figures, 1 table

We study and discuss the critical behavior of charged by nonlinear electrodynamics (NED) AdS black holes in an extended phase space. The negative cosmological constant is treated as a thermodynamic pressure and the black hole mass is considered as the chemical enthalpy. We show the analogy of thermodynamics and phase transitions studied with the Van der Waals liquid-gas system. We introduce new thermodynamic quantities conjugated to the NED coupling and a magnetic charge. It was demonstrated that the first low of black hole thermodynamics and the Smarr relation hold. We have calculated the Joule--Thomson thermodynamic coefficient and inversion temperature and the Joule--Thomson adiabatic expansion is investigated., Comment: 9 pages (two columns), 9 figures, 2 tables

Thermodynamics and phase transitions of magnetic Anti-de Sitter black holes are studied. We consider an extended phase space thermodynamics with the cosmological constant being a thermodynamic pressure and the black hole mass is treated as a chemical enthalpy. The extended phase space thermodynamics of black holes mimics the behavior of the Van der Walls liquid. Quantities conjugated to the coupling of nonlinear electrodynamics (NED) and a magnetic charge are obtained. Thermodynamic critical points of phase transitions are investigated. It is demonstrated that the first law of black hole thermodynamics and the generalized Smarr relation take place. The Joule--Thomson adiabatic expansion of NED-AdS black holes is studied. The dependence of inversion temperature on pressure and the minimum of inversion temperature are found., Comment: 24 pages, 8 figures, 1 table

We study thermodynamics and phase transitions of magnetically charged by nonlinear electrodynamics (NED) Anti-de Sitter (AdS) black holes in an extended phase space. The cosmological constant is treated as a thermodynamic pressure whereas the black hole mass is considered such as the chemical enthalpy. The horizon thermodynamics of black holes shows an analogy with the Van der Waals liquid-gas system. We define a quantities which are conjugated to the parameter of NED and a magnetic charge. It is shown that the first law of thermodynamics and the generalized Smarr relation hold. The critical exponents are the same as in the Van der Waals system. The Joule--Thomson adiabatic expansion of NED-AdS black holes is investigated because of thermodynamic behavior of black holes. The Joule--Thomson coefficient, the inversion and isenthalpic curves are discussed. The minimum of inversion temperature and the corresponding event horizon radius are obtained., Comment: 30 pages, 11 figures, 1 table

Recently Bandos, Lechner, Sorokin, and Townsend [Phys. Rev. D \textbf{102}, 121703 (2020)] proposed Modified Maxwell (ModMax) model of nonlinear duality-invariant conformal electrodynamics. Here, Generalized ModMax (GenModMax) model of nonlinear electrodynamics coupled to general relativity is studied. The metric and mass functions, and their asymptotic as $r\rightarrow\infty$ and $r\rightarrow 0$ of a magnetic black hole are obtained. Corrections to the Reissner--Nordstr\"{o}m solution are found and we show that for some model parameters the black hole is regular. The Hawking temperature and heat capacity of black holes are calculated and phase transitions are investigated. We demonstrate that black holes are not stable for certain model parameters., Comment: 14 pages, 5 figures, to be published in IJMPD

New spherically symmetric solution in 4D Einstein--Gauss--Bonnet gravity coupled with nonlinear electrodynamics is obtained. At infinity this solution has the Reissner--Nordstr\"{o}m behavior of the charged black hole. The black hole thermodynamics, entropy, shadow, energy emission rate and quasinormal modes of black holes are investigated., Comment: 19 pages, 6 figures, accepted in Universe

A modified Hayward metric of magnetically charged black hole space-time based on rational nonlinear electrodynamics with the Lagrangian ${\cal L} = -{\cal F}/(1+2\beta{\cal F})$ is considered. We introduce the fundamental length, characterizing quantum gravity effects. If the fundamental length vanishes the general relativity coupling to rational nonlinear electrodynamics is recovered. We obtain corrections to the Reissner--Nordstr\"{o}m solution as the radius approaches infinity. The metric possesses a de Sitter core without singularities as $r\rightarrow 0$. The Hawking temperature and the heat capacity are calculated. It was shown that phase transitions occur and black holes are thermodynamically stable at some event horizon radii. We demonstrate that curvature invariants are bounded and the limiting curvature conjecture takes place., Comment: 14 pages, 6 figures, accepted in IJMPA

