Dissertation and Thesis

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On Universal C∗ Algebras associated to Operator Spaces and Generalized Crossed Products
(2026-07-08) Banik, Sayan Kansa
In my thesis, we study generalized crossed product constructions of the group $ C^* $-algebra $ C^*(G) $ with respect to certain completely positive maps, where $ G $ is assumed to be a discrete amenable group. We also investigate the universal $ C^* $-algebra $ \mathcal{E}_\alpha $ introduced by Hirshberg, which is constructed from $ C^*(G) $ and a pure injective homomorphism $ \alpha \colon G \to G $. In particular, we analyze its relationship with Exel's construction of generalized crossed products associated with the endomorphism of $ C^*(G) $ induced by $ \alpha $, together with an appropriate choice of transfer operator. In addition, we study crossed products of $ C^*(G) $ arising from states and conditional expectations of the form $ E_H \colon C^*(G) \to C^*(H) $, where $ H < G $ is a proper subgroup. We examine how generalized crossed product constructions change when passing from endomorphism-based framework to to the case of completely positive maps. We study crossed products of $ C^* $-algebras with respect to states, focusing in particular on $ C^*(G) $ equipped with its canonical trace and construct a spatial representation of the system isomorphic to the universal crossed product construction. Further, we show that when $ G $ is virtually abelian, there exists no spatial representation of the system $ (C^*(G), E_H) $ inside $ B(\ell^2(G)) $. Finally, we construct a specific spatial representation of this system and show that, when $[G:H]=\infty$ , the resulting spatial $ C^* $-algebra is isomorphic to the corresponding universal crossed product In the other half, we make a detailed study of operator spaces associated to Brown's noncommutative unitary $C^\ast$-algebra $\mathcal{U}^{nc}_n$ and related $C^\ast$-algebras. Specifically, we identify the universal $C^{\ast}$-algebra $C^{\ast}\langle M_n(\C)^{\ast}\rangle$ of the operator space $M_n(\C)^{\ast}$ with the non-commutative $C^{\ast}$-algebra $\Y^{nc}_n$, the universal unital $C^{\ast}$-algebra generated by elements $u_{ij}$, $1\leq i,j\leq n$ satisfying the relations which make $[u_{ij}]$ a contractive matrix. We also exhibited several operator algebraic properties of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$- in particular, we study the Lifting property (LP), residual finite dimensionality and primitivity of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$. Further, we study the maximal and minimal operator space structure of the standard generators of $\U^{nc}_n$ as well as $\U^{nc}_{n, red}$. Finally, we discuss a natural compact quantum semigroup structure on $C^{\ast}\langle M_n(\C)^{\ast}\rangle$, characterizing invertible elements in its state space under convolution.
Consumer Welfare and Privatization in Mixed Markets: A Developing Country Perspective
(2026-07-14) Dutta, Priyanka
This thesis is presented in four chapters, each examining a distinct aspect of mixed market structures, with a consistent focus on consumer surplus as the primary metric for evaluating the desirability of privatization or private provision in the context of developing countries. The first chapter introduces a symmetric Cournot oligopoly model in which both public and private firms produce a homogeneous good and compete in quantities. We find that consumer surplus is maximized at the two extremes: when the market consists solely of public firms or solely of private firms. In contrast, mixed regimes consistently yield intermediate outcomes, never achieving either the highest or the lowest level of consumer surplus. This result is robust to the level of competition, the specific objective functions assigned to public firms, and holds across all log-concave demand functions and convex cost functions. Importantly, when cost functions are strictly convex, we show that, contrary to conventional wisdom, an increase in the number of firms does not necessarily support privatization; in fact, it may weaken the case for introducing private firms. The first chapter further extends the analysis to Bertrand competition with differentiated products, where firms compete in prices rather than quantities. Unlike in Cournot settings, firms' strategies are strategic complements under Bertrand competition. Despite this fundamental difference in the nature of competition, the key finding persists: mixed oligopolies never maximize consumer surplus. Moreover, in some cases, mixed markets can yield the lowest consumer surplus, even compared to fully public or private regimes. This counterintuitive outcome stems from the regime-contingent behavior of public firms. That is, an inefficient public firm with welfare concerns may respond to rival pricing by setting a higher price in a mixed regime than it would in a fully public one, thereby dampening consumer surplus. While the first chapter shows that mixed markets never yield the highest or lowest consumer surplus, this finding appears at odds with their widespread existence and institutional support across the globe. Several explanations may account for this disconnect. First, privatization decisions are often driven by objectives such as profitability or broader welfare considerations, rather than consumer surplus alone. Second, governments concerned with consumer surplus may still prefer a mixed regime in settings with weak competition policy, where full privatization could increase the risk of collusion. However, in the next two chapters, we demonstrate that mixed regimes can, in fact, top the consumer surplus ranking without resorting to alternative welfare metrics or relying on collusion-based explanations. What is required is to move beyond the symmetric, single-stage oligopoly framework used in the first chapter. The second chapter relaxes the assumption of symmetric firms by introducing cost heterogeneity, a key real-world feature, as firms often differ in cost structures for reasons unrelated to ownership. While privatization can