Central Values of Degree Six L-functions: The Case of Hilbert Modular Forms

In this paper we give a formula for the central value of the completed $L$-function $L(s,Sym^{2} g\times f)$, where $f$ and $g$ are Hilbert newforms, by explicitly computing the local integrals appearing in the refined Gan-Gross-Prasad conjecture for $SL_{2}\times\tilde{SL_{2}}$. We also work out the rationality of this value in some special cases and give a conjecture for the general case

ANOMALOUS TRANSPORT, QUASIPERIODICITY, AND MEASUREMENT INDUCED PHASE TRANSITIONS

With the advent of the noisy-intermediate scale quantum (NISQ) era quantum computers are increasingly becoming a reality of the near future. Though universal computation still seems daunting, a great part of the excitement is about using quantum simulators to solve fundamental problems in fields ranging from quantum gravity to quantum many-body systems. This so-called second quantum revolution rests on two pillars. First, the ability to have precise control over experimental degrees of freedom is crucial for the realization of NISQ devices. Significant progress in the control and manipulation of qubits, atoms, and ions, as well as their interactions, has not only allowed for emulation of diverse range of physical systems but has also led to realization of quantum systems in non-conventional settings such as systems out-of-equilibrium, driven by oscillating fields, and with quasiperiodic (QP) modulation. These systems often show novel properties which not only provide an interesting testbed for NISQ devices but also an opportunity to exploit them for further development of quantum computing devices. Second, the study of dynamics of quantum information in quantum systems is essential for understanding and designing better quantum computers. In addition to their practical application as resource for quantum computation, quantum information has also become an essential element for our understanding of various physical problems, such as thermalization of isolated quantum many-body systems. This interplay between quantum information and computation, and quantum many-body systems is only expected to increase with time. In this thesis, we explore these topics in two parts, corresponding respectively to the two pillars mentioned above. In the first part, we study effects of quasiperiodicity on many-body quantum systems in low dimensions. QP systems are aperiodic but deterministic, so their behavior differs from that of clean systems and disordered ones as well. Moreover, these systems can be conveniently realized in an experimental setting where it is easier to isolate them from external decoherence. %Recent advancement in experimental techniques has made it easier to design and probe quantum systems with quasi-periodic modulations. We start with the easy-plane regime of the XXZ spin chain and show that the well-known fractal behavior of the spin Drude weight implies the divergence of the low-frequency conductivity for generic values of anisotropy. We tie this to the quasi-periodic structure in the Bethe ansatz solution resulting in different species of quasiparticles getting activated along the time evolution in a quasi-periodic pattern. We then study quantum critical systems under generic quasi-periodic modulations using real-space renormalization group (RSRG) procedure. In 1d, we show that the system flows to a new fixed point with the couplings following a discrete aperiodic sequence which allows us to analytically calculate the critical properties. We dub these new classes of quasi-periodic fixed points infinite-quasiperiodicity fixed points in line with the infinite-randomness fixed point observed in random quantum systems. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains. The RSRG is not analytically tractable in 2d, but numerically implementing it for the 2d quasi-periodic $q$-state quantum Potts model, we find that it is well controlled and becomes exact in the asymptotic limit. The critical behavior is shown to be largely independent of $q$ and is controlled by an infinite-quasiperiodicity fixed point. We also provide a heuristic argument for the correlation length exponent and the scaling of the energy gap. Moving on to the second part, we study monitored quantum circuits which have recently emerged as a powerful platform for exploring the dynamics of quantum information and errors in quantum systems. Unitary evolution generates entanglement between distant particles of the system. The dynamics of entanglement has been successfully studied by replacing the Hamiltonian evolution with random quantum circuits. Recently, the robustness of unitary evolution\u27s ability to protect the entanglement against external projective measurements has received much attention. Entanglement is also a resource for quantum information, so its stability is directly related to the stability of a quantum computer against external noises. It has been observed that, in absence of any symmetry, there is a measurement induced phase transition (MIPT) in the behavior of bipartite entanglement that goes from volume law to area law as we tune the rate of measurements. Here we focus on monitored quantum circuits with U(1) symmetry which leads to the presence of a conserved charge density. These diffusive hydrodynamic modes scramble very differently than non-symmetric modes and we find that in addition to the entanglement transition, there is another transition \textit{inside} the volume phase which we call a ``charge sharpening\u27\u27 transition. The sharpening transition is a transition in the ability/inability of the measurements to detect the global charge of the system. We study this sharpening transition in a variety of settings, including an effective field theory that predicts the transition to be in a modified Kosterlitz-Thouless universality class. We provide various numerical evidence to back our predictions

Optimization of a Runge-Kutta 4th Order Method-based Airbrake Control System for High-Speed Vehicles Using Neural Networks

