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The two-dimensional material graphene possesses many impressive properties such asextraordinary carrier mobility, mechanical stiffness and optical transmittance. However,the properties of pristine graphene do not always complement the requirements of applications. Of particular interest, a band gap is needed for electronic logic devices. Recent research shows that using few-layer hexagonal boron nitride as a substrate for graphene-based electronic devices can open a band gap in graphene. Also, introducing impurities such as hydrogen atoms, transition metals or silicon atoms on or within graphene can control the electronic properties according to recent studies. Furthermore, ion irradiation is an alternative option to tailor the properties of graphene by introducing defects. In this thesis, pristine, impure or defective graphene and few-layer boron nitride were characterised by scanning transmission electron microscopy (STEM) and electron energy loss (EEL) spectroscopy. Through STEM and EEL spectroscopy, lattice structures and electronic properties of these two-dimensional materials could be investigated at the atomic scale. This thesis focuses on the frontier studies of theoretical and experimental EEL spectroscopy of graphene and few-layer boron nitride with impurities. In the EEL spectra of pristine graphene and boron nitride two prominent peaks were observed, which are attributed to the plasmon excitations of π- and π+σ-electrons. By introducing impurities such as hydrogen adatoms on graphene and substitutional oxygen and carbon atoms within single-layer boron nitride, our experimental and simulated EEL spectra show that their π-plasmon peaks are modified. The concentrations of these impurities were then evaluated via EEL spectra and atomic-resolution images. If other impurities, such as various transition metals and silicon atoms, are introduced on or within single-layer graphene, our simulated EEL spectra demonstrate that the geometry of these impurity clusters affects the π-plasmon peak in graphene and some impurities even enhance it. Finally, experiments on in-situ transmission electron microscopy and ex-situ STEM and Raman spectroscopy were conducted to investigate ion irradiated graphene. Many topological defects were, for the first time, observed in atomic-resolution STEM images of ion irradiated graphene. Simulated EEL spectra of defective graphene are also reported, which suggests that corrugations and dangling bonds in defects can modify out-of-plane EEL spectra and introduce intraband transitions, respectively.

This dissertation is a summary of the author's research work, supervised by Professor Krzysztof Ciesielski, based on three published articles in the Journal of Mathematical Analysis and Applications and one published article in the Banach Journal of Mathematical Analysis. Our work focuses on the study of paradoxical real functions regarding differentiability and generalized continuity within the foundations of real analysis. The reasons why they are paradoxical are directly connected to their definitions, which will be provided and explained in the later texts. Note that all functions discussed in the dissertation are single-variable and real valued, that is, well-defined from a proper or improper subset of the set R of real numbers into R. The material is presented in two independent chapters. Chapter 1 consists of a study of nowhere-monotone differentiable functions to which we refer as the differentiable monsters. Almost 130 years after A. Köpcke constructed the first differentiable monster and after its many simplifications, K. Ciesielski noticed a few years ago the simplest such construction by far. This construction was a shifted difference of two arbitrary strictly increasing Pompeiu-like functions, that is, of differentiable functions with their derivatives vanishing on a dense subset of their domain. However, not every differentiable monster of bounded variation admits such a Jordan-like decomposition that possesses Pompeiu-likeness. We have first characterized differentiable monsters that can be decomposed in such a ``nice'' way as those that are a difference of two increasing differentiable functions. Secondly, as Jarník's extension theorem allows a differentiable extension to be as ``good'' as being smooth on the extended parts, we work on the other direction and make a differentiable extension as ``bad'' as being nowhere-monotone on the extended parts. Since it is an easy consequence of Darboux's theorem that a differentiable monster must be Pompeiu-like, we have shown that a typical function in a designated complete metric space, which consists of all differentiable extensions that are Pompeiu-like on the extended part, is nowhere-monotone on the extended part. On the other hand, we have also shown that the family of nowhere nowhere-monotone functions is dense in this space. In Chapter 2, we additionally impose a set-theoretical axiom that the set R is not a union of less than continuum-many meager sets. A Darboux function is a function that satisfies the intermediate value property, so the classes of Darboux-like functions represent a group of functions that are continuous in a generalized sense. On the contrary, Sierpiński-Zygmund functions, first constructed in 1923 by W. Sierpiński and A. Zygmund, have as little of the standard continuity as possible. The algebra of subsets generated by these classes and the Sierpiński-Zygmund functions has nine atoms, that is, the smallest nonempty elements of the algebra. In this work, we have crafted a lemma that easily create examples in each of these nine atoms with transfinite induction. Note that the examples within the seven of the nine atoms were first discovered by K. Ciesielski and C.-H. Pan and had been included in a survey of Sierpiński-Zygmund functions in 2019 by K. Ciesielski and J. Seoane-Sepúlveda. As lineability of the main classes of Darboux-like functions, as well as of Sierpiński-Zygmund functions, has been intensively studied, our presented work has caused some on-going researches in the lineability of the nine smaller classes mentioned above.

In this thesis I report electrical transport measurements of single-walled carbonnanotubes (SWCNTs) and graphene on hexagonal boron nitride (h-BN). Chapter 1 covers the fundamental electrical properties of two-dimensional (2D) graphene and one-dimensional (1D) SWCNTs, including the energy band structure, the quantum Hall effect, quantum dot and Kondo effect. Chapter 2 includes transport data from ultra-clean suspended SWCNTs and the SU(4) Kondo effect. In Chapter 3, we developed a technique to encapsulate SWCNT with h-BN and discovered its enhanced capacity of carrying current. In Chapter 4, we investigated the Landau level crossings of bilayer graphene at high displacement field, and the quantum Hall ferromagnetism at those crossing points. Chapter 5 is about the electric field tunable Hofstadter's butterfly spectrum in the bilayer graphene/h-BN superlattice device, with a small rotation angle of 0.37. In Chapter 6, we measured non-local valley current with bilayer graphene.