improve efficiency, it may not fully eliminate these underlying cost differences. In this setting, we show that mixed regimes can deliver the highest consumer surplus. For instance, in a duopoly, privatizing the inefficient firm while retaining public ownership of the efficient one can outperform both fully public and fully private regimes. Conversely, if privatization targets the more efficient firm, leaving the less efficient one public, consumer surplus can be the lowest among all ownership structures. These results suggest that ownership design should account for firm-specific efficiency differences, and that optimal privatization policy may be highly context-dependent. The third chapter addresses another limitation of earlier models by considering a vertically related market, with upstream and downstream monopolists interacting in a two stage production process. Public firms in both sectors introduce two layers of inefficiency, while private firms in both sectors generate two rounds of markups, a classic double marginalization problem. We show that a mixed regime can yield the highest consumer surplus by eliminating one layer of inefficiency and one round of markup. This outcome is most likely when markups are moderate and inefficiencies are unevenly distributed across the two sectors. If the inefficiency of public firms is symmetric across the upstream and downstream sectors, mixed regimes tend to perform intermediately or even poorly in terms of consumer surplus. However, when public firm inefficiencies differ sufficiently across sectors, and markup levels are not too high, a mixed structure can outperform both extremes. Extending the model to a richer setting with both upstream and downstream oligopoly shows that the desirability of mixed regimes persists and, in fact, becomes stronger as competition intensifies. Thus, the interaction between vertical structure, firm efficiency, and market power plays a critical role in shaping welfare outcomes in privatization decisions. While the first three chapters evaluate privatization using consumer surplus as a welfare metric under standard oligopoly assumptions, the fourth chapter introduces a new dimension: congestion. In many markets such as healthcare, education, telecommunications, transportation etc., congestion disutility arises as firms serve more consumers, reducing individual utility. We model a Cournot oligopoly with congestion under both mixed and fully private regimes and characterize equilibrium outcomes in the presence of congestion. We conduct comparative static analysis with respect to market size and competition and determine a consumer surplus threshold: a fully private regime yields a higher consumer surplus if the relative cost inefficiency of the public firm exceeds this threshold. We show that congestion and increases in market size both lower this threshold, making privatization more favorable. However, the effect of competition is more nuanced. An increase in competition facilitates privatization only when the initial level of competition is low. Beyond a certain point, additional competition could in fact facilitate public provision. These results highlight how market structures and conditions influence optimal ownership structure in the presence of congestion. Collectively, the four chapters of this thesis underscore the importance of evaluating privatization and ownership design through the lens of consumer welfare, particularly in the context of developing economies. The results challenge simplistic assumptions about public versus private provisions and offer a nuanced framework for understanding when mixed markets can be not just a compromise, but an optimal institutional structure.
Design of reliability acceptance sampling plans
(Indian Statistical Institute, 2026-07-27) Das, Rathin
A reliability acceptance sampling plan (RASP) is used for sampling and decision-making in the acceptance or rejection of a lot of products based on lifetime data obtained from a life test. In practice, censored life tests are employed due to limitations in cost, time, and other testing resources for the collection of lifetime data. This thesis develops the design of optimal RASPs under various censoring schemes and testing environments. Design of optimal Bayesian RASPs (BRASPs) are considered under interval censoring schemes (ICS) and hybrid censoring schemes using Bayesian decision-theoretic approaches. These models incorporate the adversarial relationship between manufacturers and consumers, who differ in prior beliefs and utility functions. Global market competitiveness and rapid technological advancement have pushed manufacturers to produce products with very high reliability. For such products, the mean time to failure under normal operating conditions is often prohibitively long. To address this issue, a BRASP based on a novel adaptive simple step-stress partial accelerated life test (ASSSPALT) framework is proposed under Type-I censoring using common prior and utility functions. The adaptive scheme dynamically adjusts stress levels based on observed failures, providing a general framework that accommodates both accelerated and non-accelerated testing under Type-I censoring. For complex products, failure may occur due to multiple causes. The work considers the design of RASP for competing risk data under progressive Type-I interval censoring using the producer's and consumer's risk approaches. The asymptotic properties of maximum likelihood estimators are derived to develop optimal plans. A frailty-based model is employed to capture dependence among competing risks and evaluate its influence on sampling plan performance. Subsequently, RASP is extended to a Bayesian framework under interval censoring. Further, for complex products with very high reliability, the design of BRASP is considered based on ASSSPALT under Type-II censoring for competing risk data. This framework unifies both accelerated and non-accelerated testing scenarios for competing risk data under Type-II censoring. The proposed methodologies for designing RASPs are illustrated using real-life data.