The Runge-Kutta 4th Order (RK4) technique is extensively employed in the numerical solution of differential equations for airbrake control system design. However, its computational efficacy may encounter restrictions when dealing with high-speed vehicles that experience intricate aerodynamic forces. Using a Neural Network, a unique technique to improving the RK4-based airbrakes code is provided. The Neural Network is trained on numerous aspects of the high-speed vehicle as well as the current status of the airbrakes. This data was generated through the traditional RK4-based simulations and can predict the state of the airbrakes for any given state of the rocket in real-time. The proposed approach is demonstrated on a high-speed airbrakes control system, achieving comparable or better performance than the traditional RK4-based system while significantly reducing computational time by reducing the number of mathematical operations. The proposed method can adapt to changes in flow conditions and optimize the airbrakes system in real-time

A note on the quasiperiodic many-body localization transition in dimension d>1d>1

The nature of the many-body localization (MBL) transition and even the existence of the MBL phase in random many-body quantum systems have been actively debated in recent years. In spatial dimension $d>1$, there is some consensus that the MBL phase is unstable to rare thermal inclusions that can lead to an avalanche that thermalizes the whole system. In this note, we explore the possibility of MBL in quasiperiodic systems in dimension $d>1$. We argue that (i) the MBL phase is stable at strong enough quasiperiodic modulations for $d = 2$, and (ii) the possibility of an avalanche strongly constrains the finite-size scaling behavior of the MBL transition. We present a suggestive construction that MBL is unstable for $d \geq 3$.Comment: 5 pages, 2 figure

Developing and improving methods for robust ensemble classification: an aggregation operator and clustering-classification approach

Classification is an important technique (in pattern recognition) to categorise objects within their respective groups. In most real-world pattern recognition problems, it has become difficult to achieve best performance using an individual classifier. Ensemble algorithms, which are methods combining multiple individual classifiers, have already earned widespread approval within the machine learning community due to their ability to produce results in a wide range of applications. However, some challenges still exist in order to achieve a robust classification, in particular, classification of data points which are difficult to be assigned in one of the groups, and leveraging of existing external knowledge in order to better combine individual classifier outputs (fusion step of the ensemble). This thesis comprehensively explores these two key aspects and issues among the ensemble methods. The first challenge to generate a robust ensemble classification method is to classify data points which are difficult to label, across the applications using unlabelled datasets (and ensemble clustering frameworks). One specific problem due to this unclassified data is incomplete representation of the dataset. This limitation presents the need to introduce a new framework, which might help to improve the final classification by assigning more data to one of the groups. In this thesis, a robust two step framework is presented, which incorporates an ensemble classification stage after an ensemble clustering stage. Together, these combine to effectively identify core groups, distribute data within these groups and improve final classification through re-classifying unclustered data (that would otherwise be unassigned to any of the groups). Practical impact of the presented framework is demonstrated through application to novel real world datasets including two breast cancer datasets (breast cancer biological group stratification from the Nottingham and Edinburgh datasets), one heavy goods vehicle dataset (driving stereotypes from the Microlise dataset) and multiple standard datasets from the UCI repository (to demonstrate the robustness of the framework). Results obtained from these datasets show that our novel framework offers an improved, reliable and robust classification technique. These findings were also verified with statistical tests, visualisation techniques, cluster quality assessment and interpretation from experts (ground truth in case of the UCI repository). The second challenge focused in this thesis is leveraging external information for improving fusion step of the ensemble for a better ensemble classification performance. Insight on data offers the potentially extremely valuable prospect of leveraging external information. The use of this additional knowledge can lead to better ‘ensemble’ classification methods. One approach to capture this information is the use of aggregation operators, which combine the information from multiple sources with respect to a Fuzzy Measure (FM), which captures the worth of all the individual sources and all of their possible combinations. Several approaches to design the FMs exist in the literature; however, these methods do not leverage the external information, which could allow us to better understand the method of data fusion (or ensemble, in the case of ensemble classification). In this thesis, the concept of so called ‘A Priori’ FMs is introduced, which are generated based on external information and thus provide an alternative to the existing FM approaches (such as the algorithm-driven FMs). The thesis then proceed to develop two specific instances of such an A Priori FM in order to support the decision level fusion step in the ensemble classification. This new ensemble classification method is empirically assessed through application to multiple independent datasets. Results indicated that in cases where external information was available, the proposed ‘A Priori’ FM based ensemble classifier is a robust method achieving improved performances

Age-related hearing loss and its association with central obesity: experience at a tertiary centre

Background: Presbycusis is a slow, progressive, age-related sensorineural hearing loss, which is insidious, slow, progressive and irreversible disease and usually affects high pitch sound. It can be associated with various factors. Obesity is such a modifiable factor and its independent role with age-related hearing loss needs to be explored.Methods: This is a prospective study carried out over a period of three years in department of otorhinolaryngology at study institute. It included 1000 cases with symmetrical sensorineural hearing loss.Results: Among obese cases, high frequency hearing loss was found in significantly large number of cases. The most common audiogram in both male and female was Abrupt high tone loss type, irrespective of presence or absence of obesity.Conclusions: Obesity is a modifiable factor which has a significant association with high frequency hearing loss among the elderly population