A new generalized ModMax model of nonlinear electrodynamics with four parameters is proposed. The ModMax model and Born--Infeld-type electrodynamics are particular cases of the present model It is shown that a singularity of the electric field at the center of point-like charged particles is absent. We found corrections to Coulomb's law at $r\rightarrow\infty$ and obtain the total electrostatic and magnetic energies of point-like charges. Free electric and magnetic charges and their densities are obtained., Comment: 13 pages, 4 figures, corrected Eq. (5), journal version, accepted in Phys.Lett.B

We obtain an exact spherically symmetric and magnetically charged black hole solution in 4D Einstein-Gauss-Bonnet gravity coupled with rational nonlinear electrodynamics. The thermodynamics of our model is studied. We calculate the Hawking temperature and the heat capacity of the black hole. The phase transitions occur in the point where the Hawking temperature possesses an extremum. We show that black holes are thermodynamically stable at some range of event horizon radii when the heat capacity is positive. The logarithmic correction to the Bekenstein-Hawking entropy is obtained., Comment: 12 pages, 3 figures

We obtain an exact spherically symmetric and magnetically charged black hole solution in 4D Einstein-Gauss-Bonnet gravity coupled with nonlinear electrodynamics with the Lagrangian ${\cal L}_{NED} = -{\cal F}/(1+\sqrt{2\mid\beta{\cal F}\mid})$ (${\cal F}$ is the field strength invariant). The thermodynamics of the black hole is studied within our model. We calculate the Hawking temperature and the heat capacity of the black hole. The phase transitions occur in the point where the Hawking temperature possesses an extremum and the heat capacity diverges. It was shown that black holes are thermodynamically stable in some range of event horizon radii when the heat capacity and Hawking temperature are positive. The logarithmic correction to the Bekenstein - Hawking BH entropy is obtained which mimics a quantum correction. The dependence of the BH shadow radius on the model parameters is studied. We also investigate the particle emission rate of the BH. The stability of the BH is studied by analysing the quasinormal modes., Comment: 25 pages, 8 figures, 3 tables, published in Annals of Physics

A Hayward black hole and magnetically charged black hole based on rational nonlinear electrodynamics with the Lagrangian ${\cal L} = -{\cal F}/(1+2\beta{\cal F})$ (${\cal F}$ is a field invariant) are considered. It was shown that the metric function in both models possesses a de Sitter core without singularities as $r\rightarrow 0$. The behavior of the Hawking temperature and the heat capacity in these models are similar. The phase transitions take place when the Hawking temperature has a maximum, and black holes are thermodynamically stable at some event horizon radii when the heat capacity is positive. We show that the source of gravity in the Hayward model is questionable., Comment: 16 pages, 6 figures, 1 table, published in Gravitation and Cosmology

Nonlinear electrodynamics with two dimensional parameters is studied. The range of electromagnetic fields when principles of causality, unitarity and the classical stability hold, are obtained. A singularity of the electric field at the center of charges is absent within our model and there are corrections to the Coulomb law as $r\rightarrow\infty$. The universe inflation takes place in the background of stochastic magnetic fields. The second stage of the universe evolution is the radiation era so that the graceful exit exists. We estimated the spectral index, the tensor-to-scalar ratio, and the running of the spectral index that are in a rough accord with the PLANCK and WMAP data., Comment: 20 pages, 6 figures, accepted in Int.J.Mod.Phys.D

The source of the universe inflation is electromagnetic fields obeying rational nonlinear electrodynamics proposed earlier. Within this model the singularities of the electric field at the center of charges, the Ricci scalar, the Ricci tensor squared and the Kretschmann scalar are absent. We consider the universe which is filled by stochastic magnetic fields. It is demonstrated that the inflation lasts approximately $2$ s with the reasonable e-folding number $N\approx 63$. The Inflation starts from de Sitter spacetime and after the universe inflation end it decelerates approaching the radiation era., Comment: 10 pages, 2 figures

