The investigation of the latent geometrical space behind complex network topologies is a fervid topic in current network science and the hyperbolic space is one of the most studied, because it seems associated to the structural organization of many real complex systems. The popularity-similarity-optimization (PSO) generative model is able to grow random geometric graphs in the hyperbolic space with realistic properties such as clustering, small-worldness, scale-freeness and rich-clubness. However, it misses to reproduce an important feature of real complex systems, which is the community organization. Here, we introduce the nonuniform PSO (nPSO) generative model, a generalization of the PSO model with a tailored community structure, and we provide an efficient algorithmic implementation with a O(EN) time complexity, where N is the number of nodes and E the number of edges. Meanwhile, in recent years, the inverse problem has also gained increasing attention: given a network topology, how to provide an accurate mapping into its latent geometrical space. Unlike previous attempts based on a computationally expensive maximum likelihood optimization (whose time complexity is between O(N^3) and O(N^4)), here we show that a class of methods based on nonlinear dimensionality reduction can solve the problem with higher precision and reducing the time complexity to O(N^2).

<p>The missed-thrust problem is a modern challenge in the field of mission design. While some methods exist to quantify its effects, there still exists room for improvement for algorithms which can fully anticipate and plan for a realistic set of missed-thrust events. The present work investigates the use of machine learning techniques to provide a robust controller for a low-thrust spacecraft. The spacecraft’s thrust vector is provided by a neural network controller which guides the spacecraft to the target along a trajectory that is robust to missed thrust, and the controller does not need to re-optimize any trajectories if it veers off its nominal course. The algorithms used to train the controller to account for missed thrust are supervised learning and neuroevolution. Supervised learning entails showing a neural network many examples of what inputs and outputs should look like, with the network learning over time to duplicate the patterns it has seen. Neuroevolution involves testing many neural networks on a problem, and using the principles of biological evolution and survival of the fittest to produce increasingly competitive networks. Preliminary results show that a controller designed with these methods provides mixed results, but performance can be greatly boosted if the controller’s output is used as an initial guess for an optimizer. With an optimizer, the success rate ranges from around 60% to 96% depending on the problem.</p> <p><br></p> <p>Additionally, this work conducts an analysis of a novel hyperbolic rendezvous strategy which was originally conceived by Dr. Buzz Aldrin. Instead of rendezvousing on the outbound leg of a hyperbolic orbit (traveling away from Earth), the spacecraft performs a rendezvous while on the inbound leg (traveling towards Earth). This allows for a relatively low Delta-v abort option for the spacecraft to return to Earth if a problem arose during rendezvous. Previous work that studied hyperbolic rendezvous has always assumed rendezvous on the outbound leg because the total Delta-v required (total propellant required) for the insertion alone is minimal with this strategy. However, I show that when an abort maneuver is taken into consideration, inserting on the inbound leg is both lower Delta-v overall, and also provides an abort window which is up to a full day longer.</p>

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18769号 / 理博第4027号 / 新制||理||1580(附属図書館) / 31720 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 小野 薫, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

No description available.

No description available.

In this thesis we study two seperate yet related three parameter-families of continuously differentiable maps from the unit circle to unit circle which have a single critical point. For one of the families we show that there is a set of positive measure of parameters such that there is a set of positive measure for which all points in the latter set, the derivative experiences exponential growth. We do so by applying a similar methodology to what Michael Benedicks and Lennart Carleson used to study the quadratic family. For the other family we attempt to show a similar but weaker result using a similar method, but do not manage to do so. We expound on what difficulties the latter family provides and what features Benedicks and Carleson used for the quadratic family that we do not have available. / I den här uppsatsen studerar vi två olika men relaterede treparameterfamiljer av kontinuerligt differentierbara avbildningar från enhetscirkeln till enhetscirkeln som har exakt en kritisk punkt. For den ena familjen visar vi att det finns en mängd av positivt mått av parametrar sådana att det finns en mängd av positivt mått så att för varje punkt i den senarenämnde mängden erfar derivatan exponentiell tillväxt. Vi uppnår detta genom att använda en metod som liknar den som Michael Benedicks och Lennart Carleson använde för att studera den kvadratiska familjen. För den andra familjen försöker vi visa ett liknande men svagare resultat genom att använda en liknande metodik men misslyckas. Vi diskuterar och förklarar vilka svårigheter den senare familjen ger och vilka egenskaper som Benedicks och Carleson använder sig av hos den kvadratiska familjen som vår familj saknar

No description available.

Die extensive Zunahme von lebendigem Wachstum am Beispiel der mathematischen Fibonacci-Folge ergibt räumliche Strukturen von eigentümlichen Stülpungen und Auffaltungen, die den universellen Gesetzmäßigkeiten des Kosmos gehorchen. Solche mathematisch-abstrakten Ideen wollte ich dreidimensional in Skulpturen umsetzen und versuchte erst vergeblich, mir die daraus enstehenden räumlichen Körper rein imaginär vorzustellen und zeichnerisch oder klassisch skulptural umzusetzen – doch erst als mir die Idee kam, entlang dieser Gesetzmäßigkeiten zu häkeln, gelang es.

