Mappings between distance sets or spaces

[no abstract

Existence and uniqueness of certain automorphisms on K3 surfaces

[no abstract

Walking the tightrope: UN peacekeeping operations and durable peace: do they actually contribute

Contains fulltext : 47142.pdf (Publisher’s version ) (Open Access)Radboud Universiteit Nijmegen, 23 maart 2006Promotores : Pauwels, A., Frerks, G.417 p

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

타원 곡선의 수론에 관한 몇 가지 가설들

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 변동호.The goal of the present thesis is twofoldwe show the two conjectures concerning the arithmetic of elliptic curves: the Stein–Watkins conjecture (for 5-isogenies) and the Gross--Zagier conjecture. Essentially, Stein--Watkins conjecture tells us about the relations of optimal curves in given rational isogeny class of elliptic curves. In this thesis we show the two optimal curves differ by a 5-isogeny if and only if the isogeny class is '11a'. The Gross--Zagier conjecture provides a theoretical evidence to the strong form of Birch and Swinnerton-Dyer conjecture. We show when elliptic curves have particular types of rational torsion subgroups, the order of the torsion subgroup divides certain arithmetic invariants attached to the curve.Chapter 1. Introduction 1 Chapter 2. Elliptic curves 5 Chapter 3. Differing isogenies of optimal curves 45 Chapter 4. GrossZagier conjecture 59 Bibliography 127 Abstract (in Korean) 135Docto

Symplectic Topology of Projective Space: Lagrangians, Local Systems and Twistors

In this thesis we study monotone Lagrangian submanifolds of CPn . Our results are roughly of two types: identifying restrictions on the topology of such submanifolds and proving that certain Lagrangians cannot be displaced by a Hamiltonian isotopy. The main tool we use is Floer cohomology with high rank local systems. We describe this theory in detail, paying particular attention to how Maslov 2 discs can obstruct the differential. We also introduce some natural unobstructed subcomplexes. We apply this theory to study the topology of Lagrangians in projective space. We prove that a monotone Lagrangian in CPn with minimal Maslov number n + 1 must be homotopy equivalent to RPn (this is joint work with Jack Smith). We also show that, if a monotone Lagrangian in CP3 has minimal Maslov number 2, then it is diffeomorphic to a spherical space form, one of two possible Euclidean manifolds or a principal circle bundle over an orientable surface. To prove this, we use algebraic properties of lifted Floer cohomology and an observation about the degree of maps between Seifert fibred 3-manifolds which may be of independent interest. Finally, we study Lagrangians in CP(2n+1) which project to maximal totally complex subman- ifolds of HPn under the twistor fibration. By applying the above topological restrictions to such Lagrangians, we show that the only embedded maximal Kähler submanifold of HPn is the totally geodesic CPn and that an embedded, non-orientable, superminimal surface in S4 = HP1 is congruent to the Veronese RP2 . Lastly, we prove some non-displaceability results for such Lagrangians. In particular, we show that, when equipped with a specific rank 2 local system, the Chiang Lagrangian L∆ ⊆ CP3 becomes wide in characteristic 2, which is known to be impossible to achieve with rank 1 local systems. We deduce that L∆ and RP3 cannot be disjoined by a Hamiltonian isotopy

On the Formal Flexibility of Syntactic Categories

This dissertation explores the formal flexibility of syntactic categories. The main proposal is that Universal Grammar (UG) only provides templatic guidance for syntactic category formation and organization but leaves many other issues open, including issues internal to a single category and issues at the intercategorial, system level: these points that UG "does not care about" turn out to enrich the categorial ontology of human language in important ways. The dissertation consists of seven chapters. After a general introduction in Chapter 1, I lay out some foundational issues regarding features and categories in Chapter 2 and delineate a featural metalanguage comprising four components: specification, valuation, typing, and granularity. Based on that I put forward a templatic definition for syntactic categories, which unifies the combinatorial and taxonomic perspectives under the notion mergeme. Then, a detailed overview of the "categorial universe" I work with is presented, which shows that the syntactic category system (SCS) is an intricate web structured by five layers of abstraction divided into three broad levels of concern: the individual level (layers 1–2), the global level (layers 3–4), and the supraglobal level (layer 5). In the subsequent chapters I explore the template-flexibility pairs at each abstraction layer, with Chapters 3–4 focusing on the first layer, Chapter 5 on the second layer, and Chapter 6 on the third and fourth layers; the fifth layer is not in the scope of this dissertation. Chapter 3 examines a special type of category defined by an underspecified mergeme, the defective category, which behaves like a "chameleon" in that it gets assimilated into whatever nondefective category it merges with. This characteristic makes it potentially useful in analyzing certain adjunction structures, and I explore this potential by two case studies, one focusing on modifier-head compounds and the other on sentence-final particles. Chapter 4 examines another special type of category defined by the absence of a mergeme, the Root category. Deductive reasoning leads me to propose a generalized root syntax, according to which roots are not confined to lexical categorial environments but may legally merge with and hence "support" any non-Root category. I demonstrate the empirical consequences of this theory by a comprehensive study of the half-lexical–half-functional vocabulary items in Chinese. Chapter 5 ascends to the second abstraction layer and raises the question of whether the categorial sequences (or projection hierarchies) in human language are necessarily totally ordered, as certain analytical devices (e.g., "flavored" categories) can only be theoretically maintained if we also allow categorial sequences to be partially ordered. After a diachronic study of the flavored verbalizer $v_{BE}$ (stative) in Chinese resultative compounds, I conclude that while "flavoring" is indeed a possible type of flexibility in the SCS, it is the deviation rather than the norm due to non-UG or "third" factors and hence should be cautiously used in syntactic analyses. Chapter 6 ascends even higher on the ladder of abstraction and examines the global interconnection in the SCS ontology with the aid of mathematical Category theory. I formalize the functional parallelism across major parts of speech and the inheritance-based relations across granularity levels as Category-theoretic structures, which reveal further and more abstract templates and flexibility types in the SCS. A crucial mathematical concept in the formalization is epi-Adjunction. Finally, in Chapter 7 I summarize the main results of this dissertation and briefly discuss some potential directions of future research.My PhD is funded by Cambridge Trust and China Scholarship Council. I have also received travel grants and financial aids from Gonville and Caius College and the Faculty of Modern and Medieval Languages

Elliptic partial differential equations from an elementary viewpoint

These notes are the outcome of some courses taught to undergraduate and graduate students from the University of Western Australia, the Pontif\'{\i}cia Universidade Cat\'olica do Rio de Janeiro, the Indian Institute of Technology Gandhinagar and the Ukrainian Catholic University in 2021 and 2022