Distributed Computation of Graph Structures by Mobile Agents
(2026-07-15) Chand, Prabhat Kumar
This thesis investigates how mobile agents with no centralised control can be employed in anonymous networks to perform efficient distributed graph computations. The network is modelled as a simple, undirected, anonymous graph with n nodes and m edges, where nodes are memoryless and indistinguishable, and edges represent bidirectional communication links or traversal paths for the agents. The mobile agents are uniquely identifiable, possess limited local memory, and operate under a local communication model, in which communication is restricted to agents colocated at the same node. Under this computational model, we explore how mobile agents can collaborate effectively to solve global network problems—including dispersion in the presence of crashes, the construction of spanning trees, the identification of dominating sets, and the analysis of sub-graph hierarchies. We study the time complexity and memory usage per agent required to solve the above problems. The first contributory chapter addresses the fault-tolerant dispersion problem, aiming to evenly spread mobile agents across an anonymous graph in the presence of crash faults, from two initial configurations: rooted, where all the agents start at a single node, and arbitrary, where agents are initially scattered across the graph in multiple clusters. This chapter explores how dispersion can be achieved under both settings, ensuring that each node eventually hosts at most one functional agent despite crashes. The next chapter presents the problem of computing dominating sets using mobile agents, where two different algorithms are introduced: one for computing minimal dominating sets in O(m) time when the agents are gathered at a single node, and another for scenarios where agents start from multiple clusters. Additionally, an ln ∆ approximation algorithm for the minimum dominating set problem is provided, where ∆ is the maximum degree of the graph. The subsequent chapter focuses on subgraph analytics using mobile agents dispersed across the nodes of a graph. We present algorithms for triangle counting (3-cycles) in general graphs and butterfly counting (4-cycles) in bipartite graphs. The triangle counting framework extends to related problems such as truss decomposition, triangle centrality, and local clustering coefficient. These methods enable the distributed identification of cohesive structures and dense subgraphs, with butterfly counting being of relevance to bipartite graphs commonly found in social network analysis and recommendation systems. The final chapter focuses on constructing tree structures, specifically BFS trees and minimum spanning trees, using mobile agents. These algorithms, which assume minimal prior knowledge, improve upon existing methods by achieving better time complexity and optimal memory usage. Throughout the thesis, these graph problems are explored through the lens of the mobile agent framework, focusing on minimising the time complexity of the algorithms and memory usage per agent.
A Regression Tree Framework for Denoising and Monitoring of Image Data
(Indian Statistical Institute, 2026-07-02) Basak, Subhasish
The proliferation of advanced image acquisition technologies has led to the routine collection of large-scale image data across numerous scientific domains. This widespread reliance on image data accentuates the imperative to develop robust and efficient imaging techniques, which are essential for supporting modern applications across various scientific and industrial domains. This dissertation focuses on the development and analysis of methods for image denoising and image monitoring, two fundamental tasks in modern image analysis. A wide array of image denoising techniques exists in the literature, each tailored to handle specific types of noise or structural characteristics. However, no single method proves universally optimal, as each comes with its own advantages and trade-offs. In the first part of the dissertation, different configurations of local neighbourhoods are investigated, and an adaptive framework is proposed that combines these with local clustering-based smoothing to effectively harness the advantages of both methodologies. The dissertation then introduces a regression tree-based framework utilizing Oblique-axis Regression Trees (ORT) to estimate discontinuous regression functions in finite-dimensional spaces and applies this methodology to achieve effective image denoising. Due to an alternative set of assumptions on the underlying regression function, the overall structure of the proofs is substantially simpler than those typically found in the existing literature on regression trees. Finally, leveraging the ORT framework, the dissertation introduces an original approach to monitor drift patterns within an image sequence. Even though gradual temporal variations, known as drifts, are frequently observed in image sequences, drift monitoring remains an underexplored research area. This dissertation thus makes an effort to address that gap. Theoretical analysis and numerical studies, conducted on both simulated and real-world data, demonstrate the broad applicability and effectiveness of the proposed methods.
