Degree Sequence Index Strategy

We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the $k$-independence and the $k$-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and $K_{1,r}$-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester

Characterizations Of Zero Divisor Graphs Determined By Equivalence Classes Of Zero Divisors

We study zero divisor graphs of commutative rings determined by equivalence classes of zero divisors, specifically for a Noetherian ring R. We study the classification of these graphs. Specifically, we add more criteria to the list of characterizations that disqualify a graph as the zero divisor graph of a ring. We also briefly discuss Sage, a mathematical software, which was an aid in providing visual pictures for the graphs under study

An Algebraic Approach to Nivat's Conjecture

This thesis introduces a new, algebraic method to study multidimensional configurations, also sometimes called words, which have low pattern complexity. This is the setting of several open problems, most notably Nivat’s conjecture, which is a generalization of Morse-Hedlund theorem to two dimensions, and the periodic tiling problem by Lagarias and Wang. We represent configurations as formal power series over d variables where d is the dimension. This allows us to study the ideal of polynomial annihilators of the series. In the two-dimensional case we give a detailed description of the ideal, which can be applied to obtain partial results on the aforementioned combinatorial problems. In particular, we show that configurations of low complexity can be decomposed into sums of periodic configurations. In the two-dimensional case, one such decomposition can be described in terms of the annihilator ideal. We apply this knowledge to obtain the main result of this thesis – an asymptotic version of Nivat’s conjecture. We also prove Nivat’s conjecture for configurations which are sums of two periodic ones, and as a corollary reprove the main result of Cyr and Kra from [CK15].Algebrallinen lähestymistapa Nivat’n konjektuuriin Tässä väitöskirjassa esitetään uusi, algebrallinen lähestymistapa moniulotteisiin,matalan kompleksisuuden konfiguraatioihin. Näistä konfiguraatioista, joita moniulotteisiksi sanoiksikin kutsutaan, on esitetty useita avoimia ongelmia. Tärkeimpinä näistä ovat Nivat’n konjektuuri, joka on Morsen-Hedlundin lauseen kaksiulotteinen yleistys, sekä Lagariaksen ja Wangin jaksollinen tiilitysongelma. Väitöskirjan lähestymistavassa d-ulotteiset konfiguraatiot esitetään d:n muuttujan formaaleina potenssisarjoina. Tämä mahdollistaa konfiguraation polynomiannihilaattoreiden ihanteen tutkimisen. Väitöskirjassa selvitetään kaksiulotteisessa tapauksessa ihanteen rakenne tarkasti. Tätä hyödyntämällä saadaan uusia, osittaisia tuloksia koskien edellä mainittuja kombinatorisia ongelmia. Tarkemmin sanottuna väitöskirjassa todistetaan, että matalan kompleksisuuden konfiguraatiot voidaan hajottaa jaksollisten konfiguraatioiden summaksi. Kaksiulotteisessa tapauksessa eräs tällainen hajotelma saadaan annihilaattori-ihanteesta. Tämän avulla todistetaan asymptoottinen versio Nivat’n konjektuurista. Lisäksi osoitetaan Nivat’n konjektuuri oikeaksi konfiguraatioille, jotka ovat kahden jaksollisen konfiguraation summia, ja tämän seurauksena saadaan uusi todistus Cyrin ja Kran artikkelin [CK15] päätulokselle

Non-D-finite excursions in the quarter plane

The number of excursions (finite paths starting and ending at the origin) having a given number of steps and obeying various geometric constraints is a classical topic of combinatorics and probability theory. We prove that the sequence $(e^{\mathfrak{S}}_n)_{n\geq 0}$ of numbers of excursions in the quarter plane corresponding to a nonsingular step set $\mathfrak{S} \subseteq \{0,\pm 1 \}^2$ with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not D-finite. Moreover, we display the asymptotics of $e^{\mathfrak{S}}_n$.Comment: 17 pages, 2 figures and 2 table

Listening to the Universe through Indirect Detection

Indirect detection is the search for the particle nature of dark matter with astrophysical probes. Manifestly, it exists right at the intersection of particle physics and astrophysics, and the discovery potential for dark matter can be greatly extended using insights from both disciplines. This thesis provides an exploration of this philosophy. On the one hand, I will show how astrophysical observations of dark matter, through its gravitational interaction, can be exploited to determine the most promising locations on the sky to observe a particle dark matter signal. On the other, I demonstrate that refined theoretical calculations of the expected dark matter interactions can be used disentangle signals from astrophysical backgrounds. Both of these approaches will be discussed in the context of general searches, but also applied to the case of an excess of photons observed at the center of the Milky Way. This galactic center excess represents both the challenges and joys of indirect detection. Initially thought to be a signal of annihilating dark matter at the center of our own galaxy, it now appears more likely to be associated with a population of millisecond pulsars. Yet these pulsars were completely unanticipated, and highlight that indirect detection can lead to many new insights about the universe, hopefully one day including the particle nature of dark matter.Comment: Ph.D. thesis, MIT, April 2018; based on the work appearing in arXiv:1708.09385, arXiv:1612.05638, arXiv:1612.04814, arXiv:1511.08787, arXiv:1503.01773, and arXiv:1402.670

Split Ga vacancies and the unusually strong anisotropy of positron annihilation spectra in beta-Ga2O3

We report a systematic first-principles study on positron annihilation parameters in the beta-Ga2O3 lattice and Ga monovacancy defects complemented with orientation-dependent experiments of the Doppler broadening of the positron-electron annihilation. We find that both the beta-Ga2O3 lattice and the considered defects exhibit unusually strong anisotropy in their Doppler broadening signals. This anisotropy is associated with low symmetry of the beta-Ga2O3 crystal structure that leads to unusual kind of one-dimensional confinement of positrons even in the delocalized state in the lattice. In particular, the split Ga vacancies recently observed by scanning transmission electron microscopy produce unusually anisotropic positron annihilation signals. We show that in experiments, the positron annihilation signals in beta-Ga2O3 samples seem to be often dominated by split Ga vacancies.Peer reviewe