http://repository.iiitd.edu.in/xmlui/handle/123456789/618 2026-06-10T19:31:46Z 2026-06-10T19:31:46Z Bittu Chaubey, Sneha (Advisor) http://repository.iiitd.edu.in/xmlui/handle/123456789/1982 2026-05-30T22:18:19Z 2026-04-01T00:00:00Z

Topics in the distribution of farey sequences Bittu; Chaubey, Sneha (Advisor) In this thesis, we study the distribution of Farey sequences. Let Q be a positive integer. The Farey sequence FQ of order Q is the set of irreducible fractions in [0, 1] whose denominators do not exceed Q. The first study characterizing the behavior of sequences via equidistribution was carried out in the seminal paper of Weyl. Equidistribution refers to being evenly spaced in a measure space. Equidis- tributed sequences are particularly useful for performing numerical integration. The notion of equidistribution does not provide information about finer statistics, such as randomness, local clustering, and periodic structure of sequences. To study the fine-scale statistics of a sequence, one can study the nearest neighbor gap distri- bution, as well as ν-level correlation measure. We study the equidistribution and correlation measure for Farey sequences. The study of the Farey sequence is of independent interest because of its role in the Diophantine approximation, the circle method, and its connection to the Rie- mann Hypothesis as established by the classical work of Franel and Landau. The Farey fractions of order Q have a one-to-one correspondence with visible lattice points in the triangle with vertices (0, 0), (0, Q), and (Q, Q) through straight lines passing through the origin. The visible lattice points along polynomials have been introduced and studied by Chaubey et al. Motivated by this, we introduce polyno- mial Farey fractions as a subset of fractions a/q ∈ [0, 1] such that the point (a, q) is visible through polynomial curves and examine their distribution. In particular, we study and prove that the lim sup of the pair correlation measure of the polynomial Farey sequence is bounded. For the specific polynomial P (x) = x(x + 1), we show. that the pair correlation measure exists and establish an explicit formula for the pair correlation function which is non-Poissonian. Further, when restricting to prime denominators, the pair correlation measure is shown to be Poissonian. A sequence is said to be Poissonian if it behaves like a random uniformly distributed sequence. It is interesting to study the distribution of Farey fractions with denominators in arithmetic progression, as it is closely related to the Generalized Riemann Hypothesis. Moreover, we study an analog of Chebyshev’s bias question for polynomial Farey fractions with denominators in an arithmetic progression. Chebyshev’s bias question deals with the prime number races and states that there are more primes of the form 4n + 3 than the primes of the form 4n + 1. Furthermore, we study the distribution of the sequence of Farey fractions with k-free denominators lying in an arithmetic progression, denoted by F (m) Q,k . We prove that the sequence F (m) Q,k Q is equidistributed by establishing an estimate for a Weyl sum. Additionally, we establish an equivalent criterion for the Generalized Riemann Hypothesis in terms of the distribution of fractions in F (m) Q,k analogous to the classical results of Franel and Landau. We also investigate the correlation measure of the sequence F (m) Q,k Q and provide an explicit form for the pair correlation measure. Another effective approach to understanding the distribution of the Farey fractions is to examine their indices. We study the distribution of Farey indices by deriving asymptotic formulas for the moments of the index function of Farey fractions with B-free denominators which lie in a given arithmetic progression.

2026-04-01T00:00:00Z Singh, Karamjeet Raman, Rajiv (Advisor) http://repository.iiitd.edu.in/xmlui/handle/123456789/1817 2026-02-26T22:00:27Z 2026-01-13T00:00:00Z

On support and recognition problems for sparse hypergraphs Singh, Karamjeet; Raman, Rajiv (Advisor) A hypergraph H is a pair (V, E), where V is a set of vertices, and E is a collection of subsets of V , called hyperedges. They are used to express complex relations, and they generalize graphs where each element of E is a 2-element subset of V . Hypergraphs are one of the most important combinatorial objects of study in theoretical computer science, and have applications in several domains, including network design, scheduling problems, biology, machine learning, etc. Thus, it is important to study their structural properties. Starting with the work of Zykov [Zyk74], Voloshina and Feinberg [VF84], and John- son and Pollack [JP87], researchers have made several attempts to study the structure of a hypergraph by associating with it an appropriate graph. While their initial attempts were to introduce the planarity of a hypergraph, the notion developed in [VF84; JP87] can be generalized and is now called a support. A support for a hypergraph H = (V, E) is a graph Q = (V, F ) such that for each hyperedge E ∈ E, the induced subgraph Q[E] on the elements of E is connected. With this notion, a hypergraph is considered planar if it admits a support that is a planar graph. The concept of support has practical applications in hypergraph visualization, net- work design, and several optimization problems. Although deciding whether a hyper- graph admits a planar support is NP-hard, identifying sufficient conditions for the existence of such supports, particularly sparse or structured ones, remains a compelling research direction. Most of this thesis delves into the construction of supports for various graph classes. This thesis is divided into three parts. In Part (A), we consider hypergraphs defined by subgraphs of a given host graph. Let G = (V, E) be a graph and H be a collection of subgraphs of G. Then the pair (G, H) naturally defines a hypergraph with vertex set V and a hyperedge V (H) for each H ∈ H. We study support construction in three different settings, depending on whether the host graph G belongs to the class of graphs of (i) bounded genus, (ii) outerplanar, or (iii) bounded treewidth. We gave sufficient conditions that ensure the existence of a support from the same family of graphs as G. The results are extended to dual hypergraphs and to a more general setting- the intersection hypergraphs. We also present a fast algorithm for the construction of a planar support with straight-line embedding when the underlying hypergraph is defined by axis-parallel rectangles and points in R2. Part (B) of the thesis explores the role of supports in solving classical problems such as packing, covering, and coloring problems in hypergraphs. We study these problems for hypergraphs arising from subgraphs of a host graph as well as from geometric regions on orientable surfaces, and present approximation results to the packing and covering problems above. Finally, Part (C) turns to abstract hypergraphs and examines the computational complexity of identifying vertex orderings that forbid fixed patterns. We show NP- hardness of this problem for several vertex orderings, and we deduce implications for the recognition of hypergraphs defined by geometric regions in R2.