Dynamic Property Ordering for Efficient Multi-Property Bounded Model Checking
(Indian Statistical Institute, 2026-06-16) Kumar, Vivek
Formal verification plays a critical role in ensuring the correctness of modern hardware designs. As the complexity of digital systems increases, designs are often associated with a large number of verification properties that must be analyzed within limited computational resources. In conventional multi-property bounded model checking (BMC), all properties are verified simultaneously. While this approach enables parallel analysis, difficult properties can consume a disproportionate amount of resources, causing simpler properties to be delayed and reducing the overall efficiency of bug detection. This thesis presents dynamic property ordering techniques for efficient multi-property verification using SAT-based bounded model checking in the ABC verification framework. The central idea is to verify properties individually and dynamically prioritize them based on their observed verification progress, allowing computational resources to be directed toward properties that are more likely to yield results within a given time budget.Two dynamic property ordering algorithms are proposed. The first algorithm, ALG1, employs a round-robin style strategy in which unsolved properties are periodically reordered according to the maximum verification depth (frame) reached, prioritizing properties that demonstrate greater progress. The second algorithm, ALG2, adopts a priority-based scheduling approach where each property’s priority is determined by its verification rate, measured as frames explored per second. Properties with higher progress rates are allocated greater verification resources. The proposed approaches are evaluated on benchmark suites from the Hardware Model Checking Competition (HWMCC) 2012 and 2013 and compared against two baselines: the conventional ABC multi-property verification method and an Equal Time Bounding (ETB) strategy that distributes the available verification time equally among all properties. Experimental results demonstrate that dynamic property ordering significantly improves verification efficiency. Both ALG1 and ALG2 solve more properties and achieve greater verification depth within the same time budget, while also accelerating bug discovery. Across the benchmark set, the proposed methods provide improvements exceeding 40% over the baseline approaches in key performance metrics. The results demonstrate that dynamic property scheduling is an effective technique for improving the scalability and effectiveness of multi-property bounded model checking, offering a practical solution for faster bug detection and enhanced utilization of verification resources.
Deep Reinforcement Learning with Directed Asymmetry and Kolmogorov-Arnold Networks for Dismantling Interdependent Multiplex Networks
(Indian Statistical Institute, 2026-06-16) Dev, Soumyajit
Identifying the minimum-cost node-removal sequence that fragments a complex network - the network dismantling problem is NP-hard and central to infrastructure resilience. In interdependent multiplex networks, this difficulty is compounded by cascading cross-layer failures. While deep reinforcement learning (DRL) agents utilizing graph neural network (GNN) encoders achieve near-optimal dismantling, current state-of-the-art architectures suffer from two critical limitations. Topologically, existing agents strictly assume undirected edges, rendering them inapplicable to directed systems - such as supply chains or gene regulatory cascades - where failure propagation is fundamentally asymmetric. To resolve this, we propose Disassembling Directed Interdependent Networks (DDIN). DDIN introduces an asymmetric GraphSAGE encoder that explicitly separates incoming influence from outgoing control aggregations, paired with a multi-relational attention mechanism for cross-layer dependency fusion. Evaluated zero-shot on five real-world directed networks, DDIN achieves a 16-23% reduction in the Area Under the Dismantling Curve (AUDC) over heuristic baselines. Functionally, existing GNN encoders parameterise message-passing through multi-layer perceptrons (MLPs). These fixed-affine projections lack the capacity to model the non-smooth, high-frequency vulnerability patterns governing cascading fragility in scale-free topologies. We address this by proposing Kolmogorov-Arnold Reinforcement Learning (KARL), the first architecture to embed learnable univariate functions into an off-policy DRL combinatorial optimiser. KARL features a residual B-spline KAN encoder to stabilise gradient norms during deep message-passing, a fully KAN-parameterised cross-layer attention module (KANformer), and an orthogonal Chebyshev-KAN action-value head to mitigate boundary oscillations under non-stationary temporal-difference targets. Evaluated zero-shot on six real-world undirected multiplex networks, KARL yields a 21.96% average AUDC improvement over state-of-the-art baselines, alongside emergent structural interpretability. By independently resolving the directed topology limitation and the affine expressivity bottleneck, this dissertation establishes a robust architectural foundation for structurally faithful network dismantling models.