<p dir="ltr">In this article, we establish results concerning the nilpotent representation theory of groups. In particular, we utilize a theorem of Stallings to provide a general method that constructs pairs of groups that have isomorphic universal nilpotent quotients. We then prove by counterexample that absolute Galois groups of number fields are not determined by their universal nilpotent quotients. We also show that this is the case for residually nilpotent Kleinian groups and in fact, there exist non-isomorphic pairs that have arbitrarily large nilpotent genus. We additionally provide examples of non-isomorphic curves whose geometric fundamental groups have isomorphic universal nilpotent quotients and the isomorphisms are compatible with the outer Galois actions. </p>

Knowledge graphs (KGs) store relational information in a flexible triplet schema and have become ubiquitous for information storage in domains such as web search, e-commerce, social networks, and biology. Retrieval of information from KGs is generally achieved through logical reasoning, but this process can be computationally expensive and has limited performance due to the large size and complexity of relationships within the KGs. Furthermore, to extend the usage of KGs to non-expert users, retrieval over them cannot solely rely on logical reasoning but also needs to consider text-based search. This creates a need for multi-modal representations that capture both the semantic and structural features from the KGs. The primary objective of the proposed work is to extend the accessibility of KGs to non-expert users/institutions by enabling them to utilize non-technical textual queries to search over the vast amount of information stored in KGs. To achieve this objective, the research aims to solve four limitations: (i) develop a framework for logical reasoning over KGs that can learn representations to capture hierarchical dependencies between entities, (ii) design an architecture that can effectively learn the logic flow of queries from natural language text, (iii) create a multi-modal architecture that can capture inherent semantic and structural features from the entities and KGs, respectively, and (iv) introduce a novel hyperbolic learning framework to enable the scalability of hyperbolic neural networks over large graphs using meta-learning. The proposed work is distinct from current research because it models the logical flow of textual queries in hyperbolic space and uses it to perform complex reasoning over large KGs. The models developed in this work are evaluated on both the standard research setting of logical reasoning, as well as, real-world scenarios of query matching and search, specifically, in the e-commerce domain. In summary, the proposed work aims to extend the accessibility of KGs to non-expert users by enabling them to use non-technical textual queries to search vast amounts of information stored in KGs. To achieve this objective, the work proposes the use of multi-modal representations that capture both semantic and structural features from the KGs, and a novel hyperbolic learning framework to enable scalability of hyperbolic neural networks over large graphs. The work also models the logical flow of textual queries in hyperbolic space to perform complex reasoning over large KGs. The models developed in this work are evaluated on both the standard research setting of logical reasoning and real-world scenarios in the e-commerce domain. / Doctor of Philosophy / Knowledge graphs (KGs) are databases that store information in a way that allows computers to easily identify relationships between different pieces of data. They are widely used in domains such as web search, e-commerce, social networks, and biology. However, retrieving information from KGs can be computationally expensive, and relying solely on logical reasoning can limit their accessibility to non-expert users. This is where the proposed work comes in. The primary objective is to make KGs more accessible to non-experts by enabling them to use natural language queries to search the vast amounts of information stored in KGs. To achieve this objective, the research aims to address four limitations. Firstly, a framework for logical reasoning over KGs that can learn representations to capture hierarchical dependencies between entities is developed. Secondly, an architecture is designed that can effectively learn the logic flow of queries from natural language text. Thirdly, a multi-modal architecture is created that can capture inherent semantic and structural features from the entities and KGs, respectively. Finally, a novel hyperbolic learning framework is introduced to enable the scalability of hyperbolic neural networks over large graphs using meta-learning. The proposed work is unique because it models the logical flow of textual queries in hyperbolic space and uses it to perform complex reasoning over large KGs. The models developed in this work are evaluated on both the standard research setting of logical reasoning, as well as, real-world scenarios of query matching and search, specifically, in the e-commerce domain. In summary, the proposed work aims to make KGs more accessible to non-experts by enabling them to use natural language queries to search vast amounts of information stored in KGs. To achieve this objective, the work proposes the use of multi-modal representations that capture both semantic and structural features from the KGs, and a novel hyperbolic learning framework to enable scalability of hyperbolic neural networks over large graphs. The work also models the logical flow of textual queries in hyperbolic space to perform complex reasoning over large KGs. The results of this work have significant implications for the field of information retrieval, as it provides a more efficient and accessible way to retrieve information from KGs. Additionally, the multi-modal approach taken in this work has potential applications in other areas of machine learning, such as image recognition and natural language processing. The work also contributes to the development of hyperbolic geometry as a tool for modeling complex networks, which has implications for fields such as network science and social network analysis. Overall, this work represents an important step towards making the vast amounts of information stored in KGs more accessible and useful to a wider audience.