2026-01-13T00:00:00Z Bhavna Pandey, Ashish Kumar (Advisor) http://repository.iiitd.edu.in/xmlui/handle/123456789/1771 2025-09-20T22:00:30Z 2025-08-01T00:00:00Z

Spectral instabilities in water wave models Bhavna; Pandey, Ashish Kumar (Advisor) This thesis investigates the spectral instabilities of various nonlinear water wave models through rigorous analytical techniques. Focusing on three fundamental types of instabilities, modulational instability, transverse instability, and high-frequency instability, the work provides a unified spectral framework to study how small perturbations evolve and potentially destabilize wave solutions in dispersive systems. We begin by analyzing modulational instability, wherein a periodic traveling wave becomes unstable to long-wavelength perturbations. Using perturbation theory and spectral analysis, we characterize conditions under which modulational instability arises in generalized Ostrovsky equations. The effect of dispersion, nonlinearity, and surface tension is examined in detail. The study then turns to transverse instability, where planar wave trains destabilize due to perturbations in the transverse direction. We consider rotation-modified and surface tension-influenced variants of the Kadomtsev–Petviashvili (KP) equation, the rotation-modified KP equation, and the KD equation, and identify parameter regimes leading to transverse spectral instabilities. Finally, we explore high-frequency instability, focusing on the behavior of the spectrum. We demonstrate how high-frequency perturbations can induce instabilities in small-amplitude periodic traveling waves. Altogether, the results contribute to a deeper understanding of how wave coherence is affected by perturbations of various scales and directions. The insights gained have potential implications for the stability of wave patterns in physical settings such as oceanography, fluid mechanics, and nonlinear optics.

2025-08-01T00:00:00Z Khurana, Divya Banerjee, Shilpak (Advisor) Chaubey, Sneha (Advisor) Kunde, Philipp (Advisor) http://repository.iiitd.edu.in/xmlui/handle/123456789/1768 2025-09-06T22:00:31Z 2025-08-01T00:00:00Z

Construction of some new diffeomorphisms using a combinatorial approach: the approximation by conjugation method Khurana, Divya; Banerjee, Shilpak (Advisor); Chaubey, Sneha (Advisor); Kunde, Philipp (Advisor) Ergodic theory often involves dealing with transformations of measure spaces that preserve specific measures, such as volume or probability measures. These transformations establish a framework for studying the behavior of dynamical systems. Combinatorial methods are pivotal in ergodic theory, providing valuable techniques for understanding the behavior of dynamical systems. They are often utilized in the study of measure-preserving transformations, aiding in characterizing invariant measures, exploring ergodic decomposition, and establishing connections between different dynamical systems. Moreover, numerous scenarios exist where we can effectively characterize a measure-preserving transformation and its long-term behaviour by approaching it as the limit of specific finite objects, such as periodic processes. This thesis discusses combinatorial constructions within Ergodic Theory, employing the “Approximation by Conjugation” method. This technique facilitates the construction of maps with specific topological and measure-theoretic characteristics on the manifolds that support a non-trivial circle action. We provide examples of volume-preserving diffeomorphisms with zero topological entropy. These examples exhibit intricate ergodic properties in both smooth and analytic categories. Specifically, we present an example of a smooth diffeomorphism with an invariant measure, which is a generic but non-ergodic volume measure. This diffeomorphism satisfies various other ergodic and topological dynamic properties on the 2-torus. In ergodic theory, generic points are essential for understanding the statistical and dynamical behavior of systems, as their orbits cover most of the phase space according to the invariant measure. We examine the distinctions between generic and non-generic points in smooth dynamical systems, focusing on establishing bounds for the sizes of their respective sets. We construct an explicit collection of sets that encompass all generic points within the system and focus on determining bounds for their Hausdorff dimension, leading to more insightful and compelling conclusions. Additionally, we explore the ergodic properties of diffeomorphisms and examine their differential maps with respect to a smooth measure in the projectivization of the tangent bundle. We investigate examples of diffeomorphisms with complex ergodic properties within the projectivized tangent bundle for the smooth and analytic case. We construct a smooth diffeomorphism whose differential is weakly mixing with respect to a smooth measure in the projectivization of the tangent bundle. In the case of the 2-torus, we also obtain the analytic counterpart of such a diffeomorphism by utilizing an analytic approximation technique. We present an example of an analytic diffeomorphism whose projectivized derivative extension exhibits weak mixing with respect to a smooth measure. All of these constructions are based on the original and quantitative version of the “Approximation by Conjugation” method. This approach utilizes highly explicit well-defined conjugation maps, explicit partial partitions of the space, specific approximation strategies, and geometric and combinatorial criteria to meet specific requirements.

2025-08-01T00:00:00Z