This paper presents a majorized alternating direction method of multipliers (ADMM) with indefinite proximal terms for solving linearly constrained $2$-block convex composite optimization problems with each block in the objective being the sum of a no
n-smooth convex function and a smooth convex function, i.e., $min_{x in {cal X}, ; y in {cal Y}}{p(x)+f(x) + q(y)+g(y)mid A^* x+B^* y = c}$. By choosing the indefinite proximal terms properly, we establish the global convergence and $O(1/k)$ ergodic iteration-complexity of the proposed method for the step-length $tau in (0, (1+sqrt{5})/2)$. The computational benefit of using indefinite proximal terms within the ADMM framework instead of the current requirement of positive semidefinite ones is also demonstrated numerically. This opens up a new way to improve the practical performance of the ADMM and related methods.
One brief idea on the extended uncertainty relation and the dynamical quantization of space-time at the Planck scale is presented. The extended uncertainty relation could be a guiding principle toward the renormalizable quantum gravity. Cosmological
constant in the Universe as a quantum effect is also roughly estimated.
We investigate the properties of one--dimensional flux ``voids (connected regions in the flux distribution above the mean flux level) by comparing hydrodynamical simulations of large cosmological volumes with a set of observed high--resolution spectr
a at z ~ 2. After addressing the effects of box size and resolution, we study how the void distribution changes when the most significant cosmological and astrophysical parameters are varied. We find that the void distribution in the flux is in excellent agreement with predictions of the standard LCDM cosmology, which also fits other flux statistics remarkably well. We then model the relation between flux voids and the corresponding one--dimensional gas density field along the line--of--sight and make a preliminary attempt to connect the one--dimensional properties of the gas density field to the three--dimensional dark matter distribution at the same redshift. This provides a framework that allows statistical interpretations of the void population at high redshift using observed quasar spectra, and eventually it will enable linking the void properties of the high--redshift universe with those at lower redshifts, which are better known.
