Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants

In this paper, the global optimization problem $\min_{y\in S} F(y)$ with $S$ being a hyperinterval in $\Re^N$ and $F(y)$ satisfying the Lipschitz condition with an unknown Lipschitz constant is considered. It is supposed that the function $F(y)$ can be multiextremal, non-differentiable, and given as a `black-box'. To attack the problem, a new global optimization algorithm based on the following two ideas is proposed and studied both theoretically and numerically. First, the new algorithm uses numerical approximations to space-filling curves to reduce the original Lipschitz multi-dimensional problem to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm at each iteration applies a new geometric technique working with a number of possible H\"{o}lder constants chosen from a set of values varying from zero to infinity showing so that ideas introduced in a popular DIRECT method can be used in the H\"{o}lder global optimization. Convergence conditions of the resulting deterministic global optimization method are established. Numerical experiments carried out on several hundreds of test functions show quite a promising performance of the new algorithm in comparison with its direct competitors.Comment: 26 pages, 10 figures, 4 table

New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories

In the present paper we consider a class of extended scalar-tensor-Gauss-Bonnet (ESTGB) theories for which the scalar degree of freedom is excited only in the extreme curvature regime. We show that in the mentioned class of ESTGB theories there exist new black hole solutions which are formed by spontaneous scalarization of the Schwarzaschild balck holes in the extreme curvature regime. In this regime, below certain mass, the Schwarzschild solution becomes unstable and new branch of solutions with nontrivial scalar field bifurcate from the Schwarzschild one. As a matter of fact, more than one branches with nontrivial scalar field can bifurcate at different masses but only the first one is supposed to be stable. This effect is quite similar to the spontaneous scalarization of neutron stars. In contrast with the standard spontaneous scalarization of neutron stars which is induced by the presence of matter, in our case the scalarization is induced by the curvature of the spacetime.Comment: 13 pages, 7 figure

Tidal Love numbers of neutron stars in f(R)f(R) gravity

The recent detection of gravitational waves from a neutron star merger was a significant step towards constraining the nuclear matter equation of state by using the tidal Love numbers (TLNs) of the merging neutron stars. Measuring or constraining the neutron star TLNs allows us in principle to exclude or constraint many equations of state. This approach, however, has the drawback that many modified theories of gravity could produce deviations from General Relativity similar to the deviations coming from the uncertainties in the equation of state. The first and the most natural step in resolving the mentioned problem is to quantify the effects on the TLNs from the modifications of General Relativity. With this motivation in mind, in the present paper we calculate the TLNs of (non-rotating) neutron stars in $f(R)$ gravity. For this purpose, we first derived the equations describing both the polar and the axial stationary perturbations of neutron stars in a particular class of $f(R)$ gravity, the so-called $R^2$-gravity. Then, by solving numerically the perturbation equations, we calculate explicitly the polar and the axial $l=2$ TLNs of the neutron stars in $R^2$-gravity for three characteristic realistic equations of state. Our results show that while the polar TLNs are slightly influenced by the $R^2$ modification of General Relativity, the axial TLNs can be several times larger (in terms of the absolute value) compared to the general relativistic case.Comment: 10 pages, 3 figure

Universal I-Q relations for rapidly rotating neutron and strange stars in scalar-tensor theories

We study how rapid rotation influences the relation between the normalized moment of inertia $\bar{I}$ and quadrupole moment $\bar{Q}$ for scalarized neutron stars. The questions one has to answer are whether the EOS universality is preserved in this regime and what are the deviations from general relativity. Our results show that the $\bar{I}-\bar{Q}$ relation is nearly EOS independent for scalarized rapidly rotating stars, but the differences with pure Einstein's theory increase compared to the slowly rotating case. In general, smaller negative values of the scalar field coupling parameters $\beta$ lead to larger deviations, but these deviations are below the expected accuracy of the future astrophysical observations if one considers values of $\beta$ in agreement with the current observational constraint. An important remark is that although the normalized $\bar{I}-\bar{Q}$ relation is quite similar for scalar-tensor theories and general relativity, the unnormalized moment of inertia and quadrupole moment can be very different in the two theories. This demonstrates that although the $\bar{I}-\bar{Q}$ relations are potentially very useful for some purposes, they might not serve us well when trying to distinguish between different theories of gravity.Comment: 8 pages, 3 figure