We consider rational nonlinear electrodynamics with the Lagrangian ${\cal L} = -{\cal F}/(1+2\beta{\cal F})$ (${\cal F}=(1/4)F_{\mu\nu}F^{\mu\nu}$ is the Lorentz-invariant), proposed in [63], coupled to General Relativity. The effective geometry induced by nonlinear electrodynamics corrections are found. We determine shadow's size of regular non-rotating magnetic black holes and compare them with the shadow size of the super-massive M87* black hole imaged by the Event Horizon Telescope collaboration. Assuming that the black hole mass has a pure electromagnetic nature, we obtain the black hole magnetic charge. The size of the shadow obtained is very close to the shadow size of non-regular neutral Schwarzschild black holes. As a result, we can interpret the super-massive M87* black hole as a regular (without singularities) magnetized black hole., Comment: 12 pages, 1 figure, accepted in MPLA

A new model of nonlinear electrodynamics with dimensional parameters $\beta$ and $\gamma$ is proposed. The principles of causality and unitarity are studied. We show that a singularity of the electric field at the origin of charges is absent and the maximum of the electric field in the center is $E(0)=1/\sqrt{\beta}$. The dual symmetry is broken in our model. Corrections to the Coulomb law as $r\rightarrow\infty$ are in the order of ${\cal O}(r^{-4})$. The source of the gravitation field and inflation of the universe is electromagnetic fields. It is supposed that the universe is filled by stochastic magnetic fields. We demonstrate that after the universe inflation the universe decelerates approaching the Minkowski spacetime. The singularities of the Ricci scalar, the Ricci tensor squared and the Kretschmann scalar are absent. We calculate the speed of sound. The spectral index, the tensor-to-scalar ratio, and the running of the spectral index, which approximately agree with the PLANK and WMAP data, are evaluated., Comment: 22 pages, 7 figures. 1 table

We propose a new model of nonlinear electrodynamics with two dimensional parameters. The phenomenon of vacuum birefringence, the principles of causality and unitarity were studied. It was shown that there is no a singularity of the electric field in the center of point-like charges. Corrections to the Coulomb law as $r\rightarrow\infty$ were obtained. Dyonic and magnetized black holes are considered. We show that in the self-dual case, when the electric charge equals the magnetic charge, corrections to Coulomb's law and Reissner$-$Nordstr\"{o}m solutions are absent. The metric function and its asymptotic as $r\rightarrow\infty$ for magnetic black holes were evaluated. We calculate the magnetic mass of the black hole which is finite. The thermodynamic properties and thermal stability of black holes were investigated. We calculated the Hawking temperature, the heat capacity and the Helmholtz free energy of black holes and shown that at some parameters there are second-order phase transitions. It was demonstrated that at some range of parameters the black holes are stable., Comment: 20 pages, 5 figures, 2 tables, accepted in EPJC

Dyonic and magnetic black holes solutions with spherically symmetric configurations in general relativity are obtained. Black holes possessing electric and magnetic charges are studied where the source of the gravitational field is electromagnetic fields obeying the arcsin-electrodynamics. We find corrections to Coulomb's law and Reissner - Nordstr\"{o}m solutions. The principles of causality and unitarity are investigated. We obtain the Hawking temperature and it was shown that at some event horizons there are second-order phase transitions., Comment: 17 pages, 4 figures

Dyonic black hole solutions with spherically symmetric configurations within general relativity are investigated where the source of the gravitational field is Born - Infeld-type electrodynamics. Corrections to Coulomb's law and Reissner - Nordstrom solutions are obtained. From principles of causality and unitarity we find the restriction on electric and magnetic fields. The Hawking temperature and the heat capacity of black holes are obtained. At some event horizons second-order phase transitions take place., Comment: 17 pages, 6 figures

A new dyonic solution for black holes with spherically symmetric configurations in general relativity is obtained. We study black holes possessing electric and magnetic charges, and the source of the gravitational field is electromagnetic fields obeying the logarithmic electrodynamics. This particular form of nonlinear electrodynamics is of interest because of its simplicity. Corrections to Coulomb's law and Reissner-Nordstrom solution are found. We calculate the Hawking temperature of black holes and show that second-order phase transitions occur for some parameters of the model., Comment: 16 pages, 4 figures

We propose and investigate the modified Born$-$Infeld-type gravity model with the function $F(R) = [1-(1-\beta R/\sigma)^\sigma]/\beta$. At different values of the dimensionless parameter $\sigma$ the action is converted into some models including general relativity ($\sigma=1$), the Starobinsky model ($\sigma=2$), the exponential model of gravity ($\sigma=\infty$), and a model proposed in [9], [10] ($\sigma=0.5$). A bound on the parameters, $\beta(1-\sigma)/\sigma\leq 2\times 10^{-6}$ cm$^2$, are due to local tests. The constant curvature solutions are obtained, and we found the potential, and the mass of the scalaron in the Einstein frame. The model describes the inflation of the universe. The de Sitter spacetime is unstable and a Minkovsky spacetime is stable. We investigate the cosmological parameters and some critical points of autonomous equations., Comment: 20 pages, 10 figures, 4 tables, to appear in IJGMMP

Holographic s-wave superconductors in the framework of nonlinear Born - Infeld-type electrodynamics is investigated in the background of Schwarzschild anti-de Sitter black holes. As particular cases, at some model parameters, we obtain results for Born - Infeld and exponential electrodynamics. We explore the analytical Sturm-Liouville eigenvalue problem in the probe limit where the scalar and electromagnetic fields do not effect on the background metric. The critical temperatures of phase transitions and the order parameter are calculated which depend on the model parameters. We show that the critical exponent near the critical temperature is 1/2. Making use of the matching method we derive analytical expressions for the condensation values and the critical temperature. The conductivity by the analytical method is calculated., Comment: 22 pages, 6 figures

The Bronnikov model of nonlinear electrodynamics is investigated in general relativity. The magnetic black hole is considered and we obtain a solution giving corrections to the Reissner-Nordstr\"{o}m solution. In this model spacetime at $r\rightarrow\infty$ becomes Minkowski's spacetime. We calculate the magnetic mass of the black hole and the metric function. At some parameters of the model there can be one, two or no horizons. The Hawking temperature and the heat capacity of black holes are calculated. We show that a second-order phase transition takes place and black holes are thermodynamically stable at some range of parameters., Comment: 15 pages, 5 figures, 1 table

A model of nonlinear electrodynamics is proposed and investigated in general relativity. We consider the magnetic black hole and find a regular solution which gives corrections into the Reissner-Nordstr\"{o}m solution. At $r\rightarrow\infty$ the asymptotic spacetime becomes flat. The magnetic mass of the black hole is calculated and the metric function is obtained. At some values of the model parameter there can be one, two or no horizons. Thermodynamics of black holes is studied and we calculate the Hawking temperature and heat capacity of black holes. It is demonstrated that there is a phase transition of second order. At some parameters of the model black holes are thermodynamically stable., Comment: 12 pages, 5 figures

We investigate holographic s-wave superconductors with nonlinear arcsin-electrodynamics in the background of Schwarzschild anti-de Sitter black holes. The analytical Sturm-Liouville eigenvalue problem is explored and we assume that the scalar and electromagnetic fields do not influence on the background metric (the probe limit). The critical temperatures of phase transitions depending on the parameter of the model is obtained. We show that in our case the condensation formation becomes easier compared to Born-Infeld nonlinear electrodynamics. The critical exponent near the critical temperature is calculated which is 1/2. With the help of the matching method we derive analytic expressions for the condensation value and the critical temperature. The real and imaginary parts of the conductivity in our model, making use of an analytical method, are computed., Comment: 25 pages, 6 figures, minor corrections, to appear in Annalen Phys. (Berlin)

A new model of nonlinear electrodynamics with two parameters is proposed. We study the phenomenon of vacuum birefringence, the causality and unitarity in this model. There is no singularity of the electric field in the center of point-like charges and the total electrostatic energy is finite. We obtain corrections to the Coulomb law at $r\rightarrow\infty$. The weak, dominant and strong energy conditions are investigated. Magnetized charged black hole is considered and we evaluate the mass, metric function and their asymptotic at $r\rightarrow\infty$ and $r\rightarrow 0$. The magnetic mass of the black hole is calculated. The thermodynamic properties and thermal stability of regular black holes are discussed. We calculate the Hawking temperature of black holes and show that there are first-order and second-order phase transitions. The parameters of the model when the black hole is stable are found., Comment: 24 pages, 2 tables, 7 figures, corrected typo in Eq. (57)

A model of nonlinear electrodynamics with two parameters, coupled with general relativity, is investigated. We study the magnetized black hole and obtain solutions. The asymptotic of the metric and mass functions at $r\rightarrow\infty$ and $r\rightarrow 0$, and corrections to the Reissner-Nordstr\"{o}m solution are found. We investigate thermodynamics of black holes and calculate the Hawking temperature and heat capacity of black holes. It is shown that there are phase transitions and at some parameters of the model black holes are stable., Comment: 14 pages, 6 figures, 2 tables

We investigate a Born--Infeld-type model of nonlinear electrodynamics, possessing three parameters, coupled with general relativity. As a particular case Born--Infeld electrodynamics is reproduced. There is no singularity of the electric field at the centre of point-like charged particles and self-energy of charges is finite in this model. The magnetized black hole is studied and solutions are obtained. We demonstrate that for some parameters of the model the black hole is regular. We find the asymptotic of the metric and mass functions at $r\rightarrow\infty$ and $r\rightarrow 0$, and corrections to the Reissner--Nordstr\"{o}m solution. Thermodynamics of black holes is investigated. We calculate the Hawking temperature of black holes and show that black holes are not stable and there are phase transitions in the model under consideration., Comment: 15 pages, 6 figures, corrected Eq. (26) and Fig. 6

One of the interesting fundamental phenomenon which was observed in the last decades is the discovery of anyons, relativistic spinning particles in $2+1$ dimensions. In contrast to three-dimensional space, indistinguishable quantum particles in two-dimensional space can, in general, have anomalous statistics [1-4]. These quasiparticles carry not only a charge $q$ also the magnetic flux $\Phi_0$., Comment: 5 pages

We consider Heisenberg-Euler-type model of nonlinear electrodynamics with two parameters. Heisenberg-Euler electrodynamics is a particular case of this model. Corrections to Coulomb's law at $r\rightarrow\infty$ are obtained and energy conditions are studied. The total electrostatic energy of charged particles is finite. The charged black hole solution in the framework of nonlinear electrodynamics is investigated. We find the asymptotic of the metric and mass functions at $r\rightarrow\infty$. Corrections to the Reissner-Nordstr\"{o}m solution are obtained., Comment: 11 pages, 1 figure

A model of nonlinear electrodynamics with a dimensional parameter $\beta$ is considered. Electromagnetic fields are the source of the gravitation field and inflation of the universe. We imply that the universe is filled by stochastic magnetic fields. It is demonstrated that after the universe inflation the universe decelerates approaching the Minkowski spacetime. We evaluate the spectral index, the tensor-to-scalar ratio, and the running of the spectral index which approximately agree with the PLANK and WMAP data., Comment: 14 pages, 7 figures, to appear in IJMPA

We perform the gauge covariant quantization of the exponential model of nonlinear electrodynamics. Magnetically charged black holes, in the framework of our model are considered, and the regular black hole solution is obtained in general relativity. The asymptotic black hole solution at $r\rightarrow \infty$ is found. We calculate the magnetic mass of the black hole and the metric function which are expressed via the parameter $\beta$ of the model and the magnetic charge. The thermodynamic properties and thermal stability of regular black holes are analysed. We calculate the Hawking temperature of black holes and their heat capacity at the constant magnetic charge. We find a point where the temperature changes the sign that corresponds to the first-order phase transition. It is shown that at critical point, where the heat capacity diverges, there is a phase transition of the second-order. We obtain the parameters of the model when the black hole is stable., Comment: 19 pages, 4 figures

A modified Gibbs's rotation matrix is derived and the connection with the Euler angles, quaternions, and Cayley$-$Klein parameters is established. As particular cases, the Rodrigues and Gibbs parameterizations of the rotation are obtained. The composition law of two rotations from the quaternion representation is presented showing a convenient expression for calculating the successive rotations., Comment: 17 pages