Effective sound speed in relativistic accretion discs around rotating black holes

For axially symmetric accretion maintained in the hydrostatic equilibrium along the vertical direction in the Kerr metric, the radial Mach number does not become unity at the critical point. The sonic points are, thus, formed at a radial distance different from that where the critical points are formed. We propose that a modified dynamical sound speed can be defined through the linear perturbation of the full space-time dependent equations describing the aforementioned accretion flow structure. The linear stability analysis of such fluid equations leads to the formation of an wave equation which describes the propagation of linear acoustic perturbation. The speed of propagation of such perturbation can be used as the effective sound speed which makes the value of the Mach number to be unity when evaluated at the critical points. This allows the critical points to coalesce with the sonic points. We study how the spin angular momentum of the black hole (the Kerr parameter) influences the value of the effective sound speed

Selberg’s Central Limit Theorem for families of L-functions

In this thesis, we present a simple proof of Selberg’s Central Limit Theorem for appropriate families of L-functions. As conjectured by Selberg, his central limit theorem can only be proven for the L-functions belonging to the Selberg Class. First, we prove Selberg’s central limit theorem for classical automorphic L-functions of degree 2 associated with holomorphic cusp forms. We prove this result in the t-aspect. In Chapter 4, we prove Selberg’s central limit theorem for Dirichlet L-functions and quadratic Dirichlet L functions associated with primitive Dirichlet characters and twisted Hecke-Maass cusp forms respectively. We prove these results in the q-aspect, i.e., instead of integrating we average over Dirichlet characters. Finally, in Chapter 5, we prove that a sequence of degree 2 automorphic L-functions attached to a sequence of primitive holomorphic cusp forms form a Gaussian process. Also, any two elements from this sequence of L-functions are pair-wise independent. Additionally, we construct a random matrix that generalizes the notion of independence of the families of automorphic L-functions

Efficient Computation of Subspace Skyline over Categorical Domains

Platforms such as AirBnB, Zillow, Yelp, and related sites have transformed the way we search for accommodation, restaurants, etc. The underlying datasets in such applications have numerous attributes that are mostly Boolean or Categorical. Discovering the skyline of such datasets over a subset of attributes would identify entries that stand out while enabling numerous applications. There are only a few algorithms designed to compute the skyline over categorical attributes, yet are applicable only when the number of attributes is small. In this paper, we place the problem of skyline discovery over categorical attributes into perspective and design efficient algorithms for two cases. (i) In the absence of indices, we propose two algorithms, ST-S and ST-P, that exploits the categorical characteristics of the datasets, organizing tuples in a tree data structure, supporting efficient dominance tests over the candidate set. (ii) We then consider the existence of widely used precomputed sorted lists. After discussing several approaches, and studying their limitations, we propose TA-SKY, a novel threshold style algorithm that utilizes sorted lists. Moreover, we further optimize TA-SKY and explore its progressive nature, making it suitable for applications with strict interactive requirements. In addition to the extensive theoretical analysis of the proposed algorithms, we conduct a comprehensive experimental evaluation of the combination of real (including the entire AirBnB data collection) and synthetic datasets to study the practicality of the proposed algorithms. The results showcase the superior performance of our techniques, outperforming applicable approaches by orders of magnitude

Dependence of acoustic surface gravity on disc thickness for accreting astrophysical black holes

For axially symmetric accretion maintained in hydrostatic equilibrium along the vertical direction, we investigate how the characteristic features of the embedded acoustic geometry depends on the background Kerr metric, and how such dependence is governed by three different expressions of the thickness of the matter flow. We first obtain the location of the sonic points and stationary shock between the sonic points. We then linearly perturb the flow to obtain the corresponding metric elements of the acoustic space-time. We thus construct the causal structure to establish that the sonic points and the shocks are actually the analogue black hole type and white hole type horizons, respectively. We finally compute the value of the acoustic surface gravity as a function of the spin angular momentum of the rotating black hole for three different flow thicknesses considered in the present work. We find that for some flow models, the intrinsic acoustic geometry, although in principle may be extended up to the outer gravitational horizon of the astrophysical black hole, cannot be constructed beyond a certain truncation radius as imposed by the expressions of the thickness function of the corresponding flow.Comment: 22 pages, 9 figure