Algebraic and Computer-based Methods in the Undirected Degree/diameter Problem - a Brief Survey

This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large graphs with given degree and diameter

Event-chain Monte Carlo: foundations, applications, and prospects

This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications, and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply the method to the sampling problem in molecular simulation, that is, to real-world models of peptides, proteins, and polymers in aqueous solution.Comment: 35 pages, no figure

An upper bound for graphs of diameter 3 and given degree obtained as abelian lifts of dipoles

We derive an upper bound on the number of vertices in graphs of diameter 3 and given degree arising from Abelian lifts of dipoles with loops and multiple edges

Electronic structure of the silicon vacancy color center in diamond

This thesis investigates the electronic structure of the silicon vacancy (SiV) color center in diamond. We show detailed spectroscopic investigations and demonstrate first steps towards using the defect as quantum bit (qubit). Starting from the molecular structure of the defect, we first derive a detailed theoretical model using the concept of group theory. With this approach, we calculate the irreducible representation of the electronic states, and determine the interaction terms which lift the energetic degeneracy of these states. Owing to this level splitting, the optical emission spectrum of the defect shows a fine structure, which is observed for individual SiV centers in high quality diamond samples, using confocal microscopy at cryogenic temperatures. We apply magnetic fields in order to lift the degeneracy of magnetic sublevels and to reveal the spin state of the defect. From the excellent agreement with the proposed theoretical model, we obtain a consistent picture of the level structure and reveal, that these states can exhibit near unity spin polarization. As a first step towards spin initialization, we demonstrate spin selective excitation and discuss the results in the context of the derived electronic structure. After having determined the properties of the SiV center in an ideal environment, we extend the theoretical model to include the effect of crystal strain on the level structure. As an experimental test-bench for emitters in strained environments, we investigate nanodiamonds with single SiV centers. Again, we find an excellent agreement with the theoretical predictions.Diese Dissertation befasst sich mit der elektronischen Struktur des Silizium-Fehlstellen (SiV) Farbzentrums in Diamant. Neben detaillierten spektroskopischen Untersuchungen werden erste Experimente zur Nutzung des Defekts als Quantenbit (Qubit) gezeigt. Ausgehend von der molekularen Struktur des Defekts leiten wir unter Verwendung von Gruppentheorie ein detailliertes Modell her, für das wir die irreduziblen Darstellungen der elektronischen Zustände bestimmen. Wir berechnen Wechselwirkungsterme, welche die bestehende energetische Entartung der Zustände aufheben, und zur Ausbildung einer Feinstruktur im Emissionsspektrum des Defekts führen. Diese Feinstruktur wird experimentell an einzelnen SiV-Zentren in Diamantproben hoher kristalliner Güte untersucht. Hierzu verwenden wir konfokale Mikroskopie bei kryogenen Temperaturen. Zudem legen wir magnetische Felder an, wodurch die Entartung magnetischer Unterniveaus aufgehoben wird, und somit der Spin-Zustand des Defektzentrums eindeutig bestimmt wird. Unsere Untersuchungen ermöglichen eine widerspruchsfreie Beschreibung der SiV Niveaustruktur und zeigen, dass die Zustände eine hohe Spin-Polarisation aufweisen können. Darauf aufbauend demonstrieren wir Spin-selektive Anregung und evaluieren die Ergebnisse im Kontext des hergeleiteten Modells. Zusätzlich wird das theoretische Modell erweitert, um den Einfluss von Kristallverspannung auf die Niveaustruktur zu beschreiben. Zur experimentellen Überprüfung untersuchen wir einzelne SiV-Zentren in Nanodiamanten, welche hohe Verspannungsfelder aufweisen. Auch hier zeigt sich eine hervorragende Übereinstimmung mit den Vorhersagen des theoretischen Modells

AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications

SMARANDACHE MULTI-SPACE THEORY, Second Edition

We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multi-space came into being by purely logic. Another is the mathematical combinatorics motivated by a combinatorial speculation, i.e., a mathematical science can be reconstructed from or made by combinatorialization. Both of them contribute sciences for consistency of research with that human progress in 21st century

Spectral Edge Properties of Periodic Elliptic Operators

In this dissertation, we study some spectral problems for periodic elliptic operators arising in solid state physics, material sciences, and differential geometry. More precisely, we are interested in dealing with various effects near and at spectral edges of such operators. We use the name “threshold effects” for the features that depend only on the infinitesimal structure (e.g., a finite number of Taylor coefficients) of the dispersion relation at a spectral edge. We begin with an example of a threshold effect by describing explicitly the asymptotics of the Green’s function near a spectral edge of an internal gap of the spectrum of a periodic elliptic operator of second-order on Euclidean spaces, as long as the dispersion relation of this operator has a non-degenerate parabolic extremum there. This result confirms the expectation that the asymptotics of such operators resemble the case of the Laplace operator. Then we generalize these results by establishing Green’s function asymptotics near and at gap edges of periodic elliptic operators on abelian coverings of compact Riemannian manifolds. The interesting feature we discover here is that the torsion-free rank of the deck transformation group plays a more important role than the dimension of the covering manifold. Finally, we provide a combination of the Liouville and the Riemann-Roch theorems for periodic elliptic operators on abelian co-compact coverings. We obtain several results in this direction for a wide class of periodic elliptic operators. As a simple application of our Liouville-Riemann-Roch inequalities, we prove the existence of non-trivial solutions of polynomial growth of certain periodic elliptic operators on noncompact abelian coverings with prescribed zeros, provided that such solutions grow fast enough

Advanced Concepts in Particle and Field Theory

Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication