R+R2R + R^2 Gravity as R+R + Backreaction

Quadratic theory of gravity is a complicated constraint system. We investigate some consequences of treating quadratic terms perturbatively (higher derivative version of backreaction effects). This approach is shown to overcome some well known problems associated with higher derivative theories, i.e., the physical gravitational degree of freedom remains unchanged from those of Einstein gravity. Using such an interpretation of $R + \beta R^2$ gravity, we investigate a classical and Wheeler DeWitt evolution of $R + \beta R^2$ gravity for a particular sign of $\beta$, corresponding to non- tachyon case. Matter is described by a phenomenological $\rho \propto a(t)^{-n}$. It is concluded that both the Friedmann potential $U(a)$ (${\dot a}^2 + 2U(a) = 0$) and the Wheeler DeWitt potential $W(a)$ ($\left[-{\partial^2\over \partial a^2} + 2W(a)\right]\psi (a) =0$) develop repulsive barriers near $a\approx 0$ for $n>4$ (i.e., $p > {1\over 3}\rho$). The interpretations is clear. Repulsive barrier in $U(a)$ implies that a contracting FRW universe ($k>0, k=0, k<0$) will bounce to an expansion phase without a total gravitational collapse. Repulsive barrier in $W(a)$ means that $a \approx 0$ is a classically forbidden region. Therefore, probability of finding a universe with the big bang singularity ($a=0$ ) is exponentially suppressed.Comment: Accepted for publication in Phy. Rev. D.,18 pages, 6 figures, Latex fil

R-CNN minus R

Deep convolutional neural networks (CNNs) have had a major impact in most areas of image understanding, including object category detection. In object detection, methods such as R-CNN have obtained excellent results by integrating CNNs with region proposal generation algorithms such as selective search. In this paper, we investigate the role of proposal generation in CNN-based detectors in order to determine whether it is a necessary modelling component, carrying essential geometric information not contained in the CNN, or whether it is merely a way of accelerating detection. We do so by designing and evaluating a detector that uses a trivial region generation scheme, constant for each image. Combined with SPP, this results in an excellent and fast detector that does not require to process an image with algorithms other than the CNN itself. We also streamline and simplify the training of CNN-based detectors by integrating several learning steps in a single algorithm, as well as by proposing a number of improvements that accelerate detection

Defining R and G(R)

We show that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are essentially (nearly always) the double centralizers of corresponding root elements. In very many cases this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is finitely axiomatizable in the appropriate class of groups provided R is finitely axiomatizable in the corresponding class of rings.Comment: (1) New Theorem 1.1 generalizes earlier main theorems.(2) New version incorporates content of arXiv:2007.11440 (3) Latest version has small corrections. To appear in J. Eur. Math. So