Mathematics (Faculty of)http://hdl.handle.net/10012/99242023-06-03T12:16:10Z2023-06-03T12:16:10ZAlgorithms in Intersection Theory in the PlaneSt-Pierre, Catherinehttp://hdl.handle.net/10012/195192023-06-03T02:30:59Z2023-06-02T00:00:00ZAlgorithms in Intersection Theory in the Plane
St-Pierre, Catherine
This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal $I\subseteq \mathbb K[x,y]$, \textit{i.e.} finding the points (maximal ideals of $\mathbb K[x,y]/I$) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation.
Let $I\subseteq \mathbb K[x,y]$ be an ideal generated by $\mathcal F$, a subset of $\mathbb A[x,y]$ with $\mathbb A\hookrightarrow\mathbb K$ and $\mathbb K$ a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of $I$ or its primary component at the origin.
We introduce an $\mathfrak m$-adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of $I$ if $\mathfrak m\subseteq \mathbb A$ is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal $\mathfrak m$ suitable for our filtration.
When the field $\mathbb K$ is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete $\mathfrak m$-adic algorithm to recover $\mathcal G$, the Gr\"obner basis of $I$. We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of $n$ generators. We use this result to obtain a bound on the height of the coefficients of $\mathcal G$ and to control the probability of choosing a \textit{good} maximal ideal $\mathfrak m\subseteq\mathbb A$ to build the $\mathfrak m$-adic expansion of $\mathcal G$.
Inspired by Pardue (1994), we also give a constructive proof to
characterise a Zariski open set of $\mathrm{GL}_2(\mathbb K)$ (with action on $\mathbb K[x,y]$) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional $I$ becomes Borel-fixed when $|\mathbb K|$ is sufficiently large. This sharpens our analysis
to obtain, when $\mathbb A=\mathbb Z$ or $\mathbb A=k[t]$, a complexity less than cubic in terms of the dimension of $\mathbb Q[x,y]/\langle \mathcal G\rangle$ and softly linear in the height of the coefficients of $\mathcal G$.
We adapt the resulting method and present the analysis to find the $\langle x,y\rangle$-primary component of $I$. We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras.
2023-06-02T00:00:00ZEfficient Geo-Distributed Transaction ProcessingHildred, Joshua Thomashttp://hdl.handle.net/10012/195162023-06-01T02:30:57Z2023-05-31T00:00:00ZEfficient Geo-Distributed Transaction Processing
Hildred, Joshua Thomas
Distributed deterministic database systems support OLTP workloads over geo-replicated data. Providing these transactions with ACID guarantees requires a delay of multiple wide-area network (WAN) round trips of messaging to totally order transactions globally. This thesis presents Sloth, a geo-replicated database system that can serializably commit transactions after a delay of only a single WAN round trip of messaging. Sloth reduces the cost of determining the total global order for all transactions by leveraging deterministic merging of partial sequences of transactions per geographic region. Using popular workload benchmarks over geo-replicated Azure, this thesis shows that Sloth outperforms state-of-the-art comparison systems to deliver low-latency transaction execution.
2023-05-31T00:00:00ZImproving Cluster Scheduling Resiliency to Network FaultsQunaibi, Sarahttp://hdl.handle.net/10012/195152023-06-01T02:30:56Z2023-05-31T00:00:00ZImproving Cluster Scheduling Resiliency to Network Faults
Qunaibi, Sara
We present a comprehensive empirical study of the impact partial network partitions have on cluster managers in data analysis frameworks. Our study shows that modern scheduling approaches are vulnerable to partial network partitions. Partial partitions can lead to a complete cluster pause or a significant loss of performance.
To overcome the shortcoming of the state-of-the-art schedulers, we design the topology-aware scheduler (TAS). TAS incorporates the current network connectivity information when making a scheduling decision, to allocate fully connected nodes for a given application. TAS effectively hides partial partitions from applications. Our evaluation of a TAS prototype shows that it can tolerate partial network partitions, eliminate application halting or significant loss of performance.
2023-05-31T00:00:00ZDP-Select: Improving Utility and Privacy in Tabular Data Synthesis with Differentially Private Generative Adversarial Networks and Differentially Private SelectionEbrahimianghazani, Faezehhttp://hdl.handle.net/10012/195032023-05-30T02:31:12Z2023-05-29T00:00:00ZDP-Select: Improving Utility and Privacy in Tabular Data Synthesis with Differentially Private Generative Adversarial Networks and Differentially Private Selection
Ebrahimianghazani, Faezeh
This thesis proposes DP-Select, a novel approach to tabular data synthesis that combines DP-GAN and differentially private selection. We develop a mutual information-based selection method that is flexible and scalable for high-dimensional data and large numbers of marginals while being differentially private. We evaluate DP-Select on various datasets and demonstrate its effectiveness and utility compared to existing DP-GAN methods. Our results indicate that DP-Select significantly enhances the utility of synthesized data while maintaining strong privacy guarantees, making it a promising extension of DP-GANs for privacy-preserving data synthesis in terms of differential privacy. We also show that DP-Select performs better for smaller privacy budgets, making it an attractive option for scenarios with limited privacy budgets.
2023-05-29T00:00:00Z