Adaptive Spectral Trust Gate for Physics- Constrained Operator Learning
(2026-06-16) Chakraborty, Soham
Physics-informed machine learning improves the plausibility, data-efficiency and generalization of surrogate models by injecting prior physical knowledge into the learning process. The current approaches can be broadly divided into two main categories: soft constraints, which add a physics residual to the training loss but guarantee nothing at inference time, and hard constraints, which project the model output onto the constraint set exactly but apply the projection uniformly to every part of the signal — including parts that are dominated by noise, discretization error, or model mismatch, where the idealized physics is not actually trustworthy. This dissertation proposes the Adaptive Spectral Trust Gate (ASPINO), a mechanism that learns where to trust the physics. Operating in the Fourier domain on top of any surrogate model, a small gating network forms a per-mode convex combination of a data-driven soft path and a physics hard path. The gate is driven by features of the spectral coordinate and the spectral amplitude, so that it can apply the hard constraint in well-conditioned spectral regions and defer to the data-driven operator in regions corrupted by noise or aliasing. A single gate serves two very different hard paths - the linear Leray projection (incompressible flow) and a nonlinear rank-r SVD projection (massive-MIMO channel estimation). On the theoretical side, we give an empirical-Rademacher-complexity analysis: an unconditional safety floor — the gated class never exceeds the soft path it wraps — and, under a stated low-rank-transfer assumption, a capacity-reduction factor of 1 − ¯α (1 − √ρr), where ¯α is the fraction of capacity routed through the hard path. On Kolmogorov-flow denoising ASPINO is simultaneously the most accurate and near physical, dominating the unconstrained, hard and soft baselines; on ray-traced MIMO it improves a strong physics-informed baseline across all pilot budgets and SNRs without ever regressing. A third study, zero-shot super-resolution on the Poisson equation, confirms the discretization invariance of the gated construction. ASPINO is discretization-invariant and “plug-and-play” over the underlying operator.
Predictive importance sampling based coverage verification for multi uav trajectory planning
(Indian Statistical Institute, 2026-06-23) Ghosh, Snehashish
In next-generation wireless networks, unmanned aerial vehicle (UAV) networks are emerging as a promising solution for ultra-reliable low-latency communication (URLLC). A key challenge in millimeter-wave UAV networks is ensuring that mobile users are always in line-of-sight (LoS) coverage, since the current snapshot-based trajectory planning approach does not consider the mobility of the users during the decision interval, resulting in disastrous LoS gaps. For continuous coverage verification, standard uniform sampling is too computationally expensive, as it would need a large number of samples to estimate rare failure events that have latencies that are not suitable for real-time requirements. In this work, we introduce a Predictive Importance Sampling (PIS) framework that significantly decreases sample complexity by focusing verification efforts on regions where failure is predicted. Specifically, we propose a Long Short-Term Memory Mixture Density Network (LSTM-MDN) architecture to learn multimodal user trajectory distributions and introduce a defense approach based on mixture sampling to handle the robustness against the prediction error. We show that PIS yields unbiased failure probability estimates that have lower variance than uniform sampling. We then combine PIS with Multi-Agent Deep Deterministic Policy Gradient (MADDPG) to perform coordinated multi-UAV trajectory planning based on an energy-aware multi-objective reward function that balances throughput, coverage, fairness and energy consumption. Based on the simulation results, our proposed method improves the coverage rate, throughput and verification latency in comparison with three state-of-the-art methods, thus enabling proactive coverage management for URLLC-aware UAV networks.
Telluric Correction of M-dwarf Stars using Machine Learning
(Indian Statistical Institute, 2026-06-17) Rana, Sayak
The study of M-dwarf stars is of prime scientific interest to us because of their closer habitable zones and the favorable conditions they offer for exoplanet detection. However, telluric contamination of the ground-based spectra results in sharp absorption lines, which makes their study cumbersome. Removing this contamination is necessary for estimating key stellar parameters. The central contribution is a one-dimensional Convolutional Neural Network (CNN) that retrieves the four atmospheric parameters governing telluric absorption: pressure, temperature, humidity, and airmass. These predicted parameters are passed to Telfit which produces an estimated telluric spectrum. The observed spectrum is then divided by this estimated telluric spectrum to obtain the telluric corrected spectrum. As this network is trained exclusively on synthetic spectra, a domain gap exists at inference. Two domain-adaptation strategies are evaluated: a CycleGAN following the Cycle-StarNet framework for explicit synthetic-to-real translation and a Domain-Adversarial Autoencoder (DAAE) that learns domain-invariant spectral representations. The Domain-Adversarial Autoencoder (DAAE) achieves the lowest loss against a telluric corrected CARMENES reference spectrum outperforming all other model variants. The CycleGAN fails due to discriminator collapse under severe class imbalance between the number of real and synthetic spectra.

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