In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} in the overline {{text{MS}}} -scheme. We remind that such expansion is possible for the Adler function of the process of {{e}^{ + }}{{e}^{ - }} annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the CBK relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and overline {{text{MS}}} -scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} for all mentioned renorm-invariant perturbative quantities are discussed.

Cosmological solutions are studied in the context of the modified measure formulation of string theory, then the string tension is a dynamical variable and the string the tension is an additional dynamical degree of freedom and its value is dynamically generated. These tensions are then not universal, rather each string generates its own tension which can have a different value for each of the string world sheets and in an ensemble of strings the values of the tensions can have a certain dispersion. We consider a new background field that can couple to these strings, the “tension scalar” which is capable of changing locally along the world sheet and then the value of the tension of the string changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein’s equations and the consistency of these two Einstein’s equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein’s equation. The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit where the two string tensions are the same is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and isotropy of the cosmology disappears and the solution turns into an expanding Braneworld where the strings are confined between two expanding bubbles separated by a very small distance at large times.

General relativity (GR), created more than a century ago, has been checked in various experimental and observational tests. At an early stage of its development, GR predictions were tested in problems where the gravitational field is weak and relativistic corrections can be considered as small perturbations of the Newtonian theory of gravity. However, in recent years due to the progress of new technologies it turned out to be possible to verify the predictions of GR in the limit of a strong gravitational field, as it was done to verify predictions about the profile of the X-ray line of iron Kalpha , estimates of the gravitational wave signal during the mergers of binary black holes and/or neutron stars and during the reconstruction of the shadows of black holes in Sgr A* and M87*. Groups of astronomers using the Keck and VLT (GRAVITY) telescopes confirmed the GR predictions for the redshift of the spectral lines of the S2 star near the passage of its pericenter (these predictions were done in the first post-Newtonian approximation). It is expected that in the near future, observations of bright stars using large telescopes VLT (GRAVITY), Keck, E-ELT and TMT will allow us to verify the predictions of GR in the strong gravitational field of supermassive black holes. Observations of bright stars in the vicinity of the Galactic Center and reconstructions of the shadows of black holes allow not only to verify the predictions of the GR, but also to obtain restrictions on alternative theories of gravity.