フランス国リヨン高等師範学校との共同学位プログラムによる学位 / 京都大学 / 新制・課程博士 / 博士(理学) / 甲第23449号 / 理博第4743号 / 新制||理||1680(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 COLLINS Benoit Vincent Pierre, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Thesis advisor: Pankaj Mehta / Thesis advisor: Ziqiang Wang / Nature has revealed an astounding degree of phylogenetic and physiological diversity in natural environments -- especially in the microbial world. Microbial communities are incredibly diverse, ranging from 500-1000 species in human guts to over 1000 species in marine ecosystems. Historically, theoretical ecologists have devoted considerable effort to analyzing ecosystems consisting of a few species. However, analytical approaches and theoretical insights derived from small ecosystems consisting of a few species may not scale up to diverse ecosystems. Understanding such large complex ecosystems poses fundamental challenges to current theories and analytical approaches for modeling and understanding the microbial world. One promising approach for tackling this challenge that I develop in my thesis is to adapt and expand ideas from statistical mechanics to theoretical ecology. Statistical mechanics has helped us to understand how collective behaviors emerge from the interaction of many individual components. In this thesis, I present a unified theoretical framework for understanding complex ecosystems based on statistical mechanics, random matrix theories, and convex optimization. My thesis work has three key aspects: modeling, simulations, and theories. Modeling: Classical ecological models often focus on predator-prey relationships. However, this is not the norm in the microbial world. Unlike most macroscopic organisms, microbes relie on consuming and producing small organic molecules for energy and reproduction. In this thesis, we develop a new Microbial Consumer Resource Model that takes into account these types of metabolic cross-feeding interactions. We demonstrate that this model can qualitatively reproduce and explain statistical patterns observed in large survey data, including Earth Microbiome Project and the Human Microbiome Project. Simulations: Computational simulations are essential in theoretical ecology. Complex ecological models often involve ordinary differential equations (ODE) containing hundreds to thousands of interacting variables. Typical ODE solvers are based on numerical integration methods, which are both time and resource intensive. To overcome this bottleneck, we derived a surprising duality between constrained convex optimization and generalized consumer-resource models describing ecological dynamics. This allows us to develop a fast algorithm to solve the steady-state of complex ecological models. This improves computational performance by between 2-3 orders of magnitude compared to direct numerical integration of the corresponding ODEs. Theories:Few theoretical approaches allow for the analytic study of communities containing a large number of species. Recently, there has been considerable interest in the idea that ecosystems can be thought of as a type of disordered systems. This mapping suggests that understanding community coexistence patterns is actually a problem in "spin-glass'' physics. This has motivated physicists to use insights from spin glass theory to uncover the universal features of complex ecosystems. In this thesis, I use and extend the cavity method, originally developed in spin glass theories, to answer fundamental ecological questions regarding the stability, diversity, and robustness of ecosystems. I use the cavity method to derive new species backing bounds and uncover novel phase transitions to typicality. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

By endowing the base station with hundreds of antennas and relying on spatial multiplexing, massive multiple-input-multiple-output (MIMO) allows impressive advantages in many fronts. To reduce this promising technology to reality, thorough performance analysis has to be conducted. Along this line, this work is focused on the convenient high-dimensionality of massive MIMO’s corresponding model. Indeed, the large number of antennas allows us to harness asymptotic results from Random Matrix Theory to provide accurate approximations of the main performance metrics. The derivations yield simple closed-form expressions that can be easily interpreted and manipulated in contrast to their alternative random equivalents. Accordingly, in this dissertation, we investigate and optimize the performance of massive MIMO in different contexts. First, we explore the spectral efficiency of massive MIMO in large-scale multi-tier heterogeneous networks that aim at network densification. This latter is epitomized by the joint implementation of massive MIMO and small cells to reap their benefits. Our interest is on the design of coordinated beamforming that mitigates cross-tier interference. Thus, we propose a regularized SLNR-based precoding in which the regularization factor is used to allow better resilience to channel estimation errors. Second, we move to studying massive MIMO under Line-of-Sight (LoS) propagation conditions. To this end, we carry out an analysis of the uplink (UL) of a massive MIMO system with per-user channel correlation and Rician factor. We start by analyzing conventional processing schemes such as LMMSE and MRC under training-based imperfect-channel-estimates, and then, propose a statistical combining technique that is more suitable in LoS-prevailing environments. Finally, we look into the interplay between LoS and the fundamental limitation of massive MIMO systems, namely, pilot contamination. We propose to analyze and compare the performance using single-cell and multi-cell detection methods. In this regard, the single-cell schemes are shown to produce higher SEs as the LoS strengthens, yet remain hindered by LoS-induced interference and pilot contamination. In contrast, for multi-cell combining, we analytically demonstrate that M-MMSE outperforms both single-cell detectors by generating a capacity that scales linearly with the number of antennas, and is further enhanced with LoS.

No description available.

No / Eigenvalue based spectrum sensing schemes such as Maximum Minimum Eigenvalue (MME), Maximum Energy Detection (MED) and Energy with Minimum Eigenvalue (EME) have higher spectrum sensing performance without requiring any prior knowledge of Primary User (PU) signal but the decision hypothesis used in these eigenvalue based sensing schemes depends on the calculation of maximum eigenvalue from covariance matrix of measured signal. Calculation of the covariance matrix followed by eigenspace analysis of the covariance matrix is a resource intensive operation and takes overhead time during critical process of spectrum sensing. In this paper we propose a new blind spectrum sensing scheme based on the approximation of the maximum eigenvalue using state of the art results from Random Matrix Theory (RMT). The proposed sensing scheme has been evaluated through extensive simulations on wireless microphone signals and the proposed scheme shows higher probability of detection (Pd) performance. The proposed spectrum sensing also shows higher detection performance as compared to energy detection scheme and RMT based sensing schemes such as MME and EME.

Reliability and computational efficiency of classification error estimators are critical factors in classifier design. In a high-dimensional data setting where data is scarce, the conventional method of error estimation, cross-validation, can be very computationally expensive. In this thesis, we consider a particular discriminant analysis type classifier, the Randomly-Projected RLDA ensemble classifier, which operates under the assumption of such a ‘small sample’ regime. We conduct an asymptotic study of the generalization error of this classifier under this regime, which necessitates the use of tools from the field of random matrix theory. The main outcome of this study is a deterministic function of the true statistics of the data and the problem dimension that approximates the generalization error well for large enough dimensions. This is demonstrated by simulation on synthetic data. The main advantage of this approach is that it is computationally efficient. It also constitutes a major step towards the construction of a consistent estimator of the error that depends on the training data and not the true statistics, and so can be applied to real data. An analogous quantity for the Randomly-Projected LDA ensemble classifier, which appears in the literature and is a special case of the former, is also derived. We motivate its use for tuning the parameter of this classifier by simulation on synthetic data.

This thesis, divided into two parts, is concerned with the analysis of spectral and dynamical characteristics of certain quantum spin systems in the presence of either I) quenched disorder, or II) dynamical noise. In the first part, the quantum random energy model (QREM), a mean-field spin glass model with a many-body localisation transition, is studied. In Chapter 2, we attempt a diagrammatic perturbative analysis of the QREM from the ergodic side, proceeding by analogy to the single-particle theory of weak localisation. Whilst we are able to describe diffusion, the analogy breaks down and a description of the onset of localisation in terms of quantum corrections quickly becomes intractable. Some progress is possible by deriving a quantum kinetic equation, namely the relaxation of the one-spin reduced density matrix is determined, but this affords little insight and extension to two-spin quantities is difficult. We change our approach in Chapter 3, studying instead a stroboscopic version of the model using the formalism of quantum graphs. Here, an analytic evaluation of the form factor in the diagonal approximation is possible, which we find to be consistent with the universal random matrix theory (RMT) result in the ergodic regime. In Chapter 4, we replace the QREM's transverse field with a random kinetic term and present a diagrammatic calculation of the average density of states, exact in the large-N limit, and interpret the result in terms of the addition of freely independent random variables. In the second part, we turn our attention to noisy quantum spins. Chapter 5 is concerned with noninteracting spins coupled to a common stochastic field; correlations arising from the common noise relax only due to the spins' differing precession frequencies. Our key result is a mapping of the equation of motion of n-spin correlators onto the (integrable) non-Hermitian Richardson-Gaudin model, enabling exact calculation of the relaxation rate of correlations. The second problem, addressed in Chapter 6, is that of the dynamics of operator moments in a noisy Heisenberg model; qualitatively different behaviour is found depending on whether or not the noise conserves a component of spin. In the case of nonconserving noise, we report that the evolution of the second moment maps onto the Fredrickson-Andersen model - a kinetically constrained model originally introduced to describe the glass transition. This facilitates a rigorous study of operator spreading in a continuous-time model, providing a complementary viewpoint to recent investigations of random unitary circuits.

First, we consider the group integrals where integrands are the monomials of matrix elements of irreducible representations of classical groups. These group integrals are invariants under the group action. Based on analysis on Young tableaux, we investigate some related duality theorems and compute the asymptotics of the group integrals for fixed signatures, as the rank of the classical groups go to infinity. We also obtain the Viraosoro constraints for some partition functions, which are power series of the group integrals. Second, we show that the proof of Witten's conjecture can be simplified by using the fermion-boson correspondence, i.e., the KdV hierarchy and Virasoro constraints of the partition function in Witten's conjecture can be achieved naturally. Third, we consider the partition function involving the invariants that are intersection numbers of the moduli spaces of holomorphic maps in nonlinear sigma model. We compute the commutator of the representation of Virasoro algebra and give a fat graph(ribbon graph) interpretation for each term in the diferential operators.

This dissertation presents the applications of state-of-the-art computation techniques and data analysis algorithms in three physical and biological problems: assembling DNA pieces, optimizing self-assembly yield, and identifying correlations from large multivariate datasets. In the first topic, in-depth analysis of using Sequencing by Hybridization (SBH) to reconstruct target DNA sequences shows that a modified reconstruction algorithm can overcome the theoretical boundary without the need for different types of biochemical assays and is robust to error. In the second topic, consistent with theoretical predictions, simulations using Graphics Processing Unit (GPU) demonstrate how controlling the short-ranged interactions between particles and controlling the concentrations optimize the self-assembly yield of a desired structure, and nonequilibrium behavior when optimizing concentrations is also unveiled by leveraging the computation capacity of GPUs. In the last topic, a methodology to incorporate existing categorization information into the search process to efficiently reconstruct the optimal true correlation matrix for multivariate datasets is introduced. Simulations on both synthetic and real financial datasets show that the algorithm is able to detect signals below the Random Matrix Theory (RMT) threshold. These three problems are representatives of using massive computation techniques and data analysis algorithms to tackle optimization problems, and outperform theoretical boundary when incorporating prior information into the computation. / Engineering and Applied Sciences

No description available.