Narrow Results

In this paper, we explore the influences of the higher-order Gauss–Bonnet (GB) correction terms on the growth of perturbations at the early stage of a $(n+1)$-dimensional Friedmann–Robertson–Walker (FRW) universe. Considering a cosmological constant in the FRW background, we study the linear perturbations by adopting the spherically symmetric collapse (SC) formalism. In light of the modifications that appear in the field equations, we disclose the role of the GB coupling constant $\alpha$, as well as the extra dimensions $n>3$ on the growth of perturbations. It, in essence, is done by defining a dimensionless parameter $\tilde{\beta }=(n-2)(n-3)\alpha H_0^2$ in which $H_0$ is the Hubble constant. We find that the matter density contrast starts growing at the early stages of the universe and, as the universe expands, it grows faster compared to the standard cosmology. Besides, in the framework of GB gravity, the growth of matter perturbations in higher dimensions is faster than its standard counterpart. Further, in the presence of $\alpha$, the growth of perturbations increases as it increases. This is an expected result, since the higher-order GB correction terms increase the strength of the gravity and thus support the growth of perturbations. For the existing cosmological model, we also investigate the behavior of quantities such as density abundance, deceleration and the jerk parameter. Finally, we study the imprint of the GB parameter and the higher dimensions in the evolution of the mass function of the dark matter halos.

In this paper we discuss the problem: how large can the intermediate growth of nilpotent and relatively free semigroups be. We construct a sequence {S_n} of finitely-generated semigroups such that the growth of the semigroup S_n+1 (n = 1,2,…) is intermediate and larger than or equal to the growth of exp(m/φ_n(m)), where φ_n(m) is the nth iteration of ln m. All semigroups S_n are nilpotent in the sense of Malcev. We also find the sequence of relatively free semigroups with the same types of growth.

Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $\mathrm{\text{gr}}(H,A,B)$ be the growth function of the double cosets $AhB,h\in H$. We prove that the behavior of $\mathrm{\text{gr}}(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $\mathrm{\text{gr}}(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups $A$ and $B$ of infinite index, $\mathrm{\text{gr}}(H,A,B)$ is exponential. Moreover, there exists a constant $\lambda >0$, such that $\mathrm{\text{gr}}(H,A,B)(r)>\lambda f_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function of the group $H$. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

The problem of finding the genus of a given link type of braid index 3 is solved by constructive methods. The results depend upon a new solution to the conjugacy problem in B₃. In this solution, conjugacy classes are represented by shortest words in terms of cyclically symmetric elementary braids which are used as generators in the new presentation of B₃. By related results of Bennequin [2] and of Birman and Menasco [4], the minimal spanning surfaces are described by these shortest words. An effective algorithm is given to find these shortest words starting with an arbitrary 3-braid representative of the link. It is proved that up to an easily described family of exceptional cases, a link of braid index 3 has a unique isotopy class of minimal spanning surfaces. The growth functions of B₃ and its conjugacy classes are also given here.

In this paper, we study the distinguishability of dark energy and vacuum decay models. We start showing formally that dark energy fluids and dynamic vacuum are kinematically equivalent. Consequently, cosmological probes, such as the distances of type Ia supernovae (SNe Ia), cosmic microwave background (CMB) distance priors, baryon acoustic oscillations (BAO) distilled parameters and cosmic chronometers, are unable to allow us to distinguish such models from each other. Then, we show that, at least in principle, these models are dynamically distinguishable. So, dynamical probes, such as the evolution of linear matter perturbations, can be used to distinguish between dark energy models and dynamic vacuum models. To exemplify the kinematic equivalence between dark energy models and vacuum decay models, we study two simple models: the wCDM, commonly associated with dark energy, and the Wang–Meng model, often associated with dynamic vacuum. We test these two models with 22 measurements of $f{\sigma }_8$. First, we consider both as dark energy then we consider both as vacuum decay. We show that, although compatible with each other, when seen as dynamical vacuum both models are tightly close to the $\mathrm{\Lambda }$CDM model.

We discuss the notion of growth for discrete quantum groups, with a number of general considerations, and with some explicit computations. Of particular interest is the quantum analogue of Gromov's estimate regarding polynomial growth: we formulate the precise question, and we verify it for the duals of classical Lie groups.