Re-Scaling of Energy in the Stringy Charged Black Hole Solutions using Approximate Symmetries

This paper is devoted to study the energy problem in general relativity using approximate Lie symmetry methods for differential equations. We evaluate second-order approximate symmetries of the geodesic equations for the stringy charged black hole solutions. It is concluded that energy must be re-scaled by some factor in the second-order approximation.Comment: 18 pages, accepted for publication in Canadian J. Physic

УВЛЕЧЕНИЕ ВЯЗКОПЛАСТИЧЕСКОЙ ЖИДКОСТИ ДВИЖУЩЕЙСЯ ВЕРТИКАЛЬНО ПЛАСТИНОЙ

The liquid capture by a moving surface is the most widespread process in chemical engineering along with calendaring, extrusion moulding, pouring, and pressure moulding. The theoretical analysis of the medium capture by a moving surface, which allows revealing the fundamental physical principles and mechanisms of the process over the entire withdrawal speed range realized in practice, was performed for Newtonian, non-Newtonian, and viscoplastic liquids. However, such an analysis of the withdrawal of viscoplastic liquids with a finite yield was not made because of the features of these liquids. Shear flow of viscoplastic liquid is possible only after the stress exceeds its yield. This fact causes serious mathematical difficulties in stating and solving the problem. In the proposed work, such a theory is being developed for viscoplastic liquids.Захват жидкости движущейся поверхностью является наиболее распространённым процессом в химической технологии наряду с каландрованием, экструзионным формованием, заливкой, формованием под давлением. Теоретический анализ увлечения среды движущейся поверхностью, позволяющей вскрыть основные физические принципы и механизмы процесса во всем диапазоне скоростей извлечения, реализуемом на практике, был проведен для ньютоновских, нелинейновязких, вязкопластичных жидкостей. Однако такой анализ по увлечению вязкопластичных жидкостей, обладающих конечным пределом текучести, проведен не был в силу специфических особенностей этих жидкостей. Для вязкопластичной жидкости сдвиговое течение возможно лишь после того как напряжение превысит предел текучести. Данное обстоятельство вносит серьезные математические трудности при постановке и решении задачи. В предлагаемой работе такая теория развивается для вязкопластичных жидкостей

Properties of equations of the continuous Toda type

We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter \ep. We apply the prolongation method, the symmetry and the approximate symmetry approach. This strategy allows us to get insight into both the equations for \ep =0 and \ep \ne 0, whose properties arising in the above frameworks are mutually compared. For \ep =0, the related prolongation equations are solved by means of certain series expansions which lead to an infinite- dimensional Lie algebra. Furthermore, using a realization of the Lie algebra of the Euclidean group $E_{2}$, a connection is shown between the continuous Toda equation and a linear wave equation which resembles a special case of a three-dimensional wave equation that occurs in a generalized Gibbons-Hawking ansatz \cite{lebrun}. Nontrivial solutions to the wave equation expressed in terms of Bessel functions are determined. For \ep\,\ne\,0, we obtain a finite-dimensional Lie algebra with four elements. A matrix representation of this algebra yields solutions of the modified continuous Toda equation associated with a reduced form of a perturbative Liouville equation. This result coincides with that achieved in the context of the approximate symmetry approach. Example of exact solutions are also provided. In particular, the inverse of the exponential-integral function turns out to be defined by the reduced differential equation coming from a linear combination of the time and space translations. Finally, a Lie algebra characterizing the approximate symmetries is discussed.Comment: LaTex file, 27 page

Увлечение неньютоновской жидкости движущейся наклонной пластиной

Fluid capturing by a moving inclined surface is analyzed theoretically. A task for non-Newtonian fluid is stated in general form. The solving of this task enables revealing the basic physical principles and the mechanisms of the fluid withdrawal process over an entire range of withdrawal velocities realized in practice. The case of withdrawal of finite yield stress viscoplastic fluid is considered.Проведен теоретический анализ увлечения жидкости движущейся наклонной поверхностью. Приведена общая постановка задачи для неньютоновской жидкости. Решение этой задачи позволит вскрыть основные физические принципы и механизмы процесса во всем диапазоне скоростей извлечения, реализуемом на практике. Рассмотрен случай увлечения вязкопластической жидкости, обладающей конечным пределом текучести

Energy of Bardeen Model Using Approximate Symmetry Method

In this paper, we investigate the energy problem in general relativity using approximate Lie symmetry methods for differential equations. This procedure is applied to Bardeen model (the regular black hole solution). Here we are forced to evaluate the third-order approximate symmetries of the orbital and geodesic equations. It is shown that energy must be re-scaled by some factor in the third-order approximation. We discuss the insights of this re-scaling factor.Comment: 14 pages, no figure, accepted for publication in Physica Script

The analytical singlet αs4\alpha_s^4 QCD contributions into the e+e−e^+e^--annihilation Adler function and the generalized Crewther relations

The generalized Crewther relations in the channels of the non-singlet and vector quark currents are considered. They follow from the double application of the operator product expansion approach to the same axial vector-vector-vector triangle amplitude in two regions, adjoining to the angle sides $(x,y)$ (or $p^2,q^2$). We assume that the generalized Crewther relations in these two kinematic regimes result in the existence of the same perturbation expression for two products of the coefficient functions of annihilation and deep-inelastic scattering processes in the non-singlet and vector channels. Taking into account the 4-th order result for $S_{GLS}$ and the perturbative effects of the violation of the conformal symmetry in the generalized Crewther relation, we obtain the analytical contribution to the singlet $\alpha_s^4$ correction to the $D_A^{V}$-function. Its a-posteriori comparison with the recent result of direct diagram-by-diagram evaluation of the singlet 4-th order corrections to $D_A^{V}$- function demonstrates the coincidence of the predicted and obtained $\zeta_3^2$-contributions to the singlet term. They can be obtained in the conformal invariant limit from the original Crewther relation. On the contrary to previous belief, the appearance of $zeta_3$-terms in perturbative series in gauge models does not contradict to the property of conformal symmetry and can be considered as ragular feature. The Banks-Zaks motivated relation between our predicted and obtained 4-th order corrections is mentioned. This confirms Baikov-Chetyrkin-Kuhn expectation that the generalized Crewther relation in the channel of vector currents receives additional singlet contribution, which in this order of perturbation theory is proportional to the first coefficient of the QCD $\beta$-function.Comment: Concrete new foundations explained, abstract updated, presentation improved, 2 references added, extra acknowledgements added. This work is dedicated to K. G. Chetyrkin on the occasion of his 60th anniversary, to be published in Jetp. Lett supposedly in vol.